Quantum Field Theory 2011 Solutions

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1 Quantum Field Theory 011 Solution Yichen Shi Eater 014 Note that we ue the metric convention + ++). 1. State and prove Noether theorem in the context of a claical Lagrangian field theory defined in Minkowki pace. See Q Yukawa theory ha a Lagrangian denity L = 1 aφ a φ 1 m φ i ψγ a a + m)ψ + g ψ φ ψ. Find the Noether current and conerved charge aociated with the phae rotation ψ e iα ψ. Find the Noether current and conerved charge aociated with time tranlation ymmetry. The quetion implie that ψ e iα ψ 1 iα)ψ i a ymmetry of the theory, o we do not need to verify them. Then by Noether theorem, the conerved current αj a = L a ψ) δψ + L δ a ψ) ψ = i ψγ a iαψ) = α ψγ a ψ. And the conerved charge Q = d 3 x j 0 = d 3 x ψγ a ψ. For a pace-time tranlational ymmetry, we have x a x a + α a. Then, φx) φx) α a a φx); ψx) ψx) α a a ψx). By Noether theorem, the aociated current i the energy-momentum tenor T ab = L a ψ) b ψ L a φ) b φ + η ab L = i ψγ 1 a b ψ + a φ b φ η ab cφ c φ + 1 ) m φ + i ψγ c c + m)ψ g ψ φ ψ = i ψγ 1 a b ψ + a φ b φ η ab cφ c φ + 1 ) m φ, 1

2 by uing the equation of motion for ψ. We can do o ince a current i conerved only when the equation of motion are obeyed. So for a time tranlational ymmetry, T 00 = i ψγ 0 ψ + φ + 1 cφ c φ + 1 m φ = ψiγ i i m)ψ + 1 φ + 1 φ) + 1 m φ, where we again ued the equation of motion for ψ. And the conerved charge i the total energy, E = d 3 x T 00. Briefly decribe any additional Noether current and charge that thi theory ha. There are conerved current and charge aociated with patial tranlational invariance, i.e., total momentum. Note that there i no Noether current from the phae rotation of φ ince g ψφψ lead to a term in δl that i not a total derivative.. Free calar field theory in momentum pace i defined by a et of Schrodinger picture operator a and a atifying [a, a q ] = π)3 δ 3 q) and a Hamiltonian H = d 3 π) 3 E a a. Define what i meant by a Heienberg picture operator, and find the Heienberg picture operator a H and a H obtained from a and a. In the Heienberg picture, operator evolve with time wherea tate are time-independent, in contrat with the Schrodinger picture. For any operator, O H = e iht O S e iht. Given the Schrodinger picture field φ x) = π x) = i) d 3 1 a π) 3 e i x + a e i x) ; E d 3 E π) 3 a e i x a e i x), find the correponding Heienberg picture field. Show that φ H obey the Klein-Gordon equation. Since we have [H, a ] = E a ; [H, a ] = E a a H = eiht a e iht = e ie t a ; a H = e iht a e iht = e ie t a.

3 and Hence if we let a a H Now, in the Schodinger picture mode expanion, then φ H x) = π H x) = i) d 3 1 a π) 3 e ip x + a e ip x) ; E d 3 E π) 3 a e ip x a e ip x). φ H = ih e iht φ S e iht + e iht φ S e iht ih) = i[h, φ H ] = i d 3 x [ π y) + φ y)) + m φ y), φ x) ] = i d 3 x [ π y), φ x) ] = π x), π H = ih e iht π S e iht + e iht π S e iht ih) = i[h, π H ] = i d 3 x [ φ y)) + m φ y), φ x) ] = d 3 x φ y) δ 3 x y) + m φ y)δ 3 x y) ) = d 3 x φ y)δ 3 x y) m φ x) = φ x) m φ x). Combining the two calculation, we have which i the KG equation. φ H φ H + m φ H = 0, Define the vacuum tate 0 and find an expreion for 0 φ H x)φ H y) 0 a a 3-momentum integral. The vacuum tate 0 i defined uch that it i annihilated by an annihilation operator, i.e., a 0 = 0. 0 φ H x)φ H y) 0 = = = d 3 π) 3 d 3 π) 3 d 3 q 1 π) 3 e ip x iq y ) 0 [a p, a q] 0 E E q d 3 q 1 π) 3 e ip x iq y ) E E π) 3 δ 3 q) q d 3 π) 3 1 E e ip x y). 3. 3

