Dasgupta 08 Intro To QFT
Dasgupta 08 Intro To QFT
Dasgupta 08 Intro To QFT
Mrinal Dasgupta
1 Introduction
1.1 Lagrangian formalism in classical mechanics
1.2 Quantum mechanics
1.3 The Schrodinger picture
1.4 The Heisenberg picture
1.5 The quantum mechanical harmonic oscillator
Problems
5 Perturbation Theory
5.1 Wicks Theorem
5.2 The Feynman propagator
5.3 Two-particle scattering to O()
5.4 Graphical representation of the Wick expansion: Feynman rules
5.5 Feynman rules in momentum space
5.6 S-matrix and truncated Greens functions
Problems
6 Concluding remarks
Acknowledgements
0 Prologue
The development of Quantum Field Theory is surely one of the most important achieve-
ments in modern physics. Presently, all observational evidence points to the fact that
Quantum Field Theory (QFT) provides a good description of all known elementary parti-
cles, as well as for particle physics beyond the Standard Model for energies ranging up to
the Planck scale 1019 GeV, where quantum gravity is expected to set in and presumably
requires a new and dierent description. Historically, Quantum Electrodynamics (QED)
emerged as the prototype of modern QFTs. It was developed in the late 1940s and early
1950s chiey by Feynman, Schwinger and Tomonaga, and is perhaps the most successful
theory in physics: the anomalous magnetic dipole moment of the electron predicted by
QED agrees with experiment with a stunning accuracy of one part in 1010 !
The scope of these lectures is to provide an introduction to the formalism of Quantum
Field Theory, and as such is somewhat complementary to the other lectures of this school.
It is natural to wonder why QFT is necessary, compelling us to go through a number
of formal rather than physical considerations, accompanied by the inevitable algebra.
However, thinking for a moment about the high precision experiments, with which we
hope to detect physics beyond the Standard Model, it is clear that comparison between
theory and experiment is only conclusive if the numbers produced by either side are
water-tight. On the theory side this requires a formalism for calculations, in which
every step is justied and reproducible, irrespective of subjective intuition about the
physics involved. In other words, QFT aims to provide the bridge from the building
blocks of a theory to the evaluation of its predictions for experiments.
This program is best explained by restricting the discussion to the quantum theory
of scalar elds. Furthermore, I shall use the Lagrangian formalism and canonical quan-
tisation, thus leaving aside the quantisation approach via path integrals. Since the main
motivation for these lectures is the discussion of the underlying formalism leading to the
derivation of Feynman rules, the canonical approach is totally adequate. The physically
relevant theories of QED, QCD and the electroweak model are covered in the lectures by
Nick Evans, Sacha Davidson and Stefano Moretti.
The outline of these lecture notes is as follows: to put things into perspective, we shall
review the Lagrangian formalism in classical mechanics, followed by a brief reminder of
the basic principles of quantum mechanics in Section 1. Section 2 discusses the step from
classical mechanics of non-relativistic point particle to a classical, relativistic theory for
non-interacting scalar elds. There we will also derive the wave equation for free scalar
elds, i.e. the Klein-Gordon equation. The quantisation of this eld theory is done is
Section 3, where also the relation of particles to the quantised elds will be elucidated.
The more interesting case of interacting scalar elds is presented in Section 4: we shall
introduce the S-matrix and examine its relation with the Greens functions of the theory.
Finally, in Section 5 the general method of perturbation theory is presented, which serves
to compute the Green functions in terms of a power series in the coupling constant. Here,
Wicks Theorem is of central importance in order to understand the derivation of Feynman
rules.
1 Introduction
Let us begin this little review by considering the simplest possible system in classical
mechanics, a single point particle of mass m in one dimension, whose coordinate and
velocity are functions of time, x(t) and x(t) = dx(t)/dt, respectively. Let the particle be
exposed to a time-independent potential V (x). Its motion is then governed by Newtons
law
d2 x V
m 2 = = F (x), (1.1)
dt x
where F (x) is the force exerted on the particle. Solving this equation of motion involves
two integrations, and hence two arbitrary integration constants to be xed by initial
conditions. Specifying, e.g., the position x(t0 ) and velocity x(t0 ) of the particle at some
initial time t0 completely determines its motion: knowing the initial conditions and the
equations of motion, we also know the evolution of the particle at all times (provided we
can solve the equations of motion).
From these expressions the equations of motion can be derived by the Principle of least
Action: consider small variations of the particles trajectory, cf. Fig. 1,
x (t) = x(t) + x(t), x/x 1, (1.4)
with its initial and end points xed,
x (t1 ) = x(t1 )
x(t1 ) = x(t2 ) = 0. (1.5)
x (t2 ) = x(t2 )
x
x(t)
x(t)
Figure 1: Variation of particle trajectory with identified initial and end points.
The true trajectory the particle will take is the one for which
S = 0, (1.6)
i.e. the action along x(t) is stationary. In most systems of interest to us the stationary
point is a minimum, hence the name of the principle, but there are exceptions as well
(e.g. a pencil balanced on its tip). We can now work out the variation of the action by
doing a Taylor expansion to leading order in the variation x,
t2
d
S + S = L(x + x, x + x) dt, x = x
t dt
1t2
L L
= L(x, x) + x + x + . . . dt
t1 x x
t t2
L 2 L d L
= S+ x + x dt, (1.7)
x t1 t1 x dt x
where we performed an integration by parts on the last term in the second line. The
second and third term in the last line are the variation of the action, S, under variations
of the trajectory, x. The second term vanishes because of the boundary conditions for
the variation, and we are left with the third. Now the Principal of least Action demands
S = 0. For the remaining integral to vanish for arbitrary x is only possible if the
integrand vanishes, leaving us with the Euler-Lagrange equation:
L d L
= 0. (1.8)
x dt x
If we insert the Lagrangian of our point particle, Eq. (1.2), into the Euler-Lagrange
equation we obtain
L V (x)
= =F
x x
d L d
= mx = mx
dt x dt
V
mx = F = (Newtons law). (1.9)
x
Hence, we have derived the equation of motion by the Principal of least Action and
found it to be equivalent to the Euler-Lagrange equation. The benet is that the latter
can be easily generalised to other systems in any number of dimensions, multi-particle
systems, or systems with an innite number of degrees of freedom, such as needed for
eld theory. For example, if we now consider our particle in the full three-dimensional
Euclidean space, the Lagrangian depends on all coordinate components, L(x, x), and all
of them get varied independently in implementing Hamiltons principle. As a result one
obtains Euler-Lagrange equations for the components,
L d L
= 0. (1.10)
xi dt xi
In particular, the Lagrangian formalism makes symmetries and their physical conse-
quences explicit and thus is a convenient tool when constructing dierent kinds of theories
based on symmetries observed (or speculated to exist) in nature.
