Vacuum expectation values as sums over histories

Abstract

The way in which vacuum expectation values may be calculated as sums over histories, according to Feynman’s formulation of quantum mechanics, is discussed. It is shown that the evaluation of the functional integrals for the matrix elements of chronologically ordered operator products does not automatically lead to the propagators of quantized fields, as seemed to be so in a paper by Matthews and Salam; their work is criticized for its vagueness about boundary conditions. A limiting procedure for obtaining vacuum expectation values is defined: it involves the device of introducing an infinitesimal imaginary term into the Lagrangian and letting the epoch in which the matrix element is specified tend to infinity. It is demonstrated that the device commonly used in field theory for defining Feynman Green’s functions is of this type, but that it does not necessarily lead to vacuum expectation values when the Lagrangian is not quadratic. An alternative trick which is more generally applicable is suggested: this involves the use of a complex time parameter, which selects out the ground state in the limit of infinite epoch in a manner similar to the freezing into the ground state which occurs in thermal equilibrium at the absolute zero of temperature.