[Paper Review] Towards the Born-Weyl Quantization of Fields

Igor V. Kanatchikov|ArXiv.org|Dec 31, 1997

Spectral Theory in Mathematical Physics4 citations

TL;DR

This paper proposes a covariant quantization framework for field theories using the Born-Weyl polymomentum Hamiltonian formalism, leveraging a graded Poisson bracket on differential forms to derive a covariant Schrödinger-type equation for a hypercomplex wave function. The key contribution is showing that the quasiclassical limit reproduces the De Donder-Weyl Hamilton-Jacobi equations, offering a new geometric approach to quantum field theory with manifest spacetime covariance.

ABSTRACT

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.

Motivation & Objective

• To develop a manifestly covariant quantization scheme for field theories, addressing limitations of standard canonical quantization.
• To realize the long-standing idea of Born and Weyl (1934) on polymomentum Hamiltonian formalism in field quantization.
• To construct a covariant analogue of the Schrödinger equation using hypercomplex wave functions on spacetime-field configuration space.
• To establish a geometric bridge between the proposed quantum formalism and the functional Schrödinger picture in quantum field theory.
• To demonstrate consistency with the De Donder-Weyl Hamilton-Jacobi equations in the quasiclassical limit.

Proposed method

• Utilizes a recently developed graded Poisson bracket on differential forms in field theory to generalize canonical structures.
• Introduces a hypercomplex wave function defined on the total space of field and spacetime variables.
• Derives a covariant Schrödinger-type equation using the polymomentum formalism and the graded Poisson bracket.
• Applies quasiclassical approximation to the wave equation to recover the De Donder-Weyl Hamilton-Jacobi equations.
• Establishes a correspondence between the proposed formalism and the functional Schrödinger picture in quantum field theory.
• Employs the De Donder-Weyl theory as the classical starting point, generalizing Hamiltonian mechanics to field theories with spacetime symmetry.

Experimental results

Research questions

• RQ1 Can a manifestly covariant quantization of fields be achieved through the polymomentum Hamiltonian formalism?
• RQ2 How can a hypercomplex wave function be consistently defined in a spacetime-covariant field quantization framework?
• RQ3 Does the proposed Schrödinger-type equation reduce to the De Donder-Weyl Hamilton-Jacobi equations in the quasiclassical limit?
• RQ4 What is the relationship between the proposed formalism and the functional Schrödinger picture in quantum field theory?
• RQ5 Can the graded Poisson bracket on differential forms provide a consistent foundation for field quantization?

Key findings

• The proposed covariant Schrödinger equation is derived using a graded Poisson bracket on differential forms, ensuring spacetime covariance.
• The quasiclassical limit of the wave equation reproduces the De Donder-Weyl Hamilton-Jacobi equations, validating the classical correspondence.
• The formalism provides a geometric realization of the Born-Weyl program for field quantization, extending their 1934 idea to modern field theory.
• The hypercomplex wave function formalism allows for a unified treatment of field and spacetime variables in the quantum regime.
• The approach offers a new perspective on the functional Schrödinger picture, suggesting deeper geometric underpinnings for quantum field theory.
• The method is consistent with the principles of general covariance and provides a framework for further quantization in curved spacetime or quantum gravity.