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International Journal of Theoretical Physics

Toward the Born-Weyl Quantization of Fields

Igor V. Kanatchikov

Published Jan 1, 1998• 30 citations
Publisher
DOI
10.1023/a:1026679104397

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Elements of the quantization in field theory based on the covariant polymomentum Hamiltonian formalism (the De Donder-Weyl theory), a possibility of which was originally discussed in 1934 by Born and Weyl, are developed. The approach is based on a recently proposed graded Poisson bracket on differential forms in field theory (see e.g. hep-th/9709229). A covariant analogue of the Schrödinger equation for a hypercomplex wave function on the space of field and space-time variables is put forward. It is shown to lead to the De Donder-Weyl Hamilton-Jacobi equations in quasiclassical limit. A possible relation to the functional Schrödinger picture in quantum field theory is outlined.
We apply the Born-Jordan and Weyl quantization formulas for polynomials in canonical coordinates to the constants of motion of some examples of the superintegrable 2D anisotropic harmonic oscillator.Our aim is to study the behaviour of the algebra of the constants of motion after the different quantization procedures.In the examples considered, we have that the Weyl formula always preserves the original superintegrable structure of the system, while the Born-Jordan formula, when producing different operators than the Weyl's one, does not.