Adv QFT
Adv QFT
Adv QFT
1
Contents
1 Lecture 1: Introduction 3
8 Lecture 8: Renormalization 41
2
1 Lecture 1: Introduction
The goal for this course is to explain the current ”standard model” for particle
physics. This is too lofty of a goal for this course, so what we focus on is the
textitbuilding blocks of the standard model, such that we understand the origin
and purpose of each term of the Lagrangian.
Path Integrals
Let’s first suppose that the quantization is already done, and we have a quan-
tum system with a Hilbert space H, a Hamiltonian Ĥ, and a propagator from
integrating the Schroedinger equation, U (t) = e−iĤt .
Now work out a representation to first order for the propagator by Taylor ex-
panding
it N N
it
U (t) = e−iĤt = e− N Ĥ = lim I − Ĥ (1)
N →∞ N
Let {|ji} be a basis for the Hilbert space H and consider the transition amplitude
of evolving from an eigenstate |φi i to another eigenstate |φf i
it N
hφf | U (t) |φi i = hφf | e− N Ĥ |φi i (2)
3
Insert N − 1 Hilbert space basis completeness relations, one in between each of
the N exponentials, and note that the sum over all paths from initial state to
final state of functions of the paths
it it it
X
hφf | U (t) |φi i = hφf | e− N Ĥ |jN −1 i hjN −1 | e− N Ĥ . . . |j1 i hj1 | e− N Ĥ |φi i
j1 ,...,jN −1
(3)
X
≡ f (j1 , . . . , jN −1 ). (4)
paths
So, the transition amplitude of this state evolution is a sum over all of the
paths through the basis states j1 , . . . , jN −1 . An example schematic of a path is
visualized below.
To work out the function of the path f (j1 , . . . , jN −1 ), we need to calculate these
individual transition amplitudes between successive states |jk−1 i to some final
state hjk | in the path integral setting, and seek to write it as an exponential of
some function of the states L(jk−1 , jk ), the Lagrangian density
it it
Ĥ) |jk−1 i ' e N L(jk−1 ,jk ) .
hjk | (I − (5)
N
So, the full transition amplitude will be a product of exponentials pf Lagrangian
densities, well-known classical quantities
X X it
PN −1
L(jk−1 ,jk )
hφf | U (t) |φi i = f (j1 , . . . , jN −1 ) = eN k=2 . (6)
paths j1 ,...,jN −1
4
1. It allows the calculation of quantum quantities, the transition amplitudes,
via well-understood classical solutions and methods for handling highly
oscillatory integrals such as the saddle point method.
2. It can also be used to build nonperturbative approximation schemes, such
as Monte Carlo sampling over paths.
Turn around the path integral sum over paths to guess a quantum Hamiltonian
and Hilbert space from this classical Hamiltonian via
T
So, the transition amplitude in this case is, with ≡ δt = N
X
hqf | U (T ) |qi i = hqf | e−iĤ |qkN −1 i hqkN −1 | . . . |qk1 i hqk1 | e−iĤ |qi i .
k1 ,...,kN −1
(9)
There are three cases for the dependence of the quantum Hamiltonian on the
canonical coordinates in the expression for the propagator. It can depend purely
on position, purely on momenta, or most realistically, it can depend on both.
Y
hqk+1 | g(q̂) |qk i = g(qk ) δ(qkj − qk+1
j
) (10)
j
qk+1 + qk
Y Z dpj P j j j
k i j pk (qk+1 −qk )
=g e . (11)
2 j
2π
5
R ∞ dp ip·q
Where we used the Dirac delta distribution identity δ(q) = −∞ 2π e to in-
troduce the canonical momenta into the transition amplitude. Also note that
the Dirac delta function forces qk+1 = qk , such that f (qk ) = f ( qk+12+qk ), and
we write it in this fashion for later use.
Next, in the case that the Hamiltonian is a function purely dependent on canon-
ical momenta, such that Ĥ = h(p̂), the transition amplitude is calculated by
inserting the completeness relation for the momentum eigenbasis.
YZ
hqk+1 | h(p̂) |qk i = hqk+1 | h(p̂) · dpjk |pk i hpk |qk i (12)
j
Y Z dpj P j j j
= k
h(pk )ei j pk (qk+1 −qk ) (13)
j
2π
Where the inner product of the position and momentum eigenstates is a Fourier
1 ip·q
phase element hp|qi = 2π e , and we get the sum, since the subscript k denotes
N total canonical coordinate pairs.
Y Z dpj q +q P j j j
−iĤ(q̂,p̂) k −iH( k+12 k ,pk ) i j pk (qk+1 −qk )
hqk+1 | e |qk i = e e (15)
j
2π
Putting all this together into the propagator, which is really the transition
amplitude for a nonrelativistic quantum system,
YZ pjk i Pk Pj pjk (qk+1
Z
j j q +q
−qk )−H( k+12 k ,pk )
U (qi , qf ; T ) = dqkj e . (16)
2π
jk
Take note that there is nothing quantum on the RHS: no hats! We have used
purely classical data to define the quantum propagator, or, transition ampli-
tude, on the LHS, such that U (qi , qf ; T ) ∝ e−iH(qi ,qf ;T ) .
6
its saddle points (or critical points), which correspond to classical paths of the
system.
Monte Carlo sampling of the system can also be used to approximate the tran-
sition amplitude by building an estimator for the RHS, sampling over classical
configurations, and summing up the estimator.
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2 Lecture 2: Gaussian Path Integrals
Recall the propagator, or transition amplitude, for a nonrelativistic quantum
system
YZ Z RT j j
U (qi , qf ; T ) = Dq j (t) Dpj (t) ei 0 dt L(q ,q̇ ) . (18)
j
YZ dpjk i Pk (Pj pj (qj −qj )−H)
Z
U (qi , qf ; T ) = dqkj e k k+1 k (19)
2π
j,k
T
Evaluate these very many integrals to get an answer dependent on = N , since
we discretized, take the limit as → 0 and deal with any encountered infinities.
Key Example
Consider the classical Hamiltonian
p2
+ V (q).
H= (20)
2m
Calculate the transition amplitude (Exercise)
YZ Z j
j dp P P j j j
k i k ( j pk (qk+1 −qk )−H)
U (qi , qf ; T ) = dqk e (21)
2π
j,k
!
YZ p2
Z
dpk P k
= dqk ei k (pk (qk+1 −qk )−( 2m +V (q))) (22)
2π
k
!r
−im i Pk 2
Z q
k+1 +qk
m
(qk+1 −qk )2 −V
Y
= dqk e 2
. (23)
2π
k
8
We may also write this in the following notation, using the fact that the argu-
ment of the exponential is the discretized version of the action, now without the
p-integral
Z
lim U (qi , qf ; T ) = Dq(t)eS[q(t)] (24)
→0
1
L= (∂µ φ)2 − V (φ) (26)
2
Z
1 2 1
H = d3 x π (x) + (∇φ(x))2 + V (φ(x)) . (27)
2 2
The path integral prescription for quantum scalar fields gives the transition
amplitude, by blind application of the above, we conjecture that
Z Z R
T 4
hφb | e−iĤT |φa i = Dφ Dπ ei 0 d x (πφ̇−H(φ)) (28)
Where the boundary terms are φ(t = 0, x) = φa (x) and φ(t = T, x) = φb (x).
The field operators are discretized over a ”grid” of points xj each of width ,
such that
9
Next the derivative can be discretized via a finite difference. Note that there
are more computationally efficient symmetric differences that can be used to
discretize the derivative, but the finite difference works well for demonstration
(φ(xj + µ ) − φ(xj ))
∇µ φ(x) → (31)
|µ |
Where µ denotes the four directions in which to calculate the derivative
1 0 0 0
µ = { 00 , 10 , 01 , 00 } (32)
0 0 0 1
.
Lastly, the potential just becomes evaluated at each xj
The most important case of the scalar field is the quadratic potential, which
corresponds to the Klein-Gordon field (e.g., discretizing Klein-Gordon theory
yields the quadratic potential below), is
1 T
V (q) = q Aq. (37)
2
10
Gaussian Integrals
Consider the following integral
Z ∞ √
2
I= dx e−x = π. (38)
−∞
Proof:
Z ∞
2
I= dx e−x (39)
−∞
Z∞ Z ∞
2 2
I2 = dx e−x dy e−y (40)
−∞ −∞
Z ∞ Z 2π
2
I2 = rdr dθ e−r (41)
0 0
Z ∞
d 1 2
I 2 = 2π − e−r dr = π (42)
0 dr 2
.
This is actually a special case of the more general forms of the Gaussian integral
∞
Z r
− 21 ax2 +bx 2π b2
dx e = e 2a (43)
−∞ a
Z ∞ r
iax2 +ibx 2πi −ib2
dx e = e 2a (44)
−∞ a
(45)
.
We will later need the moments generated by the Gaussian integral
R∞ 1 2
dx xn e− 2 ax
hx i = −∞
n
R∞ 1 2
. (46)
−∞
dx e− 2 ax
Note that if n is odd, then the moment is zero and we can write the exponent
of x as 2m, where m ∈ Z, and we have the relation (Exercise)
(2m − 1)!!
hx2m i = . (47)
am
Note that the double factorial (2m − 1)!! represents the number of ways to join
2m points in pairs. – ”All science should in linear algebra or combinatorics.” –
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R∞ 1 2 !
dx e− 2 ax
2m +bx
2m d −∞
hx i= R∞ − 12 ax2
b=0
(48)
db −∞
dx e
2m
2m d b2
hx i= e 2a b=0 . (49)
db
OT DO = A (51)
Assume that B = 0 and define y = Ox. Then
Z ∞ Z ∞
T
I(A, B = 0) = dy1 · · · dyn e−y Dy
(52)
−∞ −∞
n Z ∞
Y 2
= dyj e−yj λj (53)
j=1 −∞
n r
Y π
= (54)
j=1
λj
r
πn
I(A, B = 0) = . (55)
det(A)
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3 Lecture 3: Correlation Functions and Path In-
tegrals
Recall the generating function for a single-variable Gaussian probability distri-
1 2
bution e 2 ax and the moment-generating integral
Z ∞
1 2 (2n − 1)!!
I= dx x2n e 2 ax = . (57)
−∞ an
We also derived the identity with the generating function for the multivariable
Gaussian probability distribution
Z ∞ Z ∞ r
− 12 xT Ax+J T x πn T −1
dx1 · · · dxn e = eJ A J . (58)
−∞ −∞ det(A)
The 2-point correlation function for the n-variable Gaussian is (Exercise), for
j 6= k
R∞ R∞ 1 T
−∞
dx1 · · · −∞ dxn xj xk e− 2 x Ax
hxj xk i ≡ R∞ R∞ 1 T ≡ [A−1 ]jk . (59)
−∞
dx1 · · · −∞ dxn e− 2 x Ax
Note that this is also equal to the second derivative with respect to the vector
J, evaluated at J = 0
∂2
R∞ R∞ − 12 xT Ax+J T x
∂Jj ∂Jk −∞ dx1 · · · −∞ dxn e
hxj xk i ≡ . (60)
R∞ R∞ − 21 xT Ax+J T x
−∞
dx1 · · · −∞
dxn e J=0
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[A−1 ]jk = [A−1 ]kj (63)
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And there are only 2!2!2! = 3 unique terms (products of two matrix elements),
which are (Exercise)
With the approporiate choice of A, in the context of path integrals and per-
turbative field theory, these products of correlations functions are exactly cor-
respondent to Feynman propagator, and, in turn, the Feynman diagrams, just
as we studied in Lecture 9 of the last lecture series (Quantum Field Theory) for
the 4-particle Wick contraction.
X
hxj1 . . . xjl i = [A−1 ]jπ−1 (1) jπ−1 (2) . . . [A−1 ]jπ−1 (l−1) jπ−1 (l) (64)
unique π −1
6!
(Exercise) Calculate the 6-point correlation function with 2!2!2! = 90 unique
terms
hxj1 xj2 xj3 xj4 xj5 xj6 i = [A−1 ]j1 j2 [A−1 ]j3 j4 [A−1 ]j5 j6 + . . . (65)
In short summary,
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The Matrix A for Path Integrals
Let the potential V be quadratic in the canonical position coordinate per par-
ticle qk (e.g., a one-dimensional chain of oscillators), such that the transition
amplitude, which will be discretized, evaluated, and limited → 0, from some
state qa to another qb is
!
