Section_4.pdf - Section 4 Physics 253A Alek Bedroya Fall...

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Section 4 - Physics 253AAlek BedroyaFall 20191Read off momentum-space Feynman rules from a LagrangianProblem:What are the momentum-space Feynman rules of the following Lagrangian?L=-12φ1(+m2)φ1-12φ2(+ 2m2)φ2-α2φ21φ2+βφ21φ22+γ2φ1(μφ2)(μφ2)(1)Solution:The kinetic terms give Feynman propagators as derived in class. Note that the fieldφ2has a mass term with anextra factor of two:φ1φ1=ip2-m2+φ2φ2=ip2-2m2+The interaction terms always come with a factor ofifrom the Teylor expansion of expiRd4xLint(see section 3).Derivatives give a factor ofitimes the momentum of the particle, with a plus sign if the particle is created at thevertex and a minus sign if a particle is annihilated. If the same field appearsntimes in an interaction, we usuallynormalize that term with1n!to compensate for the different ways the field can be ordered. The interaction verticesof the above Lagrangian are therefore:1
φ1φ1φ2=-iα,pqφ1φ2φ2=(ipμ)(iqμ),pqφ1φ2φ2=(-ipμ)(iqμ),φ1φ1φ2φ2=i4β,2Hamiltonian derivation of the Feynman rules2.1GoalRecall that by the LSZ reduction formula,S-matrix elements can be written ashf|S|ii=iZd4x1e-ip1x1(1+m2)· · ·iZd4xne-ipnxn(n+m2)× hΩ|T{φ1φ2· · ·φn} |Ωi,(2)whereφiφ(xi). The left hand side of this equation can be directly related to observables, recall for example thata differential cross sectioncan be written as=12E12E2|~v1-~v2||hf|S|ii|2dΠLIPS,(3)where the labels1and2refer to the incoming particles. The right hand side is what we aim to calculate in QFT,usually by using perturbation theory and Feynman diagrams. In the expressionhΩ|T{φ1φ2· · ·φn} |Ωi, the state|Ωiis the ground state of the full Hamiltonian, written in the Heisenberg picture, andφiare the particle fields inthe Heisenberg picture, evolving with the full Hamiltonian. For a general Hamiltonian, the full theory will usuallybe too complicated to solve exactly. To get expressions for theφ’s and|Ωi, we aim to write our equations in termsof thefreevacuum|0iand thefreefieldsφ0(xi), which evolve with a part of the HamiltonianH0that has a knownsolution. This will in practice usually be the free Hamiltonian, without any interactions between the fields. Notethat we really do know how the free system behaves, the free vacuum|0iwill be the state with no particles in it, andif there are any particles they will propagate freely without ever interacting or changing their paths. For example,for the free HamiltonianH0in QED, there could be states in the theory with some number of electrons, photonsand positrons, but there would not be any Coulomb force or other types of interactions between any particles. Tosummarize:The goal is to writehΩ|T{φ1φ2· · ·φn} |Ωiin terms of the free vacuum|0iand the free fieldsφ0(xi).

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