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An Elementary Introduction to
CLASSICAL FIELDS
January/2019
Contents
2 Transformations 38
2.1 Transformation Groups . . . . . . . . . . . . . . . . . . . . . . 38
2.2 Orthogonal Transformations . . . . . . . . . . . . . . . . . . . 42
2.3 The Group of Rotations . . . . . . . . . . . . . . . . . . . . . 48
2.4 The Poincaré Group . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 The Lorentz Group . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Introducing Fields 67
3.1 The Standard Prototype . . . . . . . . . . . . . . . . . . . . . 68
3.2 Non-Material Fields . . . . . . . . . . . . . . . . . . . . . . . . 75
3.2.1 Optional reading: the Quantum Line . . . . . . . . . . 77
3.3 Wavefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.4 Internal Transformations . . . . . . . . . . . . . . . . . . . . . 80
4 General Formalism 85
4.1 Lagrangian Approach . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.1 Relativistic Lagrangians . . . . . . . . . . . . . . . . . 89
4.1.2 Simplified Treatment . . . . . . . . . . . . . . . . . . . 91
4.1.3 Rules of Functional Calculus . . . . . . . . . . . . . . . 93
4.1.4 Variations . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.2 The First Noether Theorem . . . . . . . . . . . . . . . . . . . 99
4.2.1 Symmetries and Conserved Charges . . . . . . . . . . . 101
ii
4.2.2 The Basic Spacetime Symmetries . . . . . . . . . . . . 104
4.2.3 Internal Symmetries . . . . . . . . . . . . . . . . . . . 108
4.3 The Second Noether Theorem . . . . . . . . . . . . . . . . . . 110
4.4 Topological Conservation Laws . . . . . . . . . . . . . . . . . 113
iii
9 Gravitational Field 201
9.1 General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.2 The Equivalence Principle . . . . . . . . . . . . . . . . . . . . 202
9.3 Pseudo-Riemannian Metric . . . . . . . . . . . . . . . . . . . . 204
9.4 The Notion of Connection . . . . . . . . . . . . . . . . . . . . 205
9.5 Curvature and Torsion . . . . . . . . . . . . . . . . . . . . . . 206
9.6 The Levi-Civita Connection . . . . . . . . . . . . . . . . . . . 207
9.7 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
9.8 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . 209
9.9 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . 210
9.10 The Schwarzschild Solution . . . . . . . . . . . . . . . . . . . 211
Index 213
iv
Chapter 1
1.1 Introduction
The results of measurements made by an observer depend on the reference
frame of that observer. There is, however, a preferred class of frames, in
which all measurements give the same results, the so-called inertial frames.
Such frames are characterized by the following property:
This is not true if the particle is looked at from an accelerated frame. Ac-
celerated frames are non-inertial frames. It is possible to give to the laws of
Physics invariant expressions that hold in any frame, accelerated or inertial,
but the fact remains that measurements made in different general frames give
different results.
Inertial frames are consequently very special, and are used as the basic
frames. Physicists do their best to put themselves in frames which are as near
as possible to such frames, so that the lack of inertiality produce negligible
effects. This is not always realizable, not even always desirable. Any object
on Earth’s surface will have accelerations (centrifugal, Coriolis, etc). And we
may have to calculate what an astronaut in some accelerated rocket would
see.
Most of our Physics is first written for inertial frames and then, when
necessary, adapted to the special frame actually used. These notes will be
exclusively concerned with Physics on inertial frames.
We have been very loose in our language, using words with the meanings
they have for the man-in-the-street. It is better to start that way. We shall
make the meanings more precise little by little, while discussing what is
1
involved in each concept. For example, in the defining property of inertial
frames given above, the expression “moves with a constant velocity” is a
vector statement: also the velocity direction is fixed. A straight line is a
curve keeping a constant direction, so that the property can be rephrased as
But then we could ask: in which space ? It must be a space on which vectors
are well-defined. Further, measurements involve fundamentally distances and
time intervals. The notion of distance presupposes that of a metric. The
concept of metric will suppose a structure of differentiable manifold — on
which, by the way, derivatives and vectors are well defined. And so on, each
question leading to another question. The best gate into all these questions
is an examination of what happens in Classical Mechanics.
x0 = x − u t (1.1)
t0 = t . (1.2)
v0 = v − u (1.3)
2
K'
K
x'
x u
3. Newton’s force law holds in both reference systems; in fact, its expres-
sion in K,
dv k d 2 xk
m =m = F k, (1.4)
dt dt2
implies
dv 0k d2 x0k
m = m = F 0k = F k .
dt0 dt02
A force has the same value if measured in K or in K 0 . Measuring a force
in two distinct inertial frames gives the same result. It is consequently
impossible to distinguish inertial frames by making such measurements.
Also in this physical sense all inertial frames are equivalent. Of course,
the free cases F0 = F = 0 give the equation for a straight line in both
3
frames.
4. equation (1.2), put into words, states that time is absolute; given two
events, the clocks in K and K 0 give the same value for the interval of
time lapsing between them.
x0 = R x . (1.5)
Rotations are best represented in matrix language. Take the space
x1
coordinates as a column-vector x23 and the 3 × 3 rotation matrix
x
R1 1 R1 2 R1 3
R = R2 1 R2 2 R2 3 . (1.6)
R3 1 R3 2 R3 3
x03 R3 1 R3 2 R3 3 x3
The velocity and the force will rotate accordingly; with analogous vec-
tor columns for the velocites and forces, v0 = R v and F0 = R F. With
the transformed values, Newton’s law will again keep holding. Recall
that a general constant rotation requires 3 parameters (for example,
the Euler angles) to be completely specified.
4
7. Newton’s law is also preserved by translations in space and by changes
in the origin of time:
x0 = x − a (1.8)
t0 = t − a0 , (1.9)
§ 1.2 Transformations (1.1), (1.2), (1.5), (1.8) and (1.9) can be composed
at will, giving other transformations which preserve the laws of classical me-
chanics. The composition of two transformations produces another admissi-
ble transformation, and is represented by the product of the corresponding
5
matrices. There is clearly the possibility of doing no transformation at all,
that is, of performing the identity transformation
t0 1 0 0 0 0 t t
01
x 0 1 0 0 0 x1 x1
x02 = 0 0 1 0 0 x2 = x2 . (1.13)
03 3 3
x 0 0 0 1 0 x x
1 0 0 0 0 1 1 1
If a transformation is possible, so is its inverse — all matrices above are
invertible. Finally, the composition of three transformations obey the asso-
ciativity law, as the matrix product does. The set of all such transformations
constitute, consequently, a group. This is the Galilei group. For a general
transformation to be completely specified, the values of ten parameters must
be given (three for a, three for u, three angles for R, and a0 ). The trans-
formations can be performed in different orders: you can, for example, first
translate the origins and then rotate, or do it in the inverse order. Each order
leads to different results. In matrix language, this is to say that the matri-
ces do not commute. The Galilei group is, consequently, a rather involved
non-abelian group. There are many different ways to parameterize a general
transformation. Notice that other vectors, such as velocities and forces, can
also be attributed 5-component columns and will follow analogous rules.
§ 1.3 The notions of vector and tensor presuppose a group. In current lan-
guage, when we say that V is a vector in euclidean space, we mean a vector
under rotations. That is, V transforms under a rotation R according to
V 0i = Ri j V j .
T 0ij = Ri m Rj n T mn ,
6
§ 1.4 The notation used above suggests a new concept. The set of columns
t
!
x1
x2
x3
1
constitute a vector space, whose members represent all possible positions and
times. That vector space is the classical spacetime. The concept of spacetime
only acquires its full interest in Special Relativity, because this spacetime of
Classical Mechanics is constituted of two independent pieces: space itself,
and time. It would be tempting, always inspired by the notation, to write
t = x0 for the first component, but there is a problem: all components in a
column-vector should have the same dimension, which is not the case here.
To get dimensional uniformity, it would be necessary to multiply t by some
velocity. In Classical Physics, all velocities change in the same way, and
so that the 0-th component would have strange transformation properties.
In Special Relativity there exists a universal velocity, the velocity of light
c, which is the same in every reference frame. It is then possible to define
x0 = ct and build up a space of column-vectors whose components have a
well-defined dimensionality.
§ 1.5 We have said that the laws of Physics can be written as expressions
which are the same in any frame. This invariant form requires some mathe-
matics, in special the formalism of differential forms. Though it is comfort-
able to know that laws are frame-independent even if measurements are not,
the invariant language is not widely used. The reason is not ignorance of
that language. Physics is an experimental science and every time a physicist
prepares his apparatuses to take data, (s)he is forced to employ some particu-
lar frame, and some particular coordinate system. (S)he must, consequently,
know the expressions the laws involved assume in that particular frame and
coordinate system. The laws acquire different expressions in different frames
because, seen from each particular frame, they express relationships between
components of vectors, tensors and the like. In terms of components the
secret of inertial frames is that the laws are, seen from them, covariant: An
equality will have the right hand side and the left hand side changing in the
same way under transformations between them.
7
§ 1.6 The principles of Classical Mechanics can be summed up∗ in the fol-
lowing statements:
the laws of nature are the same in all of them (galilean relativity);
given an initial inertial system, all the other inertial systems are in uniform
rectilinear motion with respect to it (inertia);
the basic law says that acceleration, defined as the second time–derivative
of the cartesian coordinates, equals the applied force per unit mass:
ẍ = f (x, ẋ, t) (Newton’s law).
§ 1.8 Let us sum up what has been said, with some signs changed for the
sake of elegance. The transformations taking one into another the classical
inertial frames are:
(i) rotations R(ω) of the coordinate axis as in (1.11), where ω represents
the set of three angles necessary to determine a rotation;
∗
For a splendid discussion, see V.I. Arnold, Mathematical Methods of Classical Me-
chanics, Springer-Verlag, New York, 1968.
8
(ii) translations of the origins in space and in time: x0 = x + a and t0 =
t + ao :
1 0 0 0 a0
0 1 0 0 a1
0 0
1 0 a2 ;
(1.14)
0 0 0 1 a3
0 0 0 0 1
(iii) uniform motions (galilean boosts) with velocity u:
1 0 0 0 0
1
u 1 0 0 0
u2 0 1 0 0 . (1.15)
3
u 0 0 1 0
0 0 0 0 1
The generic element of the Galilei group can be represented as
1 0 0 0 a0
1
u R1 1 R 1 2 R 1 3 a1
2 2
G(ω, u, a) =
u R 1 R 2 2 R 2 3 a2
. (1.16)
3
u R3 1 3 3
R 2 R 3 a 3
0 0 0 0 1
Exercise 1.1 This is a particular way of representing a generic element of the Galilei
group. Compare it with that obtained by
1. multiplying a rotation and a boost;
2. the same, but in inverse order;
3. performing first a rotation, then a translation;
4. the same, but in inverse order.
Do boosts commute with each other ?
x0i = Ri j xj + ui t + ai
(1.17)
t0 = t + a0 .
In the first expression, we insist, the Einstein convention has been used.
9
Exercise 1.2 With this notation, compare what results from:
x 0 = x + u 1 t + a1 (1.18)
y 0 = y + u2 t + a2 (1.19)
z 0 = z + u3 t + a3 (1.20)
t0 = t + a0 . (1.21)
10
(1.2). To make things simpler, take the relative velocity u along the axis Ox
of K. Instead of
x0 = x − u1 t (1.22)
t0 = t , (1.23)
x − u1 t
x0 = q (1.24)
2
1 − uc2
0 t − cu2 x
t = q , (1.25)
2
1 − uc2
where c is the velocity of light. These equations call for some comments:
11
to be a universal constant, which is further the upper limit for the velocity
of propagation of any disturbance. This leads to the Poincaré principle of
relativity, which supersedes Galilei’s. There is a high price to pay: the notion
of potential must be abandoned and Mechanics has to be entirely rebuilt, with
some other group taking the role of the Galilei group. It is clear, furthermore,
that the composition of velocities (1.3) cannot hold if some velocity exists
which is the same in every frame.
§ 1.13 We have said that some other group should take the place of the
Galilei group, but that rotations should remain, as they preserve Maxwell’s
equations. Thus, the group of rotations should be a common subgroup of the
new group and the Galilei group. Rotations preserve distances between two
points in space. If these points have cartesian coordinates x = (x1 , x2 , x3 )
and y = (y 1 , y 2 , y 3 ), their distance will be
1/2
d(x, y) = (x1 − y 1 )2 + (x2 − y 2 )2 + (x3 − y 3 )2
. (1.26)
That distance comes from a metric, the Euclidean metric. Metrics are usu-
ally defined for infinitesimal distances. The Euclidean 3-dimensional metric
12
defines the distance
dl2 = δij dxi dxj
between two infinitesimally close points whose cartesian coordinates differ by
dx = (dx1 , dx2 , dx3 ).
A metric, represented by the components gij , can be represented by an
invertible symmetric matrix whose entries are precisely these components.
The Euclidean metric is the simplest conceivable one,
1 0 0
(δij ) = 0 1 0 (1.27)
0 0 1
u · v = gij ui v j ; (1.29)
orthogonality: u ⊥ v when u · v = 0;
13
the distance between two points x and y, defined as
d(x, y) = |x − y| . (1.31)
The scalar product, and consequently the norms and distances, are invariant
under rotations and translations. Notice that
14
This is invariant under rotations, transformations (1.24), (1.25) and their
generalizations. Actually, the Lorentz metric
15
or arrowed (~x) letters, while a point in spacetime is indicated by simple letters
(x).
x0 = c t
§ 1.17 The light cone Expression (1.33) is not a real distance, of course.
It is a “pseudo-distance”. Distinct events can be at a zero pseudo-distance
of each other. Fix the point y = (y 0 , y 1 , y 2 , y 3 ) and consider the set of points
x at a vanishing pseudo-distance of y. The condition for that,
ηαβ (xα − y α )(xβ − y β ) =
16
§ 1.18 Causality Notice that particles with velocities v < c stay inside
the cone. As no perturbation can travel faster than c, any perturbance at
the cone apex will affect only events inside the upper half of the cone (called
the future cone). On the other hand, the apex event can only be affected
by incidents taking place in the events inside the lower cone (the past cone).
This is the main role of the Lorentz metric: to give a precise formulation
of causality in Special Relativity. Notice that causality somehow organizes
spacetime. If point Q lies inside the (future) light cone of point P , then
P lies in the (past) light cone of Q. Points P and Q are causally related.
Nevertheless, a disturbance in Q will not affect P . In mathematical terms,
the past-future relationship is a partial ordering, “partial” because not every
two points are in the cones of each other.
The horizontal line in Figure 1.2 stands for the present 3-space. Its points
lie outside the cone and cannot be affected by whatever happens at the apex.
The reason is clear: it takes time for a disturbance to attain any other point.
Only points in the future can be affected. Classical Mechanics should be
obtained the limit c → ∞. We approach more and the classical vision by
opening the cone solid angle. If we open the cone progressively to get closer
and closer the classical case, the number of 3-points in the possible future
(and possible past) increases more and more. In the limit, the present is
included in the future and in the past: instantaneous communication becomes
possible.
17
(i) draw the complete light cones of both points (ii) look for their intersection
and (iii) choose a path joining the points while staying on the light cones.
§ 1.20 Proper time Let us go back to the interval (1.35) separating two
nearby events. Suppose two events at the same position in 3-space, so that
dl2 = d~x2 = 0. They are the same point of 3-space at different times, and
their interval reduces to
An observer fixed in 3-space will have that interval, which is pure coordinate
time. (S)he will be a “pure clock”. This time measured by a fixed observer
is its proper time. Infinitesimal proper time is just ds. Let us now attribute
coordinates x0 to this clock in its own frame, so that ds = cdt0 , and compare
with what is seen by a nearby observer, with respect to which the clock will
be moving and will have coordinates x (including a clock). Interval invariance
will give
dx2
ds2 = c2 dt02 = c2 dt2 − dx2 = c2 dt2 (1 − 2 2 ),
c dt
or 1/2
v2
0
dt = dt 1 − 2 (1.38)
c
with the velocity v = dxdt
. By integrating this expression, we can get the
relationship between a finite time interval measured by the fixed clock and
the same interval measured by the moving clock:
t2 1/2
v2
Z
t02 − t01 = dt 1 − 2 . (1.39)
t1 c
18
will age less than (his) her untravelling twin brother (the twin paradox). A
decaying particle moving fast will have a longer lifetime when looked at from
a fixed clock (time dilatation, or time dilation).
Exercise 1.4 Consider a meson µ. Take for its mean lifetime 2.2 × 10−6 s in its own
rest system. Suppose it comes from the high atmosphere down to Earth with a velocity
v = 0.9c. What will be its lifetime from the point of view of an observer at rest on Earth ?
19
transformations, or boosts. The group generalizing the rotations of E3 to 4-
dimensional spacetime, the Lorentz group, includes 3 transformations of this
kind and the 3 rotations. To these we should add the translations in 4-space,
representing changes in the origins of the four coordinates. The 10 transfor-
mations thus obtained constitute the group which replaces the Galilei group
in Special Relativity, the Lorentz inhomogeneous group or Poincaré group.
x0 + ut0
x = q (1.44)
2
1 − uc2
y0 = y
z0 = z
t0 + cu2 x0
t = q . (1.45)
2
1 − uc2
20
§ 1.24 Take again a clock at rest in K 0 , and consider two events at the same
point (x’, y’, z’) in K 0 , separated by a time interval ∆t0 = t02 − t01 . What will
be their time separation ∆t in K ? From (1.45), we have
x0 + ut0 x0 + ut0
x1 = q1 ; x2 = q2 ,
u2 u2
1 − c2 1 − c2
whose difference is
−1/2
u2
∆x = x2 − x1 = 1 − 2 ∆x0 . (1.47)
c
Thus, seen from a frame in motion, the length l = ∆x0 is always smaller than
the proper length:
1/2
u2
l = l0 1 − 2 ≤ l0 . (1.48)
c
Proper length is larger than any other. This is the Lorentz contraction,
which turns up for space lengths. The proper length is the largest length a
1/2
u2
rod can have in any frame. The ubiquitous expression 1 − c2 is called
the Lorentz contraction factor. It inverse is indicated by
1
γ=q . (1.49)
u2
1− c2
21
This notation is almost universal, and so much so that the factor is called
the gamma factor. Equations (1.44), (1.45), (1.46) and (1.48) acquire simpler
aspects,
dx = γ(dx0 + udt0 )
dy = dy 0 ; dz = dz 0
u
dt = γ(dt0 + 2 dx0 ) .
c
Dividing the first 3 equations by the last,
q q
u2 2
dx 0
dx + udt 0
dy 1 − c2 dz 1 − uc2
0 0
= 0 u 0 ; = dy 0 u 0 ; = dz 0 u 0 .
dt dt + c2 dx dt dt + c2 dx dt dt + c2 dx
Now factor dt0 out in the right hand side denominators:
q q
u2 2
0
vx + u 1 − c2 1 − uc2
0 0
vx = ; vy = vy ; vz = vz . (1.56)
1 + cu2 vx0 1 + cu2 vx0 1 + cu2 vx0
22
These are the composition laws for velocities. Recall that the velocity u is
supposed to point along the Ox axis. If also the particle moves only along
the Ox axis (vx k u, vy = vz = 0), the above formulae reduce to
v0 + u
v= 0 . (1.57)
1 + uv
c2
The galilean case (1.3) is recovered in the limit u/c → 0. Notice that we have
been forced to use all the velocity components in the above discussion. The
reason lies in a deep difference between the Lorentz group and the Galilei
group: Lorentz boosts in different directions, unlike galilean boosts, do not
commute. This happens because, though contraction is only felt along the
transformation direction, time dilatation affects all velocities — and, conse-
quently, the angles they form with each other.
§ 1.28 Angles and aberration Let us see what happens to angles. In the
case above, choose coordinate axis such that the particle velocity lies on plane
xy. In systems K and K 0 , it will have components vx = v cos θ; vy = v sin θ
and vx0 = v 0 cos θ0 ; vy0 = v 0 sin θ0 , with obvious choices of angles. We obtain
then from (1.56) r
v 0 sin θ0 u2
tan θ = 1 − . (1.58)
u + v 0 cos θ0 c2
Thus, also the velocity directions are modified by a change of frame. In the
case of light propagation, v = v 0 = c and
r
sin θ0 u2
tan θ = 1 − . (1.59)
u/c + cos θ0 c2
tan θ0 − tan θ
tan(θ0 − θ) = .
1 + tan θ0 tan θ
23
§ 1.29 Four-vectors We have seen that the column (ct, x, y, z) transforms
in a well-defined way under Lorentz transformations. That way of transform-
ing defines a Lorentz vector: any set V = (V 0 , V 2 , V 2 , V 3 ) of four quantities
transforming like (ct, x, y, z) is a Lorentz vector, or four-vector. By (1.50)
and (1.51), they will have the behavior
u 00
V 1 = γ(V 01 + V ) (1.61)
c
V2 = V 02 ; V 3 = V 03 (1.62)
u
V 0
= γ(V 00 + V 01 ) . (1.63)
c
It is usual to call V 0 the “time component” of V , and the V k ’s, “space
components”.
24
This velocity has a few special features. First, it has dimension zero. The
usual dimension can be recovered by multiplying it by c, but it is a common
practice to leave it so. Second, its components are not independent. Indeed,
it is immediate from (1.66) and (1.67) that u has unit modulus (or unit
2 2
norm): u2 = (u0 )2 − ~u2 = γ 2 − γ 2 ~vc2 = γ 2 (1 − ~vc2 ).
∴ u2 = ηαβ uα uβ = 1 . (1.69)
a · u = aα uα = ηαβ aα uβ = 0 . (1.71)
Only for emphasis: we have said in §1.14 that a metric defines a scalar
product, orthogonality, norm, etc. Both the above scalar product and the
modulus (1.69) are those defined by the metric η.
j α = e uα , (1.72)
or
vx vy vz
j = e γ 1, , , . (1.73)
c c c
Electromagnetism can be written in terms of the scalar potential φ and the
~ They are put together into the four-vector potential
vector potential A.
~ = (φ, Ax , Ay , Az ) .
A = (φ, A) (1.74)
25
~ The interaction of a charge e with a
is of the “current-potential” type, ~j · A.
static electromagnetic field is eφ. These forms of interaction are put together
in the scalar
dxα
j · A = e Aα uα = e Aα . (1.75)
ds
§ 1.33 The results of §1.8 are adapted accordingly. A 4 × 4 matrix Λ will
represent a Lorentz transformation which, acting on a 4-component column
x, gives the transformed x0 . Equations (1.17) are replaced by
x0α = Λα β xβ + aα (1.76)
The boosts are now integrated into the (pseudo-)orthogonal group, of which Λ
is a member. Translations in space and time are included in the four-vector a.
As for the Galilei group, a 5 × 5 matrix is necessary to put pseudo-rotations
and translations together. The matrix expression of the general Poincaré
transformation (1.76) has the form
x00 Λ 0 0 Λ 0 1 Λ 0 2 Λ 0 3 a0 x0
01 1
x Λ 0 Λ 1 1 Λ 1 2 Λ 1 3 a1 x 1
x0 = L x =
02 2 2 2 2 2 2
x = Λ 0 Λ 1 Λ 2 Λ 3 a x . (1.77)
03 3
x Λ 0 Λ 3 1 Λ 3 2 Λ 3 3 a3 x 3
1 0 0 0 0 1 1
These transformations constitute a group, the Poincaré group. The Lorentz
transformations are obtained by putting all the translation parameters aα =
0. In this no-translations case, a 4 × 4 version suffices. The complete, general
transformation matrix is highly complicated, and furthermore depends on
the parameterization chosen. In practice, we decompose it in a product
of rotations and boosts, which is always possible. The boosts, also called
“pure Lorentz transformations”, establish the relationship between unrotated
frames which have a relative velocity ~v = v~n along the unit vector ~n. They
are given by
−γ vc n1 −γ vc n2 −γ vc n3
γ
−γ v n1 1 + (γ − 1)n1 n1 (γ − 1)n1 n2 (γ − 1)n1 n3
c
Λ= ,
−γ vc n2 2 1
(γ − 1)n n 1 + (γ − 1)n n2 2 2 3
(γ − 1)n n
−γ vc n3 (γ − 1)n3 n1 (γ − 1)n3 n2 1 + (γ − 1)n3 n3
(1.78)
2 2 −1/2
where, as usual, γ = (1 − v /c ) .
26
1.5 Lorentz Vectors and Tensors
§ 1.34 We have said in § 1.3 that vectors and tensors always refer to a
group. They actually ignore translations. Vectors are differences between
points (technically, in an affine space), and when you do a translation, you
change both its end-points, so that its components do not change. A Lorentz
vector obeys
V 0α = Λα β V β . (1.79)
A 2nd-order Lorentz tensor transforms like the product of two vectors:
T 0αβ = Λα γ Λβ δ T γδ . (1.80)
A 3rd-order tensor will transform like the product of 3 vectors, with three
Λ-factors, and so on. Such vectors and tensors are contravariant vectors
and tensors, which is indicated by the higher indices. Lower indices sig-
nal covariant objects. This is a rather unfortunate terminology sanctioned
by universal established use. It should not be mistaken by the same word
“covariant” employed in the wider sense of “equally variant”. A covariant
vector, or covector, transforms according to
Vα0 = Λα β Vβ . (1.81)
The matrix with entries Λα β is the inverse of the previous matrix Λ. This
notation will be better justified later. For the time being let us notice that,
as indices are lowered and raised by η and η −1 , we have
so that we must have Λα γ Λα δ = δδγ in order to preserve the value of the norm.
Therefore, (Λ−1 )γ α = Λα γ and (1.81) is actually
with the last matrix acting from the right. A good picture of what
uhappens
0
1
comes as follows: conceive a (contravariant) vector u as column uu2 with
u3
four entries and a covector v as a row ( v0 ,v1 ,v2 ,v3 ). Matrices will act from the
27
left on columns, and act from the right on rows. The scalar product v · u will
be a row-column product,
u0
1
( v0 ,v1 ,v2 ,v3 ) uu2 .
u3
T 0α β γ = Λα δ Λβ Λγ φ T δ φ . (1.84)
§ 1.35 We shall later consider vector fields, which are point-dependent (that
is, event-dependent) vectors V = V (x). They will describe the states of
systems with infinite degrees of freedom, one for each point, or event. In
that case, a Lorentz transformation will affect both the vector itself and its
argument:
V 0α (x0 ) = Λα β V β (x) , (1.85)
where x0α = Λα β xβ . Tensor fields will follow suit.
Comment 1.1 The Galilei group element (1.16) is a limit when v/c → 0 of the generic
group element (1.77) of the Poincaré group. We have said “a” limit, not “the” limit,
because some redefinitions of the transformation parameters are necessary for the limit to
make sense. The procedure is called a Inönü–Wigner contraction.‡
Comment 1.2 Unlike the case of Special Relativity, there is no metric on the complete
4-dimensional spacetime which is invariant under Galilei transformations. For this reason
people think twice before talking about “spacetime” in the classical case. There is space,
and there is time. Only within Special Relativity, in the words of one of the inventors of
spacetime,“. . . space by itself, and time by itself, are condemned to fade away into mere
shadows, and only a kind of union of the two preserves an independent reality”.§
‡
R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications, J.Wiley, New
York, 1974.
