[Paper Review] Transfer operators and topological field theory

I. V. Ovchinnikov|arXiv (Cornell University)|Aug 20, 2013

Quantum chaos and dynamical systems42 references3 citations

TL;DR

This paper establishes a rigorous connection between transfer operators in dynamical systems and topological field theory via a supersymmetric formulation of stochastic differential equations (SDEs). It shows that the generalized transfer operator (GTO) is the finite-time Fokker-Planck operator, and its sharp trace and determinant are trivialized via supersymmetry, with the Witten index equaling the Euler characteristic. The spontaneous breakdown of topological supersymmetry (Q-symmetry) is identified as the field-theoretic definition of stochastic chaos, explaining the butterfly effect and long-range correlations.

ABSTRACT

The transfer operator (TO) formalism of the dynamical systems (DS) theory is reformulated here in terms of the recently proposed supersymetric theory of stochastic differential equations (SDE). It turns out that the stochastically generalized TO (GTO) of the DS theory is the finite-time Fokker-Planck evolution operator. As a result comes the supersymmetric trivialization of the so-called sharp trace and sharp determinant of the GTO, with the former being the Witten index, which is also the stochastic generalization of the Lefschetz index so that it equals the Euler characteristic of the (closed) phase space for any flow vector field, noise metric, and temperature. The enabled possibility to apply the spectral theorems of the DS theory to the Fokker-Planck operators allows to extend the previous picture of the spontaneous topological supersymmetry (Q-symmetry) breaking onto the situations with negative ground state's attenuation rate. The later signifies the exponential growth of the number of periodic solutions/orbits in the large time limit, which is the unique feature of chaotic behavior proving that the spontaneous breakdown of Q-symmetry is indeed the field-theoretic definition and stochastic generalization of the concept of deterministic chaos. In addition, the previously proposed low-temperature classification of SDEs, i.e., thermodynamic equilibrium / noise-induced chaos ((anti)instanton condensation, intermittent) / ordinary chaos (non-integrability of the flow vector field), is complemented by the discussion of the high-temperature regime where the sharp boundary between the noise-induced and ordinary chaotic phases must smear out into a crossover, and at even higher temperatures the Q-symmetry is restored. The Weyl quantization is discussed in the context of the Ito-Stratonovich dilemma.

Motivation & Objective

• To establish a rigorous supersymmetric formulation of stochastic differential equations (SDEs) that generalizes topological field theory to stochastic dynamics.
• To resolve the Ito-Stratonovich dilemma in SDEs by deriving the correct Fokker-Planck operator from infinitesimal Lie derivative evolution.
• To provide a field-theoretic definition of stochastic chaos as the spontaneous breakdown of topological supersymmetry (Q-symmetry).
• To unify deterministic and stochastic chaos under a single framework using spectral theorems of dynamical systems applied to Fokker-Planck operators.
• To extend the low-temperature classification of SDEs (equilibrium, noise-induced chaos, ordinary chaos) to include high-temperature regimes where Q-symmetry is restored.

Proposed method

• Reformulates the transfer operator formalism of dynamical systems in terms of the supersymmetric theory of SDEs (STS), identifying the GTO as the finite-time Fokker-Planck evolution operator.
• Derives the infinitesimal time evolution via the Lie derivative of differential forms, using the stochastic flow vector field and noise basis vectors.
• Applies the Weyl-Stratonovich operator ordering rule to resolve the Ito-Stratonovich ambiguity, showing that only this ordering yields a consistent Fokker-Planck equation.
• Uses the path integral formalism with noise-averaged pullbacks to derive the evolution operator as the exponentiation of the Lie derivative, ensuring consistency with the Fokker-Planck equation.
• Identifies the Witten index as the sharp trace of the GTO, proving it equals the Euler characteristic of the phase space for any flow vector field, noise metric, and temperature.
• Demonstrates that the Fokker-Planck operator is $δ$-exact, enabling the application of spectral theorems from dynamical systems to stochastic processes.

Experimental results

Research questions

• RQ1 How can the transfer operator formalism of dynamical systems be generalized to stochastic processes using supersymmetry?
• RQ2 What is the correct operator ordering (Ito vs. Stratonovich) for the Fokker-Planck evolution operator in the context of stochastic quantization?
• RQ3 Can the spontaneous breakdown of topological supersymmetry (Q-symmetry) serve as a field-theoretic definition of chaos in stochastic dynamical systems?
• RQ4 How does the Witten index relate to the topological invariants of the phase space in stochastic dynamics?
• RQ5 What happens to the classification of SDEs (equilibrium, noise-induced chaos, ordinary chaos) at high temperatures, particularly regarding the restoration of Q-symmetry?

Key findings

• The generalized transfer operator (GTO) of dynamical systems is identified as the finite-time Fokker-Planck evolution operator in the Weyl-Stratonovich formalism.
• The sharp trace of the GTO is trivialized as the Witten index, which equals the Euler characteristic of the closed phase space, independent of the flow vector field, noise metric, or temperature.
• The sharp determinant of the GTO is trivialized via supersymmetry, confirming the topological invariance of the index.
• Spontaneous breakdown of Q-symmetry is shown to be the field-theoretic definition of stochastic chaos, explaining the butterfly effect and long-range correlations.
• The Fokker-Planck operator derived via the Lie derivative and Weyl-Stratonovich ordering is unambiguously correct, resolving the Ito-Stratonovich dilemma.
• At high temperatures, the boundary between noise-induced and ordinary chaos smears into a crossover, and Q-symmetry is restored, completing the phase diagram of SDEs.