Topological field theory of dynamical systems
arXiv:1203.5162 · doi:10.1063/1.4746037
Abstract
Here, it is shown that the path-integral representation of any stochastic or deterministic continuous-time dynamical model is a cohomological or Witten-type topological field theory, i.e., a model with global topological supersymmetry (Q-symmetry). As many other supersymmetries, Q-symmetry must be perturbatively stable due to what is generically known as non-renormalization theorems. As a result, all (equilibrium) dynamical models are divided into three major categories: Markovian models with unbroken Q-symmetry, chaotic models with Q-symmetry spontaneously broken on the mean-field level by, e.g., fractal invariant sets (e.g., strange attractors), and intermittent or self-organized critical (SOC) models with Q-symmetry dynamically broken by the condensation of instanton-antiinstanton configurations (earthquakes, avalanches etc.) SOC is a full-dimensional phase separating chaos and Markovian dynamics. In the deterministic limit, however, antiinstantons disappear and SOC collapses into the "edge of chaos". Goldstone theorem stands behind spatio-temporal self-similarity of Q-broken phases known under such names as algebraic statistics of avalanches, 1/f noise, sensitivity to initial conditions etc. Other fundamental differences of Q-broken phases is that they can be effectively viewed as quantum dynamics and that they must also have time-reversal symmetry spontaneously broken. Q-symmetry breaking in non-equilibrium situations (quenches, Barkhausen effect, etc.) is also briefly discussed.
arXiv comment18 pages, 4 figures, published version
References in corpus (21)
- Non-Abelian Anyons and Topological Quantum Computation
- Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
- Making Sense of Non-Hermitian Hamiltonians
- Pseudo-Hermiticity versus PT Symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian
- Pseudo-Hermiticity versus PT-Symmetry III: Equivalence of pseudo-Her miticity and the presence of antilinear symmetries
- Dynamical synapses causing self-organized criticality in neural networks
- Pseudo-Supersymmetric Quantum Mechanics and Isospectral Pseudo-Hermi tian Hamiltonians
- The Geometry of Crumpled Paper
- Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinsky equation
- Notes on instantons in topological field theory and beyond
- Which Green Functions Does the Path Integral for Quasi-Hermitian Hamiltonians Represent?
- Kramers equation and supersymmetry
- Non-Equilibrium Thermodynamics and Topology of Currents
- Chaos and Quantum Mechanics
- Supersymmetry and fluctuation relations for currents in closed networks
- Path-Integral Formulation of Pseudo-Hermitian Quantum Mechanics and the Role of the Metric Operator
- Periodic orbit spectrum in terms of Ruelle--Pollicott resonances
- Self-Organized Criticality as Witten-type Topological Field Theory with Spontaneously Broken Becchi-Rouet-Stora-Tyutin Symmetry
- Instanton Dynamics in the Broken Phase of the Topological Sigma Model
- On field theory quantization around instantons
- New observables in topological instantonic field theories