For later purposes in eld theory we need yet another, equivalent, formal treatment,
the Hamiltonian formalism. In our 1-d system, we dene the conjugate momentum p by
L
p = mx, (1.11)
x
and the Hamiltonian H via
H(x, p) px L(x, x)
= mx2 12 mx2 + V (x)
= 1
2
mx2 + V (x) = T + V. (1.12)
The Hamiltonian H(x, p) corresponds to the total energy of the system; it is a function
of the position variable x and the conjugate momentum1 p. It is now easy to derive
Hamiltons equations
H H
= p, = x. (1.13)
x p
These are two equations of rst order, while the Euler-Lagrange equation was a single
equation of second order. Taking another derivative in Hamiltons equations and substi-
tuting one into the other, it is easy to convince oneself that the Euler-Lagrange equations
and Hamiltons equations provide an entirely equivalent description of the system. Again,
this generalises obviously to three-dimensional space yielding equations for the compo-
nents,
H H
= pi , = xi . (1.14)
xi pi
position: xi xi
momentum: pi pi = i
xi
p2 2 2
Hamiltonian: H H = + V (x) = + V (x). (1.15)
2m 2m
Secondly, one imposes commutation relations on these operators,
[xi , pj ] = i ij (1.16)
[xi , xj ] = [pi , pj ] = 0. (1.17)
The physical state of a quantum mechanical system is encoded in state vectors |, which
are elements of a Hilbert space H. The hermitian conjugate state is | = (|), and the
modulus squared of the scalar product between two states gives the probability for the
system to go from state 1 to state 2,
On the other hand physical observables O, i.e. measurable quantities, are given by the
expectation values of hermitian operators, O = O ,
Hermiticity ensures that expectation values are real, as required for measurable quantities.
Due to the probabilistic nature of quantum mechanics, expectation values correspond to
statistical averages, or mean values, with a variance
= (positive), (1.30)
j= ( ( )) (real). (1.31)
2im
Now that we have derived the continuity equation let us discuss the probability interpre-
tation of Quantum Mechanics in more detail. Consider a nite volume V with boundary
S. The integrated continuity equation is
3
dx = j d3 x
V t
V
= j dS (1.32)
S
where in the last line we have used Gausss theorem. Using Eq. (1.27) the lhs. can be
rewritten and we obtain
1t = j dS = 0. (1.33)
t S
In other words, provided that j = 0 everywhere at the boundary S, we nd that the time
derivative of 1t vanishes. Since 1t represents the total probability for nding the par-
ticle anywhere inside the volume V , we conclude that this probability must be conserved:
particles cannot be created or destroyed in our theory. Non-relativistic Quantum Me-
chanics thus provides a consistent formalism to describe a single particle. The quantity
(x, t) is interpreted as a one-particle wave function.
with
i i
OH (t) = e H(tt0 ) Oe H(tt0 ) . (1.36)
From this last equation it is now easy to derive the equivalent of the Schrodinger equation
for the Heisenberg picture, the Heisenberg equation of motion for operators:
dOH (t)
i = [OH , H]. (1.37)
dt
Note that all commutation relations, like Eq. (1.16), with time dependent operators are
now intended to be valid for all times. Substituting x, p for O into the Heisenberg equation
readily leads to
dxi H
= ,
dt pi
dpi H
= , (1.38)
dt xi
the quantum mechanical equivalent to the Hamilton equations of classical mechanics.
1.5 The quantum mechanical harmonic oscillator
Because of similar structures later in quantum eld theory, it is instructive to also briey
recall the harmonic oscillator in one dimension. Its Hamiltonian is given by
1 p2 2 2
H(x, p) = + m x . (1.39)
2 m
Employing the canonical formalism we have just set up, we easily identify the momentum
operator to be p(t) = mt x(t), and from the Hamilton equations we nd the equation of
motion to be t2 x = 2 x, which has the well known plane wave solution x exp it.
An alternative path useful for later eld theory applications is to introduce new
operators, expressed by the old ones,
1 m 1 m
a = x + i p , a = x i p . (1.40)
2 m 2 m
Problems
1.1 Starting from the denition of the Hamiltonian,
H(x, p) px L(x, x),
derive Hamiltons equations
H H
= p, = x.
x p
[Hint: the key is to keep track of what are the independent variables]
1.2 Using the Schrodinger equation for the wavefunction (x, t),
2 2
+ V (x) (x, t) = i (x, t),
2m t
show that the probability density = satises the continuity equation
+ j = 0,
t
where
j= { ( ) } .
2im
[Hint: Consider (Schr.Eq.) (Schr.Eq.)*]
1.3 Let | be a simultaneous eigenstate of two operators A, B. Prove that this implies
a vanishing commutator [A, B].
1.4 Let O be an operator in the Schrodinger picture. Starting from the denition of a
Heisenberg operator,
i i
OH (t) = e H(tt0 ) Oe H(tt0 ) ,
derive the Heisenberg equation of motion
dOH
i = [OH , H].
dt
1.5 Consider the Heisenberg equation of motion for the momentum operator p of the
harmonic oscillator with Hamiltonian
1 p2 2 2
H = + m x ,
2 m
and show that it is equivalent to Newtons law for the position operator x.