Y Z dqk 1 T
U (qa , qb ; T ) = e 2 iq Aq (66)
c()
k
Where we know the quadratic form contains a kinetic energy term plus a po-
tential energy term
!
2
X (q k+1 − q k qk+1 + qk
q T Aq = m − V ( ) . (67)
2
k
This results is A as a tridiagonal matrix for the kinetic energy term and a
potential energy term which is a matrix with elements quadratic in q
2m
−m
0 ···
− m 2m − m 0 ···
A= 0 m 2m m + [V (q 2 )] (68)
− − 0 ···
..
.. .. ..
. 0 . . .
The transition amplitude is then calculated similarly to last lecture as
∞ const.
U (qa , qb ; T ) = p (69)
det(A)
The infinite constant will not be a problem since the l-point correlation is nor-
malized, and the same exact infinite constant will appear in the denominator
and cancel the constant. So, in terms of qk , the l-point correlation reads
dqk − 21 iq T Ax
Q R
k c() qj1 . . . qjl e
hqj1 . . . qjl i ≡ Q R dqk − 1 iqT Ax . (70)
c() e
2
k
Note that with periodic boundary conditions, the elements of follow a modulo
relation Ajk = f ((j − k) mod n), where n is the number of sites/oscillators in
the chain.
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Correlations Functions and Quantum Observables
Consider the transition amplitude over 2-point spatial correlations
Z RT 4
U (qa , qb ; T ) ∝ Dφ(x) φ(x1 )φ(x2 )ei −T d x L(φ) (72)
Apply the following condition, exploiting the boundary conditions, to factor the
full field ”integral” over the individual fields and the boundary of the field
Z Z Z Z
Dφ(x) = Dφ1 (x) Dφ2 (x) Dφ(x) (75)
∂φ
So, after performing the boundary integral, we introduce quantum stuff to the
expression from the classical 2-point function above, for x02 > x01
Z Z
0
U (qa , qb ; T ) ∝ Dφ1 (x) Dφ2 (x) φ(x1 )φ(x2 ) hφb | e−iĤ(T −x2 ) |φ2 i
0 0 0
× hφ2 | e−iĤ(x2 −x1 ) |φ1 i hφ1 | e−iĤ(x1 +T ) |φa i
Now, apply the Schroedinger-picture field operator to write the classical field
operators in terms of quantum field operators. The formula is
Z Z
0
U (qa , qb ; T ) ∝ Dφ1 (x) Dφ2 (x) hφb | e−iĤ(T −x2 ) φ̂S (x) |φ2 i
0 0 0
× hφ2 | e−iĤ(x2 −x1 ) φ̂S (x) |φ1 i hφ1 | e−iĤ(x1 +T ) |φa i
This is called the time-ordered expectation value of the field operators in the
Heisenberg picture. The equation holds for x02 < x01 as well, and we can write
it as
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U (qa , qb ; T ) ∝ hφb | e−iĤT T [φ̂H (x1 )φ̂H (x2 )]e−iĤT |φa i (79)
Now, enter the limit as T → ∞, where we bring the interacting vacuum state and
the normalization for the full transition amplitude (Exercise), and introduce
the most important formula for this course
RT 4
Dφ φ(x1 )φ(x2 )ei −T d x L(φ)
R
hΩ| T [φ̂H (x1 )φ̂H (x2 )] |Ωi = lim RT
4
. (80)
Dφ ei −T d x L(φ)
T →∞(1−i)
R
So, the LHS is built of purely quantum observables equal to the classical ex-
pression of path integrals!
This expression will end up to be the propagator, which is also the inverse of
the Klein-Gordon operator, which is what we call A in the scalar quantum field
theory.
The solution to this is well-known for the case when L is quadratic in the field
operators, and one can easily discretize, evaluate the Gaussian integral, and
take the limit as → 0.
(Exercise) Calculate the l-point formula for the time-ordered expectation value
of the field operators in the Heisenberg picture.
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4 Lecture 4: Functional Quantization of the Scalar
Field
The path integral formalism for quantization of fields is an incredibly efficient
tool, but one must learn when, and when not, ot use it. Through the lectures
and many examples, we’ll develop an intuition for when to trust quantization
via path integrals.
Recall the action of the scalar field S with classical field operators φ
Z Z
4 4 1 2 1 2 2
S0 = d x L0 = d x (∂µ φ) − m φ . (81)
2 2
We will (1) discretize, tantamount to imposing a cutoff Λ, (2) evaluate the in-
tegrals, and (3) enter the continuum limit where → 0. Start the discretization
by putting the field on a lattice (a Lorentz manifold) with spacing and then
compactify the space onto a torus for periodic boundary conditions.
Now following the path integral quanitzation recipe, consider the transition
amplitude in terms of the discretized action
Z
hφf | U (qi , qf ; T ) |φi i = Dφ eiS0 (86)
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Z YZ YZ
Dφ → dφ(xj ) ≡ dqj . (87)
j j
And the discretized action for the scalar field in momentum space is (Exercise)
1 X
S0 = − (m2 − kn2 )((<φn )2 + (=φn )2 ) (91)
V 0
kn >0
Where φn ≡ φ(kn ), and the following relation for the Kronecker delta is used
to obtain this expression
n−1
1 X 2πijk
δk,0 = e n . (92)
n j=0
Our expression for the path integral for the Klein-Gordon field discretized to
a lattice (four-dimensional with periodic boundary conditions) is comprised of
Gaussian integarls over a finite number of degrees of freedom
Z Y Z Z
2 2
−i V1 )|φn |2
P
0 >0 (m −kn
I0 = Dφ eiS0 = d <φn d =φn e kn
. (93)
0 >0
kn
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Now, onto evaluating this integral, it’s just a bunch of Gaussian integrals, and
we know how to solve those. We get the following, and unrestrict kn to get the
second line (Exercise)
s s
Y −iπV −iπV
I0 = · (94)
0 >0
m2 − kn2 m2 − kn2
kn
s
Y −iπV
I0 = (95)
m2 − kn2
kn
So, we are boldly extrapolating to say that A is like the Klein-Gordon operator
and the x is like the field operator
Z
d4 x φ(x)(−∂ 2 − m2 )φ(x) ∼ xT Ax. (98)
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Operationally Well-Defined Quantities
As mentioned, I0 cancels for operationally well-defined quanities, such as the
2-point correlation function, a time-ordered expectation value of products of the
field operators. For example, using the path integral formalism
So, the discretized RHS numerator of the time-ordered expectation value above
is just a bunch of independent Gaussian integrals, quadratic in its independent
variables (Exercise)
Z Z
1 X −i(km ·x1 +kl ·x2 ) Y
numerator = 2 e d <φn d =φn
V 0
l,m kn >0
1
(m2 −k2 )((<φ )2 +(=φn )2 )
P
−i
× (<φm + i=φm )(<φl + i=φl )e V kn0 >0 n n
1 X −ikm ·(x1 −x2 ) Y −iπV −iV
= 2 e 2 − k2 2 − k 2 − i
V m 0
m n m n
kn >0
1 X −ikm ·(x1 −x2 ) −iV
= 2 e · I0 · 2
V m m − kn2 − i
Where we drastically cut down the number of integrals to evaluate, since any
integrals involving products like <φm · =φl or =φm · <φl form odd integrands
and evaluate to zero. The integeral will also be zero for terms where m 6= l and
for terms where km = kl . Integrals where km = −kl will not be zero.
Bringing this together, the RHS of the time-ordered expectation value has boiled
down to
21
So, the path integral formalism gives us exactly the propagator we wish to see.
Note that is we were to just boldy extrapolate, without discretization, etc., we
would get the same result!
For example,
1
Dφ φ(x1 )φ(x2 )eiS (∂ 2 − m2 )− 2 D(x1 − x2 )
R
R
iS
= 1 (106)
Dφ e (∂ 2 − m2 )− 2
Since if A ∼ (−∂ 2 − m2 )
Then [A−1 ]jk ∼ (−∂ 2 −m
1
2)
x
= D(x1 − x2 )
1 x2
(4) 2 2
And δ (x − y) = (−∂ − m )D(x − y).
X Z
−i...
e . . . φkl φ−kl φkq φ−kq e...
kl ,kq
22
Dφ φ(x1 )φ(x2 )ei(S0 +Sint )
R
hΩ| T [φ(x1 )φ(x2 )] |Ωi = limT →∞(1−i) R (109)
Dφ ei(S0 +Sint )
Dφ φ(x1 )φ(x2 )eiS0 (1 + Sint + 12 Sint
2
R
+ ...)
= limT →∞(1−i) R
iS 1 2
Dφ e 0 (1 + Sint + 2 Sint + . . . )
(110)
Where Sint = iλ
R 4 4
4! d z φ (z), and each term above is an integral of powers of
time-ordered quantum field operators which end up as Feynman diagrams, for
example, of the form
λm
Z Z Z
4 4
d z1 · · · d zm Dφ φ(x1 )φ(x2 )φ4 (z1 ) . . . φ4 (zm )eiS0 (111)
4!m
λm
Z Z
= m d z1 · · · d4 zm hΩ| T [φ̂(x1 )φ̂(x2 )φ̂4 (z1 ) . . . φ̂4 (zm )] |Ωi
4
(112)
4!
= Sum of Feynman diagrams (113)
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5 Lecture 5: Functional Derivatives and Gener-
ating Functionals
Here we will finish the functional quantization of the scalar field.
Recall that we can compute time-ordered correlation functions for the quantum
scalar field entirely in terms of classical quantities, which is equivalent to a sum
over all diagrams,
Functional Derivatives
The functional derivative is a tool from the calculus of variations that we now
define by an example that is the continuum analog of the standard partial
derivative
δ
J(y) = δ (4) (x − y). (118)
δJ(x)
There is a one-to-one mapping from the discrete representation to the continu-
ous with correspondences
24
x∈R → j∈Z
J(x) ∈ C(R) → Jj ∈ L2 (Z) (119)
δ ∂
δJ(x) F [J(y)] → ∂Jj F [J1 , J2 , . . . ]
Example 1
Z
δ R 4 R 4 δ
ei d y J(y)φ(y) = iei d y J(y)φ(y) d4 y J(y)φ(y)
δJ(x) δJ(x)
Z
R 4 δJ(y)
= iei d y J(y)φ(y) d4 y φ(y)
δJ(x)
R 4
Z
= iei d y J(y)φ(y) d4 y δ (4) (x − y)φ(y)
δ R 4 R 4
ei d y J(y)φ(y) = iφ(x)ei d y J(y)φ(y)
δJ(x)
Z Z
δ 4 µ δ 4 µ
d y (∂µ J(y))v (y) = boundary term − d y J(y)∂µ v (y)
δJ(x) δJ(x)
= −∂µ v µ (x)
Note that the boundary term is almost always zero, except for topologically
interesting theories.
This expression is obviously useful, since correlation functions are directly re-
lated to derivatives of Z[J]
δ δ
− δJ(x) δJ(y) Z[J] J=0
hΩ| T [φ̂(x)φ̂(y)] |Ωi = (121)
Z[J]J=0
So, if you can compute the generating functional Z[J], you have all of the n-
point correlation functiosn via derivatives for your field theory.
In free field theories, such as the Klein-Gordon field, the action is quadratic in
the field operators, and the argument of the exponential is Z[J] is
25
Z
4 1 2 2
i(S0 + J(x)φ(x) = i d x φ(x)(−∂ − m + i)φ(x) + J(x)φ(x) . (122)
2
This is analogous to the positional shift x0 = x − A−1 J, and works becuase the
Feynman propagator is the inverse of the Klein-Gordon operator, such that
Z
1 0
i(S0 + J(x)φ(x) =i d4 x φ (x)(−∂ 2 − m2 + i)φ0 (x)
2
Z Z
4 4 1
−i d x d y J(x)(−iDF (x − y))J(y) .
