| name | probability-geometry-schwinger-dyson |
| description | Score-mismatch field methodology for probing probability geometry using Schwinger-Dyson identities. Bridges statistical mechanics, information theory, and quantum field theory through geometric interpretation of equilibrium violations. |
| version | 1 |
| created | 2026-06-26T00:00:00.000Z |
| tags | ["statistics","probability","schwinger-dyson","fisher-information","equilibrium","quantum-field-theory"] |
| source | arXiv:2606.27360 |
| trigger_words | ["schwinger-dyson","score mismatch","fisher information","probability geometry","configurational temperature","equilibrium detection"] |
Probability Geometry via Schwinger-Dyson Identities
Overview
Geometric interpretation of Schwinger-Dyson identities using a universal score-mismatch field that characterizes departure from equilibrium. Connects statistical mechanics, information theory, and field theory.
Core Framework
Score-Mismatch Field
For any sampled distribution Q and equilibrium measure P_eq, define:
δs = ∇log(Q / P_eq)
This single field controls ALL Schwinger-Dyson violations.
Key Theorems
- Universality: Every Schwinger-Dyson violation = projection of δs onto a probe direction
- Fisher Identity: Relative Fisher information = ||δs||² (squared norm)
- Universal Bound: Fisher information bounds all Schwinger-Dyson violations simultaneously
Practical Formula
SD_violation(probe_direction) = <δs, probe_direction>
Fisher_information = E[||δs||²]
Applications
MCMC Equilibrium Monitoring
- Compute score-mismatch field from MCMC samples
- Fisher information gives single-number convergence diagnostic
- More sensitive than traditional autocorrelation measures
Statistical Model Validation
- Compare empirical distribution Q to model P_eq
- Score-mismatch reveals specific directions of misfit
- Fisher information quantifies overall model adequacy
Quantum Field Theory
- Check consistency of sampled field configurations
- Schwinger-Dyson violations indicate systematic errors
- Configurational temperature as observable diagnostic
Non-equilibrium Detection
- δs ≠ 0 iff system is out of equilibrium
- Direction of δs indicates which observables are most biased
- Magnitude ||δs|| gives overall departure strength
Implementation Steps
- Define equilibrium measure P_eq for your system
- Sample distribution Q from your process
- Compute score-mismatch δs = ∇log(Q/P_eq)
- Calculate Fisher information as ||δs||²
- Project onto probes to identify specific violations
- Use universal bound to certify overall consistency
Pitfalls
- Fisher information may diverge if Q has support outside P_eq
- Numerical estimation of ∇log(Q/P_eq) requires careful density estimation
- High-dimensional systems need dimension reduction for tractable δs
- Configurational temperature assumes ergodicity
References
- arXiv:2606.27360 - Probing Probability Geometry with Schwinger-Dyson Identities
