lecnotes12 - Physics 7270 Intro To Quantum Mechanics III...

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Physics 7270: Intro To Quantum Mechanics IIIMENUContractions and Wick’s theoremPicking up from last time, we showed that the -field time-ordered matrix elements we need for the LSZformula can be written in terms of time-ordered matrix elements of free fields, including an term that we series expand to do perturbation theory. We ended by arguing that thesefree-field matrix elements are always zero if they contain an odd number of fields.This heuristic argument tells us which terms are zero, but we still don’t know what the terms that survive areequal to. (We expect to get a combination of Feynman propagators, following our Schwinger-Dysonderivation.) The general result is given by Wick’s theorem, which I will state first and then explain:A contractionis a pairing of two fields within the time-ordered product, which replaces those two fields withthe Feynman propagator connecting their two points. In other words, contraction replacesThe standard notation for field contraction which you will see in some books is to connect the two fields witha square bracket above or below the expression:Schwartz doesn’t use this notation, possibly because it’s annoying to typeset. I’ll also try to avoid it for themost part, but it can be useful in counting contractions sometimes.The other as-yet undefined symbol above are the colon brackets . These represent the normal orderingoperation. Normal ordering is sort of like time ordering, in that it shoves around operators with no regard forcommutation. In particular, normal ordering takes any expression containing ladder operators and pushes all operators to the left of every . For example,When normal ordering acts on a pure number, it has no effect. The main purpose of normal ordering is that itgives objects that always vanish when taking vacuum expectation values:This is because we’ve pushed any possible creation and annihilation operators towards the side on whichthey will annihilate the vacuum.Now we see the content of Wick’s theorem: the only things that will survive the normal ordering operatorwhen the vacuum states are present are fully contractedexpressions, where all of the fields are replacedwith Feynman propagators. Acting on a two-field time ordered product gives us back our familiar result:nexp[ix]d4int{(). . .()} =:(). . .() + (all contractions) :ϕ0x1ϕ0xnϕ0x1ϕ0xnϕ()ϕ().x1x2D12::aa: (+)(+) :=+++.apapakakapakakapapakapak0| :(). . .() : |0= 0.ϕ0x1ϕ0xn{()()} =:()() : +(,)ϕ0x1ϕ0x2ϕ0x1ϕ0x2DFx1x2
(and sandwiching between the vacuum gets rid of the normal-ordered term.)I won’t prove Wick’s theorem, but the proof is very straightforward: using the definition of the Feynmanpropagator and the split fields makes it easy to show the two-field case above, and the general proof

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