### Refine

#### Document Type

- Preprint (3)

#### Language

- English (3)

#### Has Fulltext

- yes (3)

#### Is part of the Bibliography

- no (3)

#### Faculty / Organisational entity

Abstract: We aim to establish a link between path-integral formulations of quantum and classical field theories via diagram expansions. This link should result in an independent constructive characterisation of the measure in Feynman path integrals in terms of a stochastic differential equation (SDE) and also in the possibility of applying methods of quantum field theory to classical stochastic problems. As a first step we derive in the present paper a formal solution to an arbitrary c-number SDE in a form which coincides with that of Wick's theorem for interacting bosonic quantum fields. We show that the choice of stochastic calculus in the SDE may be regarded as a result of regularisation, which in turn removes ultraviolet divergences from the corresponding diagram series.

We show that the solution to an arbitrary c-number stochastic differential equation (SDE) can be represented as a diagram series. Both the diagram rules and the properties of the graphical elements reflect causality properties of the SDE and this series is therefore called a causal diagram series. We also discuss the converse problem, i.e. how to construct an SDE of which a formal solution is a given causal diagram series. This then allows for a nonperturbative summation of the diagram series by solving this SDE, numerically or analytically.

Abstract: We describe quantum-field-theoretical (QFT) techniques for mapping quantum problems onto c-number stochastic problems. This approach yields results which are identical to phase-space techniques [C.W. Gardiner, Quantum Noise (1991)] when the latter result in a Fokker-Planck equation for a corresponding pseudo-probability distribution. If phase-space techniques do not result in a Fokker-Planck equation and hence fail to produce a stochastic representation, the QFT techniques nevertheless yield stochastic di erence equations in discretised time.