The mathematical physical equations satisfied by retarded and advanced Green's functions

TL;DR

This paper derives differential equations for retarded and advanced Green's functions in mathematical physics, clarifying their properties and the role of initial conditions, and discusses the fundamental issue of irreversibility in time.

Contribution

It establishes rigorous differential equations for retarded and advanced Green's functions, incorporating initial conditions and clarifying their mathematical properties.

Findings

Differential equations for retarded and advanced GFs are rigorously derived.

The derivative of the step function involves a delta function and an infinitesimal term.

The work explains the absence of a causal Green's function in mathematical physics.

Abstract

In mathematical physics, time-dependent Green's functions (GFs) are the solutions of differential equations of the first and second time derivatives. Habitually, the time-dependent GFs are Fourier transformed into the frequency space. Then, analytical continuation of the frequency is extended to below or above the real axis. After inverse Fourier transformation, retarded and advanced GFs can be obtained, and there may be arbitrariness in such analytical continuation. In the present work, we establish the differential equations from which the retarded and advanced GFs are rigorously solved. The key point is that the derivative of the time step function is the Dirac delta function plus an infinitely small quantity, where the latter is not negligible because it embodies the meaning of time delay or time advance. The retarded and advanced GFs defined in this paper are the same as the one-body GFs defined with the help of the creation and destruction operators in many-body theory. There is no way to define the causal GF in mathematical physics, and the reason is given. This work puts the initial conditions into differential equations, thereby paving a way for solving the problem of why there are motions that are irreversible in time.