2
δ δ 1
R 4 4
e− 2 d xd y (J(x)DF (x−y)J(y)) J=0 . (127)
hΩ| T [φ̂(x)φ̂(y)] |Ωi = −
δJ(x) δJ(y)
26
δ δ δ δ
δJ1 δJ2 δJ3 δJ4 Z[J] J=0
h0| T [φˆ1 φ̂2 φ̂3 φ̂4 ] |0i = (128)
Z[J = 0]
Z
δ δ δ 1
R 4 R 4
− d4 x0 Jx0 Dx0 4 e− 2 d x d yJx Dxy Jy J=0
=
δJ1 δJ2 δJ3
(129)
Z Z
δ δ
−D34 + d4 x0 d4 y 0 Jx0 Dx0 3 Jy0 Dy0 4 × e... J=0
=
δJ1 δJ2
(130)
Z Z Z
δ 4 0 4 0 4 0
D34 d x Jx Dx 2 + D24 d y Jy Dy 3 + D23 d z Jz Dz 4 + O(J ) e... J=0
2
= 0 0 0 0 0 0
δJ1
(131)
= D34 D12 + D24 D13 + D23 D14 (132)
Interacting Fields
The time-ordered expectation value, which contains the generating functions by
Taylor expansion, for the (classical) phi-fourth interacting theory is
Fermionic Fields
For the (classical) fermionic field ψ̂, the 2-point correlation function, vacuum
expectation value, is
D ψeiS ψ(x)ψ(y)
R
hΩ| T [ψ(x)ψ(y)] |Ωi = R (134)
Dψ eiS
Rule number one for this expression (1) is to not think abou this operationally,
and rule number two (2) is to think in analogy to complex numbers, which can
provide a more clear representation and make things easier.
ψ 2 (x) = 0 (135)
ψ(x)ψ(y) = −ψ(y)ψ(x) (136)
27
Vignette: Anticommuting Numbers (Grassman Numbers)
Let V be an n-dimensional vector space with basis θa ∈ V , a = 1, . . . , n. Thus,
elements of the vector space v ∈ V have the form
n
X
v= va θ a . (137)
a=1
To build a bigger vector space G(V ) from V , we first endow V with the product
operation denoted by concatenation (e.g., θa · θb · θc = θa θb θc ).
S ∞ (V ) = span{θa , θa θb , θa θb θc , . . . } (138)
Now restrict the basis to obey the following suggestive relations
θa θb = −θb θa (139)
θa2 = 0. (140)
Then the new vector space has dimension dim(G(V )) = 2n , and is the infinite
dimensional span modulo the elements of the underlying vector space
{1, θa , θa θb , θa θb θc , . . . } (142)
With a = 1, . . . , n, followed by 1 ≤ a < b ≤ n, a < b < c, etc.
n
X X
f =α+ αj1 j2 ...jp θj1 θj2 . . . θjp (143)
p=1 1≤j1 <···<jp ≤n
28
6 Lecture 6: Grassmann Numbers
We left off with an object meant to be the classical version of a fermion: a
Grassman number. It is the object of a vector space Gn (V ), generated by basis
{1, θ1 , . . . , θn ∈ V }. Imposing the anticommutation relation
Linear functions should be linear maps from the vector space to itself, F ∈
M2n (C), the space of 2n × 2n matrices
F : Gn (V ) → Gn (V ) (148)
Nonlinear functions will be defined by analogy to the functional calculus of ma-
trices; consider a matrix M ∈ Mm (C). As long as M is diagonalizable, define
the nonlinear function f (M ) = S −1 f (D)S, where M is diagonalized by S, such
that M = S −1 DS. Since a diagonal matrix occupies a commutative algebra,
we know how to define function for D and then rotate from D to M via S.
29
Representations of Gn (V )
Consider a map from our vector space to a concrete space of matrices
π : Gn (V ) → Md (C (149)
Which must obey the anticommutation relation
+ 0 0
σ = (151)
1 0
1 0
σz = . (152)
0 −1
Recall that these Pauli operators obey the relations
{σ + , σ z } = 0 (153)
+ 2
(σ ) = 0. (154)
Construct the representation of Gn (V )
∞
X F (j) (x = 0)xj
F (x) = (159)
j=0
j!
∞
X F (j) (θ = 0)θj
F (θ) = (160)
j=0
j!
30
Example 1:
Example 2:
n
X ∂F (0, . . . , 0)
F (θ1 , . . . , θn ) = F (0) (θ1 = 0, . . . , θn = 0) + θj + O(θ2 ) (166)
j=1
∂θj
Example 1: n = 2
Example 2: n = 2
Note that even though we lose higher order terms such as θj2 , nonlinear features
are preserved in the multivariable case.
θ1 + iθ2
θ= √ (173)
2
∗ θ1 − iθ2
θ = √ . (174)
2
31
To extend to the multivariable case, define
θj1 − iθj2
θj∗ = √ (175)
2
Where θj1 , θj2 ∈ G2n (V ).
Grassman Derivatives
Define the derivative with respect to Grassman variables as the map
∂θj : Gn (V ) → Gn (V ) (176)
Which obeys the relations
The second relation follows from the anticommutation relation, and can be
thought of as bringing the corresponding θj to the front via anticommutations
and then differentiating.
For example,
Grassman Integrals
Following the analogy of the common definite integral, the Grassman integral
should be a linear map which obeys shift invariance, such that θ → θ + η,
Z
dθ : Gn (V ) → C. (178)
The only consistent definition to satisfy these two constraints for a single Grass-
man variable is
Z
dθ (a + bθ) = b, a, b ∈ C (179)
Note the very interesting property here that, by this definition, the integral and
the derivative are the exact same thing
Z
dθ (a + bθ) = ∂θ (a + bθ). (180)
This also holds for the multivariable case, where the highest order term is picked
off from the Taylor series
32
Z X
dθn . . . dθ1 (f0 + fp (j1 , . . . , jp )θj1 . . . θjp ) = fn (1, . . . , n). (181)
j1 <···<jp
This is a weird definition, but it works, behaves correctly under change of vari-
ables, and does what we need it to do.
Z Z
∗
dθ∗ dθ e−λθ θ
= dθ∗ dθ (1 − λθ∗ θ) (182)
Make a change of variables θj0 = Ujk θk , where U is a unitary matrix, such that
the product of these new Grassman variables is (Exercise)
X
θ10 θ20 . . . θn0 = U1k1 U2k2 . . . Unkn θk1 θk2 . . . θkn (186)
k1 ,...,kn
X
= U1π(1) . . . Unπ(n) sgn(π)θ1 θ2 . . . θn (187)
π∈Sn
Z
∗ ∗ θ 0∗ † 0
P
I= dθ0 1 dθ10 . . . dθ0 n dθn0 e− j,k j (U BU )jk θk det(U )det(U † ) (189)
Z
∗ ∗ 0∗ 0 0∗ 0
= dθ0 1 dθ10 . . . dθ0 n dθn0 e−λ1 θ 1 θ1 . . . e−λn θ n θn · (1) · (1) (190)
= λ1 λ2 . . . λn (191)
I = det(B) (192)
33
Recall that for the normal Gaussian integral case, we got I = (det(B))−1 .
(Exercise) The generating functional for the Grassman calculus is, where J is
a vector of Grassman numbers,
Z
†
Bθ+J † θ+θ † J
Z[J] = dθ1∗ dθ1 . . . dθn∗ dθn e−θ (193)
†
B −1 J
= eJ . (194)
Side: The mixing of regular and Grassman numbers provides the basis for the
supersymmetric method. Consider the Gaussian integral
Z Z Z
† †
dΦe−Φ MΦ
= dx1 . . . dxn dx∗1 . . . dx∗n dθ1∗ dθ1 . . . dθn∗ dθn e−Φ MΦ
(196)
= (detA)(detA)−1 (197)
Z
†
dΦe−Φ MΦ
=1 (198)
34
7 Lecture 7: Functional Quantization of the Dirac
Field
We now employ Grassmann numbers/variables to build a path integral-like ob-
ject that provides the n-point correlation functions for the Dirac (spinor) field.
Where the Grassmann variables and auxiliary fields J and J ∗ obey the anti-
commutation relations
Note that the generating functional takes a vector of Grassmann numbers, an-
ticommuting objects, as input and yields an expression quadratic in the Grass-
mann numbers which evaluates to a real number, a commuting object, since
observables correspond to Hermitian operators and real numbers as their eigen-
values; the expectation value must always be a real number, not a Grassmann
number.
Recall the Dirac spinor field which is what we mean to be classical fermions.
The classical fermion is represented by the 4-component spacetime vector
35
The Hamiltonian, or generator of time translations, that follows from this spinor
object solves the Dirac equation (i∂µ γ µ − m)ψ = 0. Recall that we defined the
conjugate-like object ψ̄ = ψ † γ 0 to induce Lorentz invariance for the Lagrangian
density L = ψ̄(−i∂/ − m)ψ, where the slash notation denotes A / = Aµ γ µ .
Thus far, we’ve been thinking about ψ = (ψ1 , ψ2 , ψ3 , ψ4 ) as the classical spinor-
valued Dirac field, but it is really the single-particle component that is needed
to build the classical Dirac field with anticommuting objects, since we guess to
employ the anticommutation relation of quantum Dirac field operators
Alternatively, we can find some classical Dirac field that we quantize via the
path integral, and use the path integral as a tool to guess the quantum theory.
Guess 1 (Wrong):
Guess 2 (Correct):
36
Consider the (0 + 1)-dimensional case, mapping continuous spacetime coordi-
nates to discrete coordinates: xj → j, where j ∈ Z/N Z, and discretize the
Grassmann numbers to the lattice to define the classical Dirac field
So, the (discrete) classical Dirac field is a list of 8·N 4 Grassmann numbers, since
N 4 is the number of lattice sites and each of the two field “operator” contains
4 components.
Note that if we work in momentum space, the Fourier coefficients will be made
to be Grassmann numbers.
The continuous classical Dirac field comes from the limit of the lattice spacing
vanishing → 0, the number of sites tending to infinity N → ∞, and the size
of the torus tending to infinity L = N → ∞.
Now, to build the quantum theory corresponding to these classical objects via
the path integral formalism, we require an action, beginning with the discretiza-
tion of the Lagrangian density L = ψ̄(i∂/ − m)ψ. For the Dirac field, the dis-
cretized Lagrangian density is
X
µ ψj+µ̂ − ψj
L(ψj , ψ̄j ) = iψ̄j γ − mψ̄j ψj . (210)
⊗4
j∈(Z/N Z)
Where ψj and ψ̄j are 4D spinors of Grassmann numbers, we’ve employed the
forward-difference to represent the partial derivative, and µ̂ is a unit vector in
the µth direction.
RT
With the Lagrangian density, we can calculate the action S = i −T dt L(ψj , ψ̄j )
and the n-point correlation functions for the Grassmann-valued quantum field
operators.
For example, for the Grassmann variables, define the 2-point correlation function
Dψ̄Dψ ψ(x)ψ̄(y)eiS
R
ˆ
h0| T [ψ̂(x), ψ̄ (y)] |0i ≡ lim R (211)
T →∞(1−i) Dψ̄Dψ eiS
To calculate the 2-point correlation function, we continue to follow the prescrip-
tion
37
Evaluate the path integral to find that the 2-point function is
d4 k ie−ik·(x−y)
Z
h0| T [ψ̂(x), ψ̄ˆ(y)] |0i = SF (x − y) = (212)
(2π)4 k/ − m + i
Side note (topic of ongiong research): Fermion doubling is a topological artifact
of incorrectly placing fermions on a lattice and taking the continuous limit.
Extra fermions, called doublers, appear in the calculation, as the dispersion
relation ω(k) becomes nonlinear and crosses the k-axis more than once. The
expected dispersion relation is linear ω(k) = ak, a > 0. In discretization,
we must accept this effect and learn how to work around it. This is done for
conveience, since without discretization, evaluating the 2-point function requires
many more tricks.
Where J and J¯ are Grassmann-valued (auxiliary) source fields that will be set
to zero after differentiation. Calculating the generating functional will yield all
n-point functions via functional derivatives, made possible by the employment of
Grassmann numbers and functional quantization versus canonical quantization.
By completing the square and simplifying the expression for the generating
functional we get (Exercise)
¯
d4 xd4 y J(x)S
R
¯ = Z0 e−
Z[J, J] F (x−y)J(y)
(214)
Where Z0 = Z[J = 0, J¯ = 0]. Recall that for Grassmann numbers, the rules of
differentiation include sign-switching and go like
d d
θη = −θ η = −θ (215)
dη dη
So the n-point correlation function is then
δ δ
h0| T [ψ (α1 ) (x1 ) . . . ψ (αn ) (xn )] |0i = Z0−1 i(−1)α1 +1 α1 . . . i(−1)αn +1 αn Z[J, J]
¯
δJ δJ
(216)
Where ψ (α) (x) = ψ(x) for α = 0 and ψ (α) (x) = ψ̄(x) for α = 1.
