The information geometry of 2-field functional integrals by Eric Smith in 2019

probabilistic inference, path integral, quantum field theory, stochastic processes, statistical physics, physics, papers, information geometry

pdf on arXiv

outline

I. Understanding the Liouville Theorems that emerge in 2-field functional integrals for dissipative systems

II. The dual geometry from cumulant-generating functions for counts on integer lattices

• A. The exponential families from generating functions for species counts

• B. The Fisher metric on the exponential family of tilted distributions

  • 1. The variance as local metric, and the Fisher distance element introduced
  • 2. Coordinate dualization, Legendre transform, and the Large-Deviation function
  • 3. Exponential families on multinomial distributions

• C. The base and the tilt: inner products between vector fields describing two sources of variation

• D. Preservation of the inner product in connection with Liouville’s theorem

II. Finite-system models as sample estimators; the Large-deviation function, and importance sampling

• A. States as samples; system scaling and sample aggregation

• B. Legendre transform and large deviations in the interpretation of importance sampling

IV. Doi-Peliti 2-field functional integrals, and dual mappings induced by time translation

• A. Time evolution of moment* and cumulant-generating functions

• B. The 2FFI representation of the identity on distributions

  • 1. Coherent states and the Peliti construction of the functional integral
  • 2. The Peliti functional integral as a statistical model

• C. Stationary paths of the 2-field action functional

• D. Canonical transformations of the field variables of integration

  • 1. An action-angle transform from coherent-state variables to number-potential variables
  • 2. Descaling with respect to the instantaneous steady-state mean number
  • 3. Time-translation along stationary paths

V. The Liouville theorem connecting dynamics to inference induced by 2-field stationary trajectories

• A. The Wigner function from the 2-field identity operator plays the role of a phase-space density

• B. The Fisher metric and cubic tensor in dual canonical coordinates

• C. The dual vector fields induced by base-distribution initial conditions and final-time tilts

  • 1. The conserved inner product of dual vector fields, and directional transport of the metric

• D. Dual connections respecting the symplectic structure of canonical transformations in the 2-field system

  • 1. Conservation of the inner product through the combined effects of two maps
  • 2. Referencing arbitrary dual connections to dually flat connections in the exponential family
  • 3. Flat connections for coherent-state coordinates

• E. On the roles of coherent-state versus number-potential coordinates in the Doi-Peliti representation

VI. A worked example: The 2-state linear system

• A. Two-argument and one-argument generating functions on distributions with a conserved quantity

• B. Generator and conserved volume element in coherent-state coordinates

  • 1. Splitting the symplectic structure between coherent-state conjugate field pairs
  • 2. Stationary-path solutions and Liouville volume element
  • 3. Invariant cumulant-generating function and the incompressible phase-space density

• C. Fisher metric

• D. Dual coordinates for base and tilt, and the additive exponential family

• E. Why coherent-state fields do not generally produce invertible coordinate transformations

• F. Flat transport in the coherent-state connection

  • 1. Connection coefficients and absorption of measure terms
  • 2. Duality of dynamics and inference in Doi-Peliti theory

• G. The Fisher information density and large-deviation ratios as sample estimators

VII. Conclusions: the duality of dynamics and inference for irreversible and reversible processes