Lagrange structure and quantization

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.

Minor corrections, format changed, 40 pages

References in corpus (25)

• Deformation quantization of Poisson manifolds, I
• Nonlinear Equations for Symmetric Massless Higher Spin Fields in $(A)d{S}_d$
• The Geometry of the Master Equation and Topological Quantum Field Theory
• Introduction to sh Lie algebras for physicists
• A path integral approach to the Kontsevich quantization formula
• Two-Dimensional Gravity and Nonlinear Gauge Theory
• Poisson Structure Induced (Topological) Field Theories
• Higher derived brackets and homotopy algebras
• Higher Spin Gauge Theories in Various Dimensions
• Relative formality theorem and quantisation of coisotropic submanifolds
• BRST theory without Hamiltonian and Lagrangian
• Target space symmetries in topological theories I
• Superfield BRST Charge and the Master Action
• On possible generalizations of field--antifield formalism
• Characteristic classes of gauge systems
• On the AKSZ formulation of the Poisson sigma model
• Algebra of Higher Antibrackets
• On the multilevel generalization of the field--antifield formalism
• Generalized Poisson sigma models
• Non-Abelian Antibrackets
• On the Multilevel Field--Antifield Formalism with the Most General Lagrangian Hypergauges
• Weak Dirac bracket construction and the superparticle covariant quantization problem
• Gauge Independence of the Lagrangian Path Integral in a Higher-Order Formalism
• On Some Generalizations of Batalin-Vilkovisky Algebras
• Second Class Constraints in a Higher-Order Lagrangian Formalism

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