Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams

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1 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) (a) Prove the Bianchi identities L = 1 4 F µνf µν, F µν = µ A ν ν A µ. µ F νρ + ν F ρµ + ρ F µν = 0. (b) Derive Maxwell s equations from L treating A µ as the dynamical variables. (c) Solve the equation of motion for A 0 assuming that A µ 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 ɛ ijk B k = F ij. (e) The canonical energy-momentum (or stress-energy) tensor of a field theory with Lagrangian L = L(φ a, µ φ a ) is defined as T µ ν = L ν φ a µ φ a δ ν µl. Construct the canonical energy-momentum tensor for the free electromagnetic field. (f) Compute T µν T νµ. (g) The canonical energy-momentum tensor is not defined uniquely. One can always modify the tensor in the following way T µ ν = T µ ν + ρ K µ νρ + O(eom), where K µ νρ = K µ 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 T µν = T µν + ρ A µ F νρ. What are K µ νρ, and O(eom) in this case? (j) Show by an explicit computation that the modified tensor is equal to T µν = F µρ F ν ρ η µνf ρσ F ρσ, and it is conserved and symmetric. To show the conservation of T use the Bianchi identities. 1

2 (k) Show that T µν yields the standard formulae for the electromagnetic energy and momentum density: 2. Maxwell s theory in the Coulomb gauge E = 1 2 (E2 + B 2 ), S = E B. Consider the free electromagnetic field with the Lagrangian (density) L = 1 4 F µνf µν, F µν = µ A ν ν A µ. (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 A 0 leads to A 0 = 0. (d) Show that momenta π i for the space components of the electromagnetic potential 4-vector are equal to the electric field: π i = E i. (e) Rewrite the action for the electromagnetic field in the form of the generalized or constrained Hamiltonian system S = dt p i q i H(p, q) + λ α ϕ α (p, q). (f) What are the Hamiltonian H and constraints ϕ α for Maxwell s theory? (g) Introduce the Poisson brackets { Ei ( x), A j ( y) } = δ j i δ( x y), and show that the constraints Poisson-commute between themselves and with the Hamiltonian. (h) Show that the constraints ϕ( x) = i E i generate gauge transformations of the vector potential with the gauge parameter α as δa i (t, x) = { A i (t, x), d 3 yϕ( y)α(t, y) }, δa 0 (t, x) = 0 α(t, x). (i) Impose the Coulomb gauge χ = i A i 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 A i and E i 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) L = 1 4 F µνf µν, F µν = µ A ν ν A µ. 2

3 (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 π i for the space components of the electromagnetic potential 4-vector are equal to the electric field: π i = E i. (d) Rewrite the action for the electromagnetic field in the form of the generalized or constrained Hamiltonian system S = dt p i q i H(p, q) + λ α ϕ α (p, q). (e) What are the Hamiltonian H and constraints ϕ α for Maxwell s theory? (f) Introduce the momentum π 0 canonically conjugate to A 0, 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 χ = d 3 x P A 0 + C( i A i + α 2 π 0) and compute the effective Lagrangian L eff = p q H eff, where p and q represent all canonically conjugated pairs of the dynamical variables (write p q explicitly in terms of A µ ), π µ and the ghost fields), and H eff = H + i Q, χ +. (h) Eliminate all momenta by using their equations of motion and derive the effective Lagrangian which depends only on A µ and C, C. Write it in a manifestly Lorentz invariant form. (i) Derive the equations of motion which follow from this effective Lagrangian. α = 1 and show that any component A µ 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 π µ 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 L eff = 1 2 µa ν µ A ν. (l) Introduce the mode expansion of A µ and π µ, and quantize Maxwell s theory in the Feynman gauge. 3 Set

4 (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 H Feynman = H Coulomb + Q, Q +, where H Coulomb 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 S = t2 t 1 dt p i q i H(p, q) + λ α ϕ α (p, q), i, j, k = 1,..., n, α, β, γ = 1,..., m < n. (a) Which conditions must ϕ α satisfy to be first-class constraints? (b) Which conditions must ϕ α satisfy to be a complete set of constraints? (c) Show that ϕ α generate gauge transformations as follows δq i = { q i, ϕ α ɛ α}, δp i = { p i, ϕ α ɛ α}, ɛ α = ɛ α (t), where ɛ α are infinitesimal parameters of gauge transformations satisfying ɛ α (t 1 ) = ɛ α (t 2 ) = 0. To this end find compensating gauge transformations of λ α from the condition of vanishing the variation δs of the action under the gauge transformations of p i, q i and λ α. (d) A unitary gauge is of the form χ α (p, q) = 0. What conditions the functions χ α (p, q) should satisfy? Show that if these conditions are satisfied then each gauge orbit close enough to the manifold χ α (p, q) = 0 intersects it. (e) The conditions ϕ α = 0 and χ α = 0 define a physical subspace Γ 2(n m) in Γ 2n. Show that dynamics in Γ 2(n m) is equivalent to those in Γ 2n. 5. Becchi-Rouet-Stora-Tyutin (BRST) quantization of constrained systems Consider the Lagrangian for the generalized or constrained Hamiltonian system L = p i q i H(p, q) + λ α ϕ α (p, q), i, j, k = 1,..., n, α, β, γ = 1,..., m < n. (a) Assume that in quantum theory ϕ α satisfy H, ϕα = 0, ϕα, ϕ β = if γ αβ ϕ γ, where f γ αβ are constants independent of p, q. What conditions must f γ αβ satisfy? (b) Enlarge the phase space by introducing momenta π α conjugate to λ α, and two canonical pairs of anticommuting Faddeev-Popov ghosts, ( P α, C α ) and ( C α, P α ): C α, P β = + iδα β, P α, C β = + iδα β, 4

