Quantization of polysymplectic manifolds

Abstract

We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition of prequantum vector bundle is obtained which incorporates in an essential way the action of the space of coefficients. We define quantization with respect to a polarization and to a spin c structure. In the presence of a complex polarization, it is shown that the polysymplectic Guillemin-Sternberg conjecture is false. We conclude with potential extensions and applications.

Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density

“…Kanatchikov had previously discussed the same type of brackets in the language of differential forms [13], which is similar to other authors use of poly symplectic Poisson brackets [72,[85][86][87][88][89][90]. Other Poisson brackets over fields lead to Dirac delta functions over space [69].…”

A global version of Gunther's polysymplectic formalism using vertical projections

“…However, it is possible to reproduce the results of the canonical commutation relations after integration; see [22] for these preliminary results in the context of Minkowski space-time. This is a more reasonable expectation for a finite dimensional, differential geometric analysis of quantum field theory, as it is not clear how operator-valued distributions would ever arise from a welldefined differential geometric structure (see, for example, [18] and [19] for similar efforts, and [4] for a more closely related, though more mathematical, perspective). This seems a promising beginning, and a full analysis of the possibilities for geometric quantization from the perspective of this particular covariant Hamiltonian framework will be an important area for future work.…”

Covariant canonical formulations of classical field theories

“…On the quantization: Concerning the quantization, we note that the lack of canonical pairs of dynamical variables in the multisymplectic approach has led to the consideration of alternative quantization procedures like geometric quantization or Schrödinger's functional approach, e.g. see references [71][72][73][74].…”