Topological field theory of dynamical systems. II

TL;DR

This paper advances the understanding of the connection between stochastic dynamical systems and topological field theories by incorporating ghost number operators to relate expectation values and interpret TFT wavefunctions as conditional probabilities.

Contribution

It introduces the necessity of including the ghost number operator in the TFT to properly relate to stochastic expectations and interprets TFT wavefunctions as conditional probabilities.

Findings

Inclusion of $(-1)^{ ext{ghost number}}$ in TFT links to stochastic expectation values.

TFT wavefunctions correspond to conditional probability densities.

Unfolds the path-integral representation of TFT to connect with stochastic partition functions.

Abstract

This paper is a continuation of the study [Chaos.22.033134] of the relation between the stochastic dynamical systems (DS) and the Witten-type topological field theories (TFT). Here, it is discussed that stochastic expectation values of a DS must be complemented on the TFT side by $(-1)^{\hat F}$, where $\hat F$ is the ghost number operator. The role of this inclusion is to unfold the natural path-integral representation of the TFT, \emph{i.e.}, the Witten index that equals up to a topological factor to the partition function of the stochastic noise, into the physical partition function of TFT/DS. It is also shown that on the DS side, the TFT's wavefunctions are the conditional probability densities.