5 and replace L with the effective Lagrangian where and Q is the BRST operator L eff = p i q i + π α λα + P α Ċ α + C α P α H eff, H eff = H + i Q, χ +. Q = C α ϕ α f γ αβ Cα Pγ C β + P α π α Assuming that q i, λ α, C α, P α are hermitian, show that Q satisfies Q = Q, Q 2 = 0, H, Q = 0, H eff, Q = 0. (c) Explain why the condition Q Ψ = 0, is not sufficient to single out a good physical subspace. (d) Let Ψ 1 and Ψ 2 differ by Q ξ : Ψ 2 = Ψ 1 + Q ξ. Show that the transition amplitudes between a state Ψ satisfying Q Ψ = 0 and Ψ i are the same. (e) Show that the transition amplitude between two states Ψ i satisfying Q Ψ i = 0 is independent of the gauge fermion operator. (f) Show that if f γ αβ = 0 then Ψ = Ψ ph + Q ξ if Q Ψ = 0, where Ψ ph does not depend on the unphysical fields, i.e. (p, q ) Γ 2(n m). (g) Take the gauge fermion operator to be χ = P α λ α + C α ( k α i a 2 δαβ π β ), where a is a constant but k α i may be a function of p, q. Compute the effective Lagrangian L eff. it depends only on (h) Eliminate the momenta π α, P α, P α by using their equations of motion and show that the resulting effective Lagrangian is of the form L eff = L + L g.f. + L gh, where L g.f. = 1 ( λα + ki α q i) 2, 2a is related to the gauge fixing condition λ α + ki α q i = 0, and L gh depends on the Faddeev-Popov ghosts. 6. Quantum electrodynamics Consider the Lagrangian describing interaction of matter with electromagnetic field L = 1 4 F µνf µν j µ A µ + L matter, F µν = µ A ν ν A µ, where j µ is an electromagnetic matter current, and L matter is the matter Lagrangian. 5

6 (a) Show that j µ must be a conserved current. (b) Let L matter = ψ(iγ µ µ m)ψ be the Dirac Lagrangian. Show that it is invariant under the internal U(1) symmetry ψ e iε ψ, ψ e iε ψ where ε is a constant real parameter of a U(1) transformation, and compute the corresponding conserved Noether s current j µ V. (c) The quantum electrodynamics Lagrangian can be written in the form L QED = 1 4 F µνf µν + ψ(iγ µ D µ m)ψ, where D µ µ + iea µ is the covariant derivative, and e is the electron charge. What is the relation between the electromagnetic electron current j µ and j µ V? (d) Derive the equations of motion for the fermions and electromagnetic field. (e) Show that L QED is invariant under gauge transformations. (f) What is the electric charge in terms of ψ, ψ 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 γ 0 I2 0 σ =, γ i i = I 2 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 S = dt p i q i H(p, q) + λ α ϕ α (p, q). (k) What are the Hamiltonian H and constraints ϕ α for QED? (l) Introduce the canonical (anti-)commutation relations A i ( x), E j ( y) = iδ i jδ( x y), ψ a ( x), ψ b ( y) + = δ ab δ( x y), and show that the constraints commute between themselves and with the Hamiltonian. (m) Show that the constraints ϕ( x) = i E i + eψ ψ generate gauge transformations of the vector potential and fermions with the gauge parameter α as δa i (t, x) = i A i (t, x), d 3 yϕ( y)α(t, y), δa 0 (t, x) = 0 α(t, x), δψ a (t, x) = i ψ a (t, x), d 3 yϕ( y)α(t, y), δψa(t, x) = i ψa(t, x), d 3 yϕ( y)α(t, y). 6

7 (n) Impose the Coulomb gauge χ = i A i, decompose the space components A i and E i 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 L = 1 4 F µνf µν + N f=1 ψ f (iγ µ µ q f γ µ A µ m f )ψ f, where q f and m f 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 q 1 = = q N q and m 1 = = m N m. Show that the Lagrangian is invariant under the internal U(N) symmetry: A µ A µ, ψ f U ff ψ f, ψf ψ f U ff where U is a constant unitary matrix U ff U gf = δ fg. Infinitesimally, U = I + ɛ, ɛ ff + ɛ f f = 0, ψ f ψ f + ɛ ff ψ f, ψ f ψ f + ψ f ɛ ff (d) Compute the corresponding U(N) conserved Noether s currents J µ ff, and check by explicit computation that they are conserved and satisfy the same reality conditions as ɛ ff do: (J µ ff ) = J µ f f. (e) Introduce the canonical anticommutation relations ψfa ( x), ψ f b( y) + = δ ff δ abδ( x y), where a, b are spinors indices. Show that the Noether charges Q ff = d 3 xj 0 ff generate the infinitesimal U(N) transformations of fermions: δψ f (t, x) = i Q gg ɛ gg, ψ f (t, x), δ ψ f (t, x) = i Q gg ɛ gg, ψ f (t, x). (f) Compute the commutator of Q ab and Q cd and show that for any choice of the indices a, b, c, d the commutator Q ab, Q cd can be written as a linear combination of the charges. They form the u(n) Lie algebra. 7