Transfer operators and topological field theory
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 finitetime FokkerPlanck evolution operator. As a result comes the supersymmetric trivialization of the socalled 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 FokkerPlanck operators allows to extend the previous picture of the spontaneous topological supersymmetry (Qsymmetry) 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 Qsymmetry is indeed the fieldtheoretic definition and stochastic generalization of the concept of deterministic chaos. In addition, the previously proposed lowtemperature classification of SDEs, i.e., thermodynamic equilibrium / noiseinduced chaos ((anti)instanton condensation, intermittent) / ordinary chaos (nonintegrability of the flow vector field), is complemented by the discussion of the hightemperature regime where the sharp boundary between the noiseinduced and ordinary chaotic phases must smear out into a crossover, and at even higher temperatures the Qsymmetry is restored. The Weyl quantization is discussed in the context of the ItoStratonovich dilemma.
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 DOI:
 10.48550/arXiv.1308.4222
 arXiv:
 arXiv:1308.4222
 Bibcode:
 2013arXiv1308.4222O
 Keywords:

 Mathematical Physics;
 High Energy Physics  Theory;
 Nonlinear Sciences  Chaotic Dynamics
 EPrint:
 51 pages, 3 figures