On the Green’s functions of quantized fields. I

The temporal development of quantized fields, in its particle aspect, is described by propagation functions, or Green’s functions. The construction of these functions for coupled fields is usually considered from the viewpoint of perturbation theory. Although the latter may be resorted to for detailed calculations, it is desirable to avoid founding the formal theory of the Green’s functions on the restricted basis provided by the assumption of expandability in powers of coupling constants. These notes are a preliminary account of a general theory of Green’s functions, in which the defining property is taken to be the representation of the fields of prescribed sources.

We employ a quantum dynamical principle for fields which has been described elsewhere.¹ This principle is a differential characterization of the function that produces a transformation from eigenvalues of a complete set of commuting operators on one space-like surface to eigenvalues of another set on a different surface,²

Here ℒ is the Lagrange function operator of the system. For the example of coupled Dirac and Maxwell fields, with external sources for each field, the Lagrange function may be taken aswhich implies the equations of motionwhereWith regard to commutation relations, we need only note the anticommutativity of the source spinors with the Dirac field components.

We shall restrict our attention to changes in the transformation function that arise from variations of the external sources. In terms of the notationthe dynamical principle can then be writtenwhereThe effect of a second, independent variation is described byin which the notation ( )₊ indicates temporal ordering of the operators. As examples we haveandThe latter result can be expressed in the notationalthough one may supplement the right side with an arbitrary gradient. This consequence of the charge conservation condition, ∂_μJ_μ = 0, corresponds to the gauge invariance of the theory.

A Green’s function for the Dirac field, in the absence of an actual spinor source, is defined byAccording to ( 9), and the anticommutativity of δη(x′) with ψ(x), we havewhere . On combining the differential equation for with ( 11), we obtain the functional differential equationAn accompanying equation for is obtained by noting thatin which the trace refers to the spinor indices, and an average is to be taken of the forms obtained with . Thus, with the special choice of gauge, , we have

The simultaneous equations ( 14) and ( 16) provide a rigorous description of G(x, x′) and 〈A_μ(x)〉.

A Maxwell field Green’s function is defined by

The differential equations obtained from ( 16) and the gauge condition are

More complicated Green’s functions can be discussed in an analogous manner. The Dirac field Green’s function defined bymay be called a “two-particle” Green’s function, as distinguished from the “one-particle” G(x, x′). It is given explicitly by

This function is antisymmetrical with respect to the interchange of x₁ and x₂, and of x₁′ and x₂′ (including the suppressed spinor indices). It obeys the differential equationwhere 𝕱 is the functional differential operator of ( 14). More symmetrically written, this equation readsin which the two differential operators are commutative.

The replacement of the Dirac field by a Kemmer field involves alterations beyond those implied by the change in statistics. Not all components of the Kemmer field are dynamically independent. Thus, if 0 refers to some arbitrary time-like direction, we havewhich is an equation of constraint expressing in terms of the independent field components , and of the external source. Accordingly, in computing we must take into account the change induced in , whence

The temporal ordering is with respect to the arbitrary time-like direction.

The Green’s function is independent of this direction, however, and satisfies equations which are of the same form as ( 14) and ( 16), save for a sign change in the last term of the latter equation which arises from the different statistics associated with the integral spin field.