White-noise calculation of annihilation and creation operators in stochastic mechanics

Abstract

A second quantization procedure is formulated in the framework of Nelson’s stochastic mechanics. It is shown that the annihilation operator is given by an integration over a Gaussian white noise Ḃ(s) with the kernel exp(iω(t−s)) and the creation operator is given by an integration over a time reversed Gaussian white noise Ḃ*(s) with the kernel exp(−iω(t−s)). The eigenfunctions on which the operators act are shown to be Gaussian random variables whose collection is dense in a Hilbert space. Employing the above operators, we can construct {N}-representation in the stochastic mechanics.