2 Classical Field Theory
2.1 From N-point mechanics to field theory
In the previous sections we have reviewed the Lagrangian formalism for a single point
particle in classical mechanics. A benet of that formalism is that it easily generalises to
any number of particles or dimensions. Let us return to one dimension for the moment but
consider an N-particle system, i.e. we have N coordinates and N momenta, xi (t), pi(t), i =
1, . . . N. For such a system we get 2N Heisenberg equations,
H dpi H dxi
= , = . (2.1)
xi dt pi dt
To make things more specic, consider a piece of a guitar string, approximated by N
coupled oscillators, as in Fig. 2. Each point mass of the string can only move in the
1
0 1
0
1
0 0
1 0
1 1
0
0
1 1
0 0
1
xi (t) 0
1 11
00 11
00
00
11
(x, t)
Figure 2: From N coupled point masses to a continuous string, i.e. infinitely many degrees
of freedom.
direction perpendicular to the string, i.e. is a particle moving in one dimension. This
approximation of a string gets better and better the more points we ll in between the
springs, and a continuous string obtains in the limit N . The displacement of the
string at some particular point x along its length is now given by a eld coordinate (x, t).
Going back to the N-point system and comparing what measures the location of a point
and its displacement, we nd the following dictionary between point mechanics and
eld theory:
In the last line we have introduced a new notation: the square brackets indicate that
L[, ] depends on the functions (x, t), (x, t) at every space-time point, but not on the
coordinates directly. Such an object is called a functional, as opposed to a function
which depends on the coordinate variables only.
Formally the above limit of innite degrees of freedom can also be taken if we are
dealing with particles in a three-dimensional Euclidean space, for which there are N three-
vectors xi specifying the positions. We then obtain a eld (x, t), dened at every point
in space and time.
x = x . (2.3)
A general function transforms as f (x) f (x ), i.e. both the function and its argument
transform. A Lorentz scalar is a function which is the same in all inertial frames,
( )( ) = ( 0 )2 ()2 . (2.7)
However, for a relativistic theory we require Lorentz invariance of the action, and this
is not obvious in the current form. The integration is over time only, rather than over
the Lorentz-invariant four-volume element d4 x = dt d3 x, and so the non-invariance of the
integration measure has to cancel against that of the Lagrange function in order to have
an invariant action. Similar reasoning applies to the arguments of the Lagrangian. In
order to have the symmetries manifest, we instead rewrite
S = d x L[, ], L[, ] = d3 x L[, ].
4
(2.9)
Again the integrand itself must vanish if S = 0 for arbitrary variations of the eld, .
This yields the Euler-Lagrange equations for a classical eld theory:
L L
= 0, (2.12)
( )
where in the second term a summation over the Lorentz index is implied.
Let us now consider the specic Lagrangian
L = 12 12 m2 2 . (2.13)
The functional derivatives yield
L L
= m2 , = , (2.14)
( )
so that
L
= = . (2.15)
( )
The Euler-Lagrange equation then implies
( + m2 )(x) = 0. (2.16)
This is the Klein-Gordon equation for a scalar eld. It is the simplest relativistic wave
equation and can be deduced from relativistic energy considerations. Here we have derived
it from the Lagrange density following our canonical formalism, in complete analogy to
point mechanics. Relativistic invariance of the equations of motion is ensured because we
started from an invariant Lagrange density. This is the power of the formalism.
In keeping the analogy with point mechanics, we can dene a conjugate momentum
through
L(, ) L(, )
(x) = = 0 (x). (2.17)
(x) (0 (x))
Note that the momentum variables p and the conjugate momentum are not the same.
The word momentum is used only as a semantic analogy to classical mechanics. Further,
we dene the Hamilton function and a corresponding Hamilton density,
H(t) = d3 x H[, ], H[, ] = L. (2.18)
For the Lagrangian density we considered, this gives
1 2
H= (x) + ((x))2 + m2 2 (x) . (2.19)
2
If we choose the positive branch of the square root then we can dene the energy as
E(k) = k2 + m2 > 0, (2.22)
E(k)t k x = k k = k k = k x (2.24)
where (k) is an arbitrary complex coecient. From the general solution one easily reads
o that is real, i.e. = .
while the conservation of energy comes from the invariance under time translations
where we have used Eq. (2.30) in the second line. The Hamiltonian density is a conserved
quantity, provided that there is no energy ow through the surface S which encloses the
volume V . In a similar manner one can show that the 3-momentum pj , which is related
to 0j , is conserved as well. It is then useful to dene a conserved energy-momentum
four-vector
P = d3 x 0 . (2.33)
In analogy to point mechanics, we thus see that invariances of the Lagrangian density
correspond to conservation laws. An entirely analogous procedure leads to conserved
quantities like anguluar mometum and spin. Furthermore one can study so-called inter-
nal symmetries, i.e. ones which are not related to coordinate but other transformations.
Examples are conservation of all kinds of charges, isospin, etc.
We have thus established the Lagrange-Hamilton formalism for classical eld theory:
we derived the equation of motion (Euler-Lagrange equation) from the Lagrangian and
introduced the conjugate momentum. We then dened the Hamiltonian (density) and
considered conservation laws by studying the energy-momentum tensor .
Problems
2.1 Given the relativistic invariance of the measure d4 k, show that the integration mea-
sure
d3 k
(2)3 2E(k)
is Lorentz-invariant, provided that E(k) = k2 + m2 .
[Hint: Start from the Lorentz-invariant expression
d4 k
(k 2 m2 ) (k0 )
(2)3
and use
1
(x2 x20 ) = ((x x0 ) + (x + x0 )).
2|x|
What is the signicance of the and functions above? If youre really keen, you
may prove the relation for (x2 x20 ).]
2.3 Show that the Hamiltonian density H for a free scalar eld is given by
1
H= (0 )2 + ()2 + m2 2 .
2
Derive the components P0 , P of the energy-momentum four-vector P in terms of
the eld operators , .
