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Path Integral Solution for Dissipative Generative Dynamics

Xidi Wang

TL;DR

The paper addresses whether mechanical systems can generate intelligent language by recasting autoregressive text as dissipative quantum dynamics within a Quantum Sequential Field (QSF) framework. It introduces a 1+1D quantum-field model, employs Koopman operator theory to linearize nonlinear dynamics, integrates Linear Attention for non-local context, and uses guided Feynman path integrals to obtain closed-form propagators with piecewise-constant dynamics. Through a four-stage progressive training pipeline, the authors lift nonlinear transformers to exact solvable QSF dynamics, achieving a substantial performance gain (val loss of $2.08$ on TinyStories) with a dissipation-aware architecture, while enforcing a Hamiltonian constraint degrades performance (val loss $3.76$), underscoring that language generation relies on controlled dissipation rather than conservation. The results provide a principled, mathematically transparent view of information flow in language models, suggesting that dissipation and non-local context are essential ingredients for coherent generation and offering a path to interpretable, physics-inspired sequence modeling. Overall, the work reframes language generation as dissipative quantum field theory, with potential implications for design principles in interpretable AI and sequence modeling.

Abstract

Can purely mechanical systems generate intelligent language? We prove that dissipative quantum dynamics with analytically tractable non-local context aggregation produce coherent text generation, while conservation laws cause fundamental failure. Employing Koopman operators with closed-form path integral propagators, we show irreversible computation fundamentally requires both controlled information dissipation and causal context aggregation. Spectral analysis reveals emergent eigenvalue structure, separating into decay modes (forgetting), growth modes (amplification), and neutral modes (preservation) -- the essential ingredients for directed information flow. Hamiltonian constraints force the elimination of these dissipative modes and degrading performance despite unchanged model capacity. This establishes language generation as dissipative quantum field theory, proving mechanical systems acquire intelligence through the combination of dissipation and non-locality, not through conservation.

Path Integral Solution for Dissipative Generative Dynamics

TL;DR

The paper addresses whether mechanical systems can generate intelligent language by recasting autoregressive text as dissipative quantum dynamics within a Quantum Sequential Field (QSF) framework. It introduces a 1+1D quantum-field model, employs Koopman operator theory to linearize nonlinear dynamics, integrates Linear Attention for non-local context, and uses guided Feynman path integrals to obtain closed-form propagators with piecewise-constant dynamics. Through a four-stage progressive training pipeline, the authors lift nonlinear transformers to exact solvable QSF dynamics, achieving a substantial performance gain (val loss of on TinyStories) with a dissipation-aware architecture, while enforcing a Hamiltonian constraint degrades performance (val loss ), underscoring that language generation relies on controlled dissipation rather than conservation. The results provide a principled, mathematically transparent view of information flow in language models, suggesting that dissipation and non-local context are essential ingredients for coherent generation and offering a path to interpretable, physics-inspired sequence modeling. Overall, the work reframes language generation as dissipative quantum field theory, with potential implications for design principles in interpretable AI and sequence modeling.

Abstract

Can purely mechanical systems generate intelligent language? We prove that dissipative quantum dynamics with analytically tractable non-local context aggregation produce coherent text generation, while conservation laws cause fundamental failure. Employing Koopman operators with closed-form path integral propagators, we show irreversible computation fundamentally requires both controlled information dissipation and causal context aggregation. Spectral analysis reveals emergent eigenvalue structure, separating into decay modes (forgetting), growth modes (amplification), and neutral modes (preservation) -- the essential ingredients for directed information flow. Hamiltonian constraints force the elimination of these dissipative modes and degrading performance despite unchanged model capacity. This establishes language generation as dissipative quantum field theory, proving mechanical systems acquire intelligence through the combination of dissipation and non-locality, not through conservation.
Paper Structure (14 sections, 17 equations, 2 figures, 2 tables)

This paper contains 14 sections, 17 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Language as Quantum Dynamics
  4. Koopman Framework for Dissipative Dynamics
  5. Linear Attention as Context-Dependent Evolution
  6. Guided Feynman Path Integrals
  7. Multi-Token Propagation
  8. Progressive Training
  9. Stage I: Spectral Foundation with Causal Structure
  10. Stage II: Koopman Operator Learning
  11. Stage III: Koopman + LinearAttention Hybrid
  12. Stage IV: Hamiltonian Constraint
  13. Experimental Validation
  14. Conclusion

Figures (2)

  • Figure 1: The mean $\zeta$ across all 30 layers decreases. The shaded region indicates one standard deviation across layers.
  • Figure 2: Eigenvalue spectrum of Koopman operators (Stage III) averaged across all 32 layers. The unit circle (dashed gray) marks the Hamiltonian constraint $|\lambda_i| = 1$. Eigenvalues with clear separation into decay modes ($|\lambda_i| < 1$, left), neutral modes ($|\lambda_i| \approx 1$, near circle), and growth modes ($|\lambda_i| > 1$, right cluster). This dissipative structure enables directed information flow essential for language generation.