4 Let u and v be arbitrary Dirac pinor. Show that ūγ a v) = vγ a u. Uing the Dirac algebra, determine Trγ a γ b ) and Trγ a γ b γ c γ d ). Since γ a = γ 0 γ a γ 0, γ 0 = γ 0, and ūγ a v) = v γ a γ 0 u = v γ 0 γ a γ 0 )γ 0 u = vγ a u. By the cyclic permutation of element in Tr), and noting that we are tracing over pinor not Lorentz) indice, Trγ a γ b ) = 1 Trγa γ b + γ b γ a ) = 1 Trηab I 4 ) = 4η ab. Trγ a γ b γ c γ d ) = Trγ a γ b η cd γ d γ c )) = η cd Trγ a γ b ) Trγ a γ b γ d γ c ) = η cd Trγ a γ b ) Trγ a η bd γ d γ b )γ c ) = η cd Trγ a γ b ) η bd Trγ a γ c ) + Trγ a γ d γ b γ c ) = η cd Trγ a γ b ) η bd Trγ a γ c ) + Trη ad γ d γ a )γ b γ c ) = η cd Trγ a γ b ) η bd Trγ a γ c ) + η ad Trγ b γ c ) Trγ d γ a γ b γ c ) = 8η cd η ab 8η bd η ac + 8η ad η bc Trγ a γ b γ c γ d ). Hence Trγ a γ b γ c γ d ) = 4η cd η ab η bd η ac + η ad η bc ) Now let {u p), v p)}, = ±1/ be an orthonormalied bai et of pinor atifying iγ a p a + m)u p) = 0, iγ a p a + m)v p) = 0; ū u r = imδ r, v v r = imδ r. Why i it neceary that p + m = 0 here? Etablih the pin um We are given But we imilarly have u p)ū p) = γ a p a im, iγ a p a + m)u p) = 0. v p) v p) = γ a p a + im. iγ b p b + m)iγ a p a + m)u p) = 0. γ b γ a p b p a + m )u p) = 0. γ a γ b p a p b + m )u p) = 0. Auming the olution i nontrivial, adding the expreion and uing the Clifford algebra, we obtain η ab p a p b + m = 0. p + m = 0. Now, ince {u p), v p)} i a bai of the four dimenional vector pace of Dirac pinor for every fixed value of p, linear operator Ap) := u p)ū p), Bp) := v p) v p) 4

5 are determined by their action on the bai pinor u r p) and v r p). Ap)u r p) = u p)ū p)u r p) = im u p)δ r Hence Similarly, Hence = imu r p) = imu r p) i iγ a p a )u r p) = γ a p a im)u r p). u p)ū p) = γ a p a im. Bp)v r p) = γ a p a + im)v r p). v p) v p) = γ a p a + im. Ue the reult above to implify the expreion ū p)γ a v r q)) ū p)γ b v r q)), where p = q = m. Decribe briefly how thi quantity could arie in a QED calculation. Note that an 1x1 matrix i equal to it trace. Alo recall the identitie Trγ a γ b γ c ) = 0 due to the antiymmetry of γ a and the cyclic permutation of trace, and Trγ a γ b ) = 4η ab, Trγ a γ b γ c γ d ) = 4η ab η cd η ac η bd + η ad η bc ). ū p)γ a v r q)) ū p)γ b v r q)) = v r q)γ a u p)ū p)γ b v r q) ) = Tr v r q)γ a u p)ū p)γ b v r q) ) = Tr v r q) v r q)γ a u p)ū p)γ b = Tr γ c q c + im)γ a γ d p d im)γ b) = Trγ c q c γ a γ d p d γ b ) + m Trγ a γ b ) = 4q a p b p qη ab + q b p a + m η ab ). Thi quantity can arie in, for example, muon-electron cattering, when finding A from A ūp 1)γ a up 1 )ūp )γ a up ). 4. 5

6 Write down the Feynman rule for a QED cattering amplitude involving only photon on external line. 1. Draw the Feynman diagram and label with momentum conitent with conervation.. For each internal fermion propagator, give a factor of i /p + m) p + m iɛ. 3. For each interaction vertex, add ieγ a, impoe vertex momentum conervation and overall momentum conervation with π) 4 δ 4 pin ) p out. 4. Integrate over momenta aociated with loop with meaure d4 k π) Add polariation vector for external photon: ɛ a incoming; ɛ a outgoing. 6. Take into account minu ign due to the commutation of fermion. 7. Divide by ymmetry factor. a) Explain the contraint on the photon polarization vector and how they arie. Uing the mode expanion of A, A = d 3 1 π) 3 ɛ r a r, e ip x + a e ip x r, ), E r the gauge condition A = 0 give ɛ = 0, hence the polariation vector are tranvere to patial momentum. b) Explain the key tep in the derivation of the photon propagator. We can imply read off the photon propagator from Maxwell Lagrangian in the Coulomb gauge: L Coulomb = 1 4 F abf ab = 1 A i A i. Fourier tranform of the propagator, being the invere of δ ij k, i, D F k) ij = δ ij k iɛ. c) Draw the lowet order Feynman diagram for two photon to two photon cattering, and give the integral expreion for the cattering amplitude to thi order. Simplification of the expreion i not required.) ie) 4 d 4 k i /p 1 + m) i /p + m) i /p 3 + m) i /p 4 + m) π) 4 γa p 1 + m iɛ γb p + m iɛ γc p 3 + m iɛ γd p m iɛ d) Draw the diagram that contribute at next to lowet order to thi cattering amplitude. 6