L = 12 12 m2 2 , (3.1)
which led to the Klein-Gordon equation in the previous section. We have seen that in eld
theory the eld (x) plays the role of the coordinates in ordinary point mechanics, and
we dened a canonically conjugate momentum, (x) = L/ = (x). We then continue
the analogy to point mechanics through the quantisation procedure, i.e. we now take our
canonical variables to be operators,
As in the case of quantum mechanis, the canonical variables commute among themselves,
but not the canonical coordinate and momentum with each other. Note that the commu-
tation relation is entirely analogous to the quantum mechanical case. There would be an
, if it hadnt been set to one earlier, and the delta-function accounts for the fact that we
are dealing with elds. It is one if the elds are evaluated at the same space-time point,
and zero otherwise.
After quantisation, our elds have turned into eld operators. Note that within the
relativistic formulation they depend on time, and hence they are Heisenberg operators.
(x y)2 = 0, light-like
y
(x y)2 < 0, space-like
space
Figure 3: The light cone about y. Events occurring at points x and y are said to be
time-like (space-like) if x is inside (outside) the light cone about y.
a nite interval |t t|. It also vanishes for t = t, as long as x = y. Only if the elds
are evaluated at an equal space-time point can they aect each other, which leads to
the equal-time commutation relations above. They can also aect each other everywhere
within the light cone, i.e. for time-like intervals. It is not hard to show that in this case
(x), (y) = [(x), (y)] = 0, for (x y)2 > 0
i d3 p ip(xy) ip(xy)
(x), (y) = e + e . (3.5)
2 (2)3
Note that, as Fourier coecients, these operators do not depend on time, even though
the right hand side does contain time variables. Having expressions in terms of the
canonical eld variables (x), (x), we can now evaluate the commutators for the Fourier
coecients. Expanding everything out and using the commutation relations Eq. (3.3), we
nd
a (k1 ), a (k2 ) = 0 (3.11)
[a(k1 ), a(k2 )] = 0 (3.12)
a(k1 ), a (k2 ) = (2)3 2E(k1 ) 3 (k1 k2 ) (3.13)
We easily recognise these for every k to correspond to the commutation relations for the
harmonic oscillator, Eq. (1.41). This motivates us to also express the Hamiltonian and
the energy momentum four-vector of our quantum eld theory in terms of these operators.
This yields
1 d3 k
H = E(k) a (k)a(k) + a(k)a (k) ,
2 (2)3 2E(k)
1 d3 k
P = k a (k)a(k) + a(k)a (k) . (3.14)
2 (2)3 2E(k)
We thus nd that the Hamiltonian and the momentum operator are nothing but a contin-
uous sum of excitation energies/momenta of one-dimensional harmonic oscillators! After
a minute of thought this is not so surprising. We expanded the solution of the Klein-
Gordon equation into a superposition of plane waves with momenta k. But of course a
plane wave solution with energy E(k) is also the solution to a one-dimensional harmonic
oscillator with the same energy. Hence, our free scalar eld is simply a collection of in-
nitely many harmonic oscillators distributed over the whole energy/momentum range.
These energies sum up to that of the entire system. We have thus reduced the problem of
handling our eld theory to oscillator algebra. From the harmonic oscillator we know al-
ready how to construct a complete basis of energy eigenstates, and thanks to the analogy
of the previous section we can take this over to our quantum eld theory.
: aa : = a a, (3.23)
i.e. the normal-ordered operators are enclosed within colons. For instance
: 12 a (p)a(p) + a(p)a (p) : = a (p)a(p). (3.24)
It is important to keep in mind that a and a always commute inside : :. This is true
for an arbitrary string of a and a . With this denition we can write the normal-ordered
Hamiltonian as
1 d3 p
: H : = : E(p) a (p)a(p) + a(p)a (p) :
2 (2)3 2E(p)
d3 p
= E(p) a (p)a(p), (3.25)
(2)3 2E(p)
and thus have the relation
H R =: H : +H vac . (3.26)
Hence, we nd that
| : H : | = |H R |, (3.27)
and, in particular, 0| : H : |0 = 0. The normal ordered Hamiltonian thus produces a
renormalised, sensible result for the vacuum energy.
k|k = 0|a(k)a (k )|0 = 0|[a(k), a (k )]|0 + 0|a (k )a(k)|0
= (2)3 2E(k) 3 (k k ), (3.29)
since the last term in the rst line vanishes (a(k) acting on the vacuum). Next we compute
the energy of this state, making use of the normal ordered Hamiltonian,
d3 k
: H : |k = E(k )a (k )a(k )a (k)|0
(2)3 2E(k )
d3 k
= 3
E(k )(2)3 2E(k)(k k )a (k)|0
(2) 2E(k )
= E(k)a (k)|0 = E(k)|k, (3.30)
and similarly one nds
: P : |k = k|k. (3.31)
Observing that the normal ordering did its job and we obtain renormalised, nite results,
we may now interpret the state |k. It is a one-particle state for a relativistic particle of
mass m and momentum k, since acting on it with the energy-momentum operator returns
the relativistic one particle energy-mometum dispersion relation, E(k) = k + m2 . The
2
a (k), a(k) are creation and annihilation operators for particles of momentum k.
In analogy to the harmonic oscillator, the procedure can be continued to higher states.
One easily checks that
and hence the state is symmetric under interchange of the two particles. Thus, the
particles described by the scalar eld are bosons.
Finally we complete the analogy to the harmonic oscillator by introducing a number
operator
N(k) = a (k)a(k), N = d3 k a (k)a(k), (3.35)
Of course the normal-ordered Hamiltonian can now simply be given in terms of this
operator,
d3 k
: H := E(k)N (k), (3.37)
(2)3 2E(k)
i.e. when acting on a Fock state it simply sums up the energies of the individual particles
to give
: H : |k1 . . . kn = (E(k1 ) + . . . E(kn )) |k1 . . . kn . (3.38)
This concludes the quantisation of our free scalar eld theory. We have followed the
canonical quantisation procedure familiar from quantum mechanics. Due to the innite
number of degrees of freedom, we encountered a divergent vacuum energy, which we had
to renormalise. The renormalised Hamiltonian and the Fock states that we constructed
describe free relativistic, uncharged spin zero particles of mass m, such as neutral pions,
for example.