§
H. Minkowski, “Space and time”, in The Principle of Relativity, New York, Dover,
28
1.6 Particle Dynamics
§ 1.36 Let us go back to Eq.(1.39). Use of Eq.(1.65) shows that time, as
indicated by a clock, is Z
1
ds ,
c α
the integral being taken along the clock’s worldline α. From the expression
√
ds = c2 dt2 − d~x2 we see that each infinitesimal contribution ds is maximal
when d~x2 = 0, that is, along the pure-time straight line, or the cone axis.
We have said in § 1.19 that there are always zero-length paths between any
two events. These paths are formed with contributions ds = 0 and stand on
the light-cones. The farther a path α stays from the light cone, the larger
R
will be the integral above. The largest value of α will be attained for α =
the pure-time straight line going through each point.
Hamilton’s minimal-action principle is a mechanical version of Fermat’s
optical minimal-time principle. Both are unified in the relativistic context,
R R
but α is a maximal time or length. Let us only retain that the integral α
is an extremal for a particle moving along a straight line in 4-dimensional
spacetime. “Moving along a straight line” is just the kind of motion a free
particle should have in an inertial frame. If we want to obtain its equation of
R
motion from an action principle, the action should be proportional to α ds.
The good choice for the action related to a motion from point P to point Q
is Z Q
S = − mc ds . (1.86)
P
The factor mc is introduced for later convenience. The sign, to make of the
action principle a minimal (and not a maximal) principle.
Let us see how to use such a principle to get the equation of motion of a
free particle. Take two points P and Q in Minkowski spacetime, and consider
the integral Z Q Z Qq
ds = ηαβ dxα dxβ .
P P
1923. From the 80th Assembly of German Natural Scientists and Physicians, Cologne,
1908.
29
Its value depends on the path chosen. It is actually a functional on the space
of paths between P and Q,
Z
F[αP Q ] = ds. (1.87)
αP Q
R
An extremal of this functional would be a curve α such that δS[α] = δds
= 0. Now,
δds2 = 2 ds δds = 2 ηαβ dxα δdxβ ,
so that
dxα
δds = ηαβ δdxβ .
ds
Thus, commuting the differential d and the variation δ and integrating by
parts,
Z Q Z Q
dxα dδxβ d dxα β
δS[α] = ηαβ ds = − ηαβ δx ds
P ds ds P ds ds
Z Q
d α β
=− ηαβ u δx ds.
P ds
The variations δxβ are arbitrary. If we want to have δS[α] = 0 for arbitrary
δxβ , the integrand must vanish. Thus, an extremal of the action (1.86) will
satisfy
d α d2 xα
mc u = mc = 0. (1.88)
ds ds2
This is the equation of a straight line, and — as it has the aspect of Newton’s
second law — shows the coherence of the velocity definition (1.64) with the
action (1.86). The solution of this differential equation is fixed once initial
conditions are given. We learn in this way that a vanishing acceleration is
related to an extremal of S[αP Q ]. In the presence of some external force F α ,
this should lead to a force law like
duα
mc = F α. (1.89)
ds
§ 1.37 To establish comparison with Classical Mechanics, let us write
Z Q
S= Ldt , (1.90)
P
30
with L the Lagrangian. By (1.65), we have
r
v2
L = − mc2 1− . (1.91)
c2
v2
Notice that, for small values of c2
,
v2 mv 2
L ≈ − mc2 (1 − ) ≈ − mc2
+ , (1.92)
2c2 2
the classical Lagrangian with the constant mc2 extracted.
δL
§ 1.38 The momentum is defined, as in Classical Mechanics, by pk = δv k,
which gives
1
p= q mv = γ mv (1.93)
2
1 − vc2
which, of course, reduces to the classical p = mv for small velocities.
The energy, again as in Classical Mechanics, is defined as E = p · v − L,
which gives the celebrated expression
mc2
E = γ mc2 = q . (1.94)
2
1 − vc2
This shows that, unlike what happens in the classical case, a particle at rest
has the energy
E = mc2 (1.95)
and justifies its subtraction to arrive at the classical Lagrangian in (1.92).
Relativistic energy includes the mass contribution. Notice that both the
energy and the momentum would become infinite for a massive particle of
velocity v = c. That velocity is consequently unattainable for a massive
particle.
Equations (1.93) and (1.94) lead to two other important formulae. The
first is
E
p = 2 v. (1.96)
c
The infinities mentioned above cancel out in this formula, which holds also
for massless particles traveling with velocity c. In that case it gives
E
|p| = . (1.97)
c
31
The second formula comes from taking the squares of both equations. It is
E 2 = p2 c2 + m2 c4 (1.98)
whose square is
p 2 = m 2 c2 . (1.101)
Force, if defined as the derivative of p with respect to proper time, will give
d du
F = p = mc ,
ds ds
just Eq.(1.89). Because u is dimensionless the quantity F , defined in this
way, has not the mechanical dimension of a force (F c would have).
§ 1.39 We have examined the case of a free particle in §1.36, where the
action Z
S = − mc ds
has been used. Let us see, through an example, what happens when a force
is present. Consider the case of a charged test particle. The coupling of a
particle of charge e to an electromagnetic potential A is given by Aα j α =
e Aa uα , as said in §1.32. The action along a curve is, consequently,
Z Z
e α e
Sem [α] = − Aα u ds = − Aα dxα .
c α c α
with a factor to give the correct dimension. The variation is
Z Z Z Z
e α e α e α e
δSem [α] = − δAα dx − Aα dδx = − δAα dx + dAβ δxβ
c α c α c α c α
dxα
Z Z Z
e β α e β α e
=− ∂β Aα δx dx + ∂α Aβ δx dx = − [∂β Aα −∂α Aβ ]δxβ ds
c α c α c α ds
32
Z
e
=− Fβα uα δxβ ds ,
c α
where we have defined the object
Fαβ = ∂α Aβ − ∂β Aα . (1.102)
is Z Q
d α e
δS = ηαβ mc u − Fβα uα δxβ ds.
P ds c
The extremal satisfies
d α e α β
mc
u = F β u . (1.104)
ds c
This is the relativistic version of the Lorentz force law. It has the general
form (1.89).
Indices are here raised and lowered with the euclidean metric.
B Calculate
1. ijk imn
33
2. ijk ijn
3. ijk ijk .
Exercise 1.8 (Facultative: supposes some knowledge of electromagnetism and vector cal-
culus) Tensor (1.102) is Maxwell’s tensor, or electromagnetic field strength. If we compare
~ and the magnetic field B
with the expressions of the electric field E ~ in vacuum, we see
that
F0i = ∂0 Ai − ∂i A0 = ∂ct Ai − ∂i φ = Ei ;
~ k = ijk Bk ,
Fij = ∂i Aj − ∂j Ai = ijk (rotA)
vj
1
F i β uβ = F i 0 u0 + F i j uj = E i γ + i jk Bk γ = γ E i + i jk v j Bk
c c
i
~ + 1 ~v × B
=γ E ~ . (1.107)
c
Thus, for the space components, Eq.(1.104) is
d e ~ 1 ~
p~ = γ E + ~v × B .
ds c c
Using Eq.(1.65),
d 1
F~ = p~ = e ~ ~
E + ~v × B , (1.108)
dt c
which is the usual form of the Lorentz force felt by a particle of charge e in a electromag-
netic field. The time component gives the time variation of the energy:
d 0 e
mc u = F 0 i ui
ds c
d d ~ · ~v ∴ d E = e E
~ · ~v .
∴ (γmc2 ) = γ E =eγ E (1.109)
ds dt dt
admitting now the possibility of a mass which changes along the path (think of a rocket
spending its fuel along its trajectory).
34
1. Taking into account the possible variation of m, find, by the same procedure pre-
viously used, the new equation of force:
d ∂
[mc uα ] = mc, (1.110)
ds ∂xα
with a force given by the mass gradient turning up;
3. the force is orthogonal to the path, that is, to its velocity at each point;
Inertial frame a reference frame such that free (that is, in the absence
to any forces) motion takes place with constant velocity is an inertial
frame;
(a) in Classical Physics, Newton’s force law in an inertial frame is
dv k
m = F k;
dt
35
(b) in Special Relativity, the force law in an inertial frame is
duα
mc = F α.
ds
Incidentally, we are stuck to cartesian coordinates to discuss forces:
the second time-derivative of a coordinate is an acceleration only if
that coordinate is cartesian.
36
but it has in common with gravity that universal character. Locally —
that is, in a small enough domain of space a gravitational force cannot
be distinguished from that kind of “inertial” force.
37
Chapter 2
Transformations
x00 x0 x0
1 0 0 0
x01 0 −1 0 0 x1 −x1
= =
x02 x2 −x2
0 0 −1 0
x03 0 0 0 −1 x3 −x3
38
One can also conceive the specular inversion of only one of the coordinates,
as the x-inversion or the x–and–y-inversion:
1 0 0 0 1 0 0 0
0 −1 0 0 0 −1 0 0
; . (2.3)
0 0 1 0 0 0 −1 0
0 0 0 1 0 0 0 1
39
Ti and Tj commute. If Ti · Tj = Tj · Ti is true for all pairs of members of G, G
is said to be a commutative, or abelian group. A subgroup of G is a subset
H of elements of G satisfying the same rules.
§ 2.3 The transformations P and T above do form a group, with the com-
position represented by the matrix product. P and T , if applied twice, give
the identity, which shows that they are their own inverses. They are actually
quite independent and in reality constitute two independent (and rather triv-
ial) groups. They have, however, something else in common: they cannot be
obtained by a step–by–step addition of infinitesimal transformations. They
are “discrete” transformations, in contraposition to the “continuous” trans-
formations, which are those that can be obtained by composing infinitesimal
transformations step–by–step. Notice that the determinants of the matrices
representing P and T are −1. The determinant of the identity is +1. Adding
an infinitesimal contribution to the identity will give a matrix with determi-
nant near to +1. Groups of transformations which can be obtained in this
way from the identity, by adding infinitesimal contributions, are said to be
“continuous” and “connected to the identity”. P and T are not connected
to the identity.
0 0 1 0 0 0 0 0 1
40
It gives an infinitesimal rotation in the plane (x1 , x2 ):
0
x1 1 −δφ 0 x1 x1 − δφ x2
20
x = δφ 1 0 x2 = x2 + δφ x1 .
30
x 0 0 1 x3 x3
0 0 0
x3
§ 2.5 But there is more. The set of N ×N matrices, for any integer N , forms
also a vector space. In a vector space of matrices, we can always choose a
linear base, a set {Ja } of matrices linearly independent of each other. Any
other matrix can be written as a linear combination of the Ja ’s: W = wa Ja .
We shall suppose that all the elements of a matrix Lie group G can be written
in the form M = exp[wa Ja ], with a fixed and limited number of Ja ’s. The
matrices Ja are called the generators of G. They constitute an algebra with
the operation defined by the commutator, which is the Lie algebra of G.
41
2.2 Orthogonal Transformations
§ 2.6 We have said in § 1.22 that a group of continuous transformations
preserving a symmetric bilinear form η on a vector space is an orthogonal
group or, if the form is not positive–definite, a pseudo–orthogonal group.
Rotations preserve the distance d(x, y) of E3 because R(ω) is an orthogo-
nal matrix. Let us see how this happens. Given a transformation represented
by a matrix M ,
0
X 0
xi = M i j xj ,
j
The matrix form of this condition is, for each group element Λ,
ΛT η Λ = η , (2.7)
42
where ΛT is the transpose of Λ. It follows clearly that det Λ = ±1.
When η is the Lorentz metric, the above condition defines a Λ belonging
to the Lorentz group.
(Λ−1 )α β 0 = Λβ 0 α .
Comment 2.2 Actually, it follows from the formal identity det M = exp[tr ln M ] that
det M = 1 ⇒ tr ln M = 0.
We shall need some notions of algebra, Lie groups and Lie algebras. They
are introduced through examples in what follows, in a rather circular and
repetitive way, as if we were learning a mere language.
43
§ 2.7 The invertible N × N real matrices constitute the real linear group
GL(N, R). Members of this group can be obtained as the exponential of
some K ∈ gl(N, R), the set of all real N × N matrices. GL(N, R) is thus a
Lie group, of which gl(N, R) is the Lie algebra. The generators of the Lie
algebra are also called, by extension, generators of the Lie group.
Consider the set of N × N matrices. This set is, among other things, a
vector space. The simplest matrices will be those ∆α β whose entries are all
zero except for that of the α-th row and β-th column, which is 1:
(∆α β )δ γ = δ δ α δ β γ . (2.9)
δ
= δ β φ ∆α ξ (2.10)
where, in (∆α β )δ γ , γ is the column index.
§ 2.8 Algebra This type of operation, taking two members of a set into
a third member of the same set, is called a binary internal operation. A
binary internal operation defined on a vector space V makes of V an algebra.
The matrix product defines an algebra on the vector space of the N × N
44
matrices, called the product algebra. Take now the operation defined by
the commutator: it is another binary internal operation, taking each pair
(A, B) into the matrix [A, B] = AB − BA. Thus, the commutator turns the
vector space of the N × N matrices into another algebra. But, unlike the
simple product, the commutator defines a very special kind on algebra, a Lie
algebra.
§ 2.9 Lie algebra A Lie algebra comes up when, in a vector space, there is
an operation which is antisymmetric and satisfies the Jacobi identity. This
is what happens here, because [A, B] = −[B, A] and
This Lie algebra, of the N × N real matrices with the operation defined by
the commutator, is called the real N -linear algebra, denoted gl(N, R). A
theorem (Ado’s) states that any Lie algebra can be seen as a subalgebra of
gl(N, R), for some N .
The members of a vector base for the underlying vector space of a Lie
algebra are the generators of the Lie algebra. {∆α β } is called the canonical
base for gl(N, R). A Lie algebra is summarized by its commutation table.
For gl(N, R), the commutation relations are
β δ
∆α , ∆φ ζ = f(α )(φ ) (γ ) ∆γ δ .
(2.11)
β ζ
The constants appearing in the right-hand side are the structure coefficients,
whose values in the present case are
δ
f(α )(φ ) (γ ) = δφ β δα γ δδ ζ − δα ζ δφ γ δδ β . (2.12)
β ζ
w2 = wα β wφ ξ ∆α β ∆φ ξ = wα β wβ ξ ∆α ξ = (w2 )α ξ ∆α ξ
w3 = wα β wφ ξ wγ δ ∆α β ∆φ ξ ∆γ δ = (w3 )α ξ ∆α ξ , etc,
45
and M will have entries
∞
!r0 ∞
r0
wα β
r 0 X wn X 1 0
M s = e β ∆α
s = s = (wn )r s .
n=0
n! n=0
n!
The generators of so(η) will then be Jαβ = ∆αβ - ∆βα , with commutation
relations
[Jαβ , Jγδ ] = ηαδ Jβγ + ηβγ Jαδ − ηβδ Jαγ − ηαγ Jβδ . (2.14)
These are the general commutation relations for the generators of orthogonal
and pseudo–orthogonal groups. We shall meet many cases in what follows.
Given η, the algebra is fixed up to conventions. The usual group of rota-
tions in the 3-dimensional Euclidean space is the special orthogonal group,
denoted by SO(3). Being “special” means connected to the identity, that is,
represented by 3 × 3 matrices of determinant = +1.
Exercise 2.5 When η is the Lorentz metric, (2.14) is the commutation table for the
generators of the Lorentz group. Use Exercise 2.4 to prove (2.14).
46
particular, the group O(3) is formed by the orthogonal 3 × 3 real matrices.
The group U (2) is the group of unitary 2 × 2 complex matrices. SO(3) is
formed by all the matrices of O(3) which have determinant = +1. SU (2) is
formed by all the matrices of U (2) which have determinant = +1.
Comment 2.3 If a group SO(p, q) preserves η(p, q), so does the corresponding affine
group, which includes the translations. We should be clear on this point. When we write
xj , for example, we mean xj − 0, that is, the coordinate is counted from the origin. The
translation indicated by aj is a global translation of the whole space — of the point and
of the origin together. xj − 0 goes into (xj + aj ) - (0 + aj ). Vectors are not changed.
All points are changed in the same way, so that differences remain invariant. This would
perhaps become more evident if we noticed that the (squared) distances between two point
x and x0 in E3 are actually always η(x0 − x, x0 − x) = (x0 − x)2 + (y 0 − y)2 + (z 0 − z)2 .
Translations lead simultaneously x → x + a and x0 → x0 + a. The affine group related to
SO(r, s) is frequently called “inhomogeneous SO(p, q)” and denoted ISO(r, s). In cases
given below, the Euclidean group on E3 can be indicated also by ISO(3), and the Poincaré
group, also called inhomogeneous Lorentz group, by ISO(3, 1).
47
2.3 The Group of Rotations
§ 2.12 The rotation group in Euclidean 3-dimensional space can actually be
taken either as the special orthogonal group SO(3) or as the special unitary
group SU (2). Their Lie algebras are the same. The distinction lies in their
manifolds, and has important consequences. For example, SU (2) has twice
more representations than SO(3). In particular, the Pauli representation
given below (Eq. 2.23) is a representation of SU (2), but not of SO(3). As
far as space only in concerned, it is SO(3) which is at work, but some fields
— spinors — can “feel” the difference. In this section we shall be using the
ordinary metric gij = δij for the Euclidean space, so that there will be no
difference at all between raised and lowered indices.
Comment 2.4 Lie groups are differentiable manifolds. The SU (2) manifold is a 3-
dimensional sphere S 3 , while the SO(3) manifold is like half the sphere S 3 . The topology
of SU (2) is consequently simpler. It all happens as if SU (2) “covered” SO(3) twice.
Technically, SU (2) is indeed the double covering of SO(3).
[Jb , Jc ] = if a bc Ja (2.15)
will be that in which their matrix elements are just the structure coefficients,
Given a Lie algebra, the representation whose matrix elements are just the
structure coefficients is called the “adjoint representation” of the algebra.
It is more deeply concerned with the group geometry than any other rep-
resentation. For SO(3), this representation coincides with the fundamental
one, which is not the case for most groups.
For the rotation group, the structure constants are given by f a bc = abc ,
the Kronecker completely antisymmetric symbol given in Exercise 1.7. Thus,
(2.15) is just the usual table of commutators
48
(if we take hermitian matrices for the Jk ’s; if antihermitian, just drop the i
factors). The matrices are
0 0 0 0 0 i 0 −i 0
J1 = 0 0 −i ; J2 = 0 0 0 ; J3 = i 0 0 .
0 i 0 −i 0 0 0 0 0
(2.18)
Without the i factors, the generators constitute also a basis for the underlying
vector space of the Lie algebra. In terms of the group parameters, which can
be taken as 3 angles collectively represented by the vector ω = (w1 , w2 , w3 ),
a generic member of the Lie algebra will be
0 −ω3 ω2
W = − i Ja wa = ω3 0 −ω1 .
−ω2 ω1 0
are particularly useful for computations. The intuitive meaning of the expo-
nential matrix becomes clear when we apply it to a vector x:
xk = (u · x)u
49
and a piece orthogonal to u,
x⊥ = (u × x) × u cos w = x − (u · x)u,
A complete rotation around the 3rd axis is given by ω = (0, 0, 2π). It leads
to R(0, 0, 2π)x = xk + x⊥ = x. As expected, a complete rotation around an
axis corresponds to the identity transformation: R(0, 0, 2π) = I. We shall
see below, in the discussion of the Pauli representation, that this result is
not as trivial as it may seem.
Notice that these transformations can be seen as acting on the “position
vector” x, or any other vector. We can give here an important step towards
abstraction: to define a vector as an object transforming as above. Still
better: an object transforming as above is said to belong to the “vector
representation” of the group.
ω1 ω3 ω2 ω3 −ω2 2 − ω1 2
50
Comment 2.5 γab is actually a metric on the group manifold, called the Killing–Cartan
metric [more about that far below, see Eq.(8.64)]. We have chosen the factors so that, for
the rotation group, it coincides with the Euclidean metric (1.27) in cartesian coordinates,
1 0 0
(δij ) = 0 1 0 . (2.22)
0 0 1
Exercise 2.6 For the squared angular momentum, it is usual to write the eigenvalue in
the form J 2 = j(j + 1). Show that, for the vector representation given by matrices (2.18),
J 2 = 2 and j = 1.
§ 2.15 For general Lie groups and algebras, more invariants can be found,
each one related to an invariant multilinear form. The number of independent
invariants will be the rank of the Lie algebra. The rotation group has rank 1:
the above invariant is the only one. The Euclidean group, the group SU (3),
the Lorentz group and the Poincaré group are rank-2 groups and will have
two independent invariants. As the invariants commute with every operator,
they have the same eigenvalues for all the states of a given representation.
An important result from group theory is the following: a representation
is completely characterized by the eigenvalues of the independent invariant
operators. As said above, the number of independent invariant operators is
the rank of the group. Their eigenvalues are consequently used to label the
representations. Of course, any function of the invariants is itself invariant.
It is then possible to choose the independent invariants to be used — with
obvious preference for those with a clear physical meaning.
§ 2.16 The action of the group transformations on its own Lie algebra is
called the adjoint representation of the group. The algebra members are
transformed by similarity: given any element M = Ja M a in the algebra, it
will transform according to M → M 0 = g −1 M g. Because of its role in the
differential structure of the Lie group, this is the most important represen-
tation, on which all the other representations somehow mirror themselves.
Here, for an arbitrary element M ,
sin ω 1 − cos ω
M 0 = R−1 M R = M cos ω + [M, W ] + W (M · ω) .
ω ω2
51
Comment 2.6 We see easily why γab = 12 tr(Ja Jb ) is an invariant: it is a trace, and
traces are invariant under similarities because tr(AB) = tr(BA) and consequently trM 0
= tr(R−1 M R) = tr(RR−1 M ) = trM .
Comment 2.7 The form (2.19) is one of many possible different parameterizations for a
given rotation. It has been used because it gives special emphasis to the role of the Lie al-
gebra and the related adjoint representation. The Euler angles, much used in the problem
of the rigid body, provide another. Lie groups are manifolds of a very special kind and,
roughly speaking, parameters are coordinates on the group manifold. Changing parame-
terizations corresponds to changing coordinate systems. As with coordinates in general, a
special parameterization can ease the approach to a particular problem. The Cayley-Klein
parameters, for example, are more convenient to solve some gyroscope problems.
It is easily checked that the generators of the Lie algebra, satisfying (2.17),
are actually { 12 σa }. The general element W in the Lie algebra representation
is now given by
!
ω3 ω 1 − iω 2
W = 12 σa ωa = 12 .
ω1 + iω2 − ω3
Exercise 2.7 In a way analogous to Exercise 2.6, show that for the spinor representation
given by the Pauli matrices (2.23), the values are indeed J 2 = 3/4 and j = 1/2.
§ 2.19 We have seen above that R(0, 0, 2π) = I for the vector representation.
Here, however, we find immediately that R(0, 0, 2π) = I cos π = − I. A
52
complete rotation around an axis does not lead back to the original point.
Only a double complete rotation, like R(0, 0, 4π), does. Objects belonging to
this representation are deeply different from! vectors. They
! are called (Weyl
1 0
or Pauli) spinors. The column vectors and , eigenvectors of σ3 ,
0 1
can be used as a basis: any such spinor can be written in the form
! ! !
1 0 ψ↑
ψ = ψ↑ + ψ↓ = . (2.25)
0 1 ψ↓
Comment 2.8 The reason for the “arrow” notation is the following: σ3 will appear later
as the operator giving the spin eigenvalues along the axis 0z: eigenvalue +1 for spin “up”,
eigenvalue -1 for spin “down”. The general Pauli spinor will be a superposition of both.
Comment 2.9 Non–matrix representations can be of great help in some cases. For ex-
ample, take the functions Ψ(x) defined on the Euclidean 3-dimensional space. Acting on
them, the set of differential operators Jm = imrs xr ∂ s satisfy the rules (2.17) and lead to
the invariant value j = 1. They provide thus a representation equivalent to that of matri-
ces (2.18). Two distinct kinds of operators, acting on different spaces but with the same
value of the invariants are said to provide different “realizations” of the corresponding
representation.
§ 2.20 We have above defined the vector representation of the group. A vec-
tor, or a member of the vector representation, is any object v = (v 1 , v 2 , v 3 )T
0 0 0
transforming according to v = R(ω) v, or , (v 1 , v 2 , v 3 )T = R(ω) (v 1 , v 2 , v 3 )T .
We shall use the compact version
0 0
v i = Ri j v j ,
0
giving each component. The position vector is a particular case, with xi =
0
Ri j xj . A question may come to the mind: how would a composite object,
such as one with components like xi xj , transform? Well, each piece will
0 0 0 0
transform separately, and xi xj = Ri m Rj n xm xn . Any object transforming
in this way,
0 0 0 0
T i j = Ri m Rj n T mn ,
is said to be a second–order tensor, or to belong to a tensor representation of
the group. Notice that there is no need at all that it be a product like xi xj :
the only thing which matters is the way it undergoes the transformations of
53
interest. In the same way, higher–order tensors are defined: a tensor of order
r is any object transforming according to
0 0 0 ... ir 0 0 0 0 0
T i1 i2 i3 = R i1 j1 R i2 j2 R i3 j3 . . . R ir jr T j1 j2 j3 ... jr
.
54
2.4 The Poincaré Group
§ 2.22 Different observers, as said, are supposed to be attached to different
inertial reference frames. The field equations must be valid and the same
in any inertial reference frame. This means that they must be written as
equalities with the right– and left–hand sides transforming in exactly the
same way. We say then that they are “in covariant form”. To obtain covariant
equations, it is enough that they come as extrema of an action functional
which is invariant under the group of transformations of inertial frames.
The transformations of the Poincaré group are, by definition, those con-
tinuous transformations under which is invariant the interval between two
events, or points on Minkowski spacetime.
The Poincaré transformations are of two types, each constituting a sub–
group: translations and Lorentz transformations. The latter leave the above
expression invariant because they are the (pseudo) rotations in Minkowski
space, on which the interval represents a (pseudo) distance. Spacetime trans-
lations constitute the so–called inhomogeneous part of the Poincaré group.
They leave the interval invariant because they change both the origin and
other points in the same way. Mathematically speaking, this means that
spacetime should be seen not as a vector space, but as an affine space:
Minkowski spacetime has no preferred origin. The interval is also invariant
under some discrete transformations: inversions (2.2), (2.3) of the space axis
and of the time axis (2.4). Some authors round up these transformations and
those constituting the Lorentz group into the “full Lorentz group”. Other
withdraw from this denomination the time inversion. We shall consider only
the continuous transformations here, and consider a Poincaré transformation
as a Lorentz transformation plus a translation. Given cartesian coordinates
on spacetime,
x0α = Λα β xβ + aα . (2.26)
The interval will remain invariant if
55
to indicate (2.26) by the compact notation
L = (Λ, a) . (2.28)
Comment 2.10 In (2.26), a Lorentz transformation is performed first, and then transla-
tions are applied: (Λ, a) = (1, a)·(Λ, 0). This is the most widely–used convention. Another
is possible: we could have written x0a = Λα β (xβ + aβ ) and (Λ, a) = (Λ, 0) · (1, a). In the
latter parameterization, a Lorentz transformation is applied to the already translated
point.
56
original frame. It is in this sense that a configuration is invariant in Special
Relativity: it is impossible to discover, by the sole analysis of the states, on
which frame the system stands. Thus, given an L as in (1.77), there exists
acting on the configuration space an operator U (L) such that
φ0 = U (L) φ .