Check (Exercise) that the quantum 2-point correlation function comes out to
be the expected
38
Interactions of Fermions and Bosons
The path integral is a great tool for guessing Feynman rules as well, since we
can expand in a Taylor series and recognize patterns that represent certain sym-
metries and diagrams. WIthout needing to introduce too much gauge theory,
we introduce massive quantum electrodynamics (QED), a quantum field theory
that models the interaction of fermions and (massive) bosons. We expect the
photon (boson) mass to be zero, but consider it massive for now, and note that
the upper bounds on the mass of the photon have been calculated to be nonzero
( 10−20 ).
Following the path integral quantization, the classical 2-point correlation func-
tion is
DψDψ̄DA ψ(x)ψ̄(y)eiS
R
h0| T [ψ(x)ψ̄(y)] |0i = lim R . (219)
T →∞(1−i) DψDψ̄DA eiS
Write the action in terms of the free theory and the interacting theory
S = S0 + Sint
Z
4 1 µν 1 2 µ
S= /
d x (ψ̄(i∂ − mf )ψ − F Fµν + mb Aµ A
4 2
Z
+ −ie d4 x ψ̄Aµ γ µ ψ .
Then the quantized 2-point correlation function for massive QED with the
Taylor expansion is an infinite series of n-point correlation functions for the
Grassmann-valued Gaussian path integrals
1. Draw a straight
line
from a to b with momenta p for each fermion and
associate p/−mif +i
ab
39
2. Draw squiggly line
from α to β with momenta q for each boson and asso-
−i
ciate k2 −m2 +i δαβ
b
40
8 Lecture 8: Renormalization
This is an incredibly important lecture where we ask, “How do we do physics?”
and “How do we make progress in physics?”
1. Observation
→ Empirical data
2. Explanation
→ Data compression
3. Understanding
→ Models
E.g., Hamiltonian with Hilbert space, neural network, list of data
4. Prediction
→ New observation
5. Repeat from Step 1
This algorithm yields an increasingly smaller list of plausible models that corre-
spond with the observed data. To the physicist, a model is often a Hamiltonian
(or Lagrangian), with an associated Hilbert space HΛ , that depends on some
list of unknown parameters zj ∈ R, for j = 1, . . . , n
For each model ĤΛ (z1 , . . . , zn ) that we make a prediction for hAj i, if the pa-
rameters zj do not yield the correct expectation value, then we reject that set
of parameters for the model, and end up with a map
41
We say that a model ĤΛ (z1 , . . . , zn ) is “simpler” than another model ĤΛ0 (z10 , . . . , zn0 0 )
if one or both of the following conditions are satisfied: n < n0 and/or |Λ| > |Λ0 |.
The parameters z1 , . . . , zn are essentially coupling constants, and are not di-
rectly observable and not operationally well-defined.
The degrees of freedom that we wish to explain in this context Λ are the mo-
mentum modes (of interacting bosons).
To tame these infinities, we first impose a cutoff |Λ| < ∞, where Λ is an arbi-
trary parameter. So, predictions will change with respect to the chosen cutoff,
since hAj i depends on Λ, such that hAj i = fj (z1 , . . . , zn ; Λ).
We can declare victory if we can invert the prediction fj (z1 , . . . , zn ; Λ) and move
the Λ-dependence onto the parameters, such that zj = zj (Λ).
A theory which allows hAj i = fj (z1 (Λ), . . . , zn (Λ); Λ), ∀Λ and fixed n, is called
renormalizable.
Side note about parameters: Note that the mass of an electron is the measured
value, but in the model it is defined by the imposed cutoff. For a different
cutoff, the coupling constant m may be different than the actual mass of the
electron, but in the “correct”, or “most correct”, model, we call it the “mass of
the electron”.
42
Note on things to come: the combinatorial proof that φ4 , and other models, is
renormalizable.
Recall that the scattering S-matrix for φ4 theory with no cutoff, or |Λ| = ∞,
blows up to infinity
The first term/diagram that rescaling λ does not work for introduces a loga-
rithmic divergence and has the form
(−iλ)2 d4 k
Z
i i
I= . (225)
2 (2π)4 k 2 − m2 + i (k1 + k2 − k)2 − m2 + i
Impose a cutoff on the momenta |k| < kc ∈ R. Then the integral above becomes
(Exercise)
43
(−iλ)2 i2 d4 k
Z
i i
I= 4 k 2 − m2 + i (k + k − k)2 − m2 + i
(226)
2 Λ (2π) 1 2
kc2
I = 2iC log . (227)
(k1 + k2 )2
To solve for λ(kc ), let the scattering amplitude be the experimental value
M(kc , λ) = Mexp and solve the differential equation that allows λ to vary
with respect to kc and match up to Mexp
dλ
kc = 6Cλ2 + O(λ3 ). (229)
dkc
44
9 Lecture 9: Renormalizability (of φ4 Theory)
This topic covers results of Bogoliubov, Parasiuk, Hepp, Zimmermann, or the
BPHZ renormalization scheme.
Ĥ(m, λ, z; Λ) (232)
Where z is the field strength renormalization parameter.
Degree of Divergence
Consider a diagram with BE external lines. The diagram has a superficial degree
of divergence D if it diverges with the cutoff as kcD . For D = 0, we say that the
diagram has logarithmic divergence: log (kc ).
D = 4 − BE . (233)
Examples:
45
1. BE = 2 =⇒ D = 2 =⇒ ∼ kc2
2. BE = 4 =⇒ D = 0 =⇒ ∼ log (kc )
1
3. BE = 6 =⇒ D = −2 =⇒ ∼ kc2
Note that as BE and kc increase, the divergences become increasingly less ob-
servable. Each pair of incoming/outgoing particles contributes a propagator
R d4 k B2E
−2 i
proportional to kc to the 4D momentum integral ∼ (2π)4 k2 −m2 +i .
1. BI = 3, V = 2, L = 2, D = 2
2. BI = 3, V = 2, L = 2, D = 2
46
3. BI = 5, V = 4, L = 2, D = −2
Proof :
It seems that there are BI such integrals, but momentum conservation reduces
the total number of loop integrals to
L = BI − (V − 1). (234)
Each vertex has four lines and each line connects two vertices, such that
4V = BE + 2BI . (235)
Now, recall that for each loop there is a factor ∼ kc4
from the integral, and for
each line there is a factor kc−2 from the propagator. Then
D = 4L − 2BI = 4 − BE . (236)
Exercise: Prove this result for n-dimensional spacetime.
1 2 λ
L = L(z1 = m, z2 = λ, z3 = z) = (z (∂µ φ)2 − z 2 m2 φ2 ) − z 4 φ4 . (237)
2 4!
Rewrite this in terms of the physical Lagrangian, the one that has been success-
ful in corresponding with experimental data, and counter terms dependent on
three new parameters A, B, and C,
The Feynman rules for the renormalized φ4 theory are the same rules plus
two more due to an additional type of vertex that depend of the “additional
interaction” parameters.
47
i
Top-left: k2 −m2phys +i
Top-right: −iλphys
Bottom-left: 2i(Ak 2 + B)
Bottom-right: 4! · iC
So, counter terms are added to the Lagrangian as “additional interactions”,
which introduce new Feynman diagrams. The parameters are determined iter-
atively to order λN
phys , at which we call them AN , BN , and CN , and nothing
depends on the cutoff.
Non-Renormalizable Theories
A non-renormalizable theory requires an infinite number of parameters to en-
sure that operationally-defined quantities do not depend on the cutoff.
L = L0phys + Lint
phys (λ) + (counter terms). (240)
Calculate the scattering amplitude to order O(λN phys ), and we will see that we
need counter terms to eliminate dependence on the cutoff kc , but as we go to
higher and higher order in λphys , we need more and more counter terms, and
this will continue and diverge, requiring an infinite number of counter terms
and associated parameters to eliminate the cutoff dependence.
48
10 Lecture 10: Abelian Gauge Theory (Quan-
tum Electrodynamics)
Why study abelian gauge theory?
An example of an abelian gauge theory is quantum electrodynamics and the
electromagnetic field, as well as SU (2), and SU (3) gauge bosons.
In developing a gauge theory, we will follow the same route of specifying sym-
metries, giving rise to invariants, of the theory and then, via quantization, look
for (projective) unitary representations of the group which are local.
As opposed to other field theories, the gauge theory should be symmetric under
a local gauge group G which acts independently at each location in spacetime.
For example, consider the circle group U (1), which consists of all complex num-
bers with absolute value equal to 1 under multiplication, the roots of unity
Consider the DIrac field, where we want a U (1) gauge-invariant quantum field
theory of electrons. Elements of G act independently on each spacetime locationx ∈
M1+3 . Equivalently, there is a copy of U (1) attached to each x acting indepen-
dently of each other.
Which theories are invariant under the Poincaré group and the local
gauge group G?
As it stands, we have an empty set of theories that are invariant under the
local gauge group. Begin populating it by building a Lagrangian density, with
a classical, continuous spacetime, and find which kinds of terms will be invariant.
Terms of the Lagrangian density for the Dirac field that we already know to be
invariant are ψ̄ψ and (ψ̄γ µ ψ)2 (contracted with itself).
49
/ is not invariant, as it is not well-defined! Why?
Note that the quantity ψ̄ ∂ψ
Yes, they exist and are made rigorous in the formalism of fibre bundles and
principal bundles. Note that U (y, x) is a not a field and is nonlocal object, but
it is expressable in terms of local objects and local data.
So, ψ(x + nµ ) and U (x + nµ , x)ψ(x) transform the same way under the local
gauge group G, and we can introduce dynamics to the theory and define the
covariant derivative as
50
ψ(x + nµ ) − U (x + nµ , x)ψ(x)
Dµ ψ(x) ≡ lim (250)
→0
What about the parallel transporter U (y, x)?
Where we tried U (x, x) = 1, since 1 transforms correctly under the gauge group,
and it is traditional to call ∂µ U (x, x) = −iαAµ (x), where α is the fine structure
constant. Now, Aµ (x) is not arbitrary, and must satisfy some constraints.
µ
eiα(x+n ) U (x + nµ , x)e−iα(x) = 1 − ienµ Aµ (x) + inµ ∂µ α(x) + . . . (254)
Furthermore, the covariant derivative of the field transforms under the local
gauge group by introducing the phase factor, the same as the action of G on the
field ψ itself
51
/ − mψ̄ψ + auxiliary field term(s)
L = ψ̄ Dψ (258)
To include the auxiliary field Aµ (x) in the quantization we need to endow it
with dynamics.
To first order,
µ
Aµ (x+ 12 nµ )
U (x + nµ , x) ∼ e−ien . (259)
Use the parallel transporter to build a plaquette operator, which transverses an
object around a square of dimension
U (x) = U (x, x + 1̂) · U (x + 1̂, x + (1̂ + 2̂)) · U (x + (1̂ + 2̂), x + 2̂) · U (x + 2̂, x).
(260)
The plaquette operator U is gauge invariant, and, working out the Taylor
series, we can write it in terms of the auxiliary field (Exercise)
1 1 1 1 3
U (x) = e−iα(−A2 (x+ 2 2̂)−A1 (x+ 2 1̂+2̂)+A2 (x+1̂+ 2 2̂)+A1 (x+ 2 1̂))+O( ) . (261)
Expand in (Exercise)
52
1
− F µν Fµν . (264)
4
Then the first nontrivial Langrangian density we can construct is the exact
Lagrangian for quantum electrodynamics: an electromagnetic field minimally
coupled to the Dirac field.
1
/ − m)ψ − F µν Fµν .
L = ψ̄(D (265)
4
The term “minimally coupled” means that the theory is renormalizable.
Alternative Derivation of F µν
Since Dµ is gauage invariant, the commutator with itself [Dµ , Dν ] is also gauge
invariant. So, under the local gauge group, the commutator transforms as
53
11 Lecture 11: Nonabelian Gauge Theory (Yang-
Mills)
As a recap of abelian gauge theory, a gauge theory is a theory that is invariant
under a group G of local symmetry transformations, which act independently
at each point in spacetime M1,3 .
In the context of the Dirac spinor and fermionic field theories, consider the local
phase transformation
The new, larger, more constrained symmetry group that we are building an
invariant theory under is Poincaré group plus the local gauge group G. The
only terms invaraint under this new symmetry group that we found fit for the
Lagrangian density were
∂µ → ∂µ − ieAµ . (272)
Call Dµ = ∂µ − ieAµ , and then we have the additional invariant term ψ̄ Dψ / to
include in the Lagrangian density under this new representation of the derivative
/ + F µν Fµν .
L = ψ̄ Dψ (273)
Where F µν is the spacetime curvature tensor that includes derivatives of the
gauge field Aµ .
54
Lie groups.
1
X
|vjk |2 = 2
j,k=0
V † (x)V (x) = I
det(V (x)) = 1
We need to choose how V (x) acts on a field. Introduce two independent spinor
fields ψ0 (x) and ψ1 (x) that form a new basis, under which the new theory must
be invariant,
1
X
ψj (x) = vjk (x)ψk (x). (275)
k=0
v00 (x) · I4×4 v01 (x) · I4×4 ψ0 (x)
g : Ψ(x) → . (277)
v10 (x) · I4×4 v11 (x) · I4×4 ψ1 (x)
The invariant terms we can construct from the doublet field are
1
γµ
X 0
Ψ̄Ψ = ψ̄j ψj and (Ψ̄Γµ Ψ)2 , where Γµ = (278)
0 γµ
j=0
As in the abelian case, build the covariant derivative by introducing the parallel
transporter U (y, x) ∈ SU (2) and going to a representation of the local gauge
group. Under the local gauge transformation
55
The covariant derivative is then defined to be
1
nµ Dµ Ψ ≡ lim (Ψ(x + n) − U (x + n, x)Ψ(x)) (280)
→0
Stepping back, suppose that we have some element of the gauge group U ∈
SU (2), which we assume is differentiable. Note that this U is not the parallel
transporter yet.
U = eiA (281)
†
Where A is a Hermitian matrix, such that A = A and Tr(A) = 0.
1 0 1 1 0 −i 1 1 0
σ1 = , σ2 = , and σ 3 = (283)
2 1 0 2 i 0 2 0 −1
[σ j , σ k ] = ijkl σ l . (284)
j
So, to specify the Hermitian matrix A, we need three real numbers α ∈ R.
Then, in the chosen Pauli basis, the covariant derivative is an 8 × 8 matrix and
is defined as
56
1
nµ Dµ Ψ ≡ lim (Ψ(x + n) − U (x + n, x)Ψ(x)) (286)
→0
Where
σj
Dµ = ∂µ − igAjµ (x)
. (287)
2
The coefficient field Ajµ (x) is not arbitrary and must obey transformation laws
of the local gauge group, as well as give a representation of the local gauge group
determined by the action of the local gauge group on the parallel transporter
Calculating the action of V (x + n) and V † (x), the first order term in becomes
(Exercise)
σj σj
i
L.G. : Ajµ (x) → V (x) Ajµ + ∂µ V † (x). (290)
2 2 2
Hint: V (x + n)V † (x) = [(1 + nµ ∂µ )V (x)] V † (x) + O(2 ).
Next, to compute V (x)∂µ V † (x), we will take the infinitesimal approach. Recall
that for the abelian case, we just had the phase factor α(x)∂µ α† (x), which was
just a number, but now with V (x) we have a 2 × 2 matrix with an 8 × 8 repre-
sentation.
σj σj
V (x)∂µ V † (x) = (I + iαj )∂µ (I − iαj ) (292)
2 2
∂αj σ j
= −i µ + O(α2 ) (293)
∂x 2
Then, under the local gauge transformation, the gauge field and sigma matrices
transform as
j j
σj σj k σk
σ σ 1
L.G. : Ajµ (x) → Ajµ (x) j j
+ (∂µ α (x)) + i α (x) , Aµ (x) . (294)
2 2 g 2 2 2
57
Now we can see the infinitesimal local gauge transformation does to the covariant
derivative of the doublet spinor field Ψ(x)
σj
j
L.G. : Ψ(x) → I + iα (x) Ψ(x) (295)
2
σj σj σj k σk σj
j j j j
L.G. : Dµ Ψ(x) → ∂µ − igAµ (x) − i(∂µ α (x)) + g α (x) , Aµ (x) 1 + iα (x) Ψ(x)
2 2 2 2 2
(296)
σj
L.G. : Dµ Ψ(x) → 1 + iαj (x) Dµ Ψ(x) (297)
2
= V (x)Dµ Ψ(x) (298)
Where, for the physicist, ignoring issues of connectivity with the local gauge
group (will need gauge fixing), we make the “big” gauge transformation (last
line) by exponentiating (e.g., (1 + nx )n = ex ).
Now we need to build a Langrangian density term that gives dynamics to the
gauge field Ajµ (x). Recall the commutator [Dµ , Dν ], the curvature of the SU (2)
fibre bundle, which is local gauge invariant, and involves derivatives of Ajµ . This
transforms under local gauge as
σj
[Dµ , Dν ] = igFµνj (300)
2
j j j k
j σ j σ j σ kσ
= ig ∂µ Aν − ∂ν Aµ − ig Aµ , Aν (301)
2 2 2 2
So, in the nonabelian case, the curvature tensor Fµν depends quadratically on
Ajµ , whereas in the abelian case it was linearly dependent. Therefore, the invari-
ant term ∼ F µν Fµν yields cubic and quartic terms in the Lagrangian density,
making an “interacting” theory.
σj
L.G. : Fµνj → V (x)(Fµνj )V † (x). (302)
2
58
By the similarity transformation, we can build an invariant Lorentz scalar with
the trace of the invariant term (Exercise, and note that σ j σ k contributes δjk )
σj σk
1
Tr Fµνj Fµνk = (Fµνj )2 . (303)
2 2 8
The (classical) Langrangian density for the nonabelian gauge theory, invariant
under local gauge and Poincaré transformations is
1
L = Ψ̄(iD/ − m)Ψ − (Fµνj )2 . (304)
4
Note that in the abelian case (theory of QED), the dynamics in the Lagrangian
density of the gauge field were quadratic. Considering only the gauge field term,
with no matter (fermions), essentially results in the wave equation.
2
Fµν = (∂µ Aν − ∂n uAµ )2 (305)
In the nonabelian case, they are cubic and quartic, since there is the commuta-
tor of the gauge fields.
59
12 Lecture 12: Quantization of Gauge Theories
Peskin and Schroeder, page 294
Recall the classical Lagrangian density L = ψ̄(iD/ − m)ψ − 41 Fµνa F µν, a , where a
denotes individual fields, that we crafted to be invariant under the local gauge
symmetry group SU (2). We introduced the “helper” gauge field Aaµ which man-
ifests in the terms of the spacetime curvature tensor , or the curvature of the
a
SU (2) fibre bundle, Fµν = −i[Dµ , Dν ], where Dµ = ∂µ − igAaµ σ2 is the covari-
ant derivative.
We now quantize this gauge theoryto build a quantum theory that is invariant
under the Poincaré and local gauge symmetry groups by finding the correct
representation that has this Lagrangian density as its effective classical limit.
Two problems that arise for the gauge theories are (1) the classical theory (L0 )
is already nonlinear, and (2) there are lots of symmetries, global and local. Lo-
cal symmetries are represented by copies of SU (2) acting independently of each
other at each spacetime location.
60
a list of three numbers. Now, SU (2) acts independently on Aaµ at each point
in M0+1 . This action is tantamount to multiplying by a phase on a sphere S 3
at each spacetime location, since SU (2) is parameterized by four numbers, the
coefficients of the quaternions with norm equal to one. Below is a schematic of
the space of all Aaµ ’s.
61
Schematic of gauge field configurations and equivalence classes represented by
rotations in SO(2).
Note that we only pick one representative per equivalence class, but there can
be more than one depending on the choice of gauge fixing condition, where the
gauge fixing condition if a function that crosses the equivalence class circle more
than once. This is called the Gribov ambiguity, and commonly happens in the
Coulomb gauge.
62
configurations. By applying a gauge fixing condition, we isolate the intersting
part of the integral and count each distinct physical configuration only once.
Finding the right gauge fixing function G allows us to separate out this over-
counting in the path integral and throw it away. Note that we are free to “throw
it away” since we are guessing a quantum theory.
δG(Aα )
Z
1 = Dα δ(G(Aα )) det . (307)
δα
Breaking equation this down, we are performing a path integral over all possible
gauge transformations, and picking out only the G(Aα ) that equals zero, obeying
the gauge fixing condition, choosing a single representative of the equivalence
class. The determinant is called the Faddeev-Popov determinant. The notation
Aα indicates the locally gauge transformed gauge field
1
(Aα )aµ = Aaµ + ∂µ αa + f abc Abµ αc (308)
2
1
= Aµ + D µ α a
a
(309)
g
Where f abc are the structure constants from the Pauli spin matrix commutation
relations
a b
σ σ σc
, = if abc . (310)
2 2 2
This way of writing 1 is the continuous, functional generalization of that for
discrete, many-variable n-dimensional vectors
n Z
Y ∂gj
1= daj δ (n) (g(a))det (311)
j=1
∂ak
63
Inserting the contiuous, functional version of one into the path integral, we now
have an expression that integrates over the equivalence classes but “sucks out”
the overcounting to just one representative of the class
δG(Aα ) iS
Z Z
Dα DADψDψ̄ δ(G(Aα ))det e . (313)
δα
Evaluating the Faddeev-Popov determinant in our choice of the Lorentz gauge
(Exercise)
δG(Aα )
1 µ
det = det ∂ Dµ (314)
δα g
Z R 4
1 µ
= DcDc̄ ei g d x c̄(−∂ Dµ )c (315)
Where, recall from the study of fermions, we have used auxiliary Grassman-
valued, scalar, spin-0 fields c and c̄. These fields are non-physical and must
disappear form the final results: Faddeev-Popov ghosts or ghosts.
Inserting the ghost expression for the determinant into the path integral, we
now have
Z Z Z Z
1
d4 x c̄(−∂ µ Dµ )c)
R
Dα DA DcDc̄ DψDψ̄ δ(∂ µ Aaµ − ω a )ei(S+ g . (316)
Integrate out the delta functional, since ω a is arbitrary, using Ran Gaussian inte-
gral over ω a with coefficient ξ ∈ [0, 1], and calling S 0 = S + g1 d4 x c̄(−∂ µ Dµ )c,
Z Z
0
R 4 1 a 2
Dω e−i d x 2 ξ(ω ) D... δ(∂ µ Aaµ − ω a )eiS (317)
1 a 2 1 1
L0 = ψ̄(iD
/ − m)ψ − (Fµν ) + ξ(∂ µ Aaµ )2 + c̄a (−∂ µ Dµab )cb . (319)
4 2 g
Note that the integral over α blows up to infinity, but in correlation functions
we always have ratios of the path integrals and the N (ξ) · ∞’s will cancel out.
64
• Does the path integral above even define a quantum theory, and is it in-
variant under Poincaré and local gauge symmetry group transformations?
See the work of ‘t Hooft.
• Do the ghosts c and c̄ vanish from the processes?
Feynman diagrams will clear up this concern.
• Is the Lagrangian density (theory) L0 renormalizable?
Also see the work of ‘t Hooft.
65
13 Lecture 13: Quantization of Nonabelian Gauge
Theory
In the path integral quantization of gauge theories, we started by guessing a
quantum theory by integrating the action functional over fermion fields ψ, ψ̄
and gauge boson fields A to calculate transition amplitudes.
1 1
L = − (∂µ Aaν − ∂ν Aaµ )2 + ξ(∂ µ Aaµ )2 + ψ̄(i∂/ − m)ψ + c̄a (−∂ µ ∂µ )ca + Lint (g).
4 2
(321)
Note that the interacting bits, including the bits from the covariant derivative
Dµ are absorbed into Lint (g), and everything else in the expression above is the
free theory with partial derivatives ∂µ .
