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� � � �2 V (σ, π) = − 12 µ 2 σ 2 + π 2 + 14 λ σ 2 + π 2 0.1 Thursday 3 June 2010 (µ ∈ R , λ > 0) 1. Find Summer sheet all QFTthe I symmetries of the lagrangian and obtain all the conserved charges corresponding FIELD to the continuous symmetries THEORY 1. Problem 1 2. Find the extremal configurations of the hamiltonian and discuss Consider the following lagrangian1 involving two real scalar fields σ(t, x) and Suppose the vacuum of the quantum theory is such that π(t, x) on the that Minkowski spacetime 1 x) | 0 µ� = υ1 �L 0 |=σ(t, x) π) |0� = 0 ∂ σ ∂ σ + 2 ∂ µ π ∂ �µ0π |−π(t, V (σ, 2 µ whereand set σ �(t, x) υ� � 0� | σ �(t, x) | 0 � = 0 � 2≡ σ(t, �x) − 2 2 2 2 1 2 1 V (σ, π) = − 2 µ σ + π + 4 λ σ + π (µ ∈ R , λ > 0) 3. Write the Feynman rules for the quantum theory of the real scalar fields 1. σ theπ(t, symmetries of the lagrangian �Find (t, x)all and x) and comment the result.and obtain all the conserved charges corresponding to the continuous symmetries FIELD THEORY 2. 2. Find the extremal configurations of the hamiltonian and discuss Suppose that theselfenergy vacuum ofatthe theory is such that Calculate the electron onequantum loop �| 0 � = � 0 | σ(t, x) � 0 | π(t, x) | 0 � = 0 d4 kυ µ 2 ν Σ2 (p/) = i(ie) γ D (k) S(p + k) γ µν (2π)4 and set σ �(t, x) ≡ σ(t, x) dimensional −υ � 0regularisation, |σ �(t, x) | 0 � = where 0 in quantum electrodynamics using D µν (k) is the photon propagator in the Feynman gauge, while S(p) is the Dirac 3. Write the Feynman rules for the quantum theory of the real scalar fields propagator. Segregate the structure of the divergent and finite parts. σ �(t, x) and π(t, x) and comment the result. FIELD THEORY 2. 1 Calculate the electron selfenergy one loop Jeoffrey Goldstone (1961) Il Nuovo at Cimento, Volume 9, p.154 � 4 HINT: The extrema of the hamiltonian correspond to space-time dfunctional k µ 2 ν 34 γthat Σ (p /) = i(ie) Dyou S(pto +find k) γthe µν (k) independent σ- and2 π-configurations such have extrema of the (2π) potential functional V(σ, π). in quantum electrodynamics using dimensional regularisation, where D µν (k) is the photon propagator in the Feynman gauge, while S(p) is the Dirac propagator. Segregate the structure of the divergent and finite parts. 1 Jeoffrey Goldstone (1961) Il Nuovo Cimento, Volume 9, p.154 3 0.2 Thursday 18 February 2010 Summer sheet QFT I FIELD THEORY 1. Problem 2 Consider the following classical lagrangian involving a Dirac bispinor field L = ψ̄ (x) ( i ∂/ − M − a / − b/ γ5 ) ψ (x) where, as usual, ψ̄ (x) = ψ † (x) γ 0 while a / = γ µ a µ , b/ = γ µ b µ with the constant and fixed real quantities given by a µ = (a, − a) b0 = b , b1 = b2 = b3 = 0 the same for all the inertial observers 2 . 1. Show that the term ψ̄ (x) a / ψ (x) in the lagrangian can be eliminated by means of a gauge transformation on the bispinor field ψ (x) �→ ψ � (x) = exp{− i f (x)} ψ (x) with a suitable choice of the real and dimensionless function f (x) . 2. Find all the continuous symmetry groups of invariance for the above classical Lagrange density and verify if the Action is invariant under the discrete parity tranformation (t, x) �→ (t, − x) ψ (x) �→ ψ � (x � ) = γ 0 ψ (t, − x) 3. Find the explicit form of the Feynman propagator in the momentum space, that is, the opposite of the inverse of the kinetic term S F ( p, b ) = [ − i (p/ − M − b/γ5 ) ] −1 4. Verify that in the limit b → 0 one recovers the customary well known Feynman propagator of the Dirac field: namely, S F(p) = 2 i ( p/ + M ) p2 − M 2 + i ε For example one can realistically assume b < 3 × 10 −17 eV. 11 0.4 Thursday 10 September 2009 Summer sheet QFT I Problem 3 FIELD THEORY 1. The minimal Goldstone model : consider the classical Lagrangian for a real scalar field L = 12 ∂ µ φ (x) ∂ µ φ (x) + 12 µ 2 φ 2 (x) − 14 λ φ 4 (x) with µ ∈ R and λ > 0 1. Find all the symmetries of the above field theoretic model 2. Write the energy momentum tensor as well as the total energy and total momentum for this model and comment about it 3. Determine all the degenerate classical field configurations of minimal energy and specify their symmetry 4. Choose one of the degenerate minimum field configurations φ min for the classical Hamiltonian of the minimal Goldstone model and expand the Lagrangian around it : which are the symmetries of the resulting new classical Action? Comment about the classical Hamiltonian as a function of the shifted field ϕ = φ − φ min FIELD THEORY Consider the Compton scattering e− γ → e− γ . The initial state contains a photon of momentum k, energy k = | k |, and an electron at rest of mass me . In the final state we have the scattered photon of momentum k � , energy k � = | k � |, and an electron which has received the recoil momentum p � = k − k � and with the energy � � p0� = E � = ( k − k � )2 + m2e = k2 + k � 2 − 2k · k � cos θ + m2e where θ is the angle between the vectors k and k � . The electron spin states � are u r and ū r � (p � ) while the photon