Quantum Field Theory I
Examination questions will be composed from those below and from questions in
the textbook and previous exams
III. Quantization of constrained systems and Maxwell’s theory
1. The energy-momentum tensor for the free electromagnetic field
Consider the free electromagnetic field with the Lagrangian (density)
1
L = − Fµν F µν ,
4
Fµν = ∂µ Aν − ∂ν Aµ .
(a) Prove the Bianchi identities
∂µ Fνρ + ∂ν Fρµ + ∂ρ Fµν = 0 .
(b) Derive Maxwell’s equations from L treating Aµ as the dynamical variables.
(c) Solve the equation of motion for A0 assuming that Aµ decrease fast enough at space
infinity.
(d) Write the Bianchi identities and the equations in the standard form by identifying
E i = −F 0i and ijk B k = −F ij .
(e) The canonical energy-momentum (or stress-energy) tensor of a field theory with
Lagrangian L = L(φa , ∂µ φa ) is defined as
Tµ ν =
∂L
∂µ φa − δµν L .
∂∂ν φa
Construct the canonical energy-momentum tensor for the free electromagnetic field.
(f) Compute Tµν − Tνµ .
(g) The canonical energy-momentum tensor is not defined uniquely. One can always
modify the tensor in the following way
Tµ ν = Tµ ν + ∂ρ Kµ νρ + O(eom) ,
where Kµ νρ = −Kµ ρν ,
and O(eom) is any term vanishing on-shell, i.e. on a solution to the equations of
motion.
(h) Why is the modified energy-momentum tensor conserved ?
(i) Modify the canonical energy-momentum tensor for the free electromagnetic field as
follows
Tµν = Tµν + ∂ ρ Aµ Fνρ .
What are Kµ νρ , and O(eom) in this case?
(j) Show by an explicit computation that the modified tensor is equal to
1
Tµν = −Fµρ Fν ρ + ηµν Fρσ F ρσ ,
4
and it is conserved and symmetric. To show the conservation of T use the Bianchi
identities.
1
(k) Show that Tµν yields the standard formulae for the electromagnetic energy and momentum density:
1
~=E
~ ×B
~.
E = (E 2 + B 2 ) , S
2
2. Maxwell’s theory in the Coulomb gauge
Consider the free electromagnetic field with the Lagrangian (density)
1
L = − Fµν F µν ,
4
Fµν = ∂µ Aν − ∂ν Aµ .
(a) What is the gauge symmetry of the theory?
(b) Show that the Coulomb gauge is a “good” gauge, that is any gauge orbit contains a
representative which satisfies the Coulomb gauge condition.
(c) Show that in the Coulomb gauge the equation of motion for A0 leads to A0 = 0.
(d) Show that momenta πi for the space components of the electromagnetic potential
4-vector are equal to the electric field: πi = Ei .
(e) Rewrite the action for the electromagnetic field in the form of the generalized or
constrained Hamiltonian system
Z
S = dt pi q̇ i − H(p, q) + λα ϕα (p, q) .
(f) What are the Hamiltonian H and constraints ϕα for Maxwell’s theory?
(g) Introduce the Poisson brackets
Ei (~x), Aj (~y ) = δij δ(~x − ~y ) ,
and show that the constraints Poisson-commute between themselves and with the
Hamiltonian.
(h) Show that the constraints ϕ(~x) = ∂ i Ei generate gauge transformations of the vector
potential with the gauge parameter α as
Z
δAi (t, ~x) = Ai (t, ~x), d3 yϕ(~y )α(t, ~y ) , δA0 (t, ~x) = ∂0 α(t, ~x) .
(i) Impose the Coulomb gauge χ = ∂i Ai and show that it is a unitary gauge, that is
{χ(~x), χ(~y )} = 0, and {ϕ(~x), χ(~y )} is an invertible (infinite dimensional) matrix.
(j) Decompose the space components Ai and Ei into transverse and longitudinal parts,
and derive the reduced action in the Coulomb gauge which depends only on the
physical degrees of freedom.