If we want to describe charged pions as well, we need to introduce complex scalar
elds, the real and imaginary parts being necessary to describe opposite charges. For
particles with spin we need still more degrees of freedom and use vector or spinor elds,
which have the appropriate rotation and Lorentz transformation properties. Moreover, for
fermions there is the Pauli principle prohibiting identical particles with the same quantum
numbers to occupy the same state, so the state vectors have to be anti-symmetric under
interchange of two particles. This is achieved by imposing anti-commutation relations,
rather than commutation relations, on the corresponding eld operators. Apart from
these complications which account for the nature of the particles, the formalism and
quantisation procedure is the same as for the simpler scalar elds, to which we shall stick
for this reason.
Problems
3.1 Using the expressions
for and in terms of a and a , show that the unequal time
commutator (x), (x ) is given by
i d3 p
ip(xx ) ip(xx )
(x), (x ) = e + e .
2 (2)3
3.2 Being time-dependent Heisenberg operators, the operators O = (x, t), (x, t) of
scalar eld theory obey the Heisenberg equation
i O = [O, H].
t
In analogy to what you did in problem 1.5, demonstrate the equivalence of this
equation with the Klein-Gordon equation.
to show that
L = L0 + Lint (4.1)
where
L0 = 12 12 m2 2 (4.2)
is the free Lagrangian density discussed before. The Hamiltonian density of the interaction
is related to Lint simply by
Hint = Lint , (4.3)
which follows from its denition. We shall leave the details of Lint unspecied for the
moment. What we will be concerned with mostly are scattering processes, in which two
initial particles with momenta p1 and p2 scatter, thereby producing a number of particles
in the nal state, characterised by momenta k1 , . . . , kn . This is schematically shown in
Fig. 4. Our task is to nd a description of such a scattering process in terms of the
underlying quantum eld theory.
p1 k1
k2
p2
kn
Figure 4:
Scattering of two initial particles with momenta p1 and p2 into n particles with
momenta k1 , . . . , kn in the nal state.
which acts on a corresponding basis of |in states. Long after the collision the particles
in the nal state evolve again like in the free theory, and the corresponding operator is
acting on states |out. The elds in , out are the asymptotic limits of the Heisenberg
operator . They both satisfy the free Klein-Gordon equation, i.e.
Operators describing free elds can be expressed as a superposition of plane waves (see
Eq. (3.6)). Thus, for in we have
d3 k ikx ikx
in (x) = e ain (k) + e ain (k) , (4.7)
(2)3 2E(k)
with an entirely analogous expression for out (x). Note that the operators a and a also
carry subscripts in and out.
We can now use the creation operators ain and aout to build up Fock states from the
vacuum. For instance
We must now distinguish between Fock states generated by ain and aout , and therefore we
have labelled the Fock states accordingly. In eqs. (4.8) and (4.9) we have assumed that
there is a stable and unique vacuum state:
S is called the S-matrix or S-operator. Note that the plane wave solutions of in and out
also imply that
ain = S aout S , ain = S aout S . (4.14)
By comparing in with out states one can extract information about the interaction
this is the very essence of detector experiments, where one tries to infer the nature of the
interaction by studying the products of the scattering of particles that have been collided
with known energies. As we will see below, this information is contained in the elements
of the S-matrix.
By contrast, in the absence of any interaction, i.e. for Lint = 0 the distinction between
in and out is not necessary. They can thus be identied, and then the relation between
dierent bases of the Fock space becomes trivial, S = 1, as one would expect.
What we are ultimately interested in are transition amplitudes between an initial
state i of, say, two particles of momenta p1 , p2 , and a nal state f , for instance n particles
of unequal momenta. The transition amplitude is then given by
f, out| i, in = f, out| S |i, out = f, in| S |i, in Sfi . (4.15)
The S-matrix element Sfi therefore describes the transition amplitude for the scattering
process in question. The scattering cross section, which is a measurable quantity, is then
proportional to |Sfi |2 . All information about the scattering is thus encoded in the S-
matrix, which must therefore be closely related to the interaction Hamiltonian density
Hint . However, before we try to derive the relation between S and Hint we have to take a
slight detour.
At t = the interaction vanishes, i.e. Hint = 0, and hence the elds in the Interaction
and Heisenberg pictures are identical, i.e. H (x, t) = I (x, t) for t . The relation
between H and I can be worked out easily:
The eld H (x, t) contains the information about the interaction, since it evolves over
time with the full Hamiltonian. In order to describe the in and out eld operators,
we can now make the following identications:
Furthermore, since the elds I evolve over time with the free Hamiltonian H0 , they
always act in the basis of in vectors, such that
We have thus derived a formal expression for the S-matrix in terms of the operator U(t),
which tells us how operators and state vectors deviate from the free theory at time t,
measured relative to t0 = , i.e. long before the interaction process.
An important boundary condition for U(t) is
What we mean here is the following: the operator U actually describes the evolution
relative to some initial time t0 , which we will normally suppress, i.e. we write U(t)
instead of U(t, t0 ). We regard t0 merely as a time label and x it at , where the
interaction vanishes. Equation (4.27) then simply states that U becomes unity as t t0 ,
which means that in this limit there is no distinction between Heisenberg and Dirac elds.
