On the Green’s functions of quantized fields. II

Abstract

Communicated May 22, 1951 In all of the work of the preceding note there has been no explicit reference to the particular states on σ1 and σ2 that enter in the definitions of the Green’s functions. This information must be contained in boundary conditions that supplement the differential equations. We shall determine these boundary conditions for the Green’s functions associated with vacuum states on both σ1 and σ2. The vacuum, as the lowest energy state of the system, can be defined only if, in the neighborhood of σ1 and σ2, the actual external electromagnetic field is constant in some time-like direction (which need not be the same for σ1 and σ2). In the Dirac one-particle Green’s function, for example,the temporal variation of Ψ(x) in the vicinity of σ1 can then be represented bywhere P0 is the energy operator and X is some fixed point. Therefore,in which P0vac is the vacuum energy eigenvalue. Now P0 – P0vac has no negative eigenvalues, and accordingly G …