Instituto Superior T´ecnico Lectures in Quantum Field Theory – Lecture 2 Jorge C. Rom˜ao Instituto Superior T´ecnico, Departamento de F´ısica & CFTP A. Rovisco Pais 1, 1049-001 Lisboa, Portugal January 2014 Jorge C. Rom˜ao IDPASC School Braga – 1 IST Summary for Lecture 2: QED as an Example Summary ❐ QED as a gauge theory QED as a gauge theory Propagators & GF ❐ Propagators and Green functions How to find QED F.R.? − Coulomb scattering e ❐ Feynman rules for QED Coulomb scattering e+ γ in internal lines ◆ Electrons and positrons in external lines Higher Orders ◆ Photons in internal lines γ in external lines QED Feynman Rules ◆ Photons in external lines Simple Processes ◆ Higher orders Compton Scattering − − e e+ → µ µ+ ❐ Example 1: Compton scattering QFT Computations ❐ Example 2: e− + e+ µ− + µ+ in QED → Jorge C. Rom˜ao IDPASC School Braga – 2 IST Local invariance of Dirac equation Summary ❐ We start with Dirac Lagrangian QED as a gauge theory • Local invariance • QED Lagrangian = ψ(i∂/ m)ψ Propagators & GF L − How to find QED F.R.? − ❐ Coulomb scattering e It is invariant under global phase transformations Coulomb scattering e+ ′ iα in internal lines ψ = e ψ, α infinitesimal δψ = iαψ, δψ = iαψ γ → − Higher Orders γ in external lines ❐ What happens if the transformations are local, α = α(x)? QED Feynman Rules Simple Processes δψ = iα(x)ψ ; δψ = iα(x)ψ Compton Scattering − − − e e+ → µ µ+ QFT Computations ❐ We have then δ = ψγµψ ∂ α(x) L − µ and the Lagrangian is no longer invariant Jorge C. Rom˜ao IDPASC School Braga – 3 IST Local invariance of Dirac equation ... Summary ❐ We see that the problem is connected with the fact that ∂µψ do not QED as a gauge theory • Local invariance transform as ψ. We are then led to the concept of covariant derivative Dµ • QED Lagrangian that transforms as the fields, Propagators & GF How to find QED F.R.? − δDµψ = iα(x)Dµψ Coulomb scattering e Coulomb scattering e+ γ in internal lines ❐ For the Dirac field we define, in analogy with minimal prescription, Higher Orders γ in external lines Dµ = ∂µ + ieAµ QED Feynman Rules Simple Processes ❐ Compton Scattering The vector field Aµ is a field that ensures that we can choose the phase − − e e+ → µ µ+ locally. Its transformation is chosen to compensate the term proportional to QFT Computations ∂µα 1 δA = ∂ α(x) µ − e µ ❐ We are then led to the introduction of the electromagnetic field Aµ satisfying the usual gauge invariance. Jorge C. Rom˜ao IDPASC School Braga – 4 IST QED Lagrangian Summary ❐ This new vector field Aµ needs a kinetic term. The only term quadratic that QED as a gauge theory • Local invariance is invariant under the local gauge transformations is • QED Lagrangian Propagators & GF Fµν = ∂µAµ ∂ν Aµ, δFµν = 0 How to find QED F.R.? − − Coulomb scattering e ❐ µ Coulomb scattering e+ A mass term of the form A Aµ is not gauge invariant, so the field Aµ γ in internal lines (photon) is massless Higher Orders ❐ γ in external lines The final Lagrangian is QED Feynman Rules Simple Processes 1 µν QED = F F + ψ(iD/ m)ψ + Compton Scattering L −4 µν − ≡Lfree Linteraction − − e e+ → µ µ+ QFT Computations where 1 = F F µν + ψ(i∂/ m)ψ, = eψγ ψAµ Lfree −4 µν − Linteraction − µ ❐ This Lagrangian is invariant under local gauge transformations and describes the interactions of electrons (and positrons) with photons. The theory is called Quantum Electrodynamics (QED) Jorge C. Rom˜ao IDPASC School Braga – 5 IST The non-relativistic propagator Summary ❐ We will follow the method of Richard Feynman to arrive at the rules for QED as a gauge theory calculations in QED. Propagators & GF • Non-relativistic Prop. • GF as propagators ❐ As a warm up exercise we start with the non-relativistic Schr¨odinger equation • S Matrix • Relativistic Prop. • New processes ∂ • Green Function i H ψ(~x, t) = 0, H = H0 + V • S Matrix elements ∂t − • In & Out states How to find QED F.R.? where H0 is the free particle Hamiltonian − Coulomb scattering e 2 Coulomb scattering e+ H0 = ∇ γ in internal lines −2m Higher Orders γ in external lines ❐ We can rewrite the equation in the form QED Feynman Rules Simple Processes ∂ Compton Scattering i H0 ψ = V ψ − − ∂t − e e+ → µ µ+ QFT Computations ❐ For arbitrary V this equation can normally only be solved in perturbation theory Jorge C. Rom˜ao IDPASC School Braga – 6 IST The non-relativistic propagator ... Summary ❐ For the scattering problems we are interested we will develop a perturbative QED as a gauge theory expansion using the technique of the Green’s Functions (GF). We introduce Propagators & GF • Non-relativistic Prop. the GF for the free Schr¨odinger equation with retarded boundary condition • GF as propagators • S Matrix • ∂ Relativistic Prop. i H (~x′) G (x′,x) = δ4(x′ x), G (x′,x) = 0 for t′ <t • New processes ∂t′ − 0 0 − 0 • Green Function • S Matrix elements • In & Out states ❐ How to find QED F.R.? If φi(~x, t) is a solution of the free Schr¨odinger equation, − Coulomb scattering e + ∂ Coulomb scattering e i H0 φi(~x, t) = 0 γ in internal lines ∂t − Higher Orders γ in external lines the most general solution of the original equation QED Feynman Rules Simple Processes ∂ i H0 ψ = V ψ Compton Scattering ∂t − − − e e+ → µ µ+ QFT Computations is ′ ′ ′ ′ 4 ′ ψ(~x ,t ) = φi(~x ,t ) + d x G0(x ,x)V (x)ψ(x) Z Jorge C. Rom˜ao IDPASC School Braga – 7 IST The non-relativistic propagator ... Summary ❐ We can use this integral equation to establish a perturbative series. Consider QED as a gauge theory that the interaction is localized, that is V (~x, t) 0 as t . Then due Propagators & GF → → −∞ • Non-relativistic Prop. to the retarded GF properties we have • GF as propagators • S Matrix ′ ′ ′ ′ • Relativistic Prop. ′ lim ψ(~x ,t ) = φi(~x ,t ) • New processes t →−∞ • Green Function • S Matrix elements • In & Out states that is in the remote past we have a plane wave. How to find QED F.R.? − ❐ Coulomb scattering e Now if V is small (in some sense) we can solve the integral equation Coulomb scattering e+ perturbatively γ in internal lines Higher Orders ′ ′ ′ ′ 4 ′ ψ(~x ,t ) =φi(~x ,t ) + d x1 G0(x ,x1)V (x1)φi(x1) γ in external lines Z QED Feynman Rules 4 4 ′ Simple Processes + d x1d x2 G0(x ,x1)V (x1)G0(x1,x2)V (x2)φi(x2) Compton