Lagrange structure and quantization

Abstract

A path-integral quantization method is proposed for dynamical systems whose classical equations of motion do 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.

A minimal BV action for Vasiliev’s four-dimensional higher spin gravity

“…Further developments of the AKSZ formalism can be found in [22,23,24] and [25,26,27,28,29,30,31,32,33], and its close ties to unfolded dynamics have been stressed in [34,35,36,37,38,39]. For related treatments of more general dynamical systems, not necessarily based on differential algebras, see [40,41] and references therein.…”

Fedosov quantization of Lagrange–Finsler and Hamilton–Cartan spaces and Einstein gravity lifts on (co) tangent bundles

“…There were proposed different sophisticate constructions with formal and partial solutions for quantum gravity and field interactions theories. We cite here the BRST quantization methods for non-Abelian and open gauge algebras [1,2,3,4], deformation quantization [5,6,7,8], quantization of general Lagrange structures and, in general, BRST quantization without Lagrangians and Hamiltonians [9,10], W -geometry and Moyal deformations of gravity via strings and branes [11,12,13] and quantum loops and spin networks [14,15,16].…”

Consistent interactions and involution

“…The gauge algebra of the general (not necessarily Lagrangian) system is known in the same details as in the Lagrangian case, and the corresponding BRST complex is also well studied [24], [25] that allows one to systematically control all the compatibility conditions.…”

“…These relations have further compatibility conditions involving higher structure functions (see for details [24], [25]). The existence of all the higher structure functions and their locality have been proven in [26] under the condition that the generators L, R and the structure function U involved in (14) are all local.…”