Suppose then that, acting on the configuration space, there are operators
U (L1 ), U (L2 ), U (L3 ), etc, corresponding to the transformations L1 , L2 , L3 ,
etc. And suppose furthermore that those operators respect the group condi-
tions:
We say then that U (L) “represents” L, and that the set of all U (L) constitutes
a representation of the group on the configuration space.
Comment 2.11 This scheme is quite general, holding for any group acting on some space,
though not every space accepts a representation of a given group.
57
would provide linear transformations. They are said to constitute a linear
representation. Notice that non–linear representations are quite possible, but
the formalism they are involved in is far more complicated. For simplicity,
we shall devote ourselves almost exclusively to linear representations, which
are realized by fields with components.
As said, we should specify the group whenever we use the expressions
“vectors” and “tensors”. The tensors used in General Relativity are tensors
of the group of general coordinate transformations. We shall be interested,
by now, in the Lorentz group tensors (which are unaffected by translations).
A tensor of a generic order N (or: a field belonging to a tensor representation)
will transform according to
0 0 0 0 0 0
T µ1 µ2 ...µN = Λµ1 ν1 Λµ2 ν2 . . . ΛµN νN T ν1 ν2 ...νN (2.30)
§ 2.24 A first important particular case is the scalar field which, because it
has only one component, is invariant:
Comment 2.12 When a field belongs to a representation but has only one component,
we say that it is a singlet. This terminology holds for other groups: whenever a field
ignores a symmetry group, we put it into a singlet representation. The Lagrangian of any
theory is a scalar with respect to all the symmetries.
58
When talking about relativistic fields, we classify them as scalars, vectors,
2nd-order tensors, spinors, etc. All these names refer to their behavior under
the Lorentz group.
§ 2.25 A surprise comes out in this story: the tensor representations do not
cover all the linear representations. This is due to the fact that the mappings
Λ → U (Λ) defining the representations are not necessarily single–valued. We
have seen that this happens for the SU (2) Pauli representation: a rotation
of an angle 2π, which in E3 is the same as a rotation of 4π, is taken into +1
or -1.
Comment 2.13 For the time being, only particles corresponding to low order tensors
and spinors (that is, small values of j) have been discovered in Nature.
| < φ(x)|ψ(x) > |2 = | < φ0 (x0 )|ψ 0 (x0 ) > |2 = | < φ(x)|U † (L)U (L)|ψ(x) > |2 ,
that is,
U † (L) U (L) = ±1 .
The operators U (L) must, consequently, be either unitary or antiunitary —
which, by the way, is true for any symmetry. In the Hamiltonian case, for
example, two systems are equivalent when related by a canonical transfor-
mation. Here, two systems are equivalent when related by a unitary or antiu-
nitary transformation. A discrete transformation can, in principle, be repre-
sented by an antiunitary operator. For continuous transformations, however,
which can, when small enough, be seen as infinitesimally close to the identity,
the operators must be actually unitary, as for small transformations both U
59
and U † are both close to the identity. Consequently, an U (L) connected to
the identity will have the form
U (L) = eiJ ,
where aµ and αµν = − ανµ are the parameters of the transformations and
P µ , M µν their generators, matrices in the linear representations. The com-
mutation relations of the generators are characteristic of the group itself and,
consequently, independent of the representation. The generators themselves
belong to a vector space and, with the operation defined by the commuta-
tor, constitute an algebra. The commutators are antisymmetric and satisfy
the Jacobi identity, defining the Lie algebra of the group. The Poincaré
commutation rules are
[P µ , P ν ] = 0 (2.37)
µν λ
= − i P µ η νλ − P ν η µλ
M ,P (2.38)
[M µν , M ρσ ] = i (M µρ η νσ + M νσ η µρ − M νρ η µσ − M µσ η νρ ) . (2.39)
P µ = i ∂µ (2.40)
M µν = − i (xµ ∂ ν − xν ∂ µ ) (2.41)
60
whose operators act on functions defined on spacetime (when no place for
∂
confusion exists, we shall be using the notation ∂α = ∂x α ). We have said
61
them. These are special combinations of the momentum and the angular
momentum. We shall see later how to obtain the momentum and angular
momentum for fields (this is the role of the first Noether theorem), and here
only illustrate the ideas in the case of a mechanical particle. The invariant
best related to mass is precisely Pµ P µ , whose eigenvalues are well–known
from Special Relativity: Pµ P µ = m2 c2 , m being the rest mass of the particle
of 4-momentum P µ . In realization (2.40), this invariant operator is (minus)
the D’Alembertian. Another invariant operator is Wµ W µ , where Wµ is the
Pauli–Lubanski operator
Indices are here raised and lowered with the Lorentz metric, so that 0123 = − 1. In
4-dimensional space there is no more a relationship between vectors and antisymmetric
matrices. However, the symbol allows the definition of the dual: given a 2nd order anti-
symmetric tensor F ρσ , its dual is defined as
1
F̃µν = 2 µνρσ F ρσ . (2.45)
Notice that, in order to prepare for the contraction, indices must be raised. The dual,
consequently, depends on the metric. Some identities come out from the contractions of
the symbols themselves:
1. once contracted:
µνρσ µαβγ = δνα δρβ δσγ − δνα δσβ δργ − δσα δρβ δνγ + δσα δνβ δργ + δρα δσβ δνγ − δρα δνβ δσγ (2.46)
62
2. twice contracted:
µνρσ µνβγ = 2 (δρβ δσγ − δσβ δργ ) (2.47)
3. thrice contracted:
µνρσ µνργ = 3! δσγ = 6 δσγ (2.48)
4. totally contracted:
µνρσ µνρσ = 4! = 24 . (2.49)
63
can take, for the rotations, those of (2.18) transmuted into 4 × 4 matrices:
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 i 0 0 −i 0
J1 = ; J2 = ; J3 = ;
0 0 0 −i 0 0 0 0 0 i 0 0
0 0 i 0 0 −i 0 0 0 0 0 0
The relationship is given by Jk = 12 kij Jij ; Jk0 = iKk ; J0k = − iKk . The
Lorentz matrices can then be written as exponentials,
µ
Λµ ν = exp[ 2i ω αβ Jαβ ] ν ,
Notice that the generators Jαβ generate the complete Lorentz transformation,
including the change x → x0 . In effect, it is impossible to effectuate a Lorentz
64
transformation on the functional form φν alone, as the argument is itself a
Lorentz vector. The infinitesimal transformations, to first order in a small
parameter δω αβ , follow directly:
that is,
Equating the two expressions for φ0µ (x0 ), we obtain the infinitesimal trans-
formation at fixed point x:
65
66
Chapter 3
Introducing Fields
67
continuous infinity of degrees of freedom is what we shall call a classical
field. Instead of the above q k , the system is represented by a function φ(x),
with φ replacing q and the continuous variable x replacing the tag k. The
simplest among the systems of this kind is the so-called 1-dimensional solid,
or better, the vibrating line. Each point of the continuous elastic solid takes
part in the dynamics and the whole motion can only be described if all their
positions are specified.
φ1 φ2 φj φj+1 φj+3
z}|{ z}|{ z}|{ z}|{ z}|{
0 1 2 3 j j+1 j+3 N −1 N
•—
| —
•—|———–|–• ——| - - - |—•——
—|——•—|——•—| - - - |—
———•|
|← →|
a
The discrete vibrating line
Let φj be the displacement of the j-th atom with respect to its equilibrium
position and suppose (here dynamics comes in) harmonic forces to be acting
between (only) nearest neighbors, all with the same elastic constant K. This
j-th atom will obey Newton’s equation of motion
Vj = K4 (φj+1 − φj )2 + (φj−1 − φj )2 .
(3.2)
∗
E. M. Henley and W. Thirring, Elementary Quantum Field Theory, McGraw-Hill,
New York, 1962.
68
The total Lagrangian function for all the atoms will be
1
PN h 2 2
i
L = 2 j=1 mφ̇j − K (φj+1 − φj ) , (3.3)
§ 3.4 The Hamilton equations are particular cases of the Liouville equation
Ḟ = {F, H},
which governs the evolution of a general dynamical function F (φ, π). The
curly bracket is the Poisson bracket:
X ∂A ∂B ∂A ∂B
{A, B} = − .
j
∂φj ∂πj ∂πj ∂φj
The only nonvanishing Poisson brackets involving the degrees of freedom and
their momenta are {φi , πj } = δij .
We have thus the three main approaches to the one-dimensional system of
coupled oscillators, whose complete description requires the knowledge of all
69
the displacements φi with respect to the equilibrium positions φi = 0. Two
questions remain: (i) the Lagrangian (3.3) is not as yet completely specified,
as the summation requires the knowledge of φ0 and φN +1 ; (ii) the physical
problem is not well characterized, as the boundary conditions are missing.
Both problems are solved by taking periodic conditions: φi+N = φi . This
corresponds to making the extremities join each other (N = 0). In fact, we
had been cheating a bit when we wrote Eq.(3.3). As the summation runs
from i = 1 to i = N , that expression only acquires a meaning after a periodic
condition φ0 = φN , φN +1 = φ1 is imposed.
§ 3.5 The degrees of freedom are coupled in Eqs. (3.1). To solve the equa-
tions, it is highly convenient to pass into the system of normal coordinates,
in terms of which the degrees of freedom decouple from each other. Such
coordinates φ̃i , and their conjugate momenta π̃i , will be such that
N/2 N/2
i 2πnj 2πnj
X X
φj = √1
N
e N φ̃n ; πj = √1
N
ei N π̃n . (3.5)
n=−N/2 n=−N/2
to give
N N
−i 2πnm 2πnm
X X
φ̃m = √1
N
e N φn ; π̃m = √1
N
e−i N πn . (3.7)
n=1 n=1
Though we have passed from the real φi , πi to the complex variables φ̃i ,
π̃i , the number of independent variables is the same because φ̃−j = φ̃∗j and
π̃−j = π̃j∗ . We find easily that the only nonvanishing Poisson brackets can
be summed up as {φ̃i , π̃j∗ } = δij . It is not difficult to show, with the help of
(3.6), that
N N/2
X X
πj =2
π̃j∗ π̃j ,
j=1 j=−N/2
and
N N/2
X 2
X πj
K
2
(φj − φj−1 ) = m
2
φ̃j φ̃∗j 2
4ω sin 2
.
j=1
N
j=−N/2
70
Consequently, in normal coordinates, the Hamiltonian function reads
h ∗ i
πn 2 ∗
1
PN/2 π̃n π̃n
H = 2 n=−N/2 m + m 2ω sin N φ̃n φ̃n . (3.8)
wk = 2ω sin πk
N
(3.9)
instead of ω, decouple entirely from each other. They are, of course, the
normal modes of the system. The equations of motion become simply
with solutions
1 ˙
φ̃n (t) = φ̃n (0) cos ωn t + φ̃n (0) sin ωn t . (3.11)
ωn
§ 3.6 The normal modes of vibration of the system are thus the Fourier
components of the degrees of freedom. They are “collective” degrees of free-
dom, in the sense that each mode contains information on all the original
degrees of freedom. On the space of degrees of freedom, they are “global”.
The oscillation frequency (3.9) is such that
For each one of these oscillators, we can introduce new variables ak and a∗k
as
1 h i
ak = √ mωk φ̃k + iπ̃k (3.13)
2mωk
71
1 h i
a∗k =√ ∗ ∗
mωk φ̃k − iπ̃k . (3.14)
2mωk
As a∗k 6= a−k , the total number of variables remains the same. The only
nonvanishing Poisson brackets are now {ai , a∗j } = − i δij . The equations of
motion become ȧk = − i ωk ak , with solutions
Both ak and a∗−k will have frequencies with the same sign. It is convenient
to redefine ωk as the positive object ωk = 2ω| sin(πk/N )| and take the mass
m = 1. Once this is made,
1
ak + a∗−k .
φ̃k = √
2ωk
Thus, φ̃k has only contributions of frequencies with the same sign. If we
establish by convention that these frequencies are to be called positive, it is
easy to see that φ̃−k = φ̃∗k will only have contributions of negative frequencies.
Let us substitute φ̃k in (3.5):
N/2
X 1
ak + e−i2πkj/N a∗k .
i2πkj/N
√1
φj (t) = N
√ e (3.16)
k=−N/2
2ωk
§ 3.7 We now intend to change into the continuum case. There is a well-
known recipe to do it, inspired in the trick to go from finite to continuum
Fourier transforms. We know where we want to arrive at, and we find a
procedure to accomplish it. But it should be clear that it is only that — a
practical recipe.
The prescription goes as follows. First, we take the limits a → 0 and
N → ∞ simultaneously, but in such a way that the length value L = N a
remains finite. On the same token, each intermediate label value k tends to
infinity in a way such that the product ka retains a finite value (the “distance
to the origin”). We call x this value, x = ka. Combining dimensional reasons
and the necessity to keep finite the kinetic energy, the summation N
P
n=1 and
the degree of freedom φ must behave like
N L √
Z
X 1
→ dx ; φj → a φ(x) . (3.17)
n=1
a 0
72
We have then that
1 √ φ(x + a) − φ(x) √ ∂φ
(φj+1 − φj ) = a → a .
a a ∂x
The equation of motion (3.1) will now be
2 2 1 ∂φ ∂φ
φ̈ = ω a −
a ∂x x ∂x x−a
and, in the limit, turns up as the wave equation
∂ 2φ 2
2 ∂ φ
=c , (3.18)
∂t2 ∂x2
with the parameter c = ωa as the velocity of wave propagation. The
finiteness of c requires that the frequency ω become infinite. The Lagrangian
density becomes
" 2 #
1 L
Z
∂φ
L[φ] = dx m φ̇2 − c2 , (3.19)
2 0 ∂x
arriving at
Z Z
1 ikx 1
φ(x) = √ dk e φ̃k ; π(x) = √ dk eikx π̃k . (3.22)
L L
There are two different kinds of summation limits, and differences in the
absorption of factors between φ and φ̃, as seen in (3.17) and (3.21). The
degrees, and their Fourier components, acquire dimensions in the process.†
†
The function exp[ikx] is typically a wave which repeats itself when x = 2πn/k, with
n = ±1, 2, 3, . . . . As the number of times the wave repeats itself in a cycle of length 2π, k
is called the wave-number. As an exponent can have no dimension, k must have dimension
inverse to x.
73
.
The continuum limit of (3.16), with ωk → k ω a → kc = ωk (the last
dot–equality indicating a redefinition of the symbol ωk ), is
Z
dk i(kx−ωk t)
ak (0) + e−i(kx−ωk t) a∗k (0) .
φ(x, t) = √ e (3.23)
2Lωk
§ 3.8 This example may seem a parenthesis a bit too long. We are here
using the vibrating line only as a suggestive illustration,‡ which anticipates
many points of interest. It gives due emphasis to the meaning of the position
coordinate x, transmutation of the old label i: it should not be mistaken by
a generalized coordinate. It is a parameter, appearing in the argument of
the field φ(x, t) on equal footing with the time parameter. This is manifest
in the fact that the equation of motion (3.18) does not come from (3.19) as
the usual Lagrange equation,
δL ∂L d ∂L
= − =0, (3.24)
δφ ∂φ dt ∂ φ̇
Comment 3.1 The expression in (3.24) is the Lagrange derivative of Classical Mechanics
when the degree of freedom and its first time derivative are enough to fix the problem.
When higher-order derivatives are necessary, the derivative is
δ ∂ d ∂ d2 ∂
k
= k − k
+ 2 k − ... , (3.26)
δq ∂q dt ∂ q̇ dt ∂ q̈
with alternating successive signs. This derivative takes into account the effect of a co-
ordinate transformation on the time derivatives. It is the first example of a covariant
derivative: δqδk F (q, q̇, q̈) transforms just as F (q, q̇, q̈) under coordinate transformations.
74
on one same space, “spacetime”. One aspect of this effect is the change
undergone by the action, from (3.4) to (3.20). In the latter, space and time
parameters are equally integrated over. And the Lagrange derivative changes
accordingly. There is actually more: the above wave equation is not invari-
ant under transformations analogous to those of non- relativistic Physics, the
Galilei transforms x0 = x−vt and t0 = t. It is invariant under transformations
analogous to those of relativistic Physics, alike to Lorentz transformations:
x0 = γ(x − vt) and t0 = γ(t − vx/c2 ) with γ = (1 − v 2 /c2 )−1/2 .
Let us insist: we have defined a classical field as a continuous infinity
(labeled by x) of degrees of freedom, each one described by a function. The
generalized coordinates are the very fields φ(x), one for each value of x.
Thus, the parameter x spans the space of degrees of freedom. The equation
of motion is of the kind to be found later, governing relativistic fields. It is
usual to call it “field equation” in the continuum case.
One further remark: the decomposition into Fourier components is ex-
tensively used in canonical field quantization. In that case, it is the normal
modes which are quantized as oscillators, leading to the quantization rules
for the fields themselves. There is actually a further, usually overlooked,
proviso: in order to qualify as a field, the infinite variables describing the
system must be really at work. It may happen that many, or most of them,
are quiescent. In the above use of Fourier analysis, this would show up if
most of the modes were not actually active (φ̃n = 0 for many values of n).
We might, in that case, talk of a quiescent field. An example is the field of
fluid velocities in a laminar flow, for which most of the Fourier components
are not active. They become active at the onset of turbulence.§
75
local pressure and temperature in the inhomogeneous case, and so on. The
concept of field extends to non-mechanical systems (examples: the electric
field, the gravitational field, . . . ) exhibiting a continuous infinity of degrees
of freedom. In that case, the notion of field is actually inevitable for the
description of interactions, as the alternative — the description of interac-
tions by action at a distance — has never been given a simple, satisfactory
formulation. Though implicit in the work of Galilei and Newton, the notion
has been explicitly and systematically used by Faraday and has led to the
complete description of the classical electromagnetic phenomena synthesized
in Maxwell’s equations.
In Wave Mechanics, the state of a system is characterized by a wave-
function ψ(~x, t) (or better, by the ray to which ψ(~x, t) belongs in Hilbert
space), and its time evolution is ruled by the prototype of nonrelativistic
wave equation, the Schrödinger equation. The wavefunction must be known
at each point of spacetime, so that the system requires a continuous infinity
of values to be described. Function ψ(~x, t) has, consequently, the role of a
field. It is usual to obtain it from Classical Mechanics by using the so-called
quantization rules, by which classical quantities become operators acting on
the wavefunction. Depending on the “representation”, some quantities be-
come differential operators and other are given by a simple product. Thus,
the above ψ(~x, t) corresponds to the configuration–space representation, in
which the Hamiltonian and the 3-momenta are given by
∂
H→i~ ; (3.27)
∂t
~ →
p~ → ∇, (3.28)
i
and ~x is the operator acting on ψ(~x, t) according to ψ(~x, t) → ~x ψ(~x, t).¶ In
the case of a free particle, in which H = p2 /2m, these rules lead to the free
Schrödinger equation
∂ ~2 → 2
i~ ψ(~x, t) = − ∇ ψ(~x, t) . (3.29)
∂t 2m
¶
These rules have been of great help in guessing most of the basic facts of Quantum
Mechanics. But it should be clear that they are only guides, not immune to ambiguities.
For example, they must be changed in the presence of spin.
76
We shall from now on avoid the use of arrows, indicating 3–vectors by bold–
faced characters. For example, the 3–momentum will be written p = − i ~∇.
and
†
d
â = √1 Q̂ − iP̂ = √1 Q− ,
2 2 dQ
as well as the occupation-number operator N̂ = ↠â. We find immediately
that [â, ↠] = 1 and Ĥ = ~ω(N̂ + 21 ).
One next introduces a Fock space, generated by the set {|n >} of eigen-
kets |n > of N̂ . These kets are normalized in such a way that â|n > =
k
This paragraph is, as the title above announces, optional. It supposes some knowledge
of Quantum Mechanics and introduces some notions of the so-called “second quantization”
formalism, which lies outside the scope of the present text.
77
√ √
n |n − 1 >; ↠|n > = n + 1 |n + 1 >; N̂ |n > = n |n >. The number n
is interpreted as the number of quanta. The state with zero quanta |0 >,
such that â|0 > = 0 or N̂ |0 > = 0, is that with the minimum energy and
is called the “vacuum”. Its energy, by the way, is not zero — it is Ĥ|0 > =
(1/2)~ω|0 >. Each state |n > can be obtained from the vacuum by creating
√
n quanta: |n > = (1/ n!) (↠)n |0 >. Thus, this occupation-number repre-
sentation describes the oscillator in terms of excitation quanta. When there
is only one oscillator, all the quanta are identical — they are characterized
by the energy, which is the same for all. From these states one can pass to
other representations: usual wavefunctions in configuration space are ψn (x)
= < x | n >, wavefunctions in momentum space are ψn (p) = < p | n >, etc.
Comment 3.2 Turning again to the problem of coordinates on phase space: in Quantum
Mechanics, {q k } and {pk } become operators, represented by matrices. To specify a matrix
one needs all its entries, which are, in the case, infinite.
The vibrating line has a continuous infinity of oscillators, each one char-
acterized by the momentum k. The Fock space will consist of a continuous
infinity of kets, collectively indicated, for example, by |{nk } >. The field
φ(x, t) in (3.23) becomes consequently also an operator, which is the quan-
tized field of the material line:
Z
dk i(kx−ωk t)
ak (0) + e−i(kx−ωk t) a† k (0)
φ(x, t) = √ e
2Lωk
Z
dk ikx
e ak (t) + e−ikx a† k (t) .
= √ (3.30)
2Lωk
In the last step use has been made of (3.15). Applied to the vacuum |{nk =
0} >, this field spans the one-quantum states:
Z
dk
φ(x, t)|0 >= √ exp [−i(kx − ωk t)] |0, 0, . . . , 1k , · · · > ,
2Lωk
78
where the ket in the integrand indicates the state with one quantum of mo-
mentum k. Well, the vibrating line stands for a system of material mechanic
oscillators, necessarily quantal. This “quantum line” is the simplest example
of a quantum field.
A simple direct computation gives the complete, collective quantization
rules, including all the degrees of freedom:
3.3 Wavefields
§ 3.11 Wavefunctions are precisely the kind of fields we shall be most con-
cerned with. The word “classical” acquires here a more precise meaning:
the field ψ(x, t) will be classical as long as it is an usual function, whose
values are classical (real or complex) numbers (for short, “c–numbers”). In
that case, it will belong to spaces on which hold the same algebraic proper-
ties of the complex numbers. Field ψ(x) = ψ(x, t) will no more be classical
when it belongs to function spaces with more involved algebras (for example,
when not all fields commute with each other) and can no more be treated
as an ordinary function. It is the case of quantized fields, which are cer-
tain functionals or distributions, inhabiting spaces with non-trivial (though
well-defined) internal algebras. Thus, as it stands, “classical” here means
merely “non-quantum”. The standard procedure begins with classical fields
and proceeds to quantize them by changing their algebras. And the general
structure, be it Lagrangian or Hamiltonian, is transferred to the quantum
stage, so that the preliminary study of classical fields is inescapable. It would
be highly desirable to have a means of getting at the quantum description of
a system without the previous knowledge of its classical description, not the
least reason being the possible existence of purely quantum systems with no
classical counterpart. Or with many of them, as there is no reason to believe
that the classical limit of Quantum Mechanics be unique. This would avoid
the intermediate procedure of “quantization”. For the time being, no such
79
course to a direct quantum description is in sight.
The fields appearing in relativistic field theory are not, in general, of
the material type seen above. For that reason we shall not have the same
phenomenological, immediate intuition which has conducted us to conceive,
in the example of the vibrating line, the harmonic oscillator as a first rea-
sonable trial. In other words, the access to dynamics is far more difficult.
Phenomenology gives information of a general nature on the system, basically
its symmetries. Symmetries, though important also for non-relativistic sys-
tems, become the one basic tool when the energies involved are high enough
to impose the use of a relativistic approach.
A very important fact is that every symmetry of the Lagrangian is also
a symmetry of the equations of motion. The procedure for high energies is
rather inverse to that used for a mechanical system. For the latter, in general,
the equations of motion are obtained phenomenologically and the Lagrangian
leading to them (if existent) is found afterwards. In relativistic Field Theory,
most commonly we start from a Lagrangian which is invariant under the
symmetries suggested by the phenomenological results, because then the field
equations will have the same symmetries. In particular, the Lagrangian
involving relativistic fields will be invariant under the transformations of
the Poincaré group. This is imposed by Special Relativity: the behavior of
the system does not depend on the inertial frame used to observe it. That is
where the adjective “relativistic” comes from. Furthermore, depending on the
system, other symmetries can be present, some of them “external” (parity,
conformal symmetry, . . . ), other “internal” (isospin, flavor, color, . . . ). The
Lagrangian approach is specially convenient to account for symmetries, and
is more largely used for that reason. It will be dominating in this text.
80
in a way as analogous as possible to those on spacetime.
The general arguments on symmetries and invariance of the Lagrangian
extend to these symmetries. If the theory is invariant under such internal
transformations (say, symmetries related to the conservation of isospin, fla-
vor, color, etc), the Lagrangian will be invariant and the fields will necessarily
belong to representations of the corresponding groups. Fields invariant un-
der any transformation of G, so that φ0 (x) = φ(x), are supposed to stay in a
singlet (0-dimensional) representation.
Consider then those transformations changing only the functional form
of the fields. We shall consider only a very particular kind amongst all the
possibilities of such changes: the fields will be supposed to have components
in some “interior” space, and changes will be only combinations of these com-
ponents. This means that such “interior” transformations will be supposed
to be represented by linear representations of the symmetry groups. The
fields will present indices related to such internal representations. A scalar
field (meaning: a Lorentz scalar field) φ will, for example, appear as φa , the
a indicating a direction in an internal carrier vector space. A gauge potential
is a Lorentz 4-vector belonging also to a representation of the gauge group
and will appear in the form Aa µ . There is here, of course, a physicists’ bias.
Physicists are used to calling “fields” the components of certain mathematical
objects, and we shall not fight this long–established attitude. The main ideas
are best introduced through an example. The simplest non-trivial example
of internal symmetry is provided by isospin.
Isospin has been introduced by Heisenberg in the nineteen-fifties to ac-
count for what was then called the “charge independence” of the strong
interactions. The proton and the neutron had been observed to have identi-
cal strong interactions, and almost the same mass. The mass difference was
supposed to be of electromagnetic origin. As long as we could consider strong
interactions alone, and forget about electromagnetic interactions, they were
one and the same particle. Or better: they were seen as components of a
double wavefunction, a doublet like (2.25), called the “nucleon”:
!
p
N= .
n
81
A pure proton would be the analogous to the spin up state; the neutron,
the spin down. As all that had nothing to do with spin, the name (isobaric
spin, later) isospin was coined. The Lagrangian describing strong interac-
tions would be invariant under “internal” rotations, formally identical with
the above described, but in another, “internal” space. All this talk about “in-
ternal” things (space, components, wavefunctions) is only to help intuition.
It refers only to behavior under changes in the functional form of the fields,
changes independent of their arguments (that is, of spacetime). Coming back
to isospin: we have said that the proton-neutron system, or the nucleon, was
attributed isospin 1/2. The pions appear with 3 possible charges, but their
strong interactions ignore that difference. Thus, they also exhibit charge
independence and, by an analogy with the rotation group, were accommo-
dated in an isospin = 1 representation. In this way an internal symmetry
was revealed: particles were cased in carrier vector spaces, the symmetry
says that transformations in those spaces were irrelevant to Physics. In the
isospin case the group was supposed to be just SU (2), and two representa-
tions were immediately known. A field without isospin would be cased in the
scalar representation. This was the starting point of a very powerful method.