Using the denition of U(t), Eq. (4.20), it is an easy exercise to derive the equation
of motion for U(t):
d
i U(t) = Hint (t) U(t), Hint (t) = eiH0 t Hint eiH0 t . (4.28)
dt
The time-dependent operator Hint (t) is dened in the interaction picture, and depends
on the elds in , in in the in basis. Let us now solve the equation of motion for U(t)
with the boundary condition lim U(t) = 1. Integrating Eq. (4.28) gives
t
t t
d
U(t1 ) dt1 = i Hint (t1 ) U(t1 ) dt1
dt1
t
U(t) U() = i Hint (t1 ) U(t1 ) dt1
t
U(t) = 1 i Hint (t1 ) U(t1 ) dt1 . (4.29)
The rhs. still depends on U, but we can substitute our new expression for U(t) into the
integrand, which gives
t t1
U(t) = 1 i Hint (t1 ) 1 i Hint (t2 ) U(t2 ) dt2 dt1
t t t1
= 1i Hint (t1 )dt1 dt1 Hint (t1 ) dt2 Hint (t2 ) U(t2 ), (4.30)
where t2 < t1 < t. This procedure can be iterated further, so that the nth term in the
sum is t t1 tn1
n
(i) dt1 dt2 dtn Hint (t1 ) Hint (t2 ) Hint (tn ). (4.31)
This iterative solution could be written in much more compact form, were it not for the
fact that the upper integration bounds were all dierent, and that the ordering tn <
tn1 < . . . < t1 < t had to be obeyed. Time ordering is an important issue, since one
has to ensure that the interaction Hamiltonians act at the proper time, thereby ensuring
the causality of the theory. By introducing the time-ordered product of operators, one
can use a compact notation, such that the resulting expressions still obey causality. The
time-ordered product of two elds (t1 ) and (t2 ) is dened as
(t1 )(t2 ) t1 > t2
T {(t1 ) (t2 )} =
(t2 )(t1 ) t1 < t2
(t1 t2 ) (t1 )(t2 ) + (t2 t1 ) (t2 )(t1 ), (4.32)
and since this looks like the nth term in the series expansion of an exponential, we can
nally rewrite the solution for U(t) in compact form as
t
U(t) = T exp i Hint (t ) dt , (4.34)
Once this step is completed, then for any given Lagrange density we may compute the
Greens functions of the elds, which will in turn give us the S-matrix elements providing
the link to experiment. In order to achieve this, we have to express the in/out-states in
terms of creation operators ain/out and the vacuum, then express the creation operators
by the elds in/out , and nally use the time evolution to connect those with the elds
in our Lagrangian.
Let us consider again the scattering process depicted in Fig. 4. The S-matrix element
in this case is
Sfi = k1 , k2 , . . . , kn ; outp1 , p2 ; in
= k1 , k2 , . . . , kn ; outain (p1 )p2 ; in , (4.37)
where ain is the creation operator pertaining to the in eld in. Our task is now to
express ain in terms of in , and repeat this procedure for all other momenta labelling our
Fock states.
The following identities will prove useful
a (p) = i d3 x 0 eiqx (x) eiqx (0 (x))
i d3 x eiqx 0 (x), (4.38)
a(p) = i d3 x 0 eiqx (x) eiqx (0 (x))
i d3 x eiqx 0 (x). (4.39)
where in the last line we have used Eq. (4.4) to replace in by . We can now rewrite
limt1 using the following identity, which holds for an arbitrary, dierentiable function
f (t), whose limit t exists:
+
df
lim f (t) = lim f (t) dt. (4.41)
t t+ dt
where we have used that 2 eip1 x1 = p21 eip1 x1 . For the S-matrix element one obtains
4 ip1 x1 2 2 2
Sfi = i d x1 e k1 , . . . , kn ; out 0 + m (x1 )p2 ; in
= i d4 x1 eip1 x1 x1 + m2 k1 , . . . , kn ; out(x1 )p2 ; in . (4.46)
What we have obtained after this rather lengthy step of algebra is an expression in which
the eld operator is sandwiched between Fock states, one of which has been reduced to a
one-particle state. We can now successively eliminate all momentum variables from the
Fock states, by repeating the procedure for the momentum p2 , as well as the n momenta
of the out state. The nal expression for Sfi is
n+2
Sfi = (i) d x1 d x2 d y1 d4 yn e(ip1 x1 ip2 x2 +ik1 y1 ++kn yn )
4 4 4
x1 + m2 x2 + m2 y1 + m2 yn + m2
0; outT {(y1) (yn )(x1 )(x2 )}0; in , (4.47)
where the time-ordering inside the vacuum expectation value (VEV) ensures that causality
is obeyed. The above expression is known as the Lehmann-Symanzik-Zimmermann (LSZ)
reduction formula. It relates the formal denition of the scattering amplitude to a vacuum
expectation value of time-ordered elds. Since the vacuum is uniquely the same for
in/out, the VEV in the LSZ formula for the scattering of two initial particles into n
particles in the nal state is recognised as the (n + 2)-point Greens function:
Gn+2 (y1 , y2, . . . , yn , x1 , x2 ) = 0T {(y1) (yn )(x1 )(x2 )}0 . (4.48)
You will note that we still have not calculated or evaluated anything, but merely rewritten
the expression for the scattering matrix elements. Nevertheless, the LSZ formula is of
tremendous importance and a central piece of QFT. It provides the link between elds in
the Lagrangian and the scattering amplitude Sfi2 , which yields the cross section, measurable
in an experiment. Up to here no assumptions or approximations have been made, so this
connection between physics and formalism is rather tight. It also illustrates a profound
phenomenon of QFT and particle physics: the scattering properties of particles, in other
words their interactions, are encoded in the vacuum structure, i.e. the vacuum is non-
trivial!
The elds which appear in this expression are Heisenberg elds, whose time evolution
is governed by the full Hamiltonian H0 + Hint . In particular, the s are not the in s. We
know how to handle the latter, because they correspond to a free eld theory, but not the
former, whose time evolution is governed by the interacting theory, whose solutions we
do not know. Let us thus start to isolate the dependence of the elds on the interaction
Hamiltonian. Recall the relation between the Heisenberg elds (t) and the in-elds2
We now assume that the elds are properly time-ordered, i.e. t1 > t2 > . . . > tn , so
that we can forget about writing T ( ) everywhere. After inserting Eq. (4.50) into the
denition of Gn one obtains
Now we introduce another time label t such that t t1 and t t1 . For the n-point
function we now obtain
Gn = 0U 1 (t) U(t)U 1 (t1 )in (t1 )U(t1 ) U 1 (t2 )in (t2 )U(t2 )
1 1
U (tn )in (tn )U(tn )U (t) U(t)0 . (4.52)
One can easily convince oneself that U(t, t1 ) provides the net time evolution from t1 to t.
We can now write Gn as
1
Gn = 0U (t) T in (t1 ) in(tn ) U(t, t1 ) U(t1 , t2 ) U(tn , t) U(t)0 . (4.54)
U(t, t)
2
Here and in the following we suppress the spatial argument of the fields for the sake of brevity.
