Annihilation and creation operators in stochastic quantization.

Abstract

Annihilation and creation operators in quantum mechanics are constructed in the framework of Nelson's stochastic mechanics. Employing Ito's stochastic calculus, annihilation and creation operators were found to be written as (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B\ifmmode \dot{}\else \.{}\fi{}(t)] and (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t)], respectively, where B\ifmmode \dot{}\else \.{}\fi{}(t) is a Gaussian white noise and B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t) is a time-reversed Gaussian white noise. The operators were shown to have proper commutation relations and to provide a Hamiltonian which has an eigenvalue equation desired in the {N} representation as should exist in conventional quantum mechanics. Therefore, the present formulation provides a correspondence between stochastic mechanics and conventional quantum mechanics.