Once particles (quanta of fields) were found experimentally, people tried to
accommodate them into representations (“multiplets”) of some group. One
same multiplet for particles of close masses, different components for differ-
ent charges. The strong interactions would not “see” the components, only
each multiplet as a whole. It would in this way be “independent of charge”.
The isospin rotations are formally the same as given above, though without
any spacetime realization. The indices refer to internal space. For Pauli
matrices, another notation became usual: τ1 , τ2 and τ3 instead of σ1 , σ2 and
σ3 . A rotation like (2.24) in an isospin = 1/2 spinor representation, as that
of the nucleon, will consequently be written
i kτ
N 0 (x) = e 2 ω k
N (x) . (3.32)
82
first kind). When ω a depend on the point, the transformation is a local gauge
transformation (old name: gauge transformations of the second kind). The
formalism would remain much the same, except for the derivatives. It is clear
that a derivative ∂µ , when applied to
i
ω k (x)τk
N 0 (x) = e 2 N (x)
i k0
= U (x) ∂µ + 2i Ak µ (x)τk U −1 (x) .
2
A µ (x)τk (3.35)
Exercise 3.1 Try to show that, if Ak µ (x) transforms according to (3.35), then the deriva-
tive defined in (3.33) is indeed a covariant derivative, that is, Eq.(3.34) holds. This is why
the derivative is called “covariant” derivative: applied to a field, it transforms just in the
same way as the field itself.
83
been found to mediate most of the fundamental interactions of Nature. When
related to external symmetries, they turn up as the Christoffel symbols in
gravitation theory. For internal symmetries, they appear in two distinct fam-
ilies. As the gluon fields (with gauge group SU (3)), they are the mediating
fields of chromodynamics, supposed to describe the strong interactions be-
tween the quarks. In the electroweak theory describing electrodynamics and
weak interactions, they turn up, after a certain symmetry–breaking process,
as the fields describing the photon, the Z 0 and the pair W ± . Mathematically,
they are related to connections.
84
Chapter 4
General Formalism
85
for systems with two or more degrees of freedom, it is the whole set of
equations which is invariant under a change of system of generalized
coordinates, and not each one;
7. the structural analogy resultant from this unicity in the variety is ex-
tremely useful: once a particular procedure is found to be fruitful in
a particular Lagrangian theory, its application to other cases suggests
itself immediately;
86
§ 4.2 Negative points of the Lagrangian formalism:
5. qualifying item (8) above, the Lagrangian formulation does not lead
to a unique quantization procedure; actually, even the equations of
motion, which the present discussion may seem to suggest to be more
fundamental, fail to determine a unique quantization procedure.
§ 4.3 From the Hamiltonian point of view, two physical systems are equiv-
alent when there is a canonical transformation taking coordinates and mo-
menta of one system into the coordinates and momenta of the other. The
∗
Some anholonomic cases are amenable to a Lagrangian treatment, but they are
exceptional. Because the usual quantization procedure uses, directly or indirectly, the
Lagrangian formulation, there are many constrained systems which we do not know how
to quantize.
87
superiority of the Hamiltonian formulation rests precisely in its invariance
under canonical transformations, more general than the generalized coordi-
nate transformations. The Lagrangian and Hamiltonian formulations are not
always equivalent — for them to be, in Classical Mechanics, it is necessary
that the condition 2
∂ L
det 6= 0
∂ q̇i ∂ q̇j
hold,† which forebodes difficulties in the zero–mass cases.
88
4.1 Lagrangian Approach
4.1.1 Relativistic Lagrangians
§ 4.6 Let us start thinking about the form a Lagrangian function should
have. In a relativistic theory it should, to begin with, be invariant under the
transformations of the Poincaré group. Invariance of a Lagrangian under a
transformation ensures the covariance of the Euler–Lagrange equations under
the transformation. This is the reason for which Lagrangians, in relativistic
Field Theory, are supposed to be Poincaré invariant. The fields are the
degrees of freedom, supposed to provide with their gradients (first–order
derivatives with respect to space and time coordinates, corresponding to the
“velocities”) — a complete characterization of the system. A second–order
derivative would lead to a third–order equation, and in that case the fields
and velocities would be insufficient to describe the system. We take for
Lagrangians, therefore, invariant functions of the fields and their gradients.‡
Furthermore, our Lagrangians will have no explicit dependence on space or
time coordinates, which are only parameters. These comments summarize
the underlying spirit of Field Theory: the state of the system is characterized
by the fields and their first derivatives. And, though we are heating up
to a long discussion of special–relativistic fields, these considerations would
hold for a non–relativistic (or Galilei–relativistic) theory, with the Galilei
group taking the place of the Poincaré group. Another condition which we
shall impose is that the Lagrangian be real. The reason is that no classical
system has ever been found to suggest any kind of complex energy. In the
quantum case, the Lagrangian should be self–adjoint, or simply hermitian.
A non–hermitian Lagrangian would break probability conservation, thereby
violating the scattering matrix unitarity. But it can be used in some cases
(under the form of “optical potentials”) to describe non–isolated systems.
Next, we take a Lagrangian density as simple as possible. This is a rather
loose condition. Of course, given an invariant L, also L17 , arctan(L) or
any function of L, are invariant. We take the simplest possible invariant
‡
Function of fields mean function of the forms of the fields, of the way they depend
on their arguments. Thus, what we have are actually functionals.
89
functional of the fields and their derivatives leading to results confirmed by
experiment. The theories of polynomial type, with the Lagrangian density
a polynomial in the fields and their derivatives, are always the first trial.
Models with non–polynomial terms are, however, increasingly studied. And
a last condition, this one also dictated by simplicity. The Lagrangian will be
supposed to depend, at each point of spacetime, only on the values of the fields
and their derivatives in an infinitesimal neighborhood of that point. In this
case, the Lagrangian — and the theory — is said to be “local”. Non–local
theories are in principle conceivable, but they are extremely complex and of
a rather uncontrollable diversity. We shall ignore them.
§ 4.7 Thus, given a set of fields {φi (x)}, the Lagrangian density will have
the form
L(x) = L[φ(x), ∂µ φ(x)], (4.1)
where we use the notations φ(x) = {φ1 (x), φ2 (x), φ3 (x), . . . } and ∂µ φ(x) =
{∂µ φ1 (x), ∂µ φ2 (x), ∂µ φ3 (x), . . . }. Actually, as it is defined up to a gradient,
the density has no need to be a complete invariant. Under a Poincaré trans-
formation, it can acquire a gradient. In what follows, only its integral will be
taken as invariant. From the density L(x), one could obtain directly the field
equations, simply by generalizing the procedure of Classical Mechanics so as
to abrogate the special status of the time parameter t = x0 /c with respect to
the space coordinates. We shall, however, follow a more instructive method,
arriving at the field equations through the mediation of a variational prin-
ciple, Hamilton’s principle. Here an adaptation of that of particle classical
mechanics to the continuum case, it states that the action, defined by
Z
A[φ] = d4 xL(x) , (4.2)
is minimal for all states actually occurring, that is, for all solutions of the
equations of motion.
Notice that the action contains more on the system than the Lagrangian
density or the equations. The integration in (4.2) covers the whole spacetime
region occupied by the physical system to be described by the fields. Thus,
the action includes information on this region. In this sense, it is a “global”
90
characteristic of the system. In the case of particle mechanics, what is defined
is the action of a trajectory γ,
Z
A[γ] = dt L(x, t) , (4.3)
γ
the integral being performed along γ. The action is thus a function (better:
a functional) of the trajectory. It is global, it depends on the whole of γ.
In the relativistic case, it becomes a functional of the “field configuration”
φ and depends on the domain of spacetime occupied by the system, whose
boundaries are 3-dimensional surfaces assuming the role of γ’s end-points.
In terms of fields, this region is delimited by the boundary conditions, which
are consequently incorporated in the action. In principle, the action contains
all the conceivable information on the system.
δA[φ] = 0 . (4.4)
91
fixed value of the time parameter “t”.§ The measure d4 x is also kept fixed.
We shall see later that its variation, though important in other aspects, does
not contribute to the Euler-Lagrange equations [see below, Eq.(4.29) and the
discussion leading to (4.33)]. We shall also use, for that reason,
(we shall also elaborate on this point later – see comment below equation
(4.24)). The last term in the right–hand side is
Z Z
4 ∂L 4 ∂L ∂L
dx δ (∂µ φi ) = d x ∂µ δφi − δφi ∂µ .
∂ (∂µ φi ) ∂ (∂µ φi ) ∂ (∂µ φi )
With the help of the 4-dimensional Gauss theorem, we have
Z Z
4 ∂L ∂L
d x ∂µ δφi = dσ α δφi .
∂ (∂µ φi ) ∂ (∂α φi )
This term reduces consequently to an integration over a 3-dimensional hy-
persurface, the boundary of the integration region — or the boundary of the
system. The variation δφi is arbitrary over all the interior of the system,
but on the boundary we take δφi = 0 (this is the covariant analogous to the
classical mechanical procedure of taking null variations at the trajectories
end–points). Then,
Z
4 ∂L ∂L
δA = d x − ∂µ δφi . (4.6)
∂φi ∂ (∂µ φi )
As the δφi are arbitrary, δA = 0 implies the Euler–Lagrange equations
∂L ∂L
− ∂µ =0. (4.7)
∂φi ∂ (∂µ φi )
§ 4.9 It is worth noticing that, as (4.6) holds always under the supposed
conditions, the Euler–Lagrange equations imply an extremal of the func-
tional (4.2). In functional analysis, expression (4.6) provides the definition
of the functional derivative of A[φ] with respect to φi : it is the term between
brackets in the integrand. We write
δA[φ] ∂L ∂L
= − ∂µ . (4.8)
δφi ∂φi ∂ (∂µ φi )
§
See for instance H. Goldstein, Classical Mechanics, Addison–Wesley, Reading, Mass.,
1982.
92
In the particular case of variational calculus, this functional derivative is
called Lagrange derivative. If L depends on higher–order derivatives of φ,
the Lagrange derivative takes the form
δA[φ] ∂L ∂L ∂L ∂L
= −∂µ +∂µ ∂ν −∂µ ∂ν ∂λ +· · · (4.9)
δφi ∂φi ∂ (∂µ φi ) ∂ (∂µ ∂ν φi ) ∂ (∂µ ∂ν ∂λ φi )
( + m2 ) φ(x) = 0 .
p2 = m2 c2
by using the quantization rules (3.27) and (3.28). The d’Alembertian operator is simply
the Laplace operator in 4-dimensional Minkowski space. In Cartesian coordinates {xα },
in terms of which the Lorentz metric is η = diag(1, −1, −1, −1),
= η αβ ∂α ∂β = ∂ α ∂α = ∂0 ∂0 − ∂1 ∂1 − ∂2 ∂2 − ∂3 ∂3 .
Show that the Klein-Gordon equation comes out as the Euler-Lagrange equation of
the Lagrangian
L = 12 [∂µ φ∂ µ φ − m2 φ2 ] .
§ 4.10 Consider a point y interior to the domain on which the field and the
system it describes is present, and the variation δφ(y) at that point. Instead
93
of the functional differential of A previously used, we shall take the derivative
of Z
A[φ] = d4 xL[φ(x), ∂µ φ(x)] .
We write
Z
δA[φ] 4 ∂L(x) ∂L(x) δ(∂µ φ(x))
= dx + . (4.10)
δφ(y) ∂φ(y) ∂(∂µ φ(x)) δφ(y)
As the variation is well–defined, unique at each point, we shall agree to put
δφ(x)
= δ 4 (x − y) . (4.11)
δφ(y)
This can also be written
Z
δφ(x) = d4 x δ 4 (x − y)δφ(y) .
dxi
= δji ,
dxj
which holds for Cartesian coordinates. Notice also that the variation at point
y has nothing to do with the variation at point x — they are variations of
distinct degrees of freedom, so that
δ ∂ ∂ δ ∂ 4
µ
φ(x) = µ
φ(x) = δ (x − y) . (4.13)
δφ(y) ∂x ∂x δφ(y) ∂xµ
94
At this point, (4.10) can be written
Z
δA[φ] 4 ∂L(x) 4 ∂L(x) ∂ 4
= dx δ (x − y) + δ (x − y) .
δφ(y) ∂φ(x) ∂(∂µ φ(x)) ∂xµ
Using again the four–dimensional Gauss theorem, we see that the term with
the total derivative is an integral on the boundary of the system, and — as
y is in its interior — it vanishes. The delta factors out in all the remaining
terms and
Z
δA[φ] 4 ∂L(x) ∂L(x)
= dx − ∂µ δ 4 (x − y)
δφ(y) ∂φ(x) ∂(∂µ φ(x))
∂L(y) ∂L(y)
= − ∂µ , (4.14)
∂φ(y) ∂(∂µ φ(y))
4.1.4 Variations
§ 4.11 A physical system will thus be characterized as a whole by some
symmetry-invariant action functional like (4.2),
Z
A[φ] = d4 xL[φ] , (4.15)
95
where φ represents collectively all the involved fields. Let us examine in some
more detail the total variation¶ of the action functional under the simulta-
neous change of the coordinates according to
Consider
φ0i (x0 ) = φ0i (x + δx) ≈ φ0i (x) + ∂µ φ0i (x) δxµ .
The last term is
∂µ φ0i (x) δxµ = ∂µ φi (x) δxµ + ∂µ δ̄φi (x) δxµ ≈ ∂µ φi (x) δxµ
96
∂δxσ
= δσ µ − ,
∂xµ
and consequently
∂xσ ∂ ∂δxσ ∂δxσ
∂ σ ∂ σ ∂
= = δ µ− = δ µ− ,
∂x0µ ∂x0µ ∂xσ ∂x0µ ∂xσ ∂xµ ∂xσ
which is
∂ ∂ ∂δxσ ∂
= − . (4.21)
∂x0µ ∂xµ ∂xµ ∂xσ
The derivative ∂/∂xµ of expression (4.20) is
On the other hand, if we apply (4.21) to (4.19) we find, always retaining only
the first order terms,
Thus, we can commute ∂µ and δ only if the spacetime variation δxλ is point–
independent. This is the case of equation (4.5), because there the spacetime
variable was kept fixed.
Concerning the purely functional variations,
(i) take the derivative of (4.20):
(iii) take the difference of both expressions, using (4.24) to obtain the ex-
pected result
δ̄∂µ φi (x) = ∂µ δ̄φi (x) .
The change in the functional form of the derivative of φi (x) is the derivative
of the change in the functional form of φi (x). We can spell it in commutator
form,
∂µ , δ̄ φi (x) = 0 . (4.25)
97
Let us go back to the action functional (4.15). It does not depend on
x. Its is a functional of the fields, depending on the integration domain.
Variations (4.16) and (4.17) will have effects of two kinds: changes in the
integration volume and in the Lagrangian density. We shall indicate this by
writing
Z
4
δ(d x)L + d4 x δL .
δA[φ] = (4.26)
so that
δ(d4 x) = ∂µ (δxµ ) d4 x . (4.29)
We have thus the first contribution to (4.26). The variation of the Lagrangian
density will have two contributions, one coming from the coordinate varia-
tions and another coming from the variations in the functional form of the
fields:
δL = (∂µ L) δxµ + δ̄L , (4.30)
98
where
∂L ∂L
δ̄L = δ̄φi + δ̄(∂µ φi ) . (4.31)
∂φi ∂(∂µ φi )
The dependence of L on x comes exclusively through the fields — a basic
hypothesis of Field Theory. Equation (4.25) authorizes commuting δ̄ and ∂µ ,
leading to
δL ∂L
δ̄L = δ̄φi + ∂µ δ̄φi . (4.32)
δφi ∂(∂µ φi )
with δL/δφi the Lagrangian derivative.
Attention should be paid to the different notations ∂L/∂φi and δL/δφi .
Notice that it would not be clear that we are allowed, while manipulating
the second term in (4.31), to perform an integration by parts, as here also
the integration boundaries are varying.
Finally, putting together (4.29), (4.30) and (4.32), we arrive at the ex-
pression of the action first–variation:
Z
4 δL ∂L µ
δA[φ] = d x δ̄φi + ∂µ δ̄φi + L δx . (4.33)
δφi ∂(∂µ φi )
99
being related to the former or not. Since their publication in 1918, the theo-
rem has been subjected to many extensions and adaptations. We shall here
present a version specially adapted to Field Theory.k
The transformations will be supposed to be continuous and connected
to the identity. For all that will concern us here, it will be sufficient to
consider the first-order infinitesimal case. Equation (4.33) gives the response
of the action functional to variations both in the fields and in the spacetime
coordinates:
Z
4 δL ∂L µ
δA[φ] = d x δ̄φi + ∂µ δ̄φi + L δx
δφi ∂(∂µ φi )
From this expression will come the two Noether theorems we intend to
study. The first will be concerned with global transformations, that is, with
transformations which are the same at all points of the system. In other
words, the transformation parameters (the group parameters) will be point-
independent. The second theorem is concerned with transformations which
change from point to point.
As we have seen in page 57, a well-defined behavior under transformations
require that each field belong to some representation of the corresponding
group. Suppose that Ta , for a = 1, 2, . . . , N = group dimension, are the
generators in some matrix representation to which φ belongs, and that φi are
the components. Then,
a
φ0i (x0 ) = eω Ta ij φj (x) (4.34)
100
In terms of the transformation parameters, variations (4.16) and (4.17)
will be, to first order,
δxµ a
δxµ = δω ; (4.36)
δω a
δφi (x) a
δφi (x) = δω . (4.37)
δω a
The field variation at a fixed point x will be
δφi (x) ∂φi (x) δxµ
δ̄φi (x) = a
− µ a
δω a . (4.38)
δω ∂x δω
It is clear that, from (4.35),
δφi (x)
= (Ta )ij φj (x) . (4.39)
δω a
δx µ
To get an idea on how to obtain δω a , consider again the particular case
101
but totally arbitrary. Thus, in order that the integral vanish for any δω a it is
necessary that the integrand vanishes. In other words, action invariance un-
der a global transformation imposes the vanishing of the derivative of A with
respect to the corresponding constant (but otherwise arbitrary) parameter
δω a . The condition for that is
δxµ
δL δ̄φi (x) ∂L δ̄φi
= − ∂µ +L a . (4.41)
δφi δω a ∂(∂µ φi ) δω a δω
δL
§ 4.14 When the field φi is a solution of the Euler-Lagrange equation δφ i
=
0, the current
δxµ
µ ∂L δ̄φi
Ja = − +L a . (4.42)
∂(∂µ φi ) δω a δω
will have vanishing divergence:
∂µ Ja µ = 0 . (4.43)
There will be one conserved current for each group generator. Each will
result in a conserved charge (that is, an integral of motion). To see this, take
in spacetime a volume unbounded in the space-like directions, but limited
in time by two space-like surfaces w1 and w2 . Integrating (4.43) over this
volume, we get an integral over the boundary surface, composed of w1 , w2
and the time-like boundaries supposed to be at infinity. If we now suppose
the current to be zero at infinity on these boundaries, we remain with
Z Z
µ
dσµ Ja = dσµ Ja µ . (4.44)
w1 w2
R
This means that the integral Qa = wn dσµ Ja µ , taken over a space-like (hy-
per)surface wn , is independent of which wn one takes, provided the current
vanishes at space infinity. In a more prosaic way: take an axis x0 = ct and as
spaces the planes given by t = constant; then the integral will be the same
102
on any such plane — will be time-independent. In effect, integrating (4.43)
in d3 x,
Z Z Z
d 3 0 3 i
0
d xJa (x) = − d x∂i Ja (x) = − dσi Ja i (x) = 0 . (4.45)
dx space bound
which is conserved,
d
Qa = 0 . (4.47)
dt
§ 4.15 We shall see below, when we study the main Lagrangians, applica-
tions of all that. A few comments:
(i) the (“Noether”) current is not unique; addition of the divergence of any
antisymmetric tensor, Ja µ → Ja µ + ∂λ Aa µλ , with Aa µλ = − Aa λµ , gives
another conserved current (as ∂µ ∂λ Aa µλ = 0) and the charge will not change
if the tensor Aa µλ vanishes at the space infinity.
(ii) the theorem is frequently presented in the physical literature as just
(4.47): to each transformation leaving indifferent the action (and conse-
quently the field equations) corresponds an invariant, a constant of motion.
In the mathematical literature, the theorem is (4.41). In this last, histori-
cal form, it is possible to show an inverse theorem: if there are N linearly
independent combinations of the Lagrangian derivatives reducing to diver-
gences, then the action is invariant under the transformations of some N
dimensional group. Whether there is or not some kind of inverse for the
“physical” version is not clear. If the question is whether there is a symme-
try corresponding to any integral of motion, the answer is no. The so-called
“topological invariants” are not related to symmetries — we shall see an
example below.
(iii) Equation (4.41) holds always, provided there is a symmetry of the action
functional, for fields satisfying or not the equation of motion; it provides
consequently information on the “space of states” of the system presenting
103
the symmetry. Relations of this kind, independent of the field equations,
are called “strong relations”. We shall say a little more about that in the
discussion of the second Noether theorem.
δxµ α
x0µ = xµ + δxµ = xµ + δa .
δaα
We can, in this case, take the xµ themselves as parameters,
δxµ
= δαµ . (4.48)
δaα
Fields are Lorentz tensors and spinors, and as such unaffected by translations:
δφi /δaα = 0. Consequently, from (4.38),
∂L
Θα µ := ∂α φi − δαµ L . (4.49)
∂∂µ φi
104
Exercise 4.2 Use Eq.(4.20) to rewrite the general Noether current (4.42) as
∂L δφi δxα
Ja µ = − a
+ Θα µ a . (4.50)
∂(∂µ φi ) δω δω
If we look for the analogous in Classical Mechanics, we find that this quan-
tity corresponds to the “stress–energy tensor”. In Field Theory it is usual to
call it the canonical energy–momentum tensor density. The corresponding
conserved charges will be
Z
λ
P = d3 xΘλ0 (x). (4.52)
Exercise 4.3 Again the real scalar field. From the first Lagrangian in Exercise 4.1,
L= 1
2 [∂µ φ∂ µ φ − m2 φ2 ] ,
Θλµ = ∂ λ φ∂ µ φ − η λµ L .
Show that the energy density can be put into the positive form
Θ00 = 1
2 [∂0 φ ∂0 φ + ∂i φ ∂i φ + m2 φ2 ] .
105
Lorentz Transformations and Angular Momentum
Consequently,
µ
x0µ = exp i
δω αβ xν ≈ δνµ + 2i δω αβ (Jαβ )µ ν xν =
2
Jαβ ν
= xµ + 1
2
(δω µν − δω νµ ) xν . (4.53)
We use then δω αβ = − δω βα to obtain
x0µ = xµ + δω µν xν . (4.54)
Therefore,
δxµ µ µ
= δ α x β − δ β x α . (4.55)
δω αβ
∂L δ̄φi δxµ
M µ αβ = − − L . (4.56)
∂∂µ φi δω αβ δω αβ
∂L δφi
M µ αβ = Θα µ xβ − Θβ µ xα − . (4.57)
∂∂µ φi δω αβ
106
is present for all fields and is called the orbital angular–momentum density
tensor. The last term is the spin current density,
∂L δφi
S µ αβ = − , (4.59)
∂∂µ φi δω αβ
which appears only when the field is not a Lorentz singlet. In effect, a scalar
field is defined as a field such that
δφµ (x)
= δαµ φβ − δβµ φα ; (4.62)
δω αβ
δ̄φµ (x)
= δαµ φβ − δβµ φα + [xα ∂β − xβ ∂α ] φµ . (4.63)
δω αβ
The last term is analogous to that of the scalar case and compensates
the argument change, but there is a non-vanishing net variation δφµ (x) as
a response to the Lorentz transformation. The spin density is exactly the
contribution to M µ αβ coming from this “intrinsic” response.
From ∂µ M µ αβ = 0 and ∂µ Θα µ = 0 follows
107
The antisymmetric part of the canonical energy–momentum density tensor
measures the breaking of pure–spin conservation. Of course, there is no
a priori reason for the spin to be conserved separately, but this happens
when the canonical energy–momentum density tensor is symmetric. From
the conservation of the orbital angular momentum for scalar fields, it comes
that the energy–momentum is symmetric for those fields.
where S µρσ = η ρα η σβ S µ αβ .
Exercise 4.5 (Facultative) Show that ΘB λµ is indeed symmetric, by calculating ΘB λµ
- ΘB µλ and using Eq.(4.64).
108
by
a j
φ0i (x) = [U (g)]ji φj (x) = eiω Ta i φj (x) . (4.66)
The Ta are the G generators in the U representation. In the presence of a
local gauge invariance (see section 3.4), fields like the above φi (x) appear
in physical Lagrangians in two ways. First, as free fields. Second, combined
into certain currents which couple to gauge potentials. In consequence, those
currents appear as sources in the right-hand side of the equations of motion
for the gauge fields. For this reason such φi (x) are called source fields. Gauge
potentials, on the other hand, mediate the interactions between the source
fields. They are written as Aµ = Ja Aaµ , with Ja the generators in the ad-
a
joint representation of G. Under a transformation g = eiω Ja , they change
according to (3.35),
0
Fµν (x) = gFµν (x)g −1 .
109
and δ̄φ∗ (x) = iqδαφ∗ (x). The Noether current (4.42) will then be given by
∂L δ̄φ∗
µ ∂L δ̄φ ∂L ∂L ∗
J =− + = iq φ− φ .
∂(∂µ φ) δα ∂(∂µ φ∗ ) δα ∂(∂µ φ) ∂(∂µ φ∗ )
(4.70)
From this expression we can get again (4.41) for constant parameter
transformations and the consequent Noether theorem. But we can obtain
something more: instead of using the functional differentiation, we can use
functional derivations. We could proceed in the spirit of equation (4.10),
getting directly the functional derivative of equation (4.33):
δxµ
δA[φ] δL(x) δ̄φi (x) ∂L(x) δ̄φi (x)
= + ∂µ +L a . (4.72)
δω a (y) δφi (x) δω a (y) ∂(∂µ φi (x)) δω a (y) δω (y)
110
There are two kinds of fields in those theories, “source” fields transforming
according to
δ̄φ(x) = δω a (x)Ta φ(x), (4.73)
and gauge potentials transforming according to
In both cases, fields and parameters are at the same point x. Thus, in the
simplified approach we are adopting, we use
δ̄φ(x)
a
= δ 4 (x − y) Ta φ(x);
δω (y)
δ̄Aa µ (x)
= f a bc Ac µ (x) δ 4 (x − y) − δba ∂µ δ 4 (x − y).
δω b (y)
The δ’s ensure locality. For any y interior to the system, these expressions
lead to the vanishing of the divergence term. What remains is
δA[φ, A] δL(y) δL(y) a c δL(y)
a
= Ta φ(x) + b
f bc A µ (y) + ∂µ a . (4.75)
δω (y) δφ(y) δA µ (y) δA µ (y)
The second Noether theorem says that, in the presence of a local symmetry
related to a group with N generators, that is when
δA[φ]
=0, (4.76)
δω a (y)
there are N independent relations between the Lagrange derivatives and their
derivatives. This is what we obtain from the above equation:
δL(y) δL(y) a c δL(y)
∂µ a
+ b
f bc A µ (y) = − Ta φ(x). (4.77)
δA µ (y) δA µ (y) δφi (y)
Notice that the equations of motion have not been used. These relations are
“strong”, they hold independently of the solutions, reflecting the symmetries
of the very space of possible states of the system.
