Quantum Field Theory IIPHYS P 622Radovan Dermisek Indiana UniversityNotes based on M Srednicki Quantum Field TheoryChapters 13 14 16 21 26 28 51 52 61 68 44 53 69 74 30 32 84 86 75 87 89 291Review of scalar eld theorySrednicki 5 9 102 The LSZ reduction formulabased on S 5In order to describe scattering experiments we need to construct appropriate initial and nal states and calculate scattering amplitude Summary of free theory one particle state vacuum state is annihilated by all a s then one particle state has normalization normalization is Lorentz invariant see e g Peskin Schroeder p 233Let s de ne a time independent operator that creates a particle localized in the momentum space near and localized in the position space near the origin wave packet with width go back to position space by Furier transformation is a state that evolves with time in the Schr dinger picture wave packet propagates and spreads out and so the particle is localized far from the origin in at for is a state describing two particles widely separated in the past In the interacting theory is not time independent 4 A guess for a suitable initial state Similarly let s consider a nal state The scattering amplitude is then we can normalize the wave packets so that where again and 5A useful formula Integration by parts surface term 0 particle is localized wave packet needed E g is 0 in free theory but not in interacting one 6 Thus we have or its hermitian conjugate The scattering amplitude is then given as generalized to n i and n f particles we put in time ordering without changing anything 7Lehmann Symanzik Zimmermann formula LSZ Note initial and nal states now have delta function normalization multiparticle generalization of We expressed scattering amplitudes in terms of correlation functions Now we need to learn how to calculate correlation functions in interacting quantum eld theory 8 we can always shift the eld by a constant we want Comments we assumed that creation operators of free eld theory would work comparably in the interacting theory acting on ground state is a Lorentz invariant numberso that is a single particle state otherwise it would create a linear combination of the ground state and a single particle stateso that 9since this is what it is in free eld theory creates a correctly normalized one particle state we can always rescale renormalize the eld by a constant we want one particle state is a Lorentz invariant numberso that 10 multiparticle states is a Lorentz invariant numberin general creates some multiparticle states One can show that the overlap between a one particle wave packet and a multiparticle wave packet goes to zero as time goes to in nity By waiting long enough we can make the multiparticle contribution to the scattering amplitude as small as we want see the discussion in Srednicki p 40 4111Scattering amplitudes can be expressed in terms of correlation functions of elds of an interacting quantum eld theory Summary Lehmann Symanzik Zimmermann formula LSZ provided that the elds obey these conditions might not be consistent with the original form of lagrangian 12 Consider for example After shifting and rescaling we will have instead 13Path integral for interacting eld based on S 9Let s consider an interacting phi cubed QFT with elds satisfying we want to evaluate the path integral for this theory 14 it can be also written as and for the path integral of the free eld theory we have found epsilon trick leads to additional factor to get the correct normalization we require 15 we will nd and in the limit we expect and assumes thus in the case of the perturbing lagrangian is counterterm lagrangian16 Let s look at Z J ignoring counterterms for now De ne exponentials de ned by series expansion let s look at a term with particular values of P propagators and V vertices number of surviving sources after taking all derivatives E for external isE 2P 3V3V derivatives can act on 2P sources in 2P 2P 3V different wayse g for V 2 P 3 there is 6 different terms17V 2 E 0 P 3 dx1 dx23 3 2 2 22 6 6 3 2 2 2124x1x2symmetry factor iZgg 21i x1 x2 1i x1 x2 1i x1 x2 18 V 2 E 0 P 3 dx1 dx23 3 3 2 2 6 6 3 2 2 2x1x218symmetry factor iZgg 21i x1 x1 1i x1 x2 1i x1 x1 19Feynman diagrams vertex joining three line segments stands for a line segment stands for a propagatora lled circle at one end of a line segment stands for a sourcee g for V 1 E 1What about those symmetry factors What about those symmetry factors symmetry factors are related to symmetries of Feynman diagrams 20 Symmetry factors we can rearrange three derivatives without changing diagramwe can rearrange three vertices we can rearrange two sourceswe can rearrange propagatorsthis in general results in overcounting of the number of terms that give the same result this happens when some rearrangement of derivatives gives the same match up to sources as some rearrangement of sources this is always connected to some symmetry property of the diagram factor by which we overcounted is the symmetry factor21propagators can be rearranged in 3 ways and all these rearrangements can be duplicated by exchanging the derivatives at the verticesthe endpoints of each propagator can be swapped and the effect is duplicated by swapping the two vertices22 2324 2526 2728 29All these diagrams are connected but Z J contains also diagrams that are products of several connected diagrams e g for V 4 E 0 P 6 in addition to connected diagrams we also have and also and also 30 All these diagrams are connected but Z J contains also diagrams that are products of several connected diagrams e g for V 4 E 0 P 6 in addition to connected diagrams we also have A general diagram D can be written as particular connected diagramadditional symmetry factornot already accounted for by symmetry factors of connected diagrams it is nontrivial only if D contains identical C s the number of given C in D31imposing the normalization means we can omit vacuum diagrams those with no sources thus we have thus we have found that is given by the exponential of the sum of connected diagrams Now is given by summing all diagrams D any D can be labeled by a set of n svacuum diagrams are omitted from the sum32 we used since we knowIf there were no counterterms we would be done in that case the vacuum expectation value of the eld is and we nd the source is removed by the derivative only diagrams with one source contribute which is not zero as required for the LSZ so we need counterterm33in order to satisfy we have