linear polarization vectors are ε µA (k � ) and ε νA (k) respectively, with r, r � , A, A � = 1, 2 . � � 1. Write the probability amplitude i MAA rr � (k, k ) for this process to the lowest order in the fine structure constant 2. Find the square modulus of the probability amplitude averaging over the initial and summing over the final electron polarizations 20 new classical Action? Comment about the classical Hamiltonian as a function of the shifted field ϕ = φ − φ min Summer sheet QFT I Problem 4 FIELD THEORY Consider the Compton scattering e− γ → e− γ . The initial state contains a photon of momentum k, energy k = | k |, and an electron at rest of mass me . In the final state we have the scattered photon of momentum k � , energy k � = | k � |, and an electron which has received the recoil momentum p � = k − k � and with the energy � � p0� = E � = ( k − k � )2 + m2e = k2 + k � 2 − 2k · k � cos θ + m2e where θ is the angle between the vectors k and k � . The electron spin states � are u r and ū r � (p � ) while the photon linear polarization vectors are ε µA (k � ) and ε νA (k) respectively, with r, r � , A, A � = 1, 2 . � � 1. Write the probability amplitude i MAA rr � (k, k ) for this process to the lowest order in the fine structure constant 2. Find the square modulus of the probability amplitude averaging over the initial and summing over the final electron polarizations 3. Perform the photon polarization sums (averaging over the initial ones and summing over the final ones), calculate the 20 traces and determine the unpolarized differential cross section dσ/dΩ. HINT: how many diagrams contribute to the amplitude at tree level? 0.5 Thursday 9 July 2009 Summer sheet QFT I FIELD THEORY 1. Problem 5 Consider the classical Lagrangian for the real scalar field L= 1 1 λ ∂ µ φ (x) ∂ µ φ (x) − m 2 φ 2 (x) − φ 4 (x) 2 2 4! and the dilatation trasformation abelian Lie group which acts on spacetime coordinates and field according to φ (x) → φ � (y) = e − α φ (x) xµ → y µ = eα xµ 1. Calculate the canonical energy momentum symmetric tensor T µν 2. Calculate the Noether current J µ (x) associated to the Lie group of the dilatation transformations and discuss its continuity equation 3. Define the new improved energy momentum symmetric tensor Θ µν = T µν + a ( ∂ µ ∂ ν − g µν � ) φ 2 (x) (a ∈ R) and determine the constant a in such a manner that g µν Θ µν = 0 for m = 0 (traceless condition) 4. Define a new dilatation current j µ (x) ≡ x ρ Θ µρ and show that ∂ µ j µ (x) = g λν Θ λν (x) FIELD THEORY Evaluate the differential cross section to the lowest order O(α2 ), where α is the fine structure constant, for the electron-proton collision e− p −→ e− p in the proton rest frame, disregarding the electron mass in respect to the proton mass and the proton recoil after the collision. (Nota Bene : the proton is assumed to be point-like) 25 0.7 Friday 29 May 2009 Summer sheet QFT I FIELD THEORY 1. Problem 6 Consider the planar Chern-Simons spinor electrodynamics, which is the field theory defined on the 2+1 dimensional Minkowski spacetime with x µ = (x0 , x1 , x2 ) = (ct, x, y) metric tensor g µν = g µν 1 0 0 0 −1 0 = 0 0 −1 and described by the classical the Lagrangian density L = 14 κ ε λµν A λ (x) F µν (x) + ψ̄(x) [ i∂/ + q A/(x) − M ] ψ(x) where a / ≡ γ µ a µ and ε 012 = 1 , which involves the real vector potential A λ (x) , the field strength F µν (x) = ∂ µ A ν (x) − ∂ ν A µ (x) , the two component spinor ψ (x) 1 ψ(x) = ψ2 (x) q being the charge of spinor particle, κ a real number known as the ChernSimoms coefficient and, finally, the 2 × 2 gamma matrices γ 0 = σ1 γ 1 = i σ2 γ 2 = i σ3 where σı ( ı = 1, 2, 3 ) are the Pauli matrices, in such a manner that the Clifford algebra {γ µ , γ ν } = 2g µν I holds true. 1. Determine the canonical dimensions of the gauge potential and spinor field in natural units as well as in physical units 2. Find the Euler-Lagrange field equations and make some comments 3. Specify the conditions under which the Action is gauge invariant 4. Find the normal mode decomposition of the quantized free spinor field, i.e. for q = 0 , in the massless case M = 0 Score : 1. pt. 2, 2. pt. 3, 3. pt. 10 ; 4. pt 15 34 Summer sheet QFT I Problem 7 Write the expressions for the QED Feynman amplitudes corresponding to the diagrams given in the figure. Specify the momentum routing. Of course, you are NOT supposed to perform the loop integrals. Summer sheet QFT I Problem 8: Path integral for a free particle Summer sheet QFT I Problem 9: Generating functional for fermion fields. Using the generating functional derive: 1. the two-point Green function (i.e. the Feynman propagator) for the field ψ(x); 2. the photon-fermion vertex in QED. Problem 10: The propagator for a massive vector field (using the generating functional) Use the generating functional. Problem 11: CP properties of kaon non-leptonic decays HINT: the pion is a pseudo-scalar particle. Summer sheet QFT I Problem 12: Properties of Majorana fermions Summer sheet QFT I