Quantization of Magnetic Poisson Structures

Abstract

We describe three perspectives on higher quantization, using the example of magnetic Poisson structures which embody recent discussions of nonassociativity in quantum mechanics with magnetic monopoles and string theory with non‐geometric fluxes. We survey approaches based on deformation quantization of twisted Poisson structures, symplectic realization of almost symplectic structures, and geometric quantization using 2‐Hilbert spaces of sections of suitable bundle gerbes. We compare and contrast these perspectives, describing their advantages and shortcomings in each case, and mention many open avenues for investigation.

Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

“…Nonassociativity in quantum mechanics has a long history dating back to foundational work on the theory in the 1930's. Its interest was revived in the 1980's with the realisation that the magnetic translation operators on the states of a charged particle moving in a magnetic monopole background generally form a nonassociative algebra [Jac85,GZ86]; see [Sza19a] for a mathematical introduction to the subject together with a survey and comparison of the various approaches to the quantisation of the pertinent twisted Poisson structures. The recent revived interest in these models has come about from their conjectural relevance to the low-energy dynamics of closed strings in non-geometric backgrounds, which are based on arguments invoking T-duality applied to target spaces that are tori or more generally total spaces of torus bundles [Lüs10,MSS12,BL14,MSS14], and other compact Lie groups [BP11].…”

An introduction to nonassociative physics

“…More expository technical details on the quantization of non-geometric backgrounds, their relation to double field theory and the higher structures involved, as well as of nonassociative quantum mechanics in these contexts are found in the lecture notes [65]. The more mathematically inclined reader interested in higher structures may consult the mathematical introduction [66] to the ideas presented in the following. We will give further pointers to relevant literature as we move along.…”

“…While again this approach formally captures a Hilbert space formulation of the nonassociative quantum mechanics, the physical meaning and development of the higher structures involved are at present unclear. A concise review of this approach can be found in [66].…”

“…These brackets define an associative algebra and are inverse to a symplectic 2-form which pulls back to the twisted 2-form ω B under the map which embeds the original phase space M into the extended phase space, or more concretely which restricts to ω B by eliminating the auxiliary coordinates (x i ,p i ). Some mathematical aspects of this symplectic realization are reviewed in[66].Onthe extended phase space one can then introduce the O(d, d)×O(d, d)-invariant Hamiltonian H = 1 m p I η IJ p J , (2.41)…”

Topological and Dynamical Aspects of Jacobi Sigma Models

“…The latter satisfy the Jacobi identity, but unlike Poisson brackets, violate the Leibniz rule; in other words, the Jacobi bracket still endows the algebra of functions on M with a Lie algebra structure, but it is not a derivation of the point-wise product among functions. Thus, the bi-vector field Λ may be ascribed to the family of bi-vector fields violating Jacobi identity, such as "twisted" and "magnetic" Poisson structures (see for example [11,12]) which recently received some interest in relation with the quantisation of higher structures (their Jacobiator being non-trivial) and with the description of non-trivial geometric fluxes in string theory. The violation of Jacobi identity is, however, under control, because the latter is recovered by the full Jacobi bracket, which is alternatively defined as the most general local bilinear operator on the space of real functions C ∞ (M, R) which is skewsymmetric and satisfies Jacobi identity [10], and this makes its study especially interesting to us.…”