DeDonder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory

Abstract

A quantization of field theory based on the DeDonder-Weyl covariant Hamiltonian formulation is discussed. A hypercomplex extension of quantum mechanics, in which the space-time Clifford algebra replaces that of the complex numbers, appears as a result of quantization of Poisson brackets of differential forms put forward for the DeDonder-Weyl formulation earlier. The proposed covariant hypercomplex Schrödinger equation is shown to lead in the classical limit to the DeDonder-Weyl Hamilton-Jacobi equation and to obey the Ehrenfest principle in the sense that the DeDonder-Weyl canonical field equations are satisfied for the expectation values of properly chosen operators.

LaTeX2e, 14pp., to appear in Rep. Math. Phys. (1998)

References in corpus (4)

• Canonical Structure of Classical Field Theory in the Polymomentum Phase Space
• On Field Theoretic Generalizations of a Poisson Algebra
• Short Distance Properties from Large Distance Behaviour
• Toward a finite-dimensional formulation of Quantum Field Theory

Cited by in corpus (34)

• Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories
• On the Duffin-Kemmer-Petiau Formulation of the Covariant Hamiltonian Dynamics in Field Theory
• Precanonical Quantization and the Schroedinger Wave Functional
• Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian (n-1)-forms
• Observations in Quantum Cosmology
• Precanonical quantization of Yang-Mills fields and the functional Schroedinger representation
• Hints of a New Relativity Principle from $p$-brane Quantum Mechanics
• Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space G
• Precanonical Perspective in Quantum Gravity
• De Donder-Weyl Hamiltonian formulation and precanonical quantization of vielbein gravity
• A vertical exterior derivative in multisymplectic geometry and a graded Poisson bracket for nontrivial geometries
• Covariant canonical formulations of classical field theories
• Geometric (pre)quantization in the polysymplectic approach to field theory
• Ehrenfest Theorem in Precanonical Quantization
• Schrödinger wave functional in quantum Yang-Mills theory from precanonical quantization
• Covariant canonical quantization
• Precanonical quantization and the Schrödinger wave functional revisited
• On the precanonical structure of the Schrödinger wave functional
• Extending Quantum Probability from Real Axis to Complex Plane
• Polysymplectic formulation for topologically massive Yang-Mills field theory
• Polysymplectic Hamiltonian formalism and some quantum outcomes
• A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity
• Covariant Hamiltonian field theory. Path integral quantization
• p-Mechanics and Field Theory
• De Donder-Weyl Hamiltonian formalism of MacDowell-Mansouri gravity
• Functional RG flow of the effective Hamiltonian action
• Polysymplectic formulation for BF gravity with Immirzi parameter
• Classical field theories from Hamiltonian constraint: Canonical equations of motion and local Hamilton-Jacobi theory
• Towards a "pre-canonical" quantization of gravity without the space+time decomposition
• Schrödinger Functional of a Quantum Scalar Field in Static Space-Times from Precanonical Quantization
• BV quantization of covariant (polysymplectic) Hamiltonian field theory
• Classical field theories from Hamiltonian constraint: Local symmetries and static gauge fields
• Precanonical structure of the Schrödinger wave functional in curved space-time
• p-Mechanics and De Donder-Weyl Theory