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Lagrange structure and quantization

P. O. Kazinski, S. L. Lyakhovich, A. A. Sharapov

TL;DR

The paper addresses quantization of classical dynamics that do not derive from an action by introducing a Lagrange structure with a Lagrange anchor $V$. It develops a BRST embedding and an AKSZ type topological sigma-model whose boundary dynamics reproduce the original $d$-dimensional theory, enabling standard path-integral quantization even when no action exists, and shows consistency with BV quantization when an action is present. The approach yields a systematic framework based on regularity and $S_\infty$-algebras, clarifies the role of physical observables as BRST cohomology, and provides concrete examples including non-Lagrangian systems and Maxwell theory in first-order form. This construction broadens the scope of quantizable theories and offers covariant, geometry-driven quantization applicable to a wide class of nonvariational dynamics.

Abstract

A path-integral quantization method is proposed for dynamical systems whose classical equations of motion do \textit{not} necessarily follow from the action principle. The key new notion behind this quantization scheme is the Lagrange structure which is more general than the Lagrangian formalism in the same sense as Poisson geometry is more general than the symplectic one. The Lagrange structure is shown to admit a natural BRST description which is used to construct an AKSZ-type topological sigma-model. The dynamics of this sigma-model in $d+1$ dimensions, being localized on the boundary, are proved to be equivalent to the original theory in $d$ dimensions. As the topological sigma-model has a well defined action, it is path-integral quantized in the usual way that results in quantization of the original (not necessarily Lagrangian) theory. When the original equations of motion come from the action principle, the standard BV path-integral is explicitly deduced from the proposed quantization scheme. The general quantization scheme is exemplified by several models including the ones whose classical dynamics are not variational.

Lagrange structure and quantization

TL;DR

The paper addresses quantization of classical dynamics that do not derive from an action by introducing a Lagrange structure with a Lagrange anchor . It develops a BRST embedding and an AKSZ type topological sigma-model whose boundary dynamics reproduce the original -dimensional theory, enabling standard path-integral quantization even when no action exists, and shows consistency with BV quantization when an action is present. The approach yields a systematic framework based on regularity and -algebras, clarifies the role of physical observables as BRST cohomology, and provides concrete examples including non-Lagrangian systems and Maxwell theory in first-order form. This construction broadens the scope of quantizable theories and offers covariant, geometry-driven quantization applicable to a wide class of nonvariational dynamics.

Abstract

A path-integral quantization method is proposed for dynamical systems whose classical equations of motion do \textit{not} necessarily follow from the action principle. The key new notion behind this quantization scheme is the Lagrange structure which is more general than the Lagrangian formalism in the same sense as Poisson geometry is more general than the symplectic one. The Lagrange structure is shown to admit a natural BRST description which is used to construct an AKSZ-type topological sigma-model. The dynamics of this sigma-model in dimensions, being localized on the boundary, are proved to be equivalent to the original theory in dimensions. As the topological sigma-model has a well defined action, it is path-integral quantized in the usual way that results in quantization of the original (not necessarily Lagrangian) theory. When the original equations of motion come from the action principle, the standard BV path-integral is explicitly deduced from the proposed quantization scheme. The general quantization scheme is exemplified by several models including the ones whose classical dynamics are not variational.

Paper Structure

This paper contains 24 sections, 2 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Lagrange structure: a preliminary exposition
  3. Regular Lagrange structure and $S_\infty$-algebra
  4. Lagrange structure
  5. Regularity conditions
  6. Completeness
  7. Physical observables
  8. $S_\infty$ - algebras
  9. BRST imbedding
  10. Ambient Poisson supermanifold
  11. BRST charge
  12. The unique existence of the BRST charge
  13. Exactness of Lagrange structure
  14. Physical observables
  15. Lagrange structure from the Hamiltonian viewpoint.
  16. ...and 9 more sections

Key Result

Proposition 2.1

Given a pair of sections $(T,V)$ satisfying (TV), then for any regular solution $x_0\in M$ of (T) one can find a coordinate system $(y^1,...,y^m, z^1,...,z^k)$ centered at $x_0$ together with a set of local functions $S(y)$, $E^1(y),...,E^k(y)$ such that equations $T_a(y,z)=0$ are equivalent to where $k=\mathrm{rank}(\partial_iT_a(x_0)) - \mathrm{rank}(G_{ab}(x_0))$ and $G_{ab}= V_a^i\partial_iT_

Theorems & Definitions (2)

  • Proposition 2.1
  • Proposition 4.1