Scattering Z − + → − + 4 4 4 ′ e e µ µ + d x1d x2d x3 G0(x ,x1)V (x1)G0(x1,x2)V (x2)G0(x2,x3)V (x3)φi(x3) QFT Computations Z + ··· Jorge C. Rom˜ao IDPASC School Braga – 8 IST The non-relativistic propagator ... Summary ❐ We can look at the perturbative series in another way, in terms of the full QED as a gauge theory GF of the theory with interactions, G(x′,x) Propagators & GF • Non-relativistic Prop. • GF as propagators ∂ ′ ′ ′ 4 ′ • S Matrix i H0(x ) V (x ) G(x ,x) δ (x x) • Relativistic Prop. ∂t − − ≡ − • New processes • Green Function • S Matrix elements ❐ It satisfies • In & Out states How to find QED F.R.? − ′ ′ 4 ′ Coulomb scattering e G(x ,x) = G0(x ,x) + d yG0(x , y)V (y)G(y,x) Coulomb scattering e+ Z γ in internal lines Higher Orders ❐ This leads to the perturbative series (small V ) γ in external lines QED Feynman Rules ′ ′ 4 ′ G(x ,x) =G0(x ,x) + d x1 G0(x ,x1)V (x1)G0(x1,x) Simple Processes Compton Scattering Z 4 4 ′ − − e e+ → µ µ+ + d x1d x2 G0(x ,x2)V (x2)G0(x2,x1)V (x1)G0(x1,x) QFT Computations Z + ··· Jorge C. Rom˜ao IDPASC School Braga – 9 IST Green Functions as Propagators Summary ❐ The last equation allows for suggestive graphical interpretation. We notice QED as a gauge theory ′0 0 0 0 0 that the retarded character of G0 implies x > x3 >x2 >x1 >x . Propagators & GF ··· • Non-relativistic Prop. • GF as propagators ❐ So we have the situation of the following diagrams for the first 3 terms • S Matrix • Relativistic Prop. • New processes • Green Function G(x′,x) =G (x′,x) + d4x G (x′,x )V (x )G (x ,x) • S Matrix elements 0 1 0 1 1 0 1 • In & Out states Z How to find QED F.R.? 4 4 ′ − + d x1d x2 G0(x ,x2)V (x2)G0(x2,x1)V (x1)G0(x1,x) Coulomb scattering e Z Coulomb scattering e+ + γ in internal lines ··· Higher Orders ′ ′ ′ ′ γ in external lines (~x ,t ) (~x ,t ) t ′ ′ t t QED Feynman Rules (~x ,t ) V (~x2,t2) Simple Processes ′ V (~x1,t1) Compton Scattering G0(x ,x) − − V (~x1,t1) e e+ → µ µ+ QFT Computations (~x, t) (~x, t) (~x, t) x x x Jorge C. Rom˜ao IDPASC School Braga – 10 IST Scattering processes and the S Matrix Summary ❐ We are interested in scattering processes. This means that in the past we QED as a gauge theory have a solution of the free equation, a plane wave with momentum ~ki Propagators & GF • Non-relativistic Prop. • GF as propagators 1 i~ki·~x−iωit • S Matrix φi(~x, t) = 3/2 e • Relativistic Prop. (2π) • New processes • Green Function • S Matrix elements ❐ ~ • In & Out states In the future (detector) we have another plane wave with momentum kf How to find QED F.R.? ′ ′ − 1 ~ Coulomb scattering e ′ ′ ikf ·~x −iωf t φf (~x ,t ) = 3/2 e Coulomb scattering e+ (2π) γ in internal lines Higher Orders ❐ The relevant quantity is S matrix element (transition amplitude) γ in external lines QED Feynman Rules 3 ′ ∗ ′ ′ ′ ′ Sfi =′ lim d x φf (~x ,t )ψ(~x ,t ) Simple Processes t →∞ Compton Scattering Z − − 3 ′ ∗ ′ ′ ′ ′ 4 ′ e e+ → µ µ+ = lim d x φf (~x ,t ) φi(~x ,t ) + d x1 G0(x ,x1)V (x1)φi(x1) + t′→∞ ··· QFT Computations Z Z 3 ~ ~ 3 ′ 4 ∗ ′ ′ ′ =δ (kf ki)+ lim d x d x1 φf (~x ,t )G0(x ,x1)V (x1)φi(x1) + − t′→∞ ··· Z Jorge C.