More detail will be given in the section on gauge theories, but a few
general comments can be made here. Define the object
δL(x)
Ja µ (x) = − . (4.78)
δAa µ (x)
111
The last expression above takes the form
δL(x)
∂µ Ja µ (x) − fabc Ab µ (x)J cµ (x) = − Ta φ(x). (4.79)
δφ(x)
The total operator acting on J in the left-hand side will be the covariant
derivative, actually a covariant divergence. In the right-hand side appears a
factor resembling the Euler-Lagrange form, which is zero for solutions of the
field equations. Then, a weak result would be: the “current” J has vanishing
covariant divergence. In gauge theories, J is the current produced by the
sources. The second Noether theorem does not lead to conserved quantities,
but establishes constraints on the possible sources.
There will be conserved charges under the additional proviso that the
local transformations become constant transformations outside the system.
We shall come back to these points presently.
112
4.4 Topological Conservation Laws
§ 4.21 As we have said, not every conserved quantity is related to a sym-
metry. Let us see a simple example in two–dimensional spacetime, with
coordinates x0 = vt and x1 = x. Let a field be given by a scalar function
φ(x, t) and consider the totally antisymmetric symbol in two dimensions, µν ,
01 = - 10 = 1; 00 = 11 = 0. The current defined by jµ (x, t) = µν ∂ ν φ will
be automatically conserved:
∂ µ jµ (x, t) = 0.
1 ∂ 2φ ∂ 2φ
− 2 = − sin φ(x, t). (4.80)
v 2 ∂t2 ∂x
A particular solution is the solitary wave of Figure 4.1, given by
6
f 4 4
2
2
0
-4 t
-2 0
0
x 2
4 -2
113
x − vt
φ(x, t) = 4 arctan exp √ . (4.81)
1 − v2
Let us use it to unravel the meaning of the current conservation above. For
this solution,
v 4eγ(x−vt) 1
J1 = 01 ∂ 0 φ = √ 2γ(x−vt)
= 2γv .
1−v 1+e
2 cosh[γ(x − vt)]
Sine-Gordon wave
x
-4 -2 2 4
114
many non–linear equations. The sine-Gordon equation has also solutions
with many solitons (which “grow” many times 2π), inverted solutions with
the wave decreasing (“anti–solitons”) and still solutions combining r solitons
and s anti–solitons. For such solutions, the above number n is n = r − s
= “soliton number”. We have, at the beginning of the discussion, carefully
avoided saying anything on boundary conditions, which are different for each
kind of solution. For fixed boundary conditions, however, there will be all the
solutions with a fixed n. This number n is an example of invariant related
to the topology of the fields (or of the space of solutions). This kind of
invariant is, for that reason, called “a topological number”. It is not related
to any symmetry and cannot be obtained through the Noether theorem.
Other non-linear equations exhibit solitonic solutions with non–Noetherian
conserved charges. The best known are mostly in two-dimensional space
[(1+1)-space, one dimension for space, one for time], like the Korteweg-de
Vries (KdV) equation.
There are thus two kinds of conserved quantities in Physics: those coming
from Noether’s theorem – conserved along solutions of the equations of mo-
tion – and the topological invariants, which come from the global, topological
properties of the space of states.
§ 4.22 A Note on the Hamiltonian Approach To pass into the Hamil-
tonian formalism, we must first define the momentum conjugate to each φi .
A majority of authors follow the classic analogy, putting
. ∂L
πi = ,
∂(∂0 φi )
which gives the time parameter a favored role from the start. A few others
introduce a 4-vector momentum for each φi ,
. ∂L
πiµ = , (4.83)
∂(∂µ φi )
in terms of which the field equations become compact indeed:
∂L
∂µ π i µ = . (4.84)
∂φi
Comment 4.1 We have been talking about a “minimum” of the action but it is enough
to have an extremum to arrive at the field equations. To go into the details of the principle
of minimal action, we should study also the second variation.
115
116
Chapter 5
We now proceed to a detailed discussion of the main fields which have been
found to describe elementary particles — and, consequently, the fundamental
interactions — in Nature. They are classified, as repeatedly announced, by
their behavior under transformations of the Poincaré group. It is fortunate
that, at least for the time being, only particles and fields belonging to the
lowest representations — those of small dimensions — seem to play a ba-
sic role. They also have different characters according to their spins being
integers or half-integers. Integer-spin particles are called bosons, and their
fields are bosonic fields. Half-integer-spin particles are called fermions, and
their fields are fermionic fields. Bosonic and fermionic particles have quite
distinct statistical behaviour. We start with bosons of small spins, actually
only spins 0 and 1. And first we present fields without interactions.
Comment 5.1 Schrödinger has found this wave equation even before he found his famous
nonrelativistic equation – but discarded it because it led to negative-energy solutions for
free states – for which at that time there was no interpretation.
117
The D’Alembertian operator is simply the Laplace operator in 4-dimen-
sional Minkowski space. In Cartesian coordinates {xα }, in terms of which the
Lorentz metric is η = diag (1, −1, −1, −1) and thus coordinate–independent,
it is
= η αβ ∂α ∂β = ∂ α ∂α = ∂0 ∂0 − ∂1 ∂1 − ∂2 ∂2 − ∂3 ∂3 . (5.2)
The Klein-Gordon equation describes the field in absence of any source,
that is, in absence of any interaction. It is the simplest relativistic adaptation
of the Schrödinger equation, actually the Poincaré invariant Pµ P µ = m2 of
§ 2.27, with P µ = i ∂ µ , applied to the field φ(x). This means that, as a
matter of fact,
L= 1
2
[∂µ φ∂ µ φ − m2 φ2 ] , (5.3)
Θλµ = ∂ λ φ∂ µ φ − η λµ L . (5.5)
Θ0i = ∂ 0 φ ∂ i φ (5.6)
118
and of energy
Θ00 = 1
2
[∂0 φ ∂0 φ + ∂i φ ∂i φ + m2 φ2 ] . (5.7)
Notice that, as a summation of real squares, this expression is always positive.
It is of course to be expected that the energy density be positive in the
absence of interactions.
The fact that θ00 is positive leads, by the way, to a criterion for the
presence of a field. We might ask on which region of spacetime is some
field φ(x) really present. As any contribution of the field adds up a positive
quantity, we can say that the field is present on every point x at which θ00 > 0
and absent wherever θ00 = 0.
As to the angular momentum, we have seen that it reduces to the orbital
part in this case.
Of course, a field can interact with other fields, or with itself. The study
of free fields is of interest because it allows the introduction of notions and
methods, but in itself a free field is rather empty of physical content: the
real characteristics of the system it supposedly describes can only be assessed,
measured, through interactions with other systems, described by other fields.
These characteristics are described precisely by the responses of the sys-
tem to exterior influences. The Lagrangian of a theory is the sum of free
Lagrangians, fixing the fields which are at work, and of “interaction La-
grangians”, which try to describe the interplay between them. In the case
of an isolated scalar field one tries, for reasons of simplicity, to describe the
self-interaction by monomial terms like λφ3 ,λφ4 ,λφ8 ,· · · λφn . More involved,
non-polynomial interaction Lagrangians (such as cos(αφ) and exp(αφ), which
lead to the Sine-Gordon and the Liouville equations) can be of great interest.
In 4-dimensional spacetime, only the Lagrangian Lint = λφ4 seems able to
produce a coherent theory in the quantum case. The other lead to uncon-
trollable infinities.
119
5.2 Complex Scalar Fields
§ 5.2 A complex field is equivalent to two real fields φ1 and φ2 , put together
as
φ(x) = φ1 (x) + i φ2 (x) . (5.9)
In the jargon of field theory it is usual to forget about the infinity of degrees
of freedom represented by each component and talk about each components
as if it were “one” degree. There are then “two” independent degrees of
freedom, and we can use either φ1 (x), φ2 (x) or the pair φ(x), φ∗ (x). The
Lagrangian, which leads to two independent Klein-Gordon equations, is
Θµλ = ∂ µ φ∗ ∂ λ φ + ∂ µ φ∂ λ φ∗ − η µλ L. (5.11)
~ ∗ ) · (∇φ)
Θ00 = ∂0 φ∗ ∂0 φ + (∇φ ~ + m2 φ∗ φ , (5.12)
Θ0i = ∂0 φ∗ ∂i φ + ∂i φ∗ ∂0 φ . (5.13)
§ 5.3 A question comes up naturally at the sight of (5.10): φ(x) has two
“components”, but it is a Lorentz scalar. What is the meaning of these
components ? In order to answer this question, let us begin by noting that
the Lagrangian has a supplementary invariance, absent in the real scalar case:
it does not change under the transformations
120
where α is an arbitrary constant. This transformation takes place only in the
space of fields, leaving spacetime untouched. It is a rotation in the complex
field plane, the same as
Let us see what Noether would have to say about the invariance under these
transformations (called gauge transformations of first kind, or global gauge
δxµ
transformations). Looking at the terms in the current, we shall have δω a = 0;
δ̄φ δ̄φ δ̄φ∗ δ̄φ∗ ∗
δω a
= δα = iφ; δωa = δ̄α = −iφ . Consequently, the Noether current will be
↔
J µ (x) = i [φ∗ (∂ µ φ) − (∂µ φ∗ )φ] = i φ∗ ∂ µ φ . (5.16)
Even had we guessed this expression without Noether’s help, we would know
that its divergence vanishes: from the very field equations
121
words, electrically charged fields cannot be described by a free Lagrangian.
Let us see what happens if, in (5.14), the angle becomes dependent on the
spacetime position:
A function like eiα , or eiα(x) , can be seen as a complex matrix with a sin-
gle entry. It will be, of course, a unitary matrix. The set of such unitary
1–dimensional matrices form a group, denoted U (1) or SO(2). Equations
(5.14) describe a transformation belonging to U (1) which is the same at all
points of spacetime. We try to represent this case in Figure 5.1, in which
the x-axis represents spacetime. On the other hand, Eqs.(5.19) describe a
122
Figure 5.2: Local gauge transformations: different at each point.
Comment 5.2 Nomenclature has wavered a little. Nowadays, when people say “gauge
transformations”, they mean usually gauge transformations of second kind. Because these
expressions are so telling, it would perhaps be better to call them “global” and “local”
gauge transformations.
§ 5.5 For local gauge transformations a new problem arises: the Lagrangian
(5.10) is no more invariant:
A0µ = Aµ − ∂µ α , (5.21)
123
Thus, if we want that the Lagrangian describe a charged field, we must
modify the derivatives in (5.10) to allow the presence of an electromagnetic
field. It becomes
the last term being the Lagrangian of the free electromagnetic field. And
now a very beautiful thing happens: (5.22) is invariant under the gauge
transformations described by (5.19) and (5.21) together. The potential Aµ
compensates, through its indetermination (5.21), the variance of φ. It is the
simplest known gauge potential. Its simplicity is due to the related group
U (1), which has one single generator and is, consequently, abelian.
Comment 5.3 If we proceed to quantize the theory, particles come up as quanta of each
field: neutral particles as neutral fields, charged particles as quanta of charged fields. Thus,
an approach like the above one is necessary to describe charged particles.
124
its complex conjugate describe jointly particles and antiparticles. Thus, the
field φ either creates an antiparticle or annihilates a particle; φ∗ either creates
a particle or annihilates an antiparticle. Furthermore, φ and φ∗ couple with
opposite signs with the electromagnetic field: it is enough to check the signs
in (5.22). A last comment: if φ = φ∗ , case of the real field, the current (5.16)
vanishes identically, and all additive charges are zero. Consequently, real
fields will describe particles which are equal to their antiparticles, which do
not carry any additive charge and are unable to interact electromagnetically.
§ 5.7 An electric charge creates a field, to which responds any other electri-
cally charged object. This field - the electromagnetic field - is a vector field.
The charged objects interact with each other electromagnetically and we say
that the electromagnetic field “mediates” that interaction. The same hap-
pens with other interactions. With the remarkable exception of gravitation,
all the known fundamental interactions of Nature are mediated by vector
fields.
As the name indicates, a vector field is a set of four fields transforming
as the components of a Lorentz vector. It can be indicated as
φ0 (x)
φ1 (x)
φ(x) = .
φ2 (x)
φ3 (x)
This means that the set transforms according to the vector representation
of the Lorentz group, that of the cartesian coordinates xµ :
The vector field is real or complex if each one of its components is real or
complex. The complex field is formally richer, and includes the real field as
a particular case. In Nature, we find both kinds among the mediating fields
125
of the electroweak interactions, whose quanta have been found in 1983: the
bosons Wµ± are described by a pair of complex conjugate massive vector fields
and the boson Zµ0 by a real massive vector field. The photon field Aµ is a real
vector field with vanishing mass. It is the mediating field of electromagnetism
and will be discussed in an independent chapter.
Θµν = − ∂ µ φλ ∂ ν φλ − η µν L. (5.26)
The spin density obtained from L is the first we have the opportunity to
write down:
µ
S(αβ) (x) = − φα (x)∂ µ φβ (x) + φβ (x)∂ µ φα (x). (5.27)
126
energy density which is not positive-definite:
→ 2
Θ00 = − 12 (∂ 0 φ0 )2 − 12 (∇φ0 )2 − M2 (φ0 )2
3 h
X → i
+ 21 (∂0 φj )2 + (∇φj )2 + M 2 φ2j . (5.28)
j=1
The three positive contributions are alike those of the scalar case, but there
are negative terms. A Hamiltonian which is not positive–definite is a serious
defect in a theory proposed to describe a free, non-interacting system. A
supplementary condition must be introduced in order to correct it. The only
condition which is invariant and linear in the fields is
∂µ φµ = 0, (5.29)
which should hold at each point of the system. We could put also a constant
in the right-hand side, but this would add an arbitrary constant to the theory.
The above condition reduces to three the number of independent degrees of
freedom and, when used to eliminate the miscreant φ0 , does lead to a positive-
definite energy. We shall not prove it here, as it requires the use of Fourier
analysis in detail.
The physical system is described by the Lagrangian plus the supplemen-
tary condition. This is a novel situation: the Lagrangian alone does not
determine, via the minimal principle, the acceptable conditions. There is
something amiss with the Lagrangian (5.24). The problem can be circum-
vented by using another, the Wentzel–Pauli Lagrangian
L0 = L + 1
(∂ φ )(∂ ν φµ )
2 µ ν
(5.30)
M2 ν
= − 1
4
[(∂µ φν − ∂ν φµ )(∂ µ φν − ∂ ν φµ )] + 2
φ φν . (5.31)
The Euler-Lagrange equation coming from this Lagrangian is the Proca equa-
tion
φν − ∂ ν (∂µ φµ ) + M 2 φν = 0, (5.32)
whose solutions satisfy automatically the supplementary condition if M 6=
0. In effect, taking ∂ν of the equation we obtain M 2 ∂ν φν = 0. Thus, the
Lagragian L0 automatically implements the supplementary condition. It is
127
interesting to introduce the variable Fµν = ∂µ φν − ∂ν φµ , in terms of which
the Lagrangian takes the form
M2 ν
L0 = − 41 Fµν F µν + 2
φ φν (5.33)
∂µ F µν + M 2 φν = 0. (5.34)
S 0µ αβ = S µ αβ + φα ∂β φµ − φβ ∂α φµ = φβ F µ α − φα F µ β . (5.36)
for any differentiable function f (x). This is a gauge invariance and allows
one a lot of freedom in choosing the field. Each choice of the field is called
“a gauge”. In particular, if ∂µ φµ = g(x), it is possible to implement the
supplementary condition whenever a solution f can be found for the Poisson
equation f (x) = - g(x), which would lead immediately to ∂µ φ0µ = 0. This
choice is called the Lorenz gauge (not Lorentz !). As the Lagrangian is
invariant under any “change of gauge”, the physical results found in that
particular gauge hold true in general. As already announced, the electro-
magnetic field will deserve a special chapter.
128
5.3.2 Complex Vector Fields
§ 5.9 Let us examine the case of the complex vector field, which includes
the real case. As said above, the electroweak bosons Wµ± are described by a
pair of complex conjugate massive vector fields. The Lagrangian is
L = − ∂µ φ∗ν ∂ µ φν − M 2 φ∗ν φν ,
(5.37)
∂µ φµ = 0 ; ∂µ φ∗µ = 0 . (5.38)
+ M 2 φν = 0 ; + M 2 φ∗ν = 0 .
(5.39)
and in special
129
Comment 5.4 It is a useful notation to put all antisymmetrized indices inside square
brackets. Thus, the above expression is
↔
µ
S(αβ) = φ∗[β ∂ µ φα] .
φµ = φ(1) (2)
µ + iφµ ;
∂µ F µν + M 2 φν = 0. (5.51)
Θαµ = − 1
2
(F ∗µν ∂ α φν + F µν ∂ α φ∗ν ) − η αµ L, (5.53)
130
with in particular
M2 ν ∗
Θ00 = 21 [(∂0 φk )(∂0 φ∗k ) − (∂i φ0 ∂i φ∗0 ] + 1 ∗ ij
F F
4 ij
− 2
φ φν . (5.54)
§ 5.11 A last comment: theories with massive vector fields as above inter-
acting with other fields are, in general (perturbatively) “unrenormalizable”.
This means that, once quantized, they lead to infinite values for certain fi-
nite quantities. The exceptions are the gauge fields, of which the simplest
example is the electromagnetic field.
131
Chapter 6
Electromagnetic Field
This is the best known of all fields. What follows has no pretension at all
to an introduction to electrodynamics. We shall only outline the general
aspects of the electromagnetic field, emphasizing some features which are
more specific to it while exhibiting some properties it shares with fields in
general.
132
duality: they remain invariant if we exchange
~ ⇒ H
E ~
(6.5)
~ ⇒ − E.
H ~
133
~ through any closed surface S vanishes; or, no surface can
the flux of H
contain a magnetic charge;
H 3 E 2 −E 1 0
which is obtained from the first by performing the duality transformation
(6.6) for each entry.
6.2 ~ and H
Transformations of E ~
§ 6.4 We shall here be interested in the behavior of the electric and magnetic
fields under boosts. This means, in view of Eq.(6.11), the behavior of F αβ .
It will be necessary, consequently, to examine the behavior of tensors.
134
We have in § 1.29 defined a Lorentz vector as any set V = (V 0 , V 1 , V 2 , V 3 )
of four quantities transforming like (ct, x, y, z). In the case of a boost like
that given by Eqs. (1.50) and (1.51), with a velocity v along the axis Ox (v
= vx ), they will have the behavior
v 00
A1 = γ(A01 + A )
c
A2 = A02 ; A3 = A03 (6.13)
v
A 0
= γ(A00 + A01 ) .
c
We have in § 1.32 introduced the four-vector current (1.73)
vx vy vz
j α = e uα = e γ (1, , , )
c c c
and the four-vector potential (1.74)
~ = (φ, Ax , Ay , Az ) .
A = (Aα ) = (φ, A)
135
0
Precisely, we have a transformation of type Aα = (Λ−1 )α β 0 Aβ . In detail,
Eqs. (6.13) are
0
A0 A0
γ βγ 0 0
0
A1 βγ γ 0 0 A1
= (6.15)
0
A2 A2
0 0 1 0
0
A3 0 0 0 1 A3
Exercise 6.1 Verify that the matrices appearing in Eqs. (6.15) and Eqs. (6.16)
§ 6.6 The cases above are, of course, particular boosts. Under general
Lorentz transformations (see § 1.34) vectors and covectors change as given
by Eqs. (1.79) and (1.81). Second order tensors follow (1.80). That is to
say that they transform like objects whose components are the products of
vectors and/or covector components. We are anyhow interested only in the
behavior of the antisymmetric tensor F αβ under boosts.
§ 6.7 Let us go back to the case of § 6.5. To get the behavior of a tensor T αβ ,
the simplest procedure is to consider A as in (6.13) together with another
vector
v 00
B 1 = γ(B 01 + B )
c
B2 = B 02 ; B 3 = B 03 (6.17)
v
B0 = γ(B 00 + B 01 ) .
c
136
T αβ will transform like Aα B β . Some examples are:
v 01 00 v 01 00 v 2 00 00
A1 B 1 = γ 2 (A01 B 01 + A B + B A + 2 B A )
c c c
2
v v v
∴ T 11 = γ 2 (T 011 + T 010 + T 010 + 2 T 000 ) (6.18)
c c c
v v v2
A1 B 0 = γ 2 (A01 B 00 + A01 B 01 + A00 B 00 + 2 A00 B 01 )
c c c
2
v v v
∴ T 10 = γ 2 (T 010 + T 011 + T 000 + 2 T 001 ) (6.19)
c c c
v v
A1 B 2 = γ(A01 B 02 + A00 B 02 ) and A1 B 3 = γ(A01 B 03 + A00 B 03 )
c c
012 v 002 013 v 003
∴ T 12 = γ(T + T ) and T = γ(T + T )13
(6.20)
c c
A2 B 2 = A02 B 02 ; A2 B 3 = A02 B 03 ; A3 B 3 = A03 B 03
∴ T 22 = T 022 ; and also T 33 = T 033 , T 23 = T 023 (6.21)
v 00 01 v 01 00 v 2 01 01
A0 B 0 = γ 2 (A00 B 00 + A B + A B + 2 A B )
c c c
2
v v v
∴ T 00 = γ 2 (T 000 + T 001 + T 010 + 2 T 011 ) (6.22)
c c c
v v2
A0 B 1 = γ 2 (A00 B 01 + A00 B 00 + A01 B 01 + 2 A01 B 00 )
c c
2
v v
∴ T 01 = γ 2 (T 001 + T 000 + T 011 + 2 T 010 ) (6.23)
c c
v v
A0 B 2 = γ(A00 B 02 + A01 B 02 ) ; A0 B 3 = γ(A00 B 03 + A01 B 03 )
c c
002 v 012 003 v 013
∴ T 02 = γ(T + T ) and T = γ(T + T ).03
(6.24)
c c
A few possibilities are missing, but these are more than enough to tackle
the question of tensors with well-defined symmetries in the indices. Take for
example Ex = F 01 , antisymmetric: F 01 will transform like A0 B 1 − A1 B 0 =
2
γ 2 (1 − vc2 )(A00 B 01 − A01 B 00 ) = F 001 = Ex0 . Another case: Hz = F 12 transforms
like A1 B 2 − A2 B 1 = γ(F 012 + vc F 002 ) and ∴ Hz = γ[Hz0 + vc Ey0 ]. In this way
137
we find the cases:
Ex = Ex0
v 0
Ey = γ(Ey0 + H)
c z
v
Ez = γ(Ez0 − Hy0 )
c
0
Hx = Hx
v
Hy = γ(Hy0 − Ez0 )
c
0 v 0
Hz = γ(Hz + Ey ). (6.25)
c
The left-hand side gives the fields as seen from a frame K, the right-hand
side as seen from a frame K 0 . Electric and magnetic fields depend, as we see,
on the reference frame from which they are looked at.
§ 6.8 All this is actually much simpler in the matrix version of § 6.5. The
transformation above is
0 0 0 0
F αβ = (Λ−1 )α α0 (Λ−1 )β β 0 F α β = (Λ−1 )α α0 F α β (Λ−1 )β β 0 ,
~ 0 = 0. In
§ 6.9 Suppose that the magnetic field vanishes in frame K 0 : H
138
frame K,
Ex = Ex0
Ey = γEy0
Ez = γEz0
Hx = Hx0 = 0
v v
Hy = − γ Ez0 = − Ez
c c
v 0 v
Hz = γ Ey = Ey . (6.26)
c c
Thus, in a way, an electric field in frame K 0 turns up as a magnetic field in
frame K. A magnetic field which is zero in one frame appears very effectively
in another. The relationship between E ~ and H~ can, in frame K, be summed
up as
cH~ = ~v × E.
~ (6.27)
~ and H
E ~ are clearly orthogonal to each other.
Suppose now that it is the electric field which vanishes in frame K 0 :
~ 0 = 0. This time, in frame K,
E
Hx = Hx0
Hy = γHy0
Hz = γHz0
Ex = Ex0 = 0
v v
Ey = γ Hz0 = Hz
c c
v 0 v
Ez = − γ Hy = − Hy . (6.28)
c c
~ and H
The relationship between E ~ in frame K is now encapsulated in
~ = − ~v × H.
cE ~ (6.29)
~ and H
Also in this case E ~ are orthogonal to each other. It is possible to show
~ or H
in general that, whenever a frame exists in which either E ~ vanish, there
is another frame in which they are orthogonal. And vice versa: if a frame
exists in which they are orthogonal, there exists another frame in which one
of them is zero.
139
Electric and magnetic fields cannot, of course, simply convert into each
other by a change of frames. The electric field is a true vector, while the
magnetic field is a pseudo-vector. Some other vector must be present to
avoid parity violation. This is done by the velocity v in Eqs. (6.27) and
(6.29).
~2 − H
Fαβ F αβ = − F̃αβ F̃ αβ = − 2 (E ~ 2 ). (6.30)
~ · H.
Fαβ F̃ αβ = − 4 E ~ (6.31)
These two expressions are contractions of Lorentz tensors and, consequently,
Lorentz invariants. Contraction (6.31) is not, however, invariant under parity
transformation: it is a pseudoscalar. It is, furthermore, a total derivative.
Exercise 6.2 Verify that the equations (6.32) and (6.33) are equivalent to the set (6.1)-
(6.4).
∂α j α = 0 (6.34)
140
which is just the continuity equation (6.6).
There is a certain confusion in the standard language used in this sub-
~ H)
ject. What is usually called “electromagnetic field” is the pair (E, ~ or, if
we prefer, the tensor F αβ . These are the observables of the theory. Here,
however, the role analogous to the basic fields of previous sections will be
played by the electromagnetic potential Aα , defined as that field for which
Fαβ = ∂α Aβ − ∂β Aα . (6.35)
141
§ 6.12 The wave equation We arrive at the field equation for Aα by taking
(6.35) into (6.33):
Aα − ∂ α (∂β Aβ ) = j α . (6.38)
At the same time, (6.35) makes of (6.32) an identity. Actually, (6.32) is
a 4-dimensional version of div H ~ = 0, more precisely it is div F = 0. And
(6.35) expresses F = rot A in the 4-dimensional case. Finally, taken together,
definition (6.35) and Eq.(6.38) have the same content as Maxwell’s equations.
Let us compare the sourceless case (j α = 0) with the Proca equation for
a general vector field, Eq.(5.32). We see that Aα is a real vector field with
zero mass. The difficulties with the non-positive energy turn up here again,
with an additional problem: the subsidiary condition
∂α Aα = 0 (6.39)
φ + ∂α Aα = 0 (6.40)
will lead to ∂α A0α = 0. The left-hand side of Eq. (6.38) is invariant under
transformation (6.37): this is the “gauge invariance” of the theory. When
we choose a particular A0α in (6.37), we say that we are “fixing a gauge”.
In particular, a potential satisfying Eq.(6.39) is said to be “in the Lorenz
gauge”. In that gauge, the wave equation reduces to the d’Alembert equation
Aα = j α . (6.41)
Notice that the Lorenz condition (6.39) does not fix Aα completely. To
begin with, we can pass into the Lorenz gauge from any Aα by choosing
a φ obeying (6.40). We have ∂α A0α = 0. But then a new transformation
A00α = A0α + ∂ α φ0 with φ0 = 0 (φ0 is “harmonic”) will take into the same
condition for A00α , ∂α A00α = 0. Thus, a potential in the Lorenz gauge is
determined up to a gradient of a harmonic scalar. This additional freedom
can be used to eliminate one of the components of Aα , for example A0 : choose
142
φ0 such that ∂0 φ0 = - A00 ; then we shall have A000 = 0 at any point x = (~x, t).
