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17.1) Show that the retarded field propagator for a free particle in momentum space and the time domain: given by θ(te-ty)e-i(Epte-Eqty's(3) (p-q) 17.1 The field propagntor in outline e field propagator in outline 155 he field propagator involves a simple thought experi- our interacting system in its ground state, which we interactin ent. We start with denote w 12). The thought experiment works as follows: we introdu extra particle of our choice the system. point ( in anni icle of our choice. We will use this extra particle to probe n extThe new particle is introduced (created) at a spacetime tinteracts with the system, possibly causing excitations of complicated things. Then we remove uilate) the particle at spacetime point (r, a) and ask if the syster remained in the interacting ground state 2). We're interested in the ,0 . and all manner amplitude G(a.g) for the experiment, given by G"(zw)-el( Particle annihilated ) ( Particle created )|Ω> at (y) 17.1 is, the probability amplitude that the system is still in its ground after we create a particle at y and later annihilate it at . The state amplitude for this process will depend on the complicated interaction of the probe particle with the s ystem and, it will turn out, will tell us a eat deal about the system itself. By analogy with the results of the previous chapter, the amplitude GH(a, is chapter, the amplitude G (z,g) is called the Green's function ca gator.2 So much for definitions. The mathematical object that 2We call the propagator for an interact- ing theory G(r. y) and its ground state 2). For a free theory (with no interac- (17.2) tions) we call the propagator Golz.y) or sometimes we will use the equiva tells this story of the propagator outlined above is where the + sign on the left tells us that the particle is created at y lent notation A(a. ) and the groun before being annihilated at r and the 6-function on the right guarantees state lo) this. Example 17.1 To see that this function does the job we can break down the action of the opera- tors on the states into stages. Remembering that a Heisenberg operator has a time dependence defined by ф(t,a:) e+iFtd(a)e-iHt, and substitution of this gives an expression for the propagator (17.3) Taking this one stage at a time (and temporarily ignoring the possibility of creating and annihilating antiparticles), we see that: . e-iftyon) is the state } 2) evolved to a time yo . φ"(v)e-invo Ω) is that state with a particle added at time yo at a position y t (e-i' 12) is that state time evolved to time . Now hitting this state with e () removes the particle at time ro o We terminate this string on the left with a (21 to find out how much of tie original state |2) is left in the final state. e propagator is, therefore, the amplitude that we put a particle in the system at pusition y at time y" and get it out at position æ time a 156 Propagators and fields 17.2 The Feynman propagator We don't always want to deal with the particle constrained with the 6-function so that zo agator fields. The reason is that although Gt acco purticles it misses out the essential information abour so that >) when think Example 17.2 the field creation operator for complex scalia fielg àleįpr + bpe" ip z), which creates a lates an antiparticle. If this acts on the free vacuum ψΟιο ictable can occur) then we create a particle at , but the es the vacuum bpl0) -0. Similarly, only the particle ie aj Notation for propagators to (olv(xr), so the propagator (tc) (n)10) describes a particle y to a, but tells us nothing about antiparticles In general Free propagator Free scalar fields Photon helds Fermion fielda G(a,y) or iS(r.y) Gole, v Δ(z, y) to tn mln alle le Richard Feynman struggled over how to inclhude the antip bution for some time before deciding that the most useful propagator was one that summed both particle and ant This form of the propagator makes up the guts of most quant up calculations and is called the Feynman propagator. H the propagator Feynman used which contains both particle and ticle parts? We need to introduce a new piece of machinery: the Wi time-ordering symbol T. This isn't an operator, but is nch like the normal ordering symbol N in that it is just an instructia what to do to a string in order that it makes sense. The time-orderin symbol is defined for scalar fields3 as l antiparticle ow do Gian-Carlo Wick (1909-1992) 'This form applies only to the bomonic Le. commuting) scalar fields we're con- sidering in this chapter. For an anti- commuting Fermi field ψ, we pick up s minus sign every time we swap the order of the operators, so we have so that the scalar fields are always arranged earliest on the right, la on the left. The Feynman propagator is then defined as erro)o(vo) zo >so θ(zo-yo)(12lo(x)φ"(y)IS2)+θ(yo-royolot(w)0(-10 |.unne where (2) is the interacting ground state of the system. The propagu is therefore made up of two parts. The first part applies for d than y": it creates a particle at y and propagates it to r whe destroyed. The second part applies when y is later thau an antiparticle at r and propagates the system to point y. Botl are included in the total propagator. If, as in the example above, the system doesn't contain any tions then particles just move around passing throu 17.2 The Feynman propagator 157 nd state j0) and we have the free propagator callthegnJ)or, for scalar fields, more usually just Δ(z,y) (17.6) gator is depicted in Fig. 17.1. The free propagator t of the structure of all perturbation calculations think of interactions as events that take place at fie psential par nacetine ote freely between interactions with the free the particles propagate freely betwee t derive an expression for the free Feynman propagator Fig. 17.1 The free-particle propagator Δ(z,y) is a line without an interaction blob since a free particle doesn't inter- act with any other particles on its way from y tor Example 17.3 lesgehilation field acting on the free vacuum ф"(U)10), we have for the free propagator. If we use the general expression for pars the particle creation part contributes (since bp 0) 0)a (2T) (2Ep) tipa-ioe do ave get the other half? Take a complex conjugate of eqn 17.8 (swapping Will y→ r and p → g) to get nore derin Sandwiching together eqn 17.8 and eqn 17.9 we obtain e-ip(-y) (17.10) Fig. 17.2 A third-order termin (2n)3(2Ep ) the perturbation expansion of G in terms ot Δ14. for the process is proportional to The ampli itich corresponds to a particle being created at (yol) and propagating to (20,2) vhere it is annihilated at the later time. We also need to consider the reverse order for the advanced part of the propagator: A)V(a) Here, only the antiparticle creation part contributes (since apl。) = 0), to give (17.12) Again we take the complex conjugate (and change-y and p → q): (17.13) telding,finally ip-(z-y) etroyed at the later time. Putting the two halves (eqns 17.10 we have our final answer for the free propagator Δ(r,v) an antiparticle being created at point (a, z) and propagating it is to

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