We now push forward under the belief that the perturbatively defined theory
above is representative of a quantum theory, and expand in powers of the in-
teraction term g to define processes and Feynman rules of the Yang-Mills
theory.
d4 k
Z
ˆ i
hψ̄jα (x)ψ̄lβ (y)i = δjl e−ik·(x−y) . (322)
(2π)4 k/ − m αβ
The gauge boson propagator contributes
66
d4 k e−ik·(x−y)
Z
kµ kν
hÂaµ (x)Âbν (y)i = ηµν − (1 − ξ) 2 δab . (323)
(2π)4 k k 2 + i
d4 k i
Z
hĉa (x)c̄ˆb (y)i = δab e−ik·(x−y) . (324)
(2π)4 k 2
Recall that a, b = 1, 2, 3 and µ, ν = 0, 1, 2, 3.
−ig(f abc f cde (η µρ η νσ −η µσ η νρ )+f ace f bde (η µν η ρσ −η µσ η νρ )+f ade f bce (η µν η ρσ −η µρ η νσ )).
(327)
The interaction of one gauge boson with two ghosts contributes
gf abc pµ . (328)
(Exercise) And the rest of the Feynman rules for Yang-Mills theory: symmetry
factors, signs, and conservations laws.
Big Question 1
Can we interpret this as a quantum theory? In other words, does this theory
implicitly define the Hermitian operators Âα , ψ̂, and ĉ?
In more detail, what are the possible modes of failure for the theory to not be
a valid quantum theory?
The first mode of failure that can occur is the time evolution operator not be-
ing unitary, or time translation not being a unitary process. To confirm that
time translation in this theory is a unitary process, check that the correlation
function is symmetric under the group of Poincaré transformation (Exercise).
67
This also implies that probability is conserved.
The second mode of failure occurs if, after building the Fock space, we get neg-
ative norm states, such that the inner product of the system eigenstates is less
than zero hΨ|Ψi < 0. If this happens, then the Hamiltonian is not positive
definite and we do not have a proper Hilbert space.
Note that there is a type of exception to this rule, which is a topic of current
research. Negative states may be able to be modded out by their subspaces to
produce an effective quantum theory from the remaining subspaces with positive
inner products. These remaining states are the physical, operationally defined
states, and they form a convex cone from an operator algebra. The inner prod-
uct in this state subspace forms a “Hilbert subspace”, where hΨphys |Ψphys i > 0.
Big Question 2
Is this theory, as a quantum theory, renormalizable?
Note that different cutoffs can reveal different hypotheses about reality, and it
is generally believed that physics is Lorentz and Poincaré invariant. So, a good
cutoff should retain these invariances while also taming infinities of the Feyn-
man diagram integral contributions.
dd k
Z
1
In (m) = (329)
(2π) (k − m2 + i)n
d 2
Note that in lower dimensions, these divergences are easier to handle. For ex-
ample, if d = 2, then n = 1 tames the infinity. If d = 3, n must be greater than
68
one to tame infinities of the cutoff in just three dimensions. In four dimensions,
such as our usual spacetime, n ≥ 2 is required to make the proper cancellations
(e.g., need more undetermined momenta).
Now to evaluate that integral, we separately write out √ the time component
integral and note that there are two poles at k0 = ±( k 2 + m2 − i), where k
is just the spatial components of the k-vector (e.g., three spatial components of
spacetime), and the i is Taylor expanded out of the square root,
Z d−1 Z ∞
d k 1
In (m) = dk 0 . (330)
(2π)d −∞ (k02 − k 2 − m2 + i)n
Now, rotate the contour from running along the real axis <(k0 ) to the imagi-
nary axis =(k0 ), effectively changing variables k0 → ik0 . Avoiding the poles,
everything in the region is analytic, meromorphic and we can do this rotation.
69
dd k
Z
1
In (m) = −i d (k 2 + m2 )n
(332)
Euclidean (2π)
Z ∞
rd−1
Z
i
=− d
dΩ d dr 2 (333)
(2π) 0 (r + m2 )n
d
i (2π) 2 ∞ rd−1
Z
In (m) = − dr . (334)
(2π)d Γ( d2 ) 0 (r2 + m2 )n
m2
Make the change of variables x = r 2 +m2
1
md−2n
Z
d d
In (m) = d dx xn− 2 −1 (1 − x) 2 −1 . (335)
(2π) 2 Γ( d2 ) 0
m−
I2 (m) = Γ (337)
(2π)2− 2 2
1 − log(m) 2
= − γ + O(2 ) (338)
4π 2 (1 − 2 log(2π))
Where γ is the Euler-Mascheroni constant. This diverges as → 0, as expected,
with the “bad bit” of 2 .
If we compare this with an integral that we’ve seen before, we can seee what
kind of cutoff the free parameter is. Recall the diagram where we applied the
momentum cutoff |k| < Λ. So, 1 ∼ Λ.
70
14 Lecture 14: Quantization of Nonabelian Gauge
Theory, Cont.
Working with nonabelian gauge theories, we’ve written down a Lagrangian
density for Yang-mills theory, and after the gauge-fixing procedure of the La-
grangian density via path integrals and the tricks of Faddeev and Popov, we are
ready to do some calculations.
The first topic to discuss is the beta function or renormalization group equation,
which tells us how theories behave at low and high energies. The second topic
to discuss is the departure from path integrals and gauge fixing to methods of
lattice regulators, or cutoffs, for doing calculations in nonabelian gauge theory.
Recall that a quantum theory Ĥ(z1 , . . . , zn ; Λ), where Λ is the cutoff, data
defined in a list of all the degrees of freedom of the theory, is renormalizable is
it leads to finite predictions for all operationally well-defined observables. The
expectation values of these observables must produce the same predictions for
different choices of the chosen cutoff
Apply the above equation to the Green’s function, n-point correlation functions,
where the dependency of the coupling constants and cutoff are implicit to the
vacuum state |Ωi and the dynamics of the field operators
71
quantities.
Now, to compute the relationship of G(n) to the coupling constants and the cut-
off, consider changing the (usually) continuous parameter Kc by an infinitesimal
amount. Note that a common choice of cutoff could be Kc = |pmax |.
∂G(n) ∂G(n)
dG(n) = δKc + δzj . (343)
∂Kc ∂zj
So, the coupling constants zj (Kc ) are chosen to fix G(n) with respect to trans-
formations of the cutoff of the form Kc → Kc +δKc , but G(n) should not depend
on Kc , and the above expression is equal to zero
To derive the renormalization group equation or the beta function, set the dif-
ferential dG(n) to zero, divide by δKc , and multiply by Kc
∂ dzj ∂
Kc + Kc G(n) = 0 (344)
∂Kc dKc ∂zj
∂ ∂
Kc + β(zj ) G(n) = 0. (345)
∂Kc ∂zj
This is the infinitesimal form of the statement for a renormalizable theory that
the Green’s function shouldn’t depend on the cutoff as it is changed, where
dzj dzj
β(zj ) = Kc = (346)
dKc d lnKc
Is the renormalization group or beta function, which allows us to compute the
coupling constants in terms of the cutoff zj = zj (Kc ). The behavior of β(zj (Kc ))
is often used to describe the dependence of zj on Kc .
72
Examples of Beta Functions
(1) In φ4 theory, there is one coupling constant λ, the interaction strength, and
the beta function is
3λ2
β(λ) = + O(λ3 ). (347)
16π 2
(2) In quantum electrodynamics (QED), which is an abelian gauge theory, to
first loop order, the beta function is
e3
β(e) = + O(e4 )t (348)
12π 2
Where e is the electric charge of the particle.
To tame the infinities that arise from these diagrams, apply dimensional reg-
ularization to maintain Lorentz invariance and insist that each diagram leads
to finite quantities. This then tells us what kinds of counter terms we need to
add to the Lagrangian density for our theory, and, in turn, how the coupling
constants change with respect to the cutoff.
This leads to the beta function for the SU (N ) local gauge group
g3
11N 2nf
β(g) = − − (349)
16π 2 3 3
73
Where nf is the number of fermion families. Note that if nf is small, β(g)
becomes negative, and that Kc → ∞ as g gets smaller, meaning that our theory
approaches being a free theory as g → 0, and we can use perturbation theory for
high-energy processes. This is called asymptotic freedom of nonabelian gauge
theories.
Wilson recognized that the parallel transporter is the object that allowed us to
do derivatives in the first place, and treated the parallel transporter
74
These parallel transporters are 2 × 2 unitary matrices which populate the list
of degrees of freedom, one for each link, or edge, in the classical lattice.
With this action, Rbuild the path integral DU e−iS[U ] , and Wick rotate into
R
the path integral DU e−S[U ] to work in imaginary time. Lastly, Monte Carlo
sample the path integral.
This approach is actually the best way to get nonperturbative results in quan-
tum field theory, but has its downsides:
Downside (1) is calculating processes in imaginary time is just like doing sta-
tistical mechanics with gauge theories at some defined temperature, and this is
not good for time-ordered processes (e.g., scattering). Downside (2) is that this
is not a quantum theory, as the Wilson loop is a classical configuration.
75
a Hamiltonian and Hilbert space in which the degrees of freedom live on a lat-
tice, which they argued yields Yang-Mills theory as the lattice spacing goes to
zero. This is not proven, but if you can prove it, as well as that the low-energy
limit has a mass cap, you can get a cool $1M!
The total Hilbert space of this quantum theory is a tensor product over all the
edges e in the lattice E of individual Hilbert spaces he
Z Z
LU |ψi ≡ dV ψ(V ) |U V i and RU |ψi ≡ dV ψ(V ) |V U † i . (357)
These operators are an analog of the shift operator eixp̂ . Note that |U V i is still
a member of SU (2). Differentiating these operators with respect to U will yield
momentum operators: dynamics.
76
15 Lecture 15: Hamiltonian Lattice Gauge The-
ory
Continuing on the introduction to Hamiltonian lattice gauge theory as a means
of quantization of gauge fields, we will build a microscopic formulation of gauge
theory based on the real-space lattice. In contrast to the usual way of work-
ing on the Euclidean, Wick-rotated lattices, we will begin our theory with a
Hamiltonian of classical degrees of freedom: namely, the parallel transporter
U , a 2 × 2 matrix with determinant one, such that U ∈ SU (2). Since we
are working in 4D spacetime, we will have a 4D discretized lattice with lattice
spacing a ∝ K1c . In other words, 4D spacetime is discretized up to the cutoff Kc .
Like introduced before, each link, or edge, of the lattice e ∈ E, where E is the
set of all links of the lattice, has an associated parallel transporter, correspond-
ing to the shortest, rectilinear path in between each vertex, Ue ∈ SU (2) ' S 3 .
Note that the parallel transporter is not a local object, as it is path dependent,
implicitly depending on more than one coordinate.
The classical configuration space for this lattice gauge theory is a Cartesian
product of SU (2) per link in the lattice
Recall the two operators defined in SU (2), for unitary matrix with unit deter-
minant U ∈ SU (2), that defined a right- and left-acting transformation
Where LU and RU commute, such that [LU , RU ] = 0, and form the representa-
tion defined by the relations
L†U LU = RU
†
RU = I and LU V = LU LV . (361)
To understand how the infinite-dimensional Hilbert space L2 (SU (2)) breaks up
into a direct sum of irreducible representations of SU (2), we invoke the third
77
part of the Peter-Weyl therorem, which states that the Hilbert space over SU (2)
consisting of square-integrable functions may be regarded as a representation of
a direct product of left- and right-acting operators, and the Hilbert space de-
composes into an orthogonal direct sum of all the irreducible unitary represen-
tations, with multiplicity of each irreducible representation equal to its degree,
the dimension of the underlying space of that representation (See Wikipedia
page on Peter-Weyl theorem for overview). We write this all as
M M
he ≡ L2 (SU (2)) ' Vl ⊗ Vl∗ ' C2l+1 ⊗ C2l+1 (362)
l∈ 21 Z+ l∈ 12 Z+
Where Vl ' C2l+1 is the (2l + 1)-dimensional vector space furnishing the irre-
ducible representation of SU (2) of spin, or angular momentum l and 21 Z+ =
{0, 12 , 1, 32 , 2, . . . }.