In that case, ∂0 A000 = 0 and the Lorenz condition takes the form
~ ·A
∇ ~ = 0 ; A0 = 0. (6.42)
This gauge is known as the radiation gauge, or Coulomb gauge. The choice
is not explicitly covariant, but can be made in each inertial frame. If we pass
into the momentum representation through the Fourier transform
Z
Aµ (x) = (2π)3/2 d4 k δ(k 2 )eikx Aµ (k),
1
(6.43)
143
L0 = − 41 Fµν F µν . (6.47)
Notice that only Eqs.(6.33), corresponding to the second pair, follow from
these Lagrangians. Equations (6.32), the first pair, are not really field equa-
tions — they are identities holding automatically for any Fαβ of the form
(6.35). Notice further that, as announced, it is the potential Aα which plays
the role of fundamental field: it is by taking variations with respect to it that
the field equations are got from the Lagrangians.
As a Lagrangian for the sourceless field, (6.47) has many advantages:
1 1
2
∂µ (Aν ∂ ν Aµ ) − 2
Aν ∂ ν ∂µ Aµ ,
which only reduces to a pure divergence when the Lorenz condition is satis-
fied. We shall use (6.46) only for exercise (see Comment 6.1 below).
Maxwell’s equations with sources are obtained by adding to (6.46) or
(6.47) the coupling term
LI = − jA = − jµ Aµ . (6.48)
144
Notice that, under a gauge transformation, LI acquires an extra term equal to
[− jµ ∂ µ φ], which is a total divergence [− ∂ µ (jµ φ)] due to current conservation.
This conservation is, consequently, related to gauge invariance.
Exercise 6.3 Find Eq.(6.33) from Eqs.(6.47) and (6.48).
Θαβ = − (∂ α Aγ ∂ β Aγ ) − η αβ L, (6.49)
that is,
Θ00 = − 1
2 (∂ 0 Aγ ∂ 0 Aγ ) + 1
2 (∂j Aγ ∂ j Aγ ) (6.50)
and
Θi0 = − (∂ i Aγ ∂ 0 Aγ ). (6.51)
We also obtain
S µ αβ = Aβ ∂ µ Aα − Aα ∂ µ Aβ . (6.52)
The spin density will be then given by
↔
S 0 ij = − Ai ∂ 0 Aj , (6.53)
The more usual treatment starts from Lagrangian (6.47). The canonical
energy–momentum density turns out to be
Θαβ = F νβ ∂ α Aν + 1
4
η αβ Fγδ F γδ . (6.55)
S µαβ = F µα Aβ − F µβ Aα . (6.57)
145
1
or, with Sk = 2
kij S ij , Z
~=
S d3 x E × A . (6.58)
Θαβ
B = F
αγ
Fγ β + 1
4
η αβ Fγδ F γδ . (6.59)
1
Θ00 = E 2 + H2 ,
(6.60)
8π
(for which the notation W is frequently used) and so does the momentum
density
Θi0 = (E × H)i . (6.61)
The vector
c
E×H
S= (6.62)
4π
is called the Poynting vector, and measures the energy flux of the electro-
magnetic field. This has the dimension (energy × c)/volume. Consequently,
the flux of momentum density is actually S/c2 .
146
to arrive at r
v2 ~ · ~v − eφ.
L = − mc2 1− + e
c
A (6.64)
c2
p~ ⇒ p~ + e ~
A (6.66)
c
seems to have been, historically, the first version of the so-called minimal
coupling rule.
d α
p = e
c
F αβ U β
ds
can be decomposed into three space- and one time- components. Recall (i)
the definition (1.100) of the four-momentum, p = mcU = γ(mc, mv); (ii)
that F i0 = E i and F ij = ij k H k . Then, again using Eqs.(1.68) and (1.74),
as well as Eq.(1.65) under the form ds d
= γc ddt , we find
147
the Lorentz force law proper,
d ~ + e ~v × H
p~˙ = [mγ~v ] = e E ~ (6.68)
dt c
and the energy time variation,
dE d ~ · ~v .
= γmc2 = e E (6.69)
dt dt
Notice that the particle energy remains constant in time if the field is purely
magnetic.
dE d~
p
Exercise 6.6 Verify that dt = ~v · dt . This is always valid. Here, it turns up trivially.
~ = − grad φ.
E (6.70)
Exercise 6.7 Find (6.70) from (6.71), recalling that the i-th component of the gradient
of a scalar product is given by
148
~ is a constant vector, then its relation to A
And if H ~ can be reversed:
~=
A 1 ~ × ~x.
H (6.72)
2
Exercise 6.8 Verify that (6.72) leads indeed to H ~ = rot A, ~ if H is a constant vector.
Use
[rot (~a × ~b)]k = kij ∂i (~a × ~b)j = kij ∂i (jrs ar bs ) = kij jrs ∂i (ar bs ),
and then one of the contractions of Exercise 1.7.
The next two paragraphs show two rather unrealistic exercises, intended
to fix some ideas about the probing of fields by particles. They examine
the motions of a charged particle in uniform constant electric and magnetic
fields, forgetting the current produced by those very motions.
p~˙ = e E.
~ (6.73)
p~c2
~v = ,
E
with the energy E given in the present case by
p q q
E = m c + p~ c = m c + p0 c + (eEct) = E02 + (eEt)2 c2 ,
2 4 2 2 2 4 2 2 2
149
where q
E0 = m2 c4 + p20 c2
is the particle energy at start. Then,
dx p x c2 eEtc2
vx = = =p 2 ;
dt E E0 + (eEct)2
dy p y c2 p 0 c2
vy = = =p 2 .
dt E E0 + (eEct)2
As pz (t) = 0, the motion will take place on the plane xy. The integrations
give s
2
E0 p0 c eEct
x(t) = + c2 t2 ; y(t) = arcsinh . (6.74)
eE eE E0
E0
The choice of initial conditions (x(0) = eE , y(0) = 0) has been made so
as to make simpler to get the equation of the trajectory, which is found by
eliminating t:
E0 eEy
x= cosh . (6.75)
eE p0 c
As cosh z ≈ 1 + 12 z 2 , the non-relativistic limit gives a parabola.
E~v
p~ = .
c2
The equation becomes
E d~v e ~
= ~v × H,
c2 dt c
or
d~v ec ~
= ~v × H. (6.77)
dt E
150
It will be also convenient to introduce the notation
ecH
ω= . (6.78)
E
If we now choose the axes so that H~ lies along Oz, the equations take the
forms
v̇x = ωvy ; v̇y = − ωvx ; v̇z = 0. (6.79)
A first integration gives, with convenient integration constants,
~ =0
rot E (6.81)
div E~ = 4π ρ. (6.82)
151
This is the Poisson equation. Recall that, in cartesian coordinates,
∂ 2φ ∂ 2φ ∂ 2φ
∆φ = + + 2 . (6.84)
∂x2 ∂y 2 ∂z
The sourceless case (ρ = 0) leads to the Laplace equation
∆φ = 0. (6.85)
152
The origin has been excluded. The point-like charge at ~r = 0 can be intro-
duced as eδ 3 (~r). In fact, a detailed Fourier analysis shows that
e
∆ = − 4πeδ 3 (~r) . (6.88)
r
Because the equation is linear, the field created by a set of charges
(e1 , e2 , e3 , ...) will be the sum of the fields created by the individual charges
ek . If the point at which the field is to be measured stands at a distance r1
of the charge e1 , at a distance r2 of the charge e2 , in short at a distance ri of
the charge ei , then X ei
φ= . (6.89)
i
ri
The charge in a volume element d3 x will be ρd3 x, and the variable r(x)
represents the distance between that volume element and the point at which
the field is to be measured.
We have discussed fields created by electrically-charged pointwise par-
ticles. We could have said: by electric monopoles. There is no magnetic
analogous to such fields, as there are no magnetic monopoles.
∂H~
= 0 (6.91)
∂t
~ = 0
div H (6.92)
~ = 4π ~j.
c rot H (6.93)
~ ≡ 0 implies that H
We learn from the first equation above that E ~ is constant
in time. As there are no magnetic charges, the only possible source of a pure
magnetic field is a current ~j produced by electric charges in motion. The
system is not really static, but can be made to be stationary by a device:
153
take the time average of all fields over a large lapse. We shall use the “hat”
notation H,
b E,
b etc for these averages. For example,
Z T
1
H = lim
b H(t)dt .
T →∞ T 0
Notice that the time average of a time derivative vanishes for any quantity
which remains finite. Thus,
Z T
dk
dE 1 dE k E k (T ) − E k (0)
= dt = →0
dt T 0 dt T
for T large enough.
It is better to take back the complete Maxwell’s equations. Once the
averages are taken, only remain
div H
b = 0 (6.94)
b = 4π bj.
c rot H (6.95)
rot A
b = H,
b (6.96)
c rot rot A b − c ∆A
b = c grad div A b = 4π bj.
b =− 4π b
∆A j. (6.97)
c
This is now the Poisson equation for each component of A,b and the solution
can be obtained by analogy with the electrostatic case. Equation (6.90) will
lead then to Z
4π bj
A=
b d3 x . (6.98)
c r
154
The magnetic field will be
Z
b = 4π rot
bj
H
b = rot A d3 x . (6.99)
c r(x)
~ ) = f rot V
rot (f V ~ + (grad f ) × V
~.
The rot operator acts only on r, which represents the point at which the field
is to be measured. It can consequently be introduced into the integral. The
average current
b is integrated, and can be taken as constant. Thus, rot bj = 0
j
and rot r = grad 1r × bj = bj × rr3 . We thus arrive at
Z
b = 4π
H d3 x bj ×
r
. (6.100)
c r3
This is the Biot-Savart law.
Aα = 0 , (6.101)
and have in mind (for nomenclature, for example) the most important of
electromagnetic waves, light waves. The Lorenz condition does not entirely
fix the gauge. If we use the radiation gauge (6.42)
~ = 0 ; A0 = φ = 0,
div A (6.102)
only the vector potential remains, and for it Eq.(6.101) can be written
~
1 ∂ 2A
~−
∆A =0. (6.103)
c2 ∂t2
155
What about the fields E ~ and H
~ ? Besides the inevitable H
~ = rot A,
~ we have
from (6.36) that
~
~ = − 1 ∂A .
E
c ∂t
~ and H
It is easy to see that both E ~ satisfy that same equation. Thus, the
electromagnetic potential, the electric field and the magnetic field all obey
the same equation.
∂
Exercise 6.11 Verify the statement above, by applying the operators rot and ∂t to
(6.103).
~ E
Furthermore, that equation is the same for each component of A, ~ and
~ It is consequently enough to examine the equation as holding for one
H.
component, as for a function f (~x, t):
1 ∂ 2 f (~x, t)
− ∆f (~x, t) = 0 . (6.104)
c2 ∂t2
§ 6.23 Plane waves A solution of the electromagnetic wave equation is said
~ E,
to be a plane wave when the fields (A, ~ H)
~ depend only on one of the space
coordinates: for example, when the function above is f (~x, t) = f (x, t). The
wave equation becomes
∂ 2 f (x, t) 2
2 ∂ f (x, t)
− c =0, (6.105)
∂t2 ∂x2
which is the same as
∂ ∂ ∂ ∂
−c +c f (x, t) = 0 . (6.106)
∂t ∂x ∂t ∂x
It is convenient to make a change of coordinates so that each of the above
bracketed expression becomes a simple derivative. Such coordinates are
x x
ξ =t− ; ζ =t+ ,
c c
or
1 x 1
t= 2
(ζ + ξ) ; c
= 2
(ζ − ξ).
In that case,
∂ 1 ∂ ∂
∂ 1 ∂ ∂
= 2 ∂t
−c ∂x
; ∂ζ
= 2 ∂t
+c ∂x
.
∂ξ
156
The wave equation acquires the aspect
∂ 2 f (ξ, ζ)
= 0.
∂ξ∂ζ
The solution has, consequently, the form
x x
f (x, t) = f (ξ, ζ) = f1 (ξ) + f2 (ζ) = f1 t − + f2 t + , (6.107)
c c
where the single-argument functions f1 and f1 are arbitrary. Let us examine
the meaning of this solution. Suppose first f2 (ζ) = 0, so that f (x, t) =
f1 (ξ) = f1 t − xc . Fix the plane x = constant: on that plane, the field
changes with time at each point. On the other hand, at fixed t, the field is
different for different values of x. Nevertheless, the field will have the same
value every time the variables t and xc satisfy the relation t − xc = a constant,
that is, when
x = K + ct.
If the field has a certain value at t = 0 at the point x, it will have that same
value after a time t at a point situated at a distance ct from x. Take that
value of the field: we can say that that value “propagates” along the axis
Ox with velocity c. We say, more simply, that the field propagates along the
axis Ox with the velocity of light. The solution f1 t − xc represents a plane
wave propagating with the velocity of light in the positive sense of Ox. The
same analysis leads to the conclusion that the solution f2 t + xc represents
∂Ax
= 0.
∂x
In this gauge, the component Ax is constant in space. The wave equation
2
says then that ∂∂tA2x = 0, or ∂A
∂t
x
= constant. This would say that the electric
field Ex = constant, not a wave. E ~ can only have components orthogonal
to the direction of propagation. Furthermore, Ax = 0 if we are looking
for wave solutions. It follows that the electromagnetic potential is always
157
perpendicular to the axis Ox, that is, to the direction of the plane wave
propagation.
Consider a plane wave progressing along Ox. All the field variables de-
pend only on t − xc . From the first of relations
~
~ = − ∂A
E ~ = rot A
and H ~
∂ct
we obtain
~
~ = − 1 ∂A .
E (6.108)
c ∂ξ
The second is
∂ 1j ∂ 1 1j ∂
Hk = k ij Aj (t − x/c) = k Aj (t − x/c) = − k Aj (ξ)
∂xi ∂x c ∂ξ
∴ Hk = k 1j Ej (ξ) ∴ Hz = Ey ; Hy = − Ez ; Hx = 0 . (6.109)
These relations can be put together by using the unit vector along the direc-
tion of propagation, ~n = (1, 0, 0). Then we verify that
~ = ~n × E.
H ~ (6.110)
In a plane wave, the electric and the magnetic fields are orthogonal to each
other and to the direction of propagation. These are transversal waves. We
see from (6.109) that E~ and H~ have the same absolute values.
The energy flux of a plane wave field will be given by the Poynting vector
(6.62)
~ c ~ ~
c c
S= E × ~n × E = E 2~n = H 2~n. (6.111)
4π 4π 4π
The energy flux is carried by a plane wave along its direction of propagation.
The energy density (6.60) will be
1 ~ 2 ~ 2 1 ~2 1 ~2
W = E +H = E = H
8π 4π 4π
so that
~ = c W ~n.
S (6.112)
As said below Eq.(6.62) the momentum density is S/c ~ 2 . For a plane wave,
this is W ~n/c. Thus, for an electromagnetic plane wave, the relation between
the energy W and the momentum W/c is the same as that for particles
traveling at the velocity of light, Eq.(1.97).
158
§ 6.24 Monochromatic plane waves Let us now consider solutions closer
to our simplest intuitive idea of wave: suppose the above field is a periodic
function of time. This means that all the quantities characterizing the field
— the components of A, ~ E ~ and H~ — involve time in the forms cos(ωt + α)
and/or sin(ωt + α). In that case, much can be said in a purely qualitative
discussion. The unique time multiplier ω will be the wave frequency. In
applications to Optics it appears as the light frequency. Now, light with a
single frequency means light with a single color, and for this reason such
waves are called monochromatic. For expressions of the form
Fields E~ and H~ will have analogous expressions. At fixed time and given an
initial value of the field, the wave comes back to that value at a distance x
= λ given by
2πc
λ= . (6.116)
ω
This “length of one wave” is the wavelength. And, given the above-defined
unit vector ~n along the propagation direction, the wave vector ~k is defined as
~k = ω ~n. (6.117)
c
Representation (6.115) can then be rewritten as
A ~ 0 ei(~k·~r−ωt) .
~=A (6.118)
159
The expression (~k · ~r − ωt) is the wave phase. We can actually introduce a
four-vector ω ω
k= , ~k = (1, ~n) (6.119)
c c
which will be such that kα xα = (ωt − ~k · ~r). We see that kα is a null vector,
or a light-like vector: kα k α = 0. This comes also from the fact that
A ~ 0 e−ikα xα
~=A (6.120)
must be a solution of the wave equation. Taking all this into Eqs.(6.108,
6.110), we obtain
E~ = ik A
~ ; H ~ = i~k × A.
~ (6.121)
§ 6.25 Doppler effect The behavior of the 4-vector (6.119), when seen
from different reference frames, leads to an important effect. Suppose a wave
(such as a light beam) is emitted from a (source) frame KS with a 4-vector
k(S) = ωcS (1, ~n) towards another (receptor) frame KR in the direction of
~n. Suppose ~v is the velocity of the source, or of KS , which moves towards
KR along the axis 0x. The latter will see k(R) = ωcR (1, 1, 0, 0). As the
distance separating KS and KR , as well as the unit vector ~n, lie along 0x,
k(S) = ωcS (1, 1, 0, 0).
KS KR
n v
ωS hω
R v ωR i h vi
=γ − ∴ ωS = γωR 1 −
c c c c c
160
s
v
p
1 − v 2 /c2 1+ c
∴ ωR = ωS = ωS
1 − vc 1− v
c
To consider the case in which the source, instead of moving towards the
receptor, moves away from it (always along 0x), it is enough to invert the
sign of v above. We have thus the two opposite cases:
s
1 + vc
ωR = ωS > ωS (blue shift) .
1 − vc
and s
v
1− c
ωR = ωS v < ωS (red shift) .
1+ c
161
Chapter 7
Dirac Fields
There is, however, a problem: the relativistic Hamiltonian does not lend itself
to such a simple “translation”, since
p q
H = p~ 2 c2 + m2 c4 ⇒ m2 c4 − ~2 c2 ∇ ~2 .
162
As previously said, this equation is, in a certain sense, “compulsory”, as it
states that the field is an eigenstate of the Poincaré group invariant opera-
tor Pµ P µ with eigenvalue m2 c2 . Every field corresponding to a particle of
mass m must satisfy this condition. Of course, once we use H 2 , we shall
be introducing negative energy solutions for a free system: there is no rea-
p
son to exclude H = − p2 c2 + m2 c4 . We have above (in our toy model of
Section 3.1) separated the fields into components of positive and negative
frequencies, ready to interpret the latter as related to antiparticles. This,
of course, because we now know the solution of the problem. At that time,
negative-energy solutions caused great discomfort and led Dirac to a quest
which led him, in the long run, to quite unexpected results.
§ 7.2 He started by seeking a new way to “extract the square root” of the
operator H 2 . He looked for an equation in which squaring ∂∂t and ∇~ were not
necessary. In other words, he looked for a linear, first-order equation both in
t and ~x. He began with an equation for the square root
p → →
p~2 c2 + m2 c4 = c α · p + βmc2 , (7.2)
where α
~ = (α1 , α2 , α3 ) and β are constants to be found. Taking the square,
one arrives at the conditions
(a) α1 2 = α2 2 = α3 2 = β 2 = 1;
(b) αk β + β αk = 0 for k = 1, 2, 3; (7.3)
(c) αi αj + αj αi = 0 for i, j = 1, 2, 3, but i 6= j.
163
equation corresponding to (7.2) is the matrix equation (the “Hamiltonian
form” of the Dirac equation)
∂ ~ → →
Hψ(~x, t) = i~ ψ(~x, t) = c α · ∇ ψ(~x, t) + βmc2 ψ(~x, t), (7.4)
∂t i
the wavefunction will be necessarily a column-vector, on which the matrices
act. Notice that, once conditions (7.3) are satisfied, ψ will also obey the
Klein-Gordon equation which is, as said, mandatory. We must thus look at
(7.4) as an equation involving four matrices (complex, n × n for the time
being) and the n-vector ψ. As H should be hermitian, so should αk and β
be: αk† = αk , β † = β. Take one of them (the reasoning which follows holds
for each one). Being hermitian, it has real eigenvalues and there exists a
similarity transformation which diagonalizes it. By condition (a) in (7.3),
these eigenvalues can be either + 1 or − 1. Furthermore, conditions (a) and
(b) say that
tr αk = tr (β 2 αk ) = tr (βαk β) = −tr αk → tr αk = 0;
tr β = tr (βαk2 ) = tr (αk βαk ) = −tr β → tr β = 0.
The sum of the eigenvalues vanishes, so that there must be an equal number
of eigenvalues +1 and − 1. Consequently, the number of eigenvalues is even:
n is even. The first possibility would be n = 2, but there are not four 2 × 2
matrices which are hermitian, independent and distinct from the identity.
There are only three [for example, the Pauli matrices (2.23)].
§ 7.3 The minimal possible value of n for which the αk ’s and β can be
realized is 4. We shall make the choice
! !
0 σi σ0 0
αi = ; β= , (7.5)
σi 0 0 − σ0
Comment 7.2 Notice that the argument holds as long as four matrices are needed. If
m = 0, β disappears and three 2×2 matrices (say, again the Pauli matrices) suffice. In
this case the particle is described by a Weyl spinor, or Pauli spinor.
164
§ 7.4 It is good to keep in mind that any other set of matrices obtained from
that one by similarity will also satisfy (7.3) and can be used equivalently.
Each such a set of matrices is called a “representation”. The above choice
will be called the “Dirac representation”. Equation (7.4) is now
→ →
!
∂ mc2 I ~
i
c σ · ∇
i~ ψ(~x, t) = → → ψ(~x, t), (7.6)
∂t ~
c σ · ∇ −mc 2
I
i
ψ1
ψ
2 † ∗ ∗ ∗ ∗
where ψ(~x, t) = . The hermitian conjugate ψ = ψ ψ ψ
1 2 3 4ψ
ψ3
ψ4
will obey the hermitian conjugate of the above equation,
→ ←
!
∂ † † mc2 I − ~i c σ · ∇
− i~ ψ (~x, t) = ψ (~x, t) → ← . (7.7)
∂t − ~i c σ · ∇ −mc2 I
that is,
∂ † ~ h → i
i~ ψ ψ = c div ψ † α ψ . (7.9)
∂t i
This expression is reminiscent of the continuity equation which, in non-
relativistic Quantum Mechanics, states the conservation of probability:
∂ρ →
+ div j = 0, (7.10)
∂t
where
ρ = ψ†ψ (7.11)
165
is the density of probability and
j k = c ψ † αk ψ (7.12)
is the k-th component of the probability current. From this continuity equa-
tion (and Gauss theorem) we obtain
Z
∂
d3 x ψ † (~x, t)ψ(~x, t) = 0. (7.13)
∂t
§ 7.5 Up to this point, all we have said is that the αk ’s and β are hermitian
matrices with vanishing trace. But the above continuity equation should be
put into the covariant form ∂µ j µ = 0 and this would require something else
→
of them: α = (α1 , α2 , α3 ), up to this point only a notation, must actually be
such that ψ † ψ and c ψ † αk ψ constitute a Lorentz four-vector: ρ must be the
temporal component and jk the space components. Furthermore, it suggests
that cαk be a velocity. This hint will be corroborated below.
§ 7.6 It is clear, above all, that the Dirac equation (7.4) must be covariant.
Before going into that, let us try to grasp something of the physical meaning
of the equation. We shall see later that it describes particles of spin 21 .
For that reason, we shall frequently take the liberty of referring to ψ as
the “electron wavefunction” and talk of the electron as if it were the only
→
particle in view. Let us examine the case of the electron at rest. As p = 0,
the equation reduces to
∂
i~ ψ(~x, t) = βmc2 ψ(~x, t). (7.14)
∂t
The de Broglie wavelength λ = ~/p is infinite and ψ must be uniform over
→
all the space, as p~ ψ = 0 ⇒ ∇ ψ(~x, t) = 0. This is also coherent with the
→
interpretation of c α as the velocity. Using the Dirac representation,
∂
− mc2
i~ ∂t 0 0 0 ψ1
∂
0 i~ ∂t − mc2 0 0 ψ2
= 0.
∂
+ mc2
0 0 i~ ∂t 0 ψ3
∂
0 0 0 i~ ∂t + mc2 ψ4
(7.15)
166
There are four independent solutions,
1 0
− ~i mc2 t
0
− ~i mc2 t
1
ψ1 = e ; ψ2 = e ;
0 0
0 0
(7.16)
0 0
2 0 2 0
i
i
ψ3 = e+ ~ mc t ; ψ4 = e+ ~ mc t .
1 0
0 1
We have thus a first drawback: ψ3 and ψ4 are solutions with negative en-
ergy. In quantum theory they are interpreted as wavefunctions describing
antiparticles (positrons).
§ 7.7 The electron has an electric charge, and consequently couples to the
electromagnetic field. We shall introduce in (7.4) an electromagnetic field
through the minimal coupling prescription:
e ∂
pµ ⇒ pµ − Aµ , with pµ = i~ .
c ∂xµ
The result is
∂ h e ~ i
i~ α · p~ − A
ψ(~x, t) = c~ + βmc2 + eφ ψ(~x, t), (7.17)
∂t c
where A0 = φ. Recall that an electric point–charge in an electromagnetic
~ + eφ, which appears above
field has the interaction energy HI = − ec ~v · A
under the form HI = − ec c~ α·A ~ + eφ. This validates the interpretation of
matrix c α~ as the velocity in this theory: c α
~ will be the velocity operator.
167
positive-energy and negative-energy components will have different behav-
ior, it will be convenient to introduce the two 2−component columns
! !
ψ1 ψ3
L̃ = ; S̃ = . (7.18)
ψ2 ψ4
!
L̃
Using ψ = in (7.17),
S̃
! → → → ! !
∂ L̃ [mc2 + eφ]I c σ ·( p − ec A) L̃
i~ = → → → (7.19)
∂t S̃ c σ ·( p − ec A) [− mc2 + eφ]I S̃
or → →
→
! !
∂
i~ ∂t L̃ [mc2 + eφ]L̃ + c σ ·( p − ec A)S̃
∂
= → → → (7.20)
i~ ∂t S̃ c σ ·( p − ec A)L̃ + [− mc2 + eφ]S̃
Now: the larger part of the energy will be concentrated in mc2 ; then, the
strongest time–variation will be dominated by this term. This means that,
if we look for solutions of the form
! !
L̃ i 2 L
= e− ~ mc t , (7.21)
S̃ S
!
L
most of the time variation will be isolated in the exponential, and
S
will vary slowly with t. The equation then becomes
! → → → !