(k) Introduce the mode expansion of the transverse components, and quantize Maxwell’s
theory in the Coulomb gauge.
3. Maxwell’s theory in the Lorentz gauge
Consider the free electromagnetic field with the Lagrangian (density)
1
L = − Fµν F µν ,
4
2
Fµν = ∂µ Aν − ∂ν Aµ .
(a) What is the gauge symmetry of the theory?
(b) Show that the Lorentz gauge is a “good” gauge, that is any gauge orbit contains a
representative which satisfies the Lorentz gauge condition. Is it a unique representative?
(c) Show that momenta πi for the space components of the electromagnetic potential
4-vector are equal to the electric field: πi = Ei .
(d) Rewrite the action for the electromagnetic field in the form of the generalized or
constrained Hamiltonian system
Z
S = dt pi q̇ i − H(p, q) + λα ϕα (p, q) .
(e) What are the Hamiltonian H and constraints ϕα for Maxwell’s theory?
(f) Introduce the momentum π0 canonically conjugate to A0 , and the two pairs of the
Faddeev-Popov ghosts. Define the BRST operator Q. What properties does it
satisfy?
(g) Consider the following gauge fermion operator
Z
α χ = d3 x P̄ A0 + C̄(∂i Ai + π0 )
2
and compute the effective Lagrangian
Lef f = p~ ~q˙ − Hef f ,
where p~ and ~q represent all canonically conjugated pairs of the dynamical variables
(write p~ ~q˙ explicitly in terms of Aµ ), πµ and the ghost fields), and
Hef f = H + i Q, χ + .
(h) Eliminate all momenta by using their equations of motion and derive the effective
Lagrangian which depends only on Aµ and C̄, C. Write it in a manifestly Lorentz
invariant form.
(i) Derive the equations of motion which follow from this effective Lagrangian. Set
α = 1 and show that any component Aµ satisfies the Klein-Gordon equation.
(j) To quantize Maxwell’s theory in the Feynman (α = 1) gauge, drop the ghost fields
(they decouple in electrodynamics), and shift the momenta πµ so that the resulting
effective Hamiltonian would only contain terms quadratic in the momenta.
(k) Eliminate all the momenta by using their equations of motion and show that the
resulting effective Lagrangian is
1
Lef f = − ∂µ Aν ∂ µ Aν .
2
(l) Introduce the mode expansion of Aµ and πµ , and quantize Maxwell’s theory in the
Feynman gauge.
3
(m) Compute the BRST charge and show that the physical subspace is the same as in
the Coulomb gauge.
(n) Show that the effective Hamiltonian in the Feynman gauge can be written in the
form
e +,
HFeynman = HCoulomb + Q, Q]
where HCoulomb is Maxwell’s Hamiltonian in the Coulomb gauge.
(o) Show that this implies the equality of physical quantities in the Coulomb and the
Feynman gauges.
4. Constrained systems in unitary gauges
Consider the action for the generalized or constrained Hamiltonian system
Z t2
dt pi q̇ i − H(p, q) + λα ϕα (p, q) , i, j, k = 1, . . . , n , α, β, γ = 1, . . . , m < n .
S=
t1
(a) Which conditions must ϕα satisfy to be first-class constraints?
(b) Which conditions must ϕα satisfy to be a complete set of constraints?
(c) Show that ϕα generate gauge transformations as follows
δq i = q i , ϕα α , δpi = pi , ϕα α , α = α (t) ,
where α are infinitesimal parameters of gauge transformations satisfying α (t1 ) =
α (t2 ) = 0. To this end find compensating gauge transformations of λα from the
condition of vanishing the variation δS of the action under the gauge transformations
of pi , q i and λα .
(d) A unitary gauge is of the form χα (p, q) = 0. What conditions the functions χα (p, q)
should satisfy? Show that if these conditions are satisfied then each gauge orbit close
enough to the manifold χα (p, q) = 0 intersects it.
(e) The conditions ϕα = 0 and χα = 0 define a physical subspace Γ∗2(n−m) in Γ2n . Show
that dynamics in Γ∗2(n−m) is equivalent to those in Γ2n .