Now to find a representation of SU (2) on this vector space Vl , we’ll use a piece
of Lie group representation theory not found in any textbook. Note that the
action of SU (2) generates a representation
Begin the procedure to get the matrix representation for any spin-l, take the
tensor product of n of the spin- 21 fundamental vector spaces
V 21 ⊗ V 12 ⊗ · · · ⊗ V 12 . (364)
Note that for quantum computer fans, this is the vector space of n qubits
n
V 21 ⊗ V 21 ⊗ · · · ⊗ V 12 ' C2 ⊗ · · · ⊗ C2 = C 2 . (365)
The spin-l representation Πl (U ), generated from SU (2) action on Vl , lives in thsi
tensor product space of n copies of V 21 , as long as n = 2l or l = n2 . Therefore,
we will build Vl as a subspace of this tensor product space, most fo which will
be thrown away once we find our subspace of interest, by building a set a n + 1
orthonormal vectors
78
|w n2 i = |11 . . . 1i (366)
1
|w n2 −1 i = √ (|11 . . . 10i + |11 . . . 101i + · · · + |01 . . . 1i) (367)
n
... (368)
1
|w n2 −k i = q (|1 . . . 10 . . . 0i + (all permutations of k zeros and n − k ones))
n
k
(369)
... (370)
|w− n2 i = |00 . . . 0i . (371)
Then the matrix elements of the representation Πl (U ) are simply given by the
expectation value on n copies of U in this orthonormal basis
Note that this method is good and fast for low spin representations, but clearly
gets unwieldy for working with the Hilbert space of n = 1000 qubits, and the effi-
ciency and value of the methods of addition of angular momentum and Clebsch-
Gordan coefficients, with the raising and lower operators, the highest-weight
vectors, etc.
Sticking to low spin representations for our purposes, we can start to extract
matrix representations for l = 0, l = 12 , and l = 1.
Π0 (U ) = I. (373)
1
For l = 2, we have the fundamental matrix representation
79
[Π1 (U )]11 = h11| U ⊗ U |11i = h1| U |1i h1| U |1i = a2 (376)
1
[Π1 (U )]00 = (h10| U ⊗ U |10i + h10| U ⊗ U |01i + h01| U ⊗ U |10i + h01| U ⊗ U |01i)
2
(377)
= ad + bc. (378)
Recall that the Peter-Weyl theorem tells us that the Hilbert space of square-
integrable functions on SU (2) is isomorphic to the infinite-dimensional, until we
truncate, direct sum space of (2l + 1)-dimensional vector spaces furnishing the
representation of SU (2)
M
L2 (SU (2)) ' C2l+1 ⊗ C2l+1 . (379)
l∈ 21 Z+
And SU (2) acts on L2 (SU (2)) via the operator LU : L2 (SU (2)) → L2 (SU (2))
with action
M
LU ' Πl (U ) ⊗ I. (380)
l∈ 12 Z+
The elements of the matrix representation [Πl (U )]jk ≡ tljk (U ) are square-
integrable functions from SU (2) to the complex numbers C, since SU (2) is
compact, and form an orthogonal (not orthonormal) basis for L2 (SU (2)), where
−l ≤ j, k ≤ l. So, we can expand the wavefunction ket in the orthogonal basis
of the matrix elements
XX
l
XX
l
√
|ψi = ψjk |jil |kil = ψjk 2l + 1 |tljk i . (381)
l j,k l j,k
2
The inner
R product of the Hilbert space L (SU (2)) is defined with the Haar
measure dU by
Z
(ψ, φ) ≡ dU ψ ∗ (U )φ(U ) (382)
X X X
|Ψi = Ψlj11lj22...
...k1 k2 ... |j1 il1 |k1 il1 |j2 il2 |k2 il2 . . . . (384)
l1 l2 ... j1 j2 ... k1 k2 ...
80
To give dynamics to the Hilbert space, we define some observables. Consider
j
an element of SU (2), U = ecj τ , where cj τ j are elements of the Lie algebra of
3 j
SU (2), cj ∈ R , and τ are the Pauli spin matrices multiplied by i and divided
by 2 for normalization conditions. The spin matrices obey the commuation
relations [τ j , τ k ] = −2jkl τ l . We can recover the spin matrix from the group
element via differentation
dU
= τj. (385)
dcj cj =0
ˆlj ≡ d L
. (386)
j
sτ
L
ds U =e s=0
Note that this is a factor of i away from being Hermitian, and is analogous to
the Hermitian linear momentum operator ip̂. Also, notice that we begin with
d
a unitary operator LU , apply an anti-Hermitian operator ds , and kill off the
unitarity by setting s = 0 after the derivative.
The second observable we define, similar to the first, is the right angular mo-
mentum operator
ˆlj ≡ d R
. (387)
sτ j
R
ds U =e s=0
Next, define the position observable Ûjk , which is also a map L2 (SU (2))rightarrowL2 (SU (2))
and yields the fundamental representation matrix elements when it acts on the
position eigenkets |U i of SU (2).
Z Z
l= 1
Ûjk |ψi = dU ψ(U )[Π 12 (U )]jk |U i = dU ψ(U )tjk 2 (U ) |U i . (389)
To build the Hamiltonian, kinetic energy plus potential energy, we use the pla-
quette operator to define parallel transport on the 4D lattice
We choose the convention to set arrows on each link running “left-to-right” and
“down-to-up” in the plane of the “paper”, and then walk around the plaquette
81
counterclockwise (CCW), taking the Hermitian conjugate of the parallel trans-
porter if we are traveling against the arrow. The parallel transporter for each
link is considered the observable for that link when traveling around the pla-
quette. Each link are labeled by ei , i = 1, 2, 3, 4.
Build an operator that acts on the 4D space of links, as a sum over the links of
the direct products of parallel transporters
This is then turned into an observable by summing over the last, initially-fixed
index k4
X
tr(Û ) ≡ M̂;k4 ,k4 . (391)
k4
Note that tr(Û ) is a trace operator that acts on L2 (SU (2)) ⊗ L2 (SU (2)) ⊗
L2 (SU (2)) ⊗ L2 (SU (2)), and not a number!
2 X
gH ˆlj (e)ˆlj (e)+ 1
X
ĤKS = − L L 2 a tr(Û ) + Hermitian conjugate
2a 2gH
e∈E plaquettes
(392)
Where gH is the coupling constant.
82
Mx ≡ Rx (e1 ) ⊗ Rx (e2 ) ⊗ Lx (e3 ) ⊗ Lx (e4 ) (393)
Where x ∈ SU (2). These operators obey the following commutation relations
[Mx (v), My (w)] = 0 and [Mx (v), ĤK S] = 0, for all x, y, v, w. (394)
In summary, Wilson’s formulation received much more attention at the time for
its ease of discretizstion, translation to computer programs, and use of Monte
Carlo sampling, which classical computers are good at. Kogut and Susskind
argued that the plaquette operator becomes the curvature term, as expected,
in the small lattice spacing a limit in their theory, as well as that hte kinetic
energy term becomes the kinetic energy in the timelike direction of the curvature
term. The Kogut-Susskind formulation may be very promising for quantum
simulations done by quantum computers, which are not very good at sampling
techniques, but excel in simulating the dynamics of local lattice models.
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16 Lecture 16: Spontaneous Symmetry Break-
ing
This is the last topic that we need to complete our description of the stan-
dard model of elementary particle interactions. Spontaneous symmetry breaking
(SSB) is an observed behavior within field theories, and we will begin with some
examples of classical SSB.
p2
H= + V (x). (395)
2m
This system posses Z2 symmetry in its solutions, since x → −x is a symmetry
operation. Note that the particle in this potential at x = 0 must choose a pos-
itive or negative local minimum, meaning that the ground state is degenerate
and breaks the Z2 symmetry.
The quantum analog of this classical theory does not exhibit SSB, since the
ground state wavefunction is symmetric, but the first excited state is antisym-
metric.
84
Symmetric ground state wavefunction.
This theory contains two ground states: all spins pointing up or all pointing
down, and possesses Z2 symmetry by the operation sj → −sj .
Depending on the temperature, the thermal state of this system can be in one
of two regimes: critical or non-critical. Consider the Gibbs, or mixed, state
density operator describing the system at any temperature
e−βH
ρ= (397)
Z
Where Z is the partition function and β = kB1T is the temperature factor.
Note that the Gibbs state is all Z2 symmetric, meaning that it does break any
symmetries. Now there exists a critical temperature βc , below which the system
becomes ordered due to small external magnetic fluctuations and symmetry is
broken. Above the critical temperature, the system is disordered with random
thermal fluctuations. As β → ∞,
85
1 1
ρ= − (all up states) + − (all down states). (398)
2 2
∂V (φ) λ
= −µ2 φ + φ3 = 0 (401)
∂φ 6
Yields three configurations
q that extremize the energy of the system. Namely,
6µ2
for φ = 0 and φ = ± λ . The quantum analog of this theory also exhibits
SSB.
Where the first term is the neighboring interaction and the second term is the
magnetic interaction with h as the magnetic field strength, and the Pauli spin
matrices
x 0 1 z 1 0
σ = and σ = . (403)
1 0 0 −1
The Hamiltonian exhibits Z2 symmetry, since
86
|Ω+ i ≡ |+i ⊗ |+i ⊗ · · · ⊗ |+i (405)
|Ω− i ≡ |−i ⊗ |−i ⊗ . . . |−i (406)
O : φj → [O]jk φk . (409)
What is the lowest energy configuration? Minimize the potential energy V (φj ) =
λ
− 12 µ2 (φj )2 + 4! (φj )4 with respect to φj , showing that the minimum occurs for
any constant configuration of the fields that satisfies the equation (Exercise)
µ2
(φj0 )2 = . (410)
λ
There is more than one energy configuration that leads to this solution, including
superpositions of the following vectors
µ
√
λ
0
0 õ
λ
0 , 0 ,.... (411)
. . . . . .
0 0
In the case of N = 2, the potential energy minima form the “wine bottle” or
“mexican hat” potential. The minimum energy configurations are points on this
potential and form circles around the indent of the “bottom of the bottle”. In
87
other words, any configuration that lands on the circle is a minimum energy
configuration.
Small fluctuations in the energy behave much like the harmonic oscillator.
88
0 0
j
0 0
φ0 =
. . . = . . .
(412)
ν √µ
λ
And see how this behaves with small energy fluctuations. Define shifted fields
in terms of some new coordinates where the vector of fields φ is now defined as
1 1
L = (∂µ π k )2 + (∂µ σ)2 (414)
2 2
√ √
1 λ λ λ
− 2µ2 σ 2 − λµσ 3 − λµ(π k )2 σ − σ 4 − (π k )2 σ 2 − (π k )4 .
2 4 2 4
(415)
There are N − 1 massless π k fields and one massive σ field. The second and
third terms in L correspond to an effective massive Klein-Gordon scalar field.
The N − 1 π k fields are effectively massless, as all of the other terms above
contain λ and are interaction terms.
Goldstone’s Theorem
In the O(N ) linear σ-model there are N2 independent continuous symmetries,
the dimension of the rotation group O(N ). After SSB, there are N 2−1 remain-
which is also the number of massless fields. In other words, each broken sym-
metry causes a massless excitation: the Goldstone modes or Goldstone bosons.
Proof:
89
Consider a classical field theory with fields φa (x), a = 1, 2, . . . , and the general
Lagrangian density L = (derivatives) − V (φa ).
Let φa0 be a constant (in an extrema) field that minimizes the potential such
that
∂V (φa )
= 0. (416)
∂φa φa =φa0
Then expand the potential, a function of the vector of fields φa0 ≡ φ0 , near the
minima
2
a a 1 a a b b ∂ V
V (φ ) = V (φ0 ) + (φ − φ0 )(φ − φ0 ) . (417)
2 ∂φa ∂φb φ=φ0
Call the Hessian matrix [m2 ]ab , which is symmetric and real and the eigenvalues
give the masses of the effective fields.
Where the eigenvalues correspond to the masses of the π particles, and we must
now show that there are eigenvalues equal to zero. In other words, every con-
tinuous symmetry leads to an eigenvalue equal to zero.
90
X
∆a (φa0 )[m2 ]ab = 0 (423)
a
Where a ∆a (φa0 ) = ∆T is the zero eigenvector, where the ∆a (φa0 ) are linearly
P
independent for each continuous symmetry, which follows by definition of the
general, global, continuous symmetry imposed above.
91
17 Lecture 17: The Higgs Mechanism
Here we discuss what happens when Goldstone’s theorem related to spontaneous
symmetry breaking (SSB) and local gauge invariance are combined.