∂ L eφL + c σ ·( p − ec A)S
i~ = → → → . (7.22)
∂t S c σ ·( p − ec A)L + [eφ − 2mc2 ]S
168
We see that S (“small”) is indeed very small in comparison to L (“large”):
S/L is of the order v/c. The components ψ3 and ψ4 are for that reason called
the “small components” of the Dirac wavefunction, ψ1 and ψ2 being the
“large components”. Because it associates in this way two Pauli spinors, one
large and one small, the 4-component representation is called the “bispinor
representation”. Taking the above approximated S into the equation for
∂
i~ ∂t L, we find
"→ → → → → → #
e e
∂ σ ·( p − c A) σ ·( p − c A)
i~ L = + eφ L. (7.25)
∂t 2m
This can be put into a more readable form by using the identity σ i σ j =
δ ij + i ij k σ k , which leads to
→ → → → →
→
( σ · a) ( σ · ~b) = a · ~b + i σ · a × ~b . (7.26)
→ →
Let us first look at the vector–product term. As p = ~i ∇, then
h→ e→
→ e→i jk ~ e ~ e
( p − A) × ( p − A) L = i ∂j − Aj ∂k − Ak L
c c i i c i c
~ e jk ~e
= − i (∂j Ak )L = − Bi L .
i c i c
~ (in vacuum, = our previous H)
The magnetic field B ~ turns up. Adding now
the scalar–product term,
h→ → e → i h→ → e → i → e →2 e~ →
σ ·( p − A) σ ·( p − A) = p − A − ~
σ · B. (7.27)
c c c c
The equation becomes the 2−component Pauli equation
→ 2
→
p e
∂ −cA e~ → ~
i~ L = − σ · B + eφ L, (7.28)
∂t 2m 2mc
!
ψ 1
which describes a spin– 12 electron: L = exp[ ~i mc2 t] . Under a rota-
ψ2
tion, as shown in Exercise 2.7, it transforms as a member of the representation
j = 12 .
169
7.3 Covariance
In the Hamiltonian form (7.4) of the Dirac equation, time and space play
distinct roles. To go into the so-called “covariant form”, we first define new
matrices, the celebrated Dirac’s “gamma matrices”, as
γ 0 = β ; γ i = βαi . (7.29)
β ∂ψ
Multiplying the equation by c
on the left, we find i~γ µ ∂xµ = mcψ, or
or
(6 p − mc)ψ = 0. (7.32)
In the presence of an electromagnetic field,
e
(6 p − 6 − mc)ψ = 0,
A (7.33)
c
where, of course, 6 A = γ µ Aµ . In terms of the gamma matrices, conditions
(7.3) acquire a compact form,
γ µ γ ν + γ ν γ µ = {γ µ , γ ν } = 2 η µν I. (7.34)
Multiply (g) by γ 0 on both sides and use (f ) to commute the αk ’s with γ 0 and obtain
γ i γ j + γ j γ i = − 2 δ ij I. (7.36)
Together with (e), this is just (7.34). By the way, (7.34) is the operation table of a
particular example of “Clifford algebra”.
170
The right-hand side exhibits the Lorentz metric times the unit 4 × 4 matrix
I. And, in the middle, there is a first: an anticommutator comes forth. For
i = 1, 2, 3 the matrix γ i is antihermitian [because (γ i )† = (βαi )† = (αi )† β † =
αi β = − βαi = − γ i ], whereas γ 0 (= β) remains hermitian. Other properties
are (γ i )2 = − I and (γ 0 )2 = I. In the Dirac representation the γ’s have the
forms
0 σi 0 I 0
γi = ; γ = . (7.37)
− σi 0 0 −I
Comment 7.4 The Dirac equation becomes real in the “Majorana representation”. The
µ
solutions are then superpositions of real functions. The γ’s are given by γM ajorana =
!
2
µ I σ
U γDirac U −1 , with U = U −1 = √12 .
σ 2 −I
This comes, of course, from the conditions (7.3) or (7.34), which have been
imposed just to attain this objective. It follows also that a solution of the
Dirac equation is a solution of the Klein-Gordon equation though, of course,
the Klein-Gordon equation can have solutions which do not satisfy the Dirac
equation.
171
Let us now examine the Lorentz covariance of (7.30). Recall that, in order
→
to start talking about invariance, covariance, etc, the field ψ(x) = ψ( x, t)
must belong to (the carrier space of) a linear representation of the Lorentz
group, so that a matrix U (Λ) must exist such that the field ψ 0 (x0 ), seen in
0 0 0 0
another frame (that frame in which xα = Λα β xβ , pα = Λα β pβ , etc) is
related to ψ(x) by ψ 0 (x0 ) = U (Λ)ψ(x). What we shall do is to determine
U (Λ) so as to ensure the covariance of the equation. In other words, we shall
find the representation to which ψ(x) belongs.
Multiplying the equation on the left by U (Λ) and substituting ψ(x) =
U (Λ)−1 ψ 0 (x0 ), we arrive at
0
In order to identify this with γ α pα0 − mc ψ 0 (x0 ) = 0, it will be necessary
that
0
U (Λ) γ α U (Λ)−1 = γ α Λα0 α , (7.39)
or
0 0
γ β = Λβ α U (Λ) γ α U (Λ)−1 . (7.40)
This means that the set {γ α } must constitute a 4-vector of matrices, and that
U (Λ) acts on the space of such matrices, as the representative of Λ. Notice
however that, as the Minkowski metric tensor is invariant under Lorentz
transformations,
ηab = Λc a Λd b ηcd , (7.41)
if we take into account the relation (7.34) between the gamma matrices and
the spacetime metric, we conclude that γ β does not change under such trans-
formation either. In other words, like Eq. (7.41) for ηab , the transformation
(7.40) must actually be written without the “primes”:
172
notation which became standard for historical reasons. The generators will
actually turn out to be 12 σαβ . And the double index leads to double counting,
rendering necessary an extra 21 factor. The factor 14 in the exponent owes
its origin to these two 12 factors. Now, for the specific case of the vector
representation,
i αβ
U (Λ) = exp − ω Mαβ , (7.44)
2
where the matrix representative of the Lorentz group generators in the vector
representation is
U (Λ) ≈ I − 4i δω αβ σαβ ,
and, to the first order, we find for the left-hand side of (7.39):
U γ µ U −1 ≈ γ µ − 4i δω αβ [σαβ , γ µ ].
Clearly, σαβ must be antisymmetric in the two indices, and the first idea
coming to the mind does work: the matrices
i i
σαβ = 2
(γα γβ − γβ γα ) = 2
[γα , γβ ] (7.48)
173
2. From this, [σαβ , γ ] = i[γα γβ , γ ].
[σαβ , γ ] = i(γα γβ γ − γ γα γβ )
= i(−γα γ γβ + 2γα ηβ + γα γ γβ − 2γβ ηα )
= 2i(γα ηβ − γβ ηα ).
[ 21 σαβ , 12 σγδ ] = i (ηβγ 12 σαδ − ηαγ 21 σβδ + ηαδ 12 σβγ − ηβδ 12 σαγ ), (7.50)
which shows that 21 σαβ is a Lorentz generator. As each matrix Mαβ , each
matrix 21 σαβ is a generator of a representation of the Lie algebra of the
Lorentz group. The Mαβ ’s generate the vector representation, the 21 σαβ ’s
generate the bispinor representation. Expression (7.47) is the infinitesimal
version of (7.39). It states again that the gamma’s constitute a 4–vector.
Thus, covariance of the Dirac equation requires that the Dirac field ψ(x)
belong to the bispinor representation U (Λ) generated by the above 21 σαβ ’s.
The form of the multiplication table (7.50) is general: any set of Lorentz
generators will satisfy it. It characterizes the Lie algebra of the Lorentz group.
It is clear, however, that the particular form of the matrices σαβ depend on
the “representation” we are using for the matrices γ. In the Pauli-Dirac
“representation” we are using, the σαβ ’s are particularly simple:
ijk σk 0 0i i 0 iσ i
σij = ; σ =iα = . (7.51)
0 ijk σk iσ i 0
Notice that U (Λ) is not, in general, unitary. From the hermiticity properties
of the γ’s,
(γ i )† = − γ i and (γ 0 )† = γ 0 , (7.52)
we get
(σij )† = σij and (σ0j )† = − σ0j . (7.53)
Consequently, in this “representation”, U (Λ) will be unitary for the rotations,
but not for the boosts.
Comment 7.6 Actually, there can be no unitary representation for all the members of
the Lorentz group with finite matrices. This comes from a general result from the theory
of groups: a non-compact group has no finite unitary representations.
174
A useful property comes from (7.52): sandwiching a gamma matrix be-
tween two γ 0 yields the hermitian conjugate, a property that propagates to
the σ αβ :
†
γ 0 γ α γ 0 = γ α† → γ 0 σαβ γ 0 = σαβ . (7.54)
Applying the latter order by order in the expansion of U, a result of interest
in future calculations comes out:
U −1 (Λ) = γ0 U † (Λ)γ0 . (7.55)
Before finishing this section, let us consider the specially important ex-
amples of Lorentz transformations which are rotations. Take the particular
case of a rotation of an angle φ around the axis Oz, generated by
3
z 3 12 12 σ 0
J = J = J = 21 σ = 12 . (7.56)
0 σ3
In this case,
L̃0 (x0 )
0 0 − 2i φ σ 12 L̃(x)
ψ (x ) = =e . (7.57)
S̃ 0 (x0 ) S̃(x)
Given the (diagonal) form of σ 12 , the transformation will act separately on
L̃(x) and S̃(x), and for each one we shall have the behavior of a Pauli spinor
→
under rotations. The expression for the rotation of an angle α acting on a
Pauli spinor χ(x) is
→ → i→ →
χ0 (x0 ) = e−i α · J χ(x) = e− 2 α · σ χ(x) (7.58)
→ → → →
" #
|α| α·σ |α|
= cos − i → sin χ(x). (7.59)
2 |α| 2
175
In another frame, it will be (using successively (7.55), (7.39) and (7.61))
j 0µ (x0 ) = c ψ 0† (x0 )γ 0 γ µ ψ 0 (x0 ) = c ψ † (x)U † γ 0 γ µ U ψ(x)
= c ψ † (x)γ 0 γ 0 U † γ 0 γ µ U ψ(x) = c ψ † (x)γ 0 U −1 γ µ U ψ(x)
= c ψ † (x)γ 0 Λµ ν γ ν ψ(x) = c Λµ ν ψ † (x)γ 0 γ ν ψ(x) = Λµ ν j ν (x) .
That is, the density current transforms as it should — as a Lorentz 4-vector.
In consequence, the continuity equation
∂µ j µ (x) = 0 (7.62)
is invariant.
176
also usually written in the form
h ← i
ψ i~ 6 ∂ + m c = 0, (7.68)
J µ = c ψ γµ ψ . (7.71)
To obtain the spin density (4.59), we first get the Lorentz transformations
Then, we get
J µ αβ = S µ αβ + Lµ αβ ,
with
Lµ αβ = xα θµ β − xβ θµ α
representing the orbital angular momentum density.
7.5 Parity
Another special role reserved to γ 0 is related to the parity transformation.
To keep on with a notation which became usual, we shall use
1 0 0 0
0 −1 0 0
Λ(P ) = (7.74)
0 0 −1 0
0 0 0 −1
177
for the matrix representing the parity transformation in cartesian coordinates
on spacetime, and P = U (Λ(P ) ) for its representative acting on bispinors.
Repeating what has been done to impose Lorentz covariance on the Dirac
equation, if we define
ψ 0 (x0 ) = P ψ(x),
0 (P )
Comment 7.7 From the signs in (7.74), γ β Λβ 0 α = γ 0 δ0α − γ i δiα = (γ 0 )† δ0α + (γ i )† δiα =
(γ α )† . Use then the first equality in (7.54) to find the solution below.
P = eiϕ γ 0 , (7.76)
P −1 = γ0 P † γ0 . (7.77)
178
In the Pauli-Dirac representation, γ 5 is given by
5 0 I
γ = γ5 = . (7.80)
I 0
Some of its properties are
{γ 5 , γ α } = 0, (7.81)
and
[γ 5 , σαβ ] = 0. (7.82)
This last property propagates from the spinor generators to the whole rep-
resentation U (Λ):
[γ 5 , U (Λ)] = 0. (7.83)
Notice that (7.81) says that γ 5 inverts parity:
P γ5 = − γ5 P . (7.84)
This means that, if ψ is a state with definite parity, say P ψ = +ψ, then γ 5 ψ
will have opposite eigenvalue: P γ 5 ψ = −γ 5 ψ.
It can then be shown that the following 16 matrices form a basis for the
4 × 4 matrices:
ΓS = I (7.85)
ΓVµ = γµ (7.86)
ΓTµν = σµν (7.87)
ΓA
µ = γ5 γµ (7.88)
P
Γ = γ5 . (7.89)
179
whereas the positron will satisfy
e µ
[i ~ γ µ ∂µ +γ Aµ − mc]ψc (x) = 0. (7.91)
c
Notice that in everything we have done up to now the charge sign has played
no role. We could exchange the above equation, ascribing the first to the
positron and the second to the electron. The sign of the charge is conven-
tional — only the relative sign is meaningful. What we are going to show
is the existence of a correspondence which, to each solution of one of them,
provides a solution of the other. To each electron corresponds a particle
which differs from it only by the sign of the charge. The operation C describ-
ing this correspondence is called charge conjugation. It gives the positron
wavefunction ψc (x) from the electron wavefunction ψ(x), and vice-versa:
ψc (x) = C ψ(x).
It is, like parity, an involution: C 2 = I. This operation is not given by a
simple action of a matrix on ψ(x): there is no 4 × 4 matrix leading one into
the other solution of the two equations above. Notice that we want only to
change the relative sign between the kinetic term and the charge term. The
complex conjugate of (7.90) is
h e i
i γ µ∗ (~ ∂µ − i Aµ ) + mc ψ ∗ (x) = 0 . (7.92)
c
To arrive at a solution of (7.91), we should find a matrix taking γ µ∗ into
(− γ µ ). It is traditional to write such a matrix in the form Cγ 0 :
(Cγ 0 )(γ µ∗ )(Cγ 0 )−1 = − γ µ .
If this matrix exists, then
ψc ≡ C ψ = Cγ 0 ψ ∗ ,
which would be the desired solution. Now, it so happens that the matrix
does exist, and its explicit form depends on the γ-representation used. Let
us proceed in the Pauli-Dirac representation, in which ψc ≡ C ψ = Cγ 0 ψ ∗ =
Cγ 0 (γ 0 )T ψ T = C ψ T . Furthermore, as γ 0 γ µ† γ 0 = γ µ , or γ 0 γ µ∗ γ 0 = γ µT , C
must have the effect C −1 γ µ C = − γ µT . As γ 1 and γ 3 are already equal to
minus their transposes, C must simply commute with them. As γ 0 and γ 2
are equal to their transposes, C must anticommute with them. Then, up to
a phase which will not interest us,
C = i γ 2γ 0.
Notice that C = − C −1 = − C † = − C T . Given now a solution ψ(x) of
(7.90), its charge conjugate is
ψc = i γ 2 ψ ∗ = i γ 2 γ 0 ψ T .
180
7.7 Time Reversal and CP T
The Klein-Gordon equation, being quadratic in the time variable, is automat-
ically invariant under time reversal. Such an invariance reflects our intuitive
notion by which, if we look backwards at the motion picture of the evolution
of a particle without any energy dissipation, we would see it retrace, though
in inverse order, all the points prescribed by the same equation of motion,
with inverse initial velocity. In other words, the equation of motion itself
must be invariant under time reversal.
Let us take the Dirac equation in its Hamiltonian form (7.17) (with ~ =
c = 1), in the presence of an external electromagnetic field:
∂ h→ → → i
i ψ(~x, t) = α ·(−i ∇ −e A) + βm + eφ ψ(~x, t). (7.93)
∂t
~ = ~j and
A look at the wave equations A φ = ρ, will tell us that
→ →
0 0
A (~x, − t) = − A (~x, t) and φ (~x, − t) = φ(~x, t).
One can verify that a simple matrix operation will not do the work. The first
member of the above Dirac equation, as well as the usual treatment of the
Schrödinger equation, suggest the use of the complex-conjugate equation. In
effect, what we shall look for (and find) is a 4 × 4 matrix T such that
Taking the inverse of this expression into the complex conjugate of (7.93),
we get
→
∂ 0 →∗ →
0 −1 0 0 ∗ −1 0
i 0 ψ (~x, t ) = T α T · (i∇ + e A ) + T β T m + eφ ψ 0 (~x, t0 ) .
∂t
To obtain the Dirac equation with reversed time, T must commute with β
and α2 , while anticommuting with α1 and α3 . Up to another phase which
we shall not discuss,
T = − iα1 α3 = iγ 1 γ 3 .
Thus,
T ψ(~x, t) = ψ 0 (~x, − t) = iγ 1 γ 3 ψ ∗ (~x, t).
181
The time-reversal operation is anti-unitary, and was introduced by Wigner
(“Wigner time reversal”).
Let us now examine the successive application of the operations T , C and
P:
P CT ψ(~x, t) = P Ciγ 1 γ 3 ψ ∗ (~x, t) = P iγ 2 [iγ 1 γ 3 ψ ∗ (~x, t)]∗ ,
or
ψP CT (x0 ) ≡ P CT ψ(~x, t) = ieiφ γ 5 ψ(~x, t) .
What we have just seen is a particular case of a very general theorem of
the theory of relativistic fields, which says that every possible state for a
system of particles is also possible for a system with antiparticles, though
with reversed space and time. This CP T theorem states that CP T is an
invariance of any Lorentz covariant system which is causal and local.
182
183
Chapter 8
Gauge Fields
8.1 Introduction
The study of free fields is essential to introduce the basic notions and methods
but has, by itself, small physical content. The attributes of a system can
only be discovered by studying its responses to exterior influences. The
characteristics of the system supposedly described by the field only can be
found and measured via interactions with other systems. In the spirit of
field theory, according to which everything must be ultimately described
through the mediation of fields, that would mean interactions with other
fields. Furthermore, a free field can, due to the symmetries imposed, require
the presence of another. A complex scalar field, for instance, has a charge
that, if interpreted as the electric charge, calls for (or is the cause of, or still
is the source of) another field, the electromagnetic field.
The problem of how to introduce interactions in a relativistic theory has
been the object of long discussions. The old notion of potential presented
great difficulties. There are still problems in the classical theory (with a finite
number of degrees of freedom!). We shall not be concerned with those ques-
tions. In field theory, the simplest, straightest way to introduce interaction
in a coherent way is provided by the Lagrangian formalism. What is done
in practice is to write a total Lagrangian formed by two pieces. The first —
the kinematical part — is the sum of the free Lagrangians of all the fields
involved. The second has terms representing the interactions supposed to be
at work. This takes the general form
L = Lfree + Lint . (8.1)
It is then necessary to calculate the consequences and compare with experi-
ment. In a nutshell: trial and error! For the above referred to complex scalar
184
plus electromagnetism case, the total Lagrangian is (5.22):
Basically, two criteria are used when looking for a Lagrangian: symmetry
and (if not redundant) simplicity. We select the simplest combination of fields
respecting the symmetries related to the conservation laws. The fields are
previously chosen as members of linear representations of the Lorentz group
and, to get Lint as a scalar, only their contractions are allowed. Quantum
theory adds other requirements, because not every Lint leads to well-defined
values for the ensuing calculated quantities. Many lead to infinite values for
quantities known to be finite. Actually all of them lead to infinities, but
there is a well-established procedure to make them finite (to “renormalize”
them). When this procedure fails, Lint is said to be “non-renormalizable” and
discarded. This requirement is extremely severe, and eliminates all but a few
field combinations. For example, amongst the many Lagrangians conceivable
to represent the interaction of a real scalar field with itself, such as
185
A Dirac field in the presence of an electromagnetic field can be obtained
from the free Lagrangians by the minimal coupling prescription:
h e e i
L = 2i ψγ µ {∂µ − i Aµ }ψ − ({∂µ + i Aµ }ψ)γ µ ψ −mψψ − 14 Fµν F µν , (8.7)
c c
or
µ e
i
ψγ ∂µ ψ − (∂µ ψ)γ µ ψ − mψψ − 14 Fµν F µν + ψγ µ Aµ ψ .
L= 2
(8.8)
c
This is the starting point of electrodynamics proper. The interaction La-
grangian is
e
Lint = ψγ µ ψAµ = j µ Aµ . (8.9)
c
For charged self–interacting scalar mesons, the minimal coupling prescrip-
tion yields the Lagrangian
186
8.2 The Notion of Gauge Symmetry
Gauge theories involve a symmetry group (the gauge group) and a prescrip-
tion (the minimal coupling prescription) to introduce coupling (that is, inter-
actions) between fields in such a way that the symmetry is preserved around
each point of spacetime. They account for three of the four known funda-
mental interactions of Nature (gravitation, at least for the time being, stands
apart). Namely:
Take, to fix the ideas, a scalar field endowed with supplementary degrees
of freedom (internal, alien to spacetime). If these degrees of freedom assume
N values, the field will actually be a set φ of N fields, φ = {φi }. The behavior
must be well-defined, that is, φ must belong to some representation of the
group, called the “gauge group”. For simplicity, one supposes that only
linear representations are at work. This means that each group element will
be represented by an N ×N matrix U , and the corresponding transformation
will be given by
φi (x) → φ0i (x) = Uij φj (x). (8.13)
Notice that the gauge transformation is a transformation at a fixed spacetime
point x. The number N depends on the representation, and i, j = 1, 2, . . . , N .
187
The group element U will have the form
[Ta , Tb ] = f c ab Tc . (8.15)
that is,
δ̄φi (x) ≡ φ0i (x) − φi (x) = (δαa T a )ij φj (x). (8.17)
In matrix language,
δ̄φ(x) = δαa T a φ(x). (8.18)
so that
∂L ∂L
(Tc )ij φj + (Tc )ij ∂µ φj = 0 (8.21)
∂φi ∂∂µ φi
for each generator Tc . Using the Lagrange derivative
δL ∂L ∂L
= − ∂µ ,
δφ ∂φ ∂∂µ φi
188
this is the same as
δL ∂L ∂L
(Tc )ij φj + ∂µ (Tc )ij φj + (Tc )ij ∂µ φj = 0,
δφ ∂∂µ φi ∂∂µ φi
or
δL ∂L
Tc φ + ∂µ Tc φ = 0. (8.22)
δφ ∂∂µ φ
The Noether current will be just
∂L δφi ∂L ∂L
Jcµ = − c
=− (Tc )ij φj = − Tc φ, (8.23)
∂∂µ φi δα ∂∂µ φi ∂∂µ φ
so that
δL
Tc φ = ∂µ Jcµ . (8.24)
δφ
The conservation of current comes then directly from the equations of mo-
tion δL
δφ
= 0. Such transformations, with spacetime-independent parameters,
will be the same for all events and are consequently called global trans-
formations. In the representation of φ it is always possible (almost always:
the symmetry group must be semi–simple) to define an internal scalar prod-
uct φi φi which is invariant under the group transformations. The invariant
Lagrangian will then be
L[φ] = 1
2
[(∂µ φi )† (∂ µ φi ) − m2 φ†i φi ]. (8.25)
δ̄∂µ φi (x) = δαc (x) (Tc )ij ∂µ φj (x) + ∂µ δαc (x)(Tc )ij φj (x). (8.26)
189
All this is reminiscent of what we have seen in Section 5.2, when dis-
cussing the complex scalar fields. Also there we had found a breaking in the
Lagrangian invariance when the parameters became point–dependent. Terms
in the derivatives of the parameters broke the invariance. How did we fare in
that case? We have recalled the gauge indeterminacy of the electromagnetic
potential, and found that it was possible to use that freedom to compensate
the parameter derivative by a gauge transformation. This, of course, provided
the electromagnetic potential Aµ were present. We were forced to introduce
Aµ if we wanted to restore the invariance. There, we have done it through
the minimal coupling prescription, by which the derivative is modified. That
is what we shall do here: we shall define a new covariant derivative,
Dµ φ = ∂µ φ + Aµ φ, (8.29)
Aµ = Aa µ Ta . (8.30)
Consequently,
where we have profited to exhibit some usual notations. Let us calculate the
variation of this covariant derivative:
δ̄[Dµ φi ] = δ̄(∂µ φi ) + δ̄Aa µ (Ta )ij φj + Aa µ (Ta )ij δ̄φj = δαc (Tc )ij ∂µ φj
+ (∂µ δαc )(Tc )ij φj + δ̄Aa µ (Ta )ij φj + δαc Aa µ (Ta )ij (Tc )jk φk .
Notice that (8.32) attributes to the covariant derivative (as (8.20) gave to the
usual derivative) the same behavior the fields have under transformations.
That is where the name covariant derivative comes from. Under a global
transformation, the usual derivative is already automatically covariant.
190
8.5 Local Noether Theorem
According to the minimal coupling prescription, the original Lagrangian
(8.25) has to be modified by the change
∂µ φi → Dµ φi
It then becomes
L0 ≡ L0 [φ] = 1
2
[(Dµ φi )† (Dµ φi ) − m2 φ†i φi ]. (8.34)
We are supposing a real φi . As we want (8.13) to be a unitary transformation,
the generator matrices must be anti-hermitian, Ta† = − Ta (if we want to use
hermitian matrices for the generators Ta , it is necessary to add a factor i in
the exponent of (8.14)). In consequence,
(Dµ φ)†i = ∂µ φ†i − Aa µ φ†j (Ta )ji . (8.35)
Imposing (δL0 /δφ†i ) = 0, the equation of motion comes out as
(Dµ Dµ φ)i + m2 φi = 0. (8.36)
The Lagrangian variation will be, now,
∂L0 c ∂L0
δL0 = δα (Tc )ij φj + δαc (Tc )ij (Dµ φ)j + hc ,
∂φi ∂Dµ φi
with “hc” meaning the “hermitian conjugate”. Equivalently,
0 0
0 c δL † ∂L
δL = δα (Tc )ij φj + Dµ (Tc )ij φj +
δφi ∂Dµ φ i
∂L0
(Tc )ij (Dµ φ)j + hc, (8.37)
∂Dµ φ i
where we have added the second term and subtracted it by absorption into the
Lagrange derivative (δL0 /δφi ). Writing now explicitly the covariant deriva-
tives, the last two terms give
∂L0 ∂L0
a
∂µ (Tc )ij φj − Aµ (Ta )ij (Tc )jk φk
∂Dµ φi ∂Dµ φ i
∂L0
a
+Aµ (Tc )ij (Ta )jk φk
∂Dµ φ i
∂L0 ∂L0
a
= ∂µ Tc φ − Aµ [Ta , Tc ]φ
∂Dµ φ ∂Dµ φ
∂L0 ∂L0
a
= ∂µ (Tc )φ − Aµ f b ac Tb φ
∂Dµ φ ∂Dµ φ
= − (∂µ J µ c − f b ac Aa µ J µ b ) ≡ − (∂µ J µ c + f b ca Aa µ J µ b )
= −Dµ J µ c ,
191
where
∂L0
J µc = − Tc φ (8.38)
∂Dµ φ
(compare with (8.23)). The variation of the Lagrangian is then
0
0 c δL µ
δL = δα Tc φ − Dµ J c . (8.39)
δφ
For the solutions of the field equation, that is, for δL0 /δφ = 0, the invariance
of the lagrangian gives, for each component in the algebra,
Dµ J µ c = ∂µ J µ c + f b ca Aaµ J µ b = 0. (8.40)
In terms of the matrices J µ := J µ c T c and Aµ := Aaµ Ta , the covariant
divergence becomes
Dµ J µ = ∂µ J µ + [Aµ , J µ ]. (8.41)
This is the same expression found above, if we can use the cyclic property
(f b ac = f c ba = f a cb ) of the structure constants, valid if the group is semi–
simple. In this case, the conservation law assumes the form
Dµ J µ = ∂µ J µ + [Aµ , J µ ] = 0. (8.42)
This covariant derivative differs from that of (8.29). The covariant deriva-
tive of a quantity depends on how the quantity is represented: φ is a column
vector and there Dµ acts as a matrix on a column. The current Jµ is a ma-
trix, and Dµ acts on it through a commutator. We shall see that the same
happens to Aµ . The covariant derivative depends also on the spacetime in-
dices, in a way quite analogous to the usual differentials. In the case above
we have a divergence.
Equation (8.42) is not a real conservation law. The current is not con-
served (∂µ J µ 6= 0), it has only vanishing covariant divergence. This is exactly
the concern of the second Noether theorem. It does not lead directly to a
conserved quantity. It is a constraint imposed on the current to ensure the
invariance of the modified Lagrangian.
The modified Lagrangian,
L0 [φ] = L0 [φi , Dµ φj ] (8.43)
will depend on Aaµ only through Dµ φ. The current (8.38) can then take a
simpler form. From (8.31) for fixed values of a and µ,
∂(Dµ φ)i
(Ta )ij φj = , (8.44)
∂Aa µ
so that
∂L0
J µa = − . (8.45)
∂Aa µ
192
8.6 Field Strength and Bianchi Identity
Unlike usual derivatives, covariant derivatives do not commute: it is easy to
check that
[Dµ , Dν ]φ = F a µν Ta φ, (8.46)
where
F a µν = ∂µ Aa ν − ∂ν Aa µ + f a bc Ab µ Ac ν . (8.47)
The matrix
Fµν = Ta F a µν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ] (8.48)
is the field strength. If the group is abelian, f a bc = 0, and the last term does
not exist. If, furthermore, the group has only one generator, the expression
above reduces to that holding for the electromagnetic field. Matrix (8.48),
thus, generalizes the electromagnetic field strength to the non-abelian case.
We can write (8.33) in matrix form:
This means that F is covariant. We see that, in the abelian case, Fµν is
simply invariant. This is the case, in particular, of electromagnetism. In the
general case, Fµν behaves as a matrix — it is not invariant, but covariant.
On the other hand, (8.49) is the infinitesimal version of
aT bT aT bT
A0µ = eα a
Aµ e−α b
+ eα a
∂µ e−α b
, (8.53)
which shows that Aµ is not strictly covariant. This is the expression for the
gauge transformation of Aµ in the general non-abelian case.
We have seen that the expression of the covariant derivative changes
in each case. Acting on a Lorentz scalar, which is furthermore a column
vector in internal space, it has the form (8.29). Acting on a Lorentz vector
which is furthermore a matrix in internal space, it can assume two forms,
corresponding to the divergence and the rotational in usual vector analysis.
193
That corresponding to the divergence we have seen in (8.41). As to the
rotational, it has exactly the form given in (8.48): the field strength is the
covariant derivative of the potential. This generalizes the relation B = rotA
of electromagnetism. It is frequent to write symbolically
Fµν = Dµ Aν , (8.54)
meaning by that just (8.48). The kind of derivative (divergence or rotational)
depends on the resultant indices (contracted or not), and the name covariant
derivative is used for both. This is a physicists’ practice, which actually
mixes up two quite distinct mathematical notions, that of exterior derivative
and that of coderivative (the derivative of the dual).
Gauge theories are very near to differential geometry, and the most ap-
propriate language to treat them is that of differential forms. Well, also a
tensor like Fµν has its covariant derivatives, with and without contraction.
One of them is
Dρ Fµν = Ta [∂ρ F a µν + f a bc Ab ρ F c µν ] = ∂ρ Fµν + [Aρ , Fµν ]. (8.55)
From the very definition of Fµν we obtain, by using the Jacobi identity
[Ta , [Tb , Tc ]] + [Tc , [Ta , Tb ]] + [Tb , [Tc , Ta ]] = 0, (8.56)
the following identity:
Dρ Fµν + Dν Fρµ + Dµ Fνρ = 0. (8.57)
The indices are exchanged cyclically from term to term. This Bianchi iden-
tity generalizes to the non-abelian case the so-called first pair of Maxwell’s
equations. Recall that those equations do not follow from the electromagnetic
Lagrangian, and in this sense are not dynamical.
194
µν
It so happens that F˜a F˜a µν is proportional to the above LG , so that it adds
nothing to dynamics. And also that Fa µν F˜a µν is an exact differential (the
divergence of a certain current), which does not contribute to the equations
of motion. Actually, Z
C = d4 xFa µν F˜a µν (8.60)
γab = f c ad f d bc , (8.64)
which can be used to rise and lower internal indices. We have used it implic-
itly every time some lower internal index appeared as in (8.58). A Lie group
is a differential manifold, and γab is a metric on that group manifold, which
has the special property of being invariant under the group transformations.
In this sense, all those expressions are scalar products in internal space, in-
variant under the group transformations. The Lagrangian, an invariant, can
only have indices contracted in this way.
The field equations (8.62) are called the Yang-Mills equations. They gov-
ern the field mediating all known interaction–mediating fields, if we exclude
the case of gravitation. They can be written in matrix form as
∂µ F µν + [Aµ , F µν ] = J ν . (8.65)
The left-hand side is the second form of the covariant derivative of F µν , to
which we have alluded above. This equation generalizes (the second pair of)
Maxwell’s equations to the non-abelian case with several internal degrees of
freedom. If, by analogy with (8.45), we define
∂LG
jν a = − (8.66)
∂Aa ν
195
as the current of the gauge field itself, we shall have
In matrix form, it is
∂µ F µν = j ν + J ν , (8.69)
from which
∂ν (j ν + J ν ) = 0. (8.70)
We see here what happens concerning current conservation. It is not only J µ
(the external source current) which is to be considered, but the total current,
including the gauge field current j µ itself. The meaning of the “self-current”
j µ is important: the gauge field can be its own source. This effect comes
from the non-linear character of the theory, which is a consequence of the
non-abelian character of the gauge group. Non-abelian gauge fields, even
in the absence of external sources, are highly non-trivial, because they are
self-producing. They are never actually “free”, as they are always, at least,
in interaction with each other. In the quantum case, the quanta of the gauge
fields carry themselves the charges of the theory (as if the photons carried
electric charges).
This interpretation, though satisfactory from the point of view of the
conservation law, is not without difficulties. The problem is that the to-
tal current j ν + J ν is not gauge-covariant. This is reflected in the charges
themselves, which are given by
Z Z Z
3 µ0 3 i0
Q= d x∂µ F = d x∂i F = d2 σ i Fi0 . (8.71)
V V ∂V
As F is covariant (see (8.52)), the charge will change, under a gauge trans-
formation, as Z
0
Q⇒Q = d2 σ i U (x)Fi0 U −1 (x). (8.72)
∂V
a
Thus, only if we suppose that U (x) = eα (x)Ta becomes constant on a far
enough spacelike surface ∂V , can we extract U from inside the integral and
get covariant charges, that is, charges satisfying
Q0 = U Q U −1 . (8.73)
196
This imposes a limitation on the local gauge invariance. The charges
only make sense if the transformations become global (that is, constant) at
∂V . The latter can be placed, if we like, at space infinity. This problem
is quite analogous to that of General Relativity, in which the total energy-
momentum (which plays there the role of the above current) is not covariant
and, as a consequence, the energy (one of the corresponding charges) can
only be defined for asymptotically flat spaces.
The energy-momentum tensor of a gauge field will have the same form of
that of the electromagnetic field:
Θµν = Fa µρ F aν ρ − 1
4
η µν Fa ρσ F a ρσ . (8.74)
(Ta )c b = f c ab . (8.75)
In this case, the indices i, j, k, . . . used above, vary with the same range as
the indices a, b, c, . . . . If φ belongs to the adjoint representation, it will be
a matrix φ = Ta φa instead of a column, and the covariant derivative has the
form
Dµ φ = ∂µ φ + [Aµ , φ]. (8.76)
The fields Aµ and Fµν , in particular, belong to the adjoint representation.
The expressions (8.47) and (8.48) give the covariant derivative of a vector
field in the adjoint representation. The left-hand side of (8.57) is the ex-
pression of the covariant derivative of Fµν (there, Dµ is given by (8.55)).
Thus, the true covariant derivative of an antisymmetric second-order tensor
in the adjoint representation is the cyclic sum in (8.57). Besides those covari-
ant derivatives, there are the covariant coderivatives or, roughly speaking,
the derivatives of the dual fields. The Yang-Mills equations state that the
covariant coderivative of the field strength equals the source current.
197
The reader may think that we are exchanging the roles. After all, it is
the dual who seems to be derived in (8.57), as there a cyclic sum is at work;
whereas only the field, and not its dual, appears in (8.62). The reason for
this apparent contradiction is a certain opposition between the nomencla-
tures used by physicists and mathematicians. Physicists are used to write
everything in terms of components, while mathematicians view φ, A = Aµ dxµ
and F = 12 Fµν dxµ ∧ dxν as differential forms and write things in invariant
language, with no components in sight. Differential forms inhabit integrands.
In the integration sign, also the measure is written in invariant form,Rso as
to have the same expression in any coordinate system. Instead of d4 x,
mathematicians write
Z p Z
|g| d x = J d4 x.
4
198
gauge field which mediates their interaction. There is a large evidence for
that, but it has been impossible up to now to demonstrate quark confinement
using the theory. Also the gauge quanta (“gluons”) are never seen in free
state (a phenomenon called “shielding”), another property which should be
deduced from the theory. Actually, the calculations are very, very compli-
cated, and it is not known whether the theory explains these properties or
not.
The structure of gauge theories is fairly geometric. Only the dynamic part
(Lagrangian and Yang-Mills equations) is actually Physics. All the characters
above have their mathematical counterparts, sometimes with different names.
As an interchange of names become more and more frequent in the physical
literature, we give a short glossary:
Physics Name Mathematics Name
gauge potential connection
field strength curvature
gauge group structure group
internal space fiber
external space (spacetime) base manifold
spacetime + internal space fiber bundle
Locally, around each point of spacetime, the complete space (base + fiber) is a
direct product of both spaces. But fiber bundles, globally, are not necessarily
the direct product of the base manifold and the fiber. The simplest examples:
a torus is the (global) direct product of two circles; a cylinder is a (global)
direct product of a circle and a straight line; the Möbius band is a direct
product locally, but not globally.
199
200
Chapter 9
Gravitational Field
201
9.2 The Equivalence Principle
Equivalence is a guiding principle, which inspired Einstein in his construction
of General Relativity. It is firmly rooted on experience.∗ In its most usual
form, the Principle includes three sub–principles: the weak, the strong, and
that which is called “Einstein’s equivalence principle”. Let us shortly list
them with a few comments.
• The weak equivalence principle states the universality of free fall,
or the equality inertial mass = gravitational mass. It can be stated in
the form:
202
• The strong equivalence principle (Einstein’s lift) says that
It requires that, for any and every particle, and at each point x0 , there
exists a frame in which ẍµ = 0.†
Forces equally felt by all bodies were known since long. They are the in-
ertial forces, whose name comes from their turning up in non-inertial frames.
Examples on Earth (not an inertial system!) are the centrifugal force and the
Coriolis force. Universality of inertial forces has been the first hint towards
General Relativity. A second ingredient is the notion of field. The concept
allows the best approach to interactions coherent with Special Relativity. All
known forces are mediated by fields on spacetime. Now, if gravitation is to
be represented by a field, it should, by the considerations above, be a uni-
versal field, equally felt by every particle. It should change spacetime itself.
And, of all the fields present in a space, the metric — the first fundamental
form, as it is also called — seemed to be the basic one. The simplest way
to change spacetime would be to change its metric. Furthermore, the metric
does change when looked at from a non-inertial frame, where the inertial
forces are present. The gravitational field, therefore, is represented by the
spacetime metric. In the absence of gravitation, the spacetime metric reduces
to the Minkowski metric.
†
A precise, mathematically sound formulation of the strong principle can be found in R.
Aldrovandi, P. B. Barros & J. G. Pereira: The equivalence principle revisited, Foundations
of Physics 33 (2003) 545-575 — arXiv:gr-qc/0212034.
203
9.3 Pseudo-Riemannian Metric
Each spacetime is a 4-dimensional pseudo–riemannian manifold. Its main
character is the fundamental form, or metric. For example, the spacetime
of special relativity is the flat Minkowski spacetime. Minkowski space is the
simplest, standard spacetime, and its metric, called the Lorentz metric, is
denoted
η(x) = ηab dxa dxb . (9.1)
It is a rather trivial metric. Up to the signature, the Minkowski space is an
Euclidean space, and as such can be covered by a single, global coordinate
system. This system — the cartesian system — is the father of all coordinate
systems, and just puts η in the diagonal form
+1 0 0 0
0 −1 0 0
η= . (9.2)
0 0 −1 0
0 0 0 −1
The Minkowski line element, therefore, is
ds2 = ηab dxa dxb = dx0 dx0 − dx1 dx1 − dx2 dx2 − dx3 dx3
or
ds2 = c2 dt2 − dx2 − dy 2 − dz 2 . (9.3)
On the other hand, the metric of a general 4-dimensional pseudo-rieman-
nian spacetime will be denoted by
204
9.4 The Notion of Connection
In a general pseudo-riemannian spacetime, the ordinary derivative of a ten-
sor is not covariant under a general coordinate transformation xµ → x0µ . In
order to define a covariant derivative, it is necessary to introduce a “compen-
sating field”, that is, a connection which we will denote by Γ. The covariant
derivative of a function φ (tensor of zero degree) is the usual derivative,
Dµ φ = ∂µ φ,
Dµ φν = ∂µ φν − Γλ νµ φλ . (9.7)
Let us take now a third order mixed tensor φν ρσ . Its covariant derivative will
be given by
Dµ φν ρσ = ∂µ φν ρσ + Γν λµ φλ ρσ − Γλ ρµ φν λσ − Γλ σµ φν ρλ . (9.9)
The rules to writing the covariant derivative are fairly illustrated in these
examples. Notice the signs: positive for upper indices, negative for lower
indices.
Under a general coordinate transformation xµ → x0µ , in order to yield
an appropriate behavior to the covariant derivative, the connection Γ must
transform according to
205
along all its length. If the metric is parallel–transported, Dµ gρσ = 0, and the
equation above gives the metricity condition
∂µ gρσ = Γλ ρµ gλσ + Γλ σµ gρλ = Γσρµ + Γρσµ = 2 Γ(ρσ)µ , (9.11)
where the symbol with lowered index is defined by Γρσµ = gρλ Γλ σµ and the
compact notation for the symmetrized part
1
Γ(ρσ)µ = 2
{Γρσµ + Γσρµ }, (9.12)
has been introduced. The analogous notation for the antisymmetrized part
1
Γ[ρσ]µ = 2
{Γρσµ − Γσρµ } (9.13)
is also very useful.
206
Notice that what exists is the curvature and torsion of a connection. Many
connections are defined on a given space, each one with its curvature and tor-
sion. It is common language to speak of “the curvature of space” and “torsion
of space”, but this only makes sense if a certain connection is assumed to be
included in the very definition of that space.
207
In consequence of these symmetries, the Ricci tensor (9.15) is essentially the
only contracted second-order tensor obtained from the Riemann tensor. The
scalar curvature will be now
◦ ◦
µν
R = g Rµν . (9.25)
9.7 Geodesics
As we have already seen, the action describing a free particle of mass m in
the Minkowski spacetime is
Z b
S = −mc ds, (9.26)
a
where
ds = (ηab dxa dxb )1/2 . (9.27)
In the presence of gravitation, that is, in a pseudo-riemannian spacetime, the
action describing a particle of mass m is still that given by Eq. (9.26), but
now with
ds = (gµν dxµ dxν )1/2 . (9.28)
We see from this expression that the metric tensor modifies the line element.
Taking the variation of S, the condition δS = 0 yields the equation of motion
duρ ◦ ρ
+ Γ µν uµ uν = 0, (9.29)
ds
where uρ = dxρ /ds is the particle four-velocity. The solution of this equation
of motion, called geodesic equation, gives the trajectory of the particle in the
presence of gravitation.
An important property of the geodesic equation is that it does not involve
the mass of the particle, a natural consequence of universality. Another im-
portant property is that it represents the vanishing of the covariant derivative
of the four-velocity uρ along the trajectory of the particle:
◦
◦ Duρ
uλ Dλ uρ ≡ = 0. (9.30)
ds
This is a consequence of the General Relativity approach to gravitation, in
which the gravitational interaction is geometrized, and in which the concept
of force is absent. According to this approach, gravitation produces a cur-
vature in spacetime, and the gravitational interaction is achieved by letting
(spinless) particles to follow the geodesics of this spacetime.
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9.8 Bianchi Identities
A detailed calculation gives the simplest way to exhibit curvature. Consider a
vector field U , with components U α , and take twice the covariant derivative,
◦ ◦ ◦ ◦
getting Dγ Dβ U α . Reverse then the order to obtain Dβ Dγ U α and compare.
The result is ◦ ◦ ◦ ◦ ◦
α α α
Dγ Dβ U − Dβ Dγ U = −R βγ U . (9.31)
Curvature turns up in the commutator of two covariant derivatives:
◦ ◦ ◦
[Dγ , Dβ ]U α = −Rα βγ U . (9.32)
Another one is ◦ ◦ ◦ ◦ ◦ ◦
Dµ Rκλρσ + Dσ Rκλµρ + Dρ Rκλσµ = 0 (9.34)
Notice, in both cases, the cyclic rotation of three of the indices. These
expressions are called respectively the first and the second Bianchi identities.
Now, as the metric has zero covariant derivative, it can be inserted in the
second identity to contract indices in a convenient way. Contracting with
g κρ , it comes out
◦ ◦ ◦ ◦ ◦ ◦
ρ
Dµ Rλσ − Dσ Rλµ + Dρ R λσµ = 0.
A further contraction with g λσ yields
◦ ◦ ◦ ◦ ◦ ◦
σ
Dµ R − Dσ R µ − Dρ Rρ µ = 0,
is called the Einstein tensor. Its contraction with the metric gives the scalar
curvature (up to a sign).
◦
g µν Gµν = − R. (9.37)
209
When the Ricci tensor is related to the metric tensor by
◦
Rµν = λ gµν , (9.38)
210
where T = g µν Tµν . This result can be inserted back into the Einstein equa-
tion, to give it the form
Rµν = 8πG Tµν − 12 gµν T .
c4
(9.42)
Consider the sourceless case, in which Tµν = 0. It follows from the above
equation that Rµν = 0 and, therefore, that R = 0. Notice that this does not
imply Rρ σµν = 0. The Riemann tensor can be nonvanishing even in the ab-
sence of source. Einstein’s equations are non-linear and, in consequence, the
gravitational field can engender itself. Absence of gravitation is signalled by
Rρ σµν = 0, which means a flat spacetime. This case — Minkowski spacetime
— is a particular solution of the sourceless equations. Beautiful examples of
solutions without any source at all are the de Sitter spaces.
In reality, the Einstein tensor (9.36) is not the most general parallel-trans-
ported purely geometrical second-order tensor which has vanishing covariant
derivative. The metric has the same property. Consequently, it is in principle
possible to add a term Λgµν to Gµν , with Λ a constant. Equation (9.40)
becomes
Rµν − ( 12 R + Λ)gµν = 8πGc4
Tµν . (9.43)
From the point of view of covariantly preserved objects, this equation is as
valid as (9.40). In his first trial to apply his theory to cosmology, Einstein
looked for a static solution. He found it, but it was unstable. He then
added the term Λgµν to make it stable, and gave to Λ the name cosmological
constant. Later, when evidence for an expanding universe became clear, he
called this “the biggest blunder in his life”, and dropped the term. This
is the same as putting Λ = 0. It was not a blunder: recent cosmological
evidence claims for Λ 6= 0. Equation (9.43) is the Einstein’s equation with a
cosmological term. With this extra term, Eq. (9.42) becomes
8πG
Tµν − 12 gµν T − Λgµν .
Rµν = 4 (9.44)
c
Finally, it is important to mention that Einstein’s equations can be de-
rived from an action functional, the so called Hilbert-Einstein action,
√
Z
S[g] = −g R d4 x, (9.45)
where g = det(gµν ).
211
in Classical Mechanics. It is better, in that case, to use spherical coordinates
(x0 , x1 , x2 , x3 ) = (ct, r, θ, φ). This is one of the most studied of all solutions,
and there is a standard notation for it. The interval is written in the form
dr2
2 2GM
ds = 1 − 2 c2 dt2 − r2 (dθ2 + sin2 θdφ2 ) − . (9.46)
c r 1 − 2GM2
c r
dr2
2 RS
ds = 1 − c2 dt2 − r2 (dθ2 + sin2 θdφ2 ) − . (9.48)
r 1 − RrS
212
Index
213
Hamiltonian form, 164 sine-Gordon, 113
distance, 14 Yang-Mills, 195
Doppler effect, 160 equivalence principle, 202
event, 15
effect
Doppler, 160 Fermat
Einstein principle, 29
convention, 6 fermion, 117
electrodynamics, 187 field
electromagnetic classical, 79
wave, 155 criterion for the presence, 119
electromagnetic field, 132 equation, 75
constant, 148 gauge, 184
uniform, 148 gravitational, 201
electromagnetism introduced, 75
of a charged particle, 34 mediating, 126
electrostatics, 151 non-material, 75
electroweak interactions, 187 quantum, 79
energy relativistic
of a particle, 31 scalar, 58
energy–momentum tensor, 58
and Einstein equation, 108 vector, 58
density tensor scalar, 107, 190
Belinfante, 108 complex, 120
canonical, 105 real, 117
energy-momentum vector, 107, 125
Belinfante, 210 complex, 129
of a gauge field, 197 real, 126
equation field strength, 193
d’Alembert, 142 electromagnetic, 34
Dirac, 162 force law
Euler–Lagrange, 92 Lorentz, 147
field, 75 Newton, 3
Klein-Gordon, 93, 117, 171 four-momentum
Korteweg-de Vries, 115 of a particle, 32
Laplace, 152 four-vector, 24
of motion, 75 Fourier analysis, 71
Pauli, 167 frames
Poisson, 152 accelerated, 1
Proca, 127, 142 inertial, 1, 8
Schrödinger, 76 non-inertial, 1, 36
214
functional calculus, 93 identity
fundamental interactions, 187 Bianchi, 194
Jacobi, 194
Galilei group, 6 inertial frames, 1, 8
gamma factor, 22 interval
gammalogy, 171 notion of, 15
gauge, 128 null, 17
Coulomb, or radiation, 143, 154 spacelike, 17
field, 184 timelike, 17
fixing, 142 invariant
invariance, 142 Casimir, 50
Lorenz, 128, 142 involution, 180
potential, 124 ISO(r,s), 47
theory, 110 isospin, 81
transformation, 82, 141
global, 121, 188 Jacobi identity, 194
local, 122, 189
generators KdV equation, 115
of the Lorentz group, 64 Killing-Cartan metric, 51, 195
gl(N, R), 46 Klein-Gordon equation, 93, 117
gravitational field, 201 every field obeys, 118, 163
group Korteweg-de Vries equation, 115
Galilei, 6 Kronecker
Lorentz, 55 symbol, 33, 48, 62
matrix, 45 Kronecker delta, 70
of internal transformations, 80 Lagrange
of transformations, 38 derivative
Poincaré, 20, 55 and Noether, 111
rank, 51 in Classical Mechanics, 74
rotation, 48 in Field Theory, 93
Hamilton Lagrangian
principle, 29, 90 for a particle, 31
Hamilton equations, 69 for the vibrating line, 69
Hamiltonian formulation of Mechanics, 67
for a particle, 32 free and interaction, 119
for the vibrating line, 69 Wentzel–Pauli, 127
formulation of Mechanics, 67 Lagrangian approach
harmonic, 152 and Euler–Lagrange equations, 91
Heisenberg and Hamilton principle, 90
and isospin, 81 and symmetry, 89
general aspects, 85
215
Lagrangian properties representation, 171
invariance, 89 Maxwell’s tensor, 34
locality, 90 mediating field, 126
low order, 89 metric, 13
reality, 89 Killing-Cartan, 51, 195
simplicity, 89 Lorentz, 15
Laplace equation, 152 notion of, 13
law minimal coupling
Biot-Savart, 155 prescription, 147
Coulomb, 152 Minkowski space, 15, 55
Newton, 3 momentum
Lie of a particle, 31
algebra
of a group, 41 Noether
equation, 110 current, 102, 189
light cone, 16 first theorem, 99
future and past, 17 second theorem, 110
linear norm, 13
representation, 58 normal modes
Liouville equation, 69 and Fourier analysis, 71
Lorentz normal coordinates, 70
contraction, 21 null
force law, 33, 147 interval, 17
group, 19 vector, 24
generators, 64 orthogonality, 13
metric, 15
tensor, 27 parity transformation, 38
transformation, 26, 55, 64, 75 for Dirac fields, 177
pure, or boost, 20 particle dynamics, 29
vector, 24, 27 Pauli equation, 167
Lorentz group Pauli matrices, 52
representation Pauli—Dirac
bispinor, 174 representation, 171
nonunitary if finite, 174 Pauli–Lubanski operator, 62
vector, 174 Poincaré
Lorenz group, 20, 26, 55, 63
gauge, 142 transformation, 26, 55
Lorenz gauge, 128 Poisson
brackets, 69
magnetostatics, 153 equation, 152
Majorana Poynting vector, 146
216
principle real, 117
equivalence, 202 scalar product, 13
Fermat, 29 Schrödinger, 117
Hamilton, 29, 90 Schrödinger equation, 76
of causality, 36 sine-Gordon equation, 113
of determinism, 8 SO(3), 46
of inertia, 8, 36 and SU(2), 48
of relativity, 8, 36 soliton, 114
Proca equation, 127, 142 space
proper Fock, 77
length, 21 Minkowski, 55
time, 18 spacelike
interval, 17
quantization vector, 24
vibrating line, 77 spacetime
quantization rules classical, 7
for fields, 79 Minkowski, 15
radiation special relativistic, 12
gauge, 143 spin
rank, 51, 61 density tensor, 107
red shift, 161 of a particle, 61
relativity operator, 63
galilean, 36 spinor
general, 37 Pauli,Weyl, 53
special, 36 state
representation in Classical Mechanics, 67
adjoint, 48, 51, 197 in Quantum Mechanics, 76
fundamental, 48 SU(2)
linear, 45, 58 and SO(3), 48
Lorentz vector, 125 symmetry
spinor, 53 and conserved charge, 103
tensor, 54 gauge, 108, 109
vector, 50 internal, 108
representation of gammas Lorentz, 106
chiral, 171 on spacetime, 104
Majorana, 171 translation, 104
Pauli—Dirac, 171 tensor, 6
rotations, 48 Lorentz, 27
scalar field theorem
complex, 120 Noether
217
first, 99 phase, 160
second, 110 plane, 156
time dilation, 19 propagation, 157
time reversa transversal, 158
for Dirac fields, 181 tvector, 159
time reversal, 39 wavefields, 79
Wigner, 182 wavelength, 159
timelike Weinberg–Salam theory, 187
nterval, 17 Wentzel–Pauli Lagrangian, 127
vector, 24 Wigner time reversal, 182
topological world line, 17
conservation law, 113, 195
number, 115, 195 Yang-Mills equation, 195
transformation
groups of, 38
transversality condition, 143
twin paradox, 19
U(N), 46
variation, 94, 95
action, 99
of action, 100
vector
contravariant, 135
covariant, or covector, 135
fields, 28
Lorentz, 24, 27
null, 24
representation, 125
spacelike, 24
timelike, 24
under rotations, 6
vector field, 125
vibrating line, 74
classical continuum, 72
quantum, 77
wave
electromagnetic, 155
equation, 155
monochromatic plane, 159
218