Let us now take t . The relation between U(t) and the S-matrix Eq. (4.26), as well
as the boundary condition Eq. (4.27) tell us that
which can be inserted into the above expression. We still have to work out the meaning
of 0|U 1 () in the expression for Gn . In a paper by Gell-Mann and Low it was argued
that the time evolution operator must leave the vacuum invariant (up to a phase), which
justies the ansatz
0|U 1 () = K 0|, (4.56)
with K being the phase. Multiplying this relation with |0 from the right gives
which implies
1
K= . (4.59)
0|S|0
After inserting all these relations into the expression for Gn we obtain
and thus we have nally succeeded in expressing the n-point Greens function exclusively
in terms of the in-elds. This completes the derivation of a relation between the general
denition of the scattering amplitude Sfi and the VEV of time-ordered in-elds. The
link between the scattering amplitude and the underlying eld theory is provided by the
n-point Greens function.
Problems
4.1 Using the denition U(t) = eiH0 t eiHt , derive the evolution equation for U(t):
d
i U(t) = Hint (t) U(t),
dt
where
Hint (t) = eiH0 t Hint eiH0 t .
4.2 Given that in is a free eld, obeying the Heisenberg equation of motion
[Hint: use out = S in S and out = S in S. Keep in mind that the S-matrix has
no explicit time dependence.]
5 Perturbation Theory
In this section we are going to calculate the Greens functions of scalar quantum eld
theory explicitly. We will specify the interaction Lagrangian in detail and use an approx-
imation known as perturbation theory. At the end we will derive a set of rules, which
represent a systematic prescription for the calculation of Greens functions, and can be
easily generalised to apply to other, more complicated eld theories. These are the famous
Feynman rules.
We start by making a denite choice for the interaction Lagrangian Lint . Although
one may think of many dierent expressions for Lint , one has to obey some basic principles:
rstly, Lint must be chosen such that the potential it generates is bounded from below
otherwise the system has no ground state. Secondly, our interacting theory should be
renormalisable. Despite being of great importance, the second issue will not be addressed
in these lectures. The requirement of renormalisability arises because the non-trivial vac-
uum, much like a medium, interacts with particles to modify their properties. Moreover,
if one computes quantities like the energy or charge of a particle, one typically obtains
a divergent result3 . There are classes of quantum eld theories, called renormalisable,
in which these divergences can be removed by suitable redenitions of the elds and the
parameters (masses and coupling constants).
For our theory of a real scalar eld in four space-time dimensions, it turns out that
the only interaction term which leads to a renormalisable theory must be quartic in the
elds. Thus we choose
Lint = 4 (x), (5.1)
4!
where the coupling constant describes the strength of the interaction between the scalar
elds, much like, say, the electric charge describing the strength of the interaction between
photons and electrons. The full Lagrangian of the theory then reads
1 1
L = L0 + Lint = ( )2 m2 2 4 , (5.2)
2 2 4!
3
This is despite the subtraction of the vacuum energy discussed earlier.
and the explicit expressions for the interaction Hamiltonian and the S-matrix are
Hint = Lint , Hint = d3 x 4in (x, t)
4!
4 4
S = T exp i d x in (x) . (5.3)
4!
The n-point Greens function is
Gn (x1 , . . . , xn )
r ! r "
i 1
0 T in(x1 ) in (xn ) 4 4
d y in (y) 0
4! r!
r=0
= r ! r " . (5.4)
i 1
0 T 4 4
d y in (y) 0
r=0
4! r!
This expression cannot be dealt with as it stands. In order to evaluate it we must expand
Gn in powers of the coupling and truncate the series after a nite number of terms. This
only makes sense if is suciently small. In other words, the interaction Lagrangian must
act as a small perturbation on the system. As a consequence, the procedure of expanding
Greens functions in powers of the coupling is referred to as perturbation theory.
We are now going to combine normal-ordered products with time ordering. The time-
ordered product T {(x1 )(x2 )} is given by
p2 k2
Figure 5: Scattering of two initial particles with momenta p1 and p2 into 2 particles with
momenta k1 and k2 .
and S = 1 + iT . The LSZ formula Eq. (4.47) tells us that we must compute G4 in order
to obtain Sfi . Let us work out G4 in powers of using Wicks theorem. To make life
simpler, we shall introduce normal ordering into the denition of S, i.e.
4 4
S = T exp i d x : in (x) : (5.23)
4!
Suppressing the subscripts in from now on, the expression we have to evaluate order by
order in is
Gn (x1 , . . . , xn ) (5.24)
r !
r "
i 1
0 T (x1 )(x2 )(x3 )(x4 ) d4 y : 4 (y) : 0
r=0
4! r!
= r ! r " .
i 1
0 T d4 y : 4 (y) : 0
r=0
4! r!
If r = 1, then the expression in the denominator only involves elds which are normal-
ordered. Following the discussion at the end of section 5.1 we conclude that these contri-
butions must vanish, hence
The contribution for r = 2, however, is non-zero. But then the case of r = 2 corresponds
already to O(2 ), which is higher than the order which we are working to. Therefore
x1 x3 x1 x3 x1 x3
+ +
x2 x4 x2 x4 x2 x4
But this is the same answer as if we had set = 0, so r = 0 in the numerator does not
describe scattering and is hence not a contribution to the T -matrix.
For r = 1 in the numerator we have to evaluate
! "
i
r=1: 0 T (x1 )(x2 )(x3 )(x4 ) : d y (y) : 0
4 4
4!
i
= d4 y 4! GF (x1 y)GF (x2 y)GF (x3 y)GF (x4 y), (5.29)
4!
where we have taken into account that contractions involving two elds inside : : vanish.
The factor 4! inside the integrand is a combinatorial factor: it is equal to the number of
permutations which must be summed over according to Wicks theorem and cancels the
4! in the denominator of the interaction Lagrangian. Graphically this contribution is
represented by
x1 x3
y
i d4 y
x2 x4
where the integration over y denotes the sum over all possible locations of the interaction
point y. Without normal ordering we would have encountered the following contributions
for r = 1:
x1 x3 x1 x3
+ +...
x2 x4 x2 x4
Such contributions are corrections to the vacuum and are cancelled by the denomina-
tor. This demonstrates how normal ordering simplies the calculation by automatically
subtracting terms which do not contribute to the actual scattering process.