5. Becchi-Rouet-Stora-Tyutin (BRST) quantization of constrained systems
Consider the Lagrangian for the generalized or constrained Hamiltonian system
L = pi q̇ i − H(p, q) + λα ϕα (p, q) ,
i, j, k = 1, . . . , n ,
α, β, γ = 1, . . . , m < n .
(a) Assume that in quantum theory ϕα satisfy
γ
H, ϕα = 0 ,
ϕα , ϕβ = ifαβ
ϕγ ,
γ
γ
where fαβ
are constants independent of p, q. What conditions must fαβ
satisfy?
(b) Enlarge the phase space by introducing momenta πα conjugate to λα , and two canonical pairs of anticommuting Faddeev-Popov ghosts, (P̄α , C α ) and (C̄α , P α ):
α
α
C , P̄β + = iδβα ,
P , C̄β + = iδβα ,
4
and replace L with the effective Lagrangian
Lef f = pi q̇ i + πα λ̇α + P̄α Ċ α + C̄α Ṗ α − Hef f ,
where
Hef f = H + i Q, χ + .
and Q is the BRST operator
1 γ α
C P̄γ C β + P α πα
Q = C α ϕα + fαβ
2
Assuming that qi , λα , C α , P α are hermitian, show that Q satisfies Q† = Q, Q2 = 0,
[H, Q] = 0, [Hef f , Q] = 0.
(c) Explain why the condition
Q|Ψi = 0 ,
is not sufficient to single out a “good” physical subspace.
(d) Let |Ψ1 i and |Ψ2 i differ by Q|ξi: |Ψ2 i = |Ψ1 i + Q|ξi. Show that the transition
amplitudes between a state |Ψi satisfying Q|Ψi = 0 and |Ψi i are the same.
(e) Show that the transition amplitude between two states |Ψi i satisfying Q|Ψi i = 0 is
independent of the gauge fermion operator.
γ
(f) Show that if fαβ
= 0 then
|Ψi = |Ψph i + Q|ξi if
Q|Ψi = 0 ,
where |Ψph i does not depend on the unphysical fields, i.e. it depends only on
(p∗ , q ∗ ) ∈ Γ∗2(n−m) .
(g) Take the gauge fermion operator to be
a
χ = P̄α λα + C̄α kiα − δ αβ πβ ,
2
α
where a is a constant but ki may be a function of p, q.
Compute the effective Lagrangian Lef f .
(h) Eliminate the momenta πα , P̄α , P α by using their equations of motion and show that
the resulting effective Lagrangian is of the form
Leff = L + Lg.f. + Lgh ,
where
2
1 α
λ̇ + kiα q i ,
2a
is related to the gauge fixing condition λ̇α + kiα q i = 0, and Lgh depends on the
Faddeev-Popov ghosts.
Lg.f. =
6. Quantum electrodynamics
Consider the Lagrangian describing interaction of matter with electromagnetic field
1
L = − Fµν F µν − j µ Aµ + Lmatter , Fµν = ∂µ Aν − ∂ν Aµ ,
4
µ
where j is an electromagnetic matter current, and Lmatter is the matter Lagrangian.
5
(a) Show that j µ must be a conserved current.
(b) Let Lmatter = ψ̄(iγ µ ∂µ − m)ψ be the Dirac Lagrangian. Show that it is invariant
under the internal U (1) symmetry ψ → e−iε ψ, ψ̄ → eiε ψ̄ where ε is a constant
real parameter of a U (1) transformation, and compute the corresponding conserved
Noether’s current jVµ .
(c) The quantum electrodynamics Lagrangian can be written in the form
1
LQED = − Fµν F µν + ψ̄(iγ µ Dµ − m)ψ ,
4
where Dµ ≡ ∂µ + ieAµ is the covariant derivative, and e is the electron charge.
What is the relation between the electromagnetic electron current j µ and jVµ ?
(d) Derive the equations of motion for the fermions and electromagnetic field.
(e) Show that LQED is invariant under gauge transformations.
(f) What is the electric charge in terms of ψ, ψ̄ and in terms of creation and annihilation
operators?
(g) The charge conjugation matrix is equal to C = −iγ 2 where γ µ are taken in the Weyl
or chiral representation
0 I2
0 σi
0
i
γ =
, γ =
.
I2 0
−σ i 0
Compute C and check that it is symmetric and unitary.
(h) Show that C satisfies the relations
(γ µ )∗ = −Cγ µ C ,
where (γ µ )∗ is a matrix complex-conjugate to γ µ .
(i) Show that ψ (c) ≡ Cψ ∗ satisfies the Dirac equation but with charge −e.
(j) Rewrite the QED action in the form of the generalized or constrained Hamiltonian
system
Z
S = dt pi q̇ i − H(p, q) + λα ϕα (p, q) .
(k) What are the Hamiltonian H and constraints ϕα for QED?
(l) Introduce the canonical (anti-)commutation relations
i
A (~x), Ej (~y ) = iδji δ(~x − ~y ) , [ψa (~x), ψb∗ (~y )]+ = δab δ(~x − ~y ) ,
and show that the constraints commute between themselves and with the Hamiltonian.
(m) Show that the constraints ϕ(~x) = ∂ i Ei + eψ ∗ ψ generate gauge transformations of
the vector potential and fermions with the gauge parameter α as
Z
δAi (t, ~x) = i Ai (t, ~x), d3 yϕ(~y )α(t, ~y ) , δA0 (t, ~x) = ∂0 α(t, ~x) ,
Z
Z
∗
3
∗
δψa (t, ~x) = i ψa (t, ~x), d yϕ(~y )α(t, ~y ) , δψa (t, ~x) = i ψa (t, ~x), d3 yϕ(~y )α(t, ~y ) .
6
(n) Impose the Coulomb gauge χ = ∂i Ai , decompose the space components Ai and Ei
into transverse and longitudinal parts, and derive the reduced action in the Coulomb
gauge which depends only on the physical degrees of freedom.
7. Quantum electrodynamics: several species of fermions
Consider the Lagrangian describing interaction of N species (flavors) of fermions with
electromagnetic field
N
X
1
ψ̄f (iγ µ ∂µ − qf γ µ Aµ − mf )ψf ,
L = − Fµν F µν +
4
f =1
where qf and mf are the charge and mass of the f -th fermion.
(a) Derive the equations of motion for the fermions and electromagnetic field.
(b) Show that L is invariant under gauge transformations.
(c) Assume that q1 = · · · = qN ≡ q and m1 = · · · = mN ≡ m.
Show that the Lagrangian is invariant under the internal U (N ) symmetry: Aµ → Aµ ,
ψf → Uf f 0 ψf 0 , ψ̄f → ψ̄f 0 Uf∗f 0 where U is a constant unitary matrix Uf∗f 0 Ugf 0 = δf g .
Infinitesimally, U = I + , f f 0 + ∗f 0 f = 0, ψf → ψf + f f 0 ψf 0 , ψ̄f → ψ̄f + ψ̄f 0 ∗f f 0
(d) Compute the corresponding U (N ) conserved Noether’s currents Jfµf 0 , and check by
explicit computation that they are conserved and satisfy the same reality conditions
as f f 0 do: (Jfµf 0 )∗ = −Jfµ0 f .
(e) Introduce the canonical anticommutation relations
ψf a (~x), ψf∗0 b (~y ) + = δf f 0 δab δ(~x − ~y ) ,
R
where a, b are spinors indices. Show that the Noether charges Qf f 0 = d3 xJf0f 0
generate the infinitesimal U (N ) transformations of fermions:
δψf (t, ~x) = i Qgg0 gg0 , ψf (t, ~x) , δ ψ̄f (t, ~x) = i Qgg0 gg0 , ψ̄f (t, ~x) .
(f) Compute the commutator of Qab and Qcd and show that for any choice of the indices
a, b, c, d the commutator [Qab , Qcd ] can be written as a linear combination of the
charges. They form the u(N ) Lie algebra.
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