These (massive) gauge bosons are observed in experiment, and we are inclined
to add a bosonic mass term to this theory. Naively, we can add a term of the
form 12 m2 Ajµ (x)Ajµ (x). Unfortunately, this will break local gauge invariance
since it does not obey the local gauge transformation like the gauge boson fields
do, e.g., as Ajµ (x) → Ajµ (x) − 1e ∂µ α(x).
Abelian case
We consider the abelian case for the SU (2) gauge theory combined with SSB
as the “toy” model for such a theory. Consider a theory of a complex scalar
field φ = φR + iφI (which can also be regarded as a doublet) coupled to a U (1)
gauge field Aµ
1
L = |Dµ φ|2 − Fµν F µν − V (φ) (425)
4
Where we have the covariant derivative Dµ = ∂µ + ieAµ obeying the local gauge
condition.
1
Aµ (x) → Aµ (x) − ∂µ α(x) (426)
e
iα(x)
φ(x) → e φ(x). (427)
92
For the interacting part of the theory, consider the potential
λ ∗ 2 2
V (φ) = −µ2 φ∗ φ + (φ φ) , µ > 0. (428)
2
Note that this potential is quartic in φ, leaving no hope for solving this model
straight away. To find a solution, we follow the same steps as for SSB: (1) Find
a field value φ = φ0 that minimizes the potential V (φ), and (2) expand the
Lagrangian density around the minima and study small fluctuations in the field.
µ4
V (φ) = − + µ2 φ21 + · · · + O(φ3j ) (431)
2λ
Where the . . . account for all of the other terms form substituting φ(x), includ-
ing φ2 terms and cross-terms of φ1 and φ2 . Now expand L around the minima
φ0 to get
1 1
L= |∂µ φ1 |2 + |∂µ φ2 |2 − µ2 φ21 + · · · + O(φ3j ). (432)
2 2
Note that other terms exist here, such as cross-terms, indicated by . . . , and
that we alos have shifted away the constant terms in L. The first two terms are
effectively massless, while the third term is effectively massive, since it contains
µ2 . Other terms from the covariant derivative term include
1 1 √
|Dµ φ|2 = |∂µ φ1 |2 + |∂µ φ2 |2 + 2eφ0 Aµ ∂ µ φ2 + e2 φ20 Aµ Aµ + . . . . (433)
2 2
We interpret the last term above as the bosonic mass term
1 2
∆L = e2 φ20 Aµ Aµ ≡ m Aµ Aµ . (434)
2 A
This result is in the classical, low-energy limit, since we studied only small fluc-
tuations of the field.
93
To outline the quantization of this theory, we use the path integral approach, al-
though lattice quantization also works for this theory. So, break the Lagrangian
density into free and interacting parts L = L0 + Lint , where
1 1 1
L0 = |∂µ φ1 |2 + |∂µ φ2 |2 − µ2 φ21 − Fµν F µν (435)
2 2 4
And Lint is the rest of the terms: perturbations leading to vertices in the Feyn-
man rules. For example, in momentum space, one of the vertices looks like
√
= 2ieφ0 (−ik µ ) ≡ mA k µ (436)
At the quantum level, the mass of the gauge boson fields manifest in the poles
of the Green’s functions. For example,
i
= im2A g µν + (mA k µ ) (mA k ν ) ∼ im2A (437)
k2
Nonabelian case
For the nonabelian case of SSB in SU (2) gauge theory, consider
a general,
φ1
continuous gauge group G and a set of scalar fields φ = ... that transform
φd
under the gauge group G as
94
a a
π(g) = eiα t
(439)
And the {ta } are purely imaginary matrices from the associated Lie algebra
that depend on the representation.
τ 1 = iσ x , τ 2 = iσ y , τ 3 = iσ z . (446)
Then the fields (eigenvectors) in this basis are
φ1 1
φ= and φ0 = . (447)
φ2 0
The action of the generators τ a on the minima, the vacuum state, are (Exercise)
95
1 0
τ φ0 = i (448)
1
0
τ 2 φ0 = − (449)
1
1
τ 3 φ0 = i (450)
0
τ a φ0 = 0 (452)
Then there will be zero mass terms (zero eigenvalues) in the matrix elements,
indicating that initially massless bosons become massive when symmetry is bro-
ken! Specifically in the theory of electroweak interations, these particles are the
W ± and Z bosons.
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18 Lecture 18: SSB with Gauge Theories and
Next Steps
Here we complete our discussion of SSB in the context of gauge theories (e.g.,
the Higgs mechanism), and give a broad overview of open questions and where
to go next.
Recall from the previous lecture we began workingwith a local gauge theory of
φ1
bosons with a tuple of independent scalar fields ... . Each field transforms
φd
according to the local gauge symmetry
Aaµ Abµ = Aa0 Ab0 − Aa1 Ab1 − Aa2 Ab2 − Aa3 Ab3 . (457)
97
The only non-negative term is the zeroth, timelike, longitudinal element, and
will cause the entire term to not look like a mass term. The three spatial de-
grees of freedom in this expression, if m2ab is positive semidefinite, will lead to a
mass term in the Lagrangian density L, since mass terms have a minus sign in L.
The longitudinal term does not look right and can be cancelled off since, for
the photon, momentum only has transverse components: no longitudinal com-
ponents. Consider the vacuum polarization diagram like we sketched in the
previous lecture for the Abelian Higgs model.
This diagram contributes the purely transverse term, with longitudinal contri-
butions cancelled off,
kµ kν
im2ab η µν − 2 . (458)
k
In the previous lecture, we worked out the fundamental representation of SU (2)
for the gauge theory. In this representation, work out the kinetic energy ∼
|Dµ φ|2 and find that
98
a a β
φ → eiα τ
ei 2 φ (460)
a
Where τ a = σ2 are the Pauli spin matrices from the SU (2) (gauge boson) part
of the group, and β is a scalar from the U (1) (photon) part of the group. Think
of the action of U (1) as giving charge to the field φ.
Now add the potential V (φ) to the Lagrangian density and minimize the poten-
tial such that
1 0
φ = φ0 + φ1 , where φ0 = √ (461)
2 ν.
Expand the covariant derivative around the minima φ0 , and in the fundamental
representation for this gauge group SU (2) × U (1), we get (Exercise)
i
Dµ φ = (∂µ − igAaµ τ a − g 0 Bµ )φ (462)
2
a
Aµ SU (2)
With the connection gauge field ∼ . Putting everything to-
Bµ U (1)
gether into a locally gauge invariant Lagrangian density
1
LGWS = |Dµ φ|2 − Fµν F µν + · · · + fermion term + · · · + V (φ). (463)
4
The fermion term will be discussed below.
Mass Generation
Due to the potential having a minimum configuration, we can generate mass
in this gauge theory to endow gauge bosons with mass. We want to do this,
because massive bosons are experimentally observed. Expand the Lagrangian
density around the minima φ0 and acquire a term from the kinetic energy term
that looks like (Exercise)
ν2 2 1 2
g (Aµ ) + g 0 (A2µ )2 + (igA3µ + g 0 Bµ )2 .
“mass term” ≡ (464)
8
So, A1µ ,A2µ , and the combination of fields A3µ /Bµ each act like they have a mass,
interpreted as massive gauge bosons: three degrees of freedom from SU (2). The
fourth degree of freedom is missing from this mass term, and we, therefore, con-
clude that it does not have a mass, and we interpet this as the photon: one
degree of freedom from U (1).
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1
Wµ± = √ A1µ ∓ iA2µ
(465)
2
1
Zµ0 = p gA3µ − g 0 Bµ
(466)
2
g +g 02
With masses
1
mW = gν (467)
2
1p 2
mZ = g + g 02 ν. (468)
2
The fourth field (degree of freedom), with zero mass, we call
1
Aµ = p (g 0 A3µ + gBµ ). (469)
g2 + g 02
Coupling to Fermions
Gauge bosons couple differently to left-handed and right-handed fermions. Re-
call that the chirality of fermions arose in the development of gamma γ matrices
and Weyl spinors and the kinetic energy term decouples into
/ = ψ̄L (i∂)ψ
ψ̄(i∂)ψ / L + ψ̄R (i∂)ψ
/ R. (470)
Similarly, in the gauge theory
/ = ψ̄L (iD)ψ
ψ̄(iD)ψ / L + ψ̄R (iD)ψ
/ R. (471)
To recognize this difference in chirality and coupling of gauge bosons to fermions,
recall that we have the choice of representation of the gauge group SU (2), and
the left- and right-handed fermions can be separated into different representa-
tions.
100
Next Steps
There are three ways to go from here:
Physics
• Supersymmetry (SUSY)
Upgrading symmetries of groups (e.g., Poincaré → Super Poincaré).
• Quantum gravity via field theory tools (e.g., superstring theory, loop quan-
tum gravity).
Superstring theory offers a correct effective theory to explain the high
energy physics of black holes.
• Linear quantum gravity
Does not lead to strings.
Non-renormalizable theory.
Treats the metric g µν = η µν + δg µν as the degree of freedom.
• Linear quantum gravity with the Standard Model
Only works up to some cutoff Λ, making an effective, low energy
theory of gravity.
Best theory we have to date for describing all experiments, but we also
believe that it is not fundamental, since black holes exist, which conflict
with the imposed cutoff, and this theory fails at high energy predictions.
• AdS/CFT Correspondence
Exaplins quantum gravity with field theory alone, no strings.
It states that a strongly interacting conformal quantum field theory
(CFT) on the boundary of Anti-de Sitter spacetime (AdS) is dual to a
quantum gravity theory of Anti-de Sitter spacetime (solutions to Einstein’s
equations) in its semiclassical limit, which implies that quantum gravity
is itself a quantum field theory.
We’d prefer to develop this theory in de Sitter spacetime, as that is
the spacetime that we find ourselves in (see the work of Strominger).
• Quantum information theory is becoming helpful to study the kinematics
of quantum systems (e.g., photon entanglement).
101
Mathematics
There are many rigorous formulations of quantum field theory:
• Axiomatic QFT,
• Constructive QFT,
• Functional integration, expansions, and probabilistic approaches,
• Vertex operator algebras,
• Chiral & factorization algebras,
• Topological QFT (TQFT) & n-categories.
Why is QFT so difficult to make rigorous?
This is largely in part due to perturbation theory working so darn well, and
it is used as the only standard tool in QFT at large, but it is wrong, and we
know why it is wrong. For example, consider a Gaussian R ∞path integral in a
2 4
(0 + 1)-dimensional QFT with a quartic interaction Z = −∞ dx e−x −gx . As
prescribed, we assume that g is small, do perturbation theory, get lots of terms,
and end up with a zero radius of convergence! The interaction term does im-
prove convergence, but perturbation
R theory can’t see that. Even after Wick
rotation of the path integral Z = Dφ e−S , which also improves convergence,
perturbation theory can’t calculate a finitie value for the path integral. Also
note that for spacetime dimensions d > 6, all QFTs are trivial (e.g., Gaussians).
• Axiomatic QFT
Wightman: fields are distribution-valued objects (unbounded opera-
tors) acting on a Hilbert space.
Hang-Kastler: C ∗ -algebras.
Osterwalter-Schrader: statistical mechanical foundation (Wick-rotated).
Reconstruction theorems: With all n-point correlation functions well-
behaved, we can reconstruct full Hilbert space and the unitary represen-
tations of the Poincaré symmetry group.
There are problems with local gauge theories (current research).
• Constructive QFT
Cluster expansions takes the Wick-rotated, Euclidean path integral
and trade off the low energy perturbation series expansion for estimates
of large values of the degrees of freedom that suppress bad parts of the
series to get finite results.
Successful in (1 + 1)- and (2 + 1)-dimensional spacetime, as well as
local gauge theory, but it is very intricate and has become very difficult
to communicate and check results.
102
• Algebraic QFT
Start with some axioms and abstract what the observables should be
from there.
Quantifies locality in the algebra, leading to observable (C ∗ ) algebras.
There is difficulty in finding the states (n-point function) to match
the C ∗ algebras.
• Functional integration, expansions, and probabilistic approaches
Nelson’s axioms are stronger than the Osterwalter-Schrader axioms.
• Vertex operator algebras
Very successful with conformal field theories (CFTs), but is stuck in
(1 + 1) dimensions.
• TQFT & n-categories
Exactly solvable, strongly interacting theories built on n-categories.
103