By Peskin, Schroeder.
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We will refer to the particles created by aspy as fermions and to those created by bspy as antifermions. The one-particle states p jp si 2Ep aspy j0i (3:106) are de ned so that their inner product hp rjq si = 2Ep (2 )3 (3) (p ; q) rs (3:107) is Lorentz invariant. This implies that the operator U ( ) that implements Lorentz transformations on the states of the Hilbert space is unitary, even though for boosts, 21 is not unitary. It will be reassuring to do a consistency check, to see that U ( ) implements the right transformation on (x).
48) to compute h0j arp asqy j0i = h0j arp asqy eiP x j0i = ei(p;q) x h0j eiP x arpasqy j0i = ei(p;q) x h0j arp asqy j0i : This says that if h0j apr asqy j0i is to be nonzero, p must equal q. Similarly, it can be shown that rotational invariance of j0i implies r = s. ) From these considerations we conclude that 56 Chapter 3 The Dirac Field the matrix element can be written h0j arp asqy j0i = (2 )3 (3) (p ; q) rs A(p) where A(p) is so far undetermined. Note, however, that if the norm of a state is always positive (as it should be in any self-respecting Hilbert space), A(p) must be greater than zero.
60) are, from a practical point of view, the most important results of this chapter. The Green's function DF (x ; y) is called the Feynman propagator for a Klein-Gordon particle, since it is, after all, a propagation amplitude. Indeed, the Feynman propagator will turn out to be part of the Feynman rules: DF (x ; y) (or De F (p)) is the expression that we will attach to internal lines of Feynman diagrams, representing the propagation of virtual particles. Nevertheless we are still a long way from being able to do any real calculations, since so far we have talked only about the free Klein-Gordon theory, where the eld equation is linear and there are no interactions.
Quantum field theory by Peskin, Schroeder.
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