In fact because this is true not just for free fields

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ical: all the creation of the initial states happens before the annihilation of the final states. Infact, because this is true not just for free fields, all the crazy stuff which happens at interme-diate times in an interacting theory must be time-ordered too. But the great thing is that wedon’t need to know which are the initial states and which are the final states anymore when wedo the hard part of the computation. We just have to calculate time ordered products, and theLSZ formula sorts out what is being scattered for us.2.2LSZ for operatorsFor perturbation theory in the standard model, which is mostly what we will study in these lec-tures, the LSZ formula in the above form is all that is needed. However, the LSZ formula ismore powerful than it seems and applies even if we don’t know what the particles are.If you go back through the derivation, you will see that we never needed an explicit form forthe full fieldφ(x)and its strange creation operatorsap(t)which did not necessarily evolve likecreation operators in the free theory. In fact, all we used was that the fieldφ(x)creates freeparticle states at asymptotic times. So the LSZ reduction actually implies(p3pn|S|p1p2)=bracketleftbiggiintegraldisplayd4x1eip1x1(square1+m2)bracketrightbiggbracketleftbiggiintegraldisplayd4xneipnxn(squaren+m2)bracketrightbigg(21)×(Ω|T{O1(x1)O2(x2)O3(x3)On(xn)}|Ω)(22)where theOi(x)are any operators which can create single particle states. By this we mean, that(p|O(x)|Ω)=Zeipx(23)for some numberZ. LSZ does not distinguishelementaryparticles, which we define to meanparticles which have corresponding fields appearing in the Lagrangian, from any other type ofparticle. Anything which overlaps with 1-particle states will produce an appropriate pole to becanceled by thesquare+m2factors giving a non-zeroS-matrix element. Therefore particles in theHilbert space can be produced whether or not we have elementary fields for them.2. It should not be obvious at this point that there cannot be higher order poles, such as1(p2-m2)2, comingout of time-ordered products.Such terms would signal the appearance of unphysical states known as ghosts,which violate unitarity. The fastest a correlation function can decay at largep2in a unitary theory is asp-2, aresult we will prove in Lecture III-10.4Section 2
It is probably worth saying a little more about what these operatorsOn(x)are. The opera-tors can be defined as they would be in quantum mechanics, by their matrix elements in a basisof statesψnof the theoryCnm=(ψn|O|ψm). Any such operator can be written as a sum overcreation and annihilation operatorsO=summationdisplayn,mintegraldisplaydq1dqndp1dpmaq1qqnapmap1Cnm(q1, ,pm)(24)It is not hard to prove [cf Weinberg 4.2] that theCnmare in one-to-one correspondence with thematrix elements ofOinnandmparticle states. One can turn the operator into a functional ofthe fields, using Eq.(14) and its conjugate.The most important operators in relativistic

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