To summarise, the nal answer for the scattering amplitude to O() is given by
Eq. (5.29).
(1) Draw all distinct diagrams with n external lines and m 4-fold vertices:
(4) Multiply by the number of contractions C from the Wick expansion which lead to
the same diagram.
These are the Feynman rules for scalar eld theory in position space.
Let us look at an example, namely the 2-point function. According to the Feynman
rules the contributions up to order 2 are as follows:
O(1): x1 x2 = GF (x1 x2 )
O(2 ): x1 y1 y2 x2
2
i
=C d4 y1 d4 y2 GF (x1 y1 ) [GF (y1 y2 )]3 GF (y2 x2 )
4!
The combinatorial factor for this contribution is worked out as C = 4 4!. Note that
the same graph, but with the positions of y1 and y2 interchanged is topologically distinct.
Numerically it has the same value as the above graph, and so the corresponding expression
has to be multiplied by a factor 2.
Another contribution at order 2 is
O(2 ): y1 y2
vacuum contribution;
not connected
x1 x2
This contribution must be discarded, since not all of the points are connected via a
continuous line.
Let us end this discussion with a small remark on the tadpole diagrams encountered
above. These contributions to the 2-point function are cancelled if the interaction term
is normal-ordered. However, unlike the case of the 4-point function, the corresponding
diagrams satisfy the Feynman rules listed above. In particular, the diagrams are connected
and are not simply vacuum contributions. They must hence be included in the expression
for the 2-point function.
The Feynman rules then serve to compute the Greens function G% n (p1 , . . . , pn ) order by
order in the coupling.
In every scattering process the overall momentum must be conserved, and hence
n
pi = 0. (5.32)
i=1
This can be incorporated into the denition of the momentum space Greens function one
is trying to compute:
n
Here we wont be concerned with the exact derivation of the momentum space Feynman
rules, but only list them as a recipe.
(1) Draw all distinct diagrams with n external lines and m 4-fold vertices:
i
(2)4 4 momenta ,
4!
(the delta function ensures that momentum is conserved at each vertex).
(5) Multiply by the combinatorial factor C, which is the number of contractions leading
to the same momentum space diagram (note that C may be dierent from the
combinatorial factor for the same diagram considered in position space!)
where, perhaps for obvious reasons, Gn+m is called the truncated Greens function.
x1 z1
z2
x2 G
x3 z3
Putting Eq. (5.35) back into the LSZ expression for the S-matrix element, and using
that
xi + m2 GF (xi zi ) = i 4 (xi zi ) (5.36)
one obtains
k1 , . . . , kn ; outp1 , . . . , pm ; in
m n
& m n
'
= (i)n+m d4 xi d4 yj exp i pi xi + i kj y j (5.37)
i=1 j=1 i=1 j=1
(i)n+m d4 z1 d4 zn+m 4 (x1 z1 ) 4 (yn zn+m ) Gn+m (z1 , . . . , zn+m ).
After performing all the integrations over the zk s, the nal relation becomes
k1 , . . . , kn ; outp1 , . . . , pm ; in
& '
m n m n
= d4 xi d4 yj exp i pi xi + i kj y j
i=1 j=1 i=1 j=1
Problems
5.1 Verify that
: (x1 )(x2 ) : = : (x2 )(x1 ) :
Hint: write = + + , where + and are creation and annihilation components
of .
5.3 Find the expressions corresponding to the following momentum space Feynman di-
agrams
Integrate out all the -functions but do not perform the remaining integrals.
6 Concluding remarks
Although we have missed out on many important topics in Quantum Field Theory, we got
to the point where we established contact between the underlying formalism of Quantum
Field Theory and the Feynman rules, which are widely used in perturbative calculations.
The main concepts of the formulation were discussed: we introduced eld operators,
multi-particle states that live in Fock spaces, creation and annihilation operators, the
connections between particles and elds as well as that between n-point Greens func-
tions and scattering matrix elements. Besides slight complications in accounting for the
additional degrees of freedom, the same basic ingredients can be used to formulate a quan-
tum theory for electrons, photons or any other elds describing particles in the Standard
Model and beyond. Starting from relativistic wave equations, this is discussed in the
lectures by Nick Evans at this school. Renormalisation is a topic which is not so easily
discussed in a relatively short period of time, and hence I refer the reader to standard
textbooks on Quantum Field Theory, which are listed below. The same applies to the
method of quantisation via path integrals.
Acknowledgements
I am indebted to all the previous lecturers of the QFT course at this school, who have
helped in the evolution of the course to its present form. In particular I wish to thank Owe
Philipsen from whom I inherited these lecture notes. I would like to thank Bill Murray for
running the school so successfully, as well as Margaret Evans for her friendly and ecient
organisation. Many thanks go to my fellow lecturers and the tutors for the pleasant and
entertaining collaboration, and to all the students for their interest and questions, which
made for a lively and inspiring atmosphere.
References
[1] F. Mandl and G. Shaw, Quantum Field Theory, Wiley 1984.
[2] C. Itzykson and J.B. Zuber, Quantum Field Theory, McGraw-Hill 1987.
[3] M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addi-
son Wesley 1995
x = (x0 , x) = (t, x)
x = g x = (x0 , x) = (t, x)
1 0 0 0
0 1 0 0
Metric tensor: g = g = 0
0 1 0
0 0 0 1
Scalar product:
x x = x0 x0 + x1 x1 + x2 x2 + x3 x3
= t2 x2
Gradient operators:
= ,
x t
= ,
x t
2
dAlembertian: = 2
t2
Momentum operator:
p = i = i , i = E, p (as it should be)
t
-functions:
d3 p f (p) 3 (p q) = f (q)
d3 x eipx = (2)3 3 (p)
d3 p ipx
e = 3 (x)
(2)3
(similarly in four dimensions)
Note: