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On the Green’s functions of quantized fields. II

Communicated May 22, 1951
In all of the work of the preceding note there has been no explicit reference to the particular states on σ_{1} and σ_{2} 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 σ_{1} and σ_{2}. The vacuum, as the lowest energy state of the system, can be defined only if, in the neighborhood of σ_{1} and σ_{2}, the actual external electromagnetic field is constant in some timelike direction (which need not be the same for σ_{1} and σ_{2}). In the Dirac oneparticle Green’s function, for example,the temporal variation of Ψ(x) in the vicinity of σ_{1} can then be represented bywhere P_{0} is the energy operator and X is some fixed point. Therefore,in which P_{0}^{vac} is the vacuum energy eigenvalue. Now P_{0} – P_{0}^{vac} has no negative eigenvalues, and accordingly G(x, x′), as a function of x_{0} in the vicinity of σ_{1}, contains only positive frequencies, which are energy values for states of unit positive charge. The statement is true of every timelike direction, if the external field vanishes in this neighborhood.
A representation similar to ( 26) for the vicinity of σ_{2} yieldswhich contains only negative frequencies. In absolute value, these are the energies of unit negative charge states. We thus encounter Green’s functions that obey the temporal analog of the boundary condition characteristic of a source radiating into space.^{1} In keeping with this analogy, such Green’s functions can be derived from a retarded proper time Green’s function by a Fourier decomposition with respect to the mass.
The boundary condition that characterizes the Green’s functions associated with vacuum states on σ_{1} and σ_{2} involves these surfaces only to the extent that they must be in the region of outgoing waves. Accordingly, the domain of these functions may conveniently be taken as the entire fourdimensional space. Thus, if the Green’s function G_{+}(x, x′), defined by (14), (16), and the outgoing wave boundary condition, is represented by the integrodifferential equation,the integration is to be extended over all spacetime. This equation can be more compactly written asby regarding the spacetime coordinates as matrix indices. The mass operator M is then symbolically defined byIn these formulae, A_{+} and δ/δJ are considered to be diagonal matrices,There is some advantage, however, in introducing “photon coordinates” explicitly (while continuing to employ matrix notation for the “particle coordinates”). Thuswhere γ(ξ) is defined byThe differential equation for A_{+}(ξ) can then be writtenwhere Tr denotes diagonal summation with respect to spinor indices and particle coordinates. The associated photon Green’s function differential equation is
To express the variational derivatives that occur in ( 31) and ( 36) we introduce an auxiliary quantity defined byThusfrom which we obtainandWith the introduction of matrix notation for the photon coordinates, this Green’s function equation becomesand the polarization operator P is given byIn this notation, the mass operator expression readswhere Tρ denotes diagonal summation with respect to the photon coordinates, including the vector indices.
The twoparticle Green’s functioncan be represented by the integrodifferential equationthereby introducing the interaction operator I_{12}. The unit operator 1_{12} is defined by the matrix representationOn comparison with (21) we find that the interaction operator can be characterized symbolically bywhere G_{1} and G_{2} are the oneparticle Green’s functions of the indicated particle coordinates.
The various operators that enter in the Green’s function equations, the mass operator M, the polarization operator P, the interaction operator I_{12}, can be constructed by successive approximation. Thus, in the first approximation,whereand the Green’s functions that appear in these formulae refer to the 0th approximation (M = m, P = 0). We also have, in the first approximation,Perturbation theory, as applied in this manner, must not be confused with the expansion of the Green’s functions in powers of the charge. The latter procedure is restricted to the treatment of scattering problems.
The solutions of the homogeneous Green’s function equations constitute the wave functions that describe the various states of the system. Thus, we have the oneparticle wave equationand the two particle wave equationwhich are applicable equally to the discussion of scattering and to the properties of bound states. In particular, the total energy and momentum eigenfunctions of two particles in isolated interaction are obtained as the solutions of ( 52) which are eigenfunctions for a common displacement of the two spacetime coordinates. It is necessary to recognize, however, that the mass operator, for example, can be largely represented in its effect by an alteration in the mass constant and by a scale change of the Green’s function. Similarly, the major effect of the polarization operator is to multiply the photon Green’s function by a factor, which everywhere appears associated with the charge. It is only after these renormalizations have been performed that we deal with wave equations that involve the empirical mass and charge, and are thus of immediate physical applicability.
The details of this theory will be published elsewhere, in a series of articles entitled “The Theory of Quantized Fields.”
References
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Green’s functions of this variety have been discussed by Stueckelberg, E. C. G., Helv. Phys. Acta, 19, 242 (1946), and by Feynman, R. P., Phys. Rev., 76, 749 (1949).