Deformation quantization of framed presymplectic manifolds
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DEFORMATION QUANTIZATION OF FRAMED PRESYMPLECTIC MANIFOLDS N. D. Gorev,∗ B. M. Elfimov,∗ and A. A. Sharapov∗
We consider the problem of deformation quantization of presymplectic manifolds in the framework of the Fedosov method. A class of special presymplectic manifolds is distinguished for which such a quantization can always be constructed. We show that in the general case, the obstructions to quantization can be identified with some special elements of the third cohomology group of a differential ideal associated with the presymplectic structure.
Keywords: presymplectic manifold, deformation quantization, Fedosov quantization DOI: 10.1134/S0040577920080085
1. Introduction In [1], Fedosov proposed a remarkable geometric construction of deformation quantization of general symplectic manifolds. In addition to explicit recurrence relations for calculating the ∗-product, this construction gives an expression for the trace on the algebra of quantum observables and also underlies the algebraic index theorem [2]. Finding a wide resonance among both mathematicians and physicists, Fedosov quantization was rethought and generalized by many authors. For instance, a connection was established in [3], [4] between Fedosov quantization and the BVF–BRST quantization method for constrained Hamiltonian systems. A generalization to supermanifolds was formulated in [5]. There is an adaptation of the construction to the cases of cotangent bundles [6] and almost K¨ ahler manifolds [7]. Finally, the Fedosov quantization method was extended to the class of irregular Poisson structures (called quasisymplectic structures) including the Poisson r-bracket. This paper is devoted to another natural application of the Fedosov method: the problem of deformation quantization of presymplectic manifolds. Until now, the question of quantizing presymplectic manifolds was considered mainly in the context of geometric quantization (see, e.g., [10], [11]). Having a common goal, the mentioned approaches focus on different aspects of the quantum mechanical description of systems: while the primary object in the geometric quantization method is the state space (a geometrized version of the Schr¨odinger representation), deformation quantization is aimed at constructing the algebra of quantum observables (Heisenberg representation). We prefer the second approach because our interest in this subject is primarily aroused by possible applications to problems in quantum field theory.1 ∗
Tomsk State University, Tomsk, Russia, e-mail: [email protected].
This research was performed in the framework of a government assignment by the Russian Ministry of Education and Science (Project No. 0721-2020-0033). 1 By all accounts, the Heisenberg representation reputedly has a definite advantage over the Schr¨ odinger picture in quantum field theory.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 280–296, August, 2020. Received March 15, 2020. Revised March 15, 2020. Accepted April 1, 2020. c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2042-1
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