[Paper Review] Reconsideration of De Donder-Weyl theory by covariant analytic mechanics

Satoshi Nakajima|arXiv (Cornell University)|Feb 15, 2016

Cosmology and Gravitation Theories11 references3 citations

TL;DR

This paper establishes a rigorous equivalence between covariant analytic mechanics (CAM) and an improved version of the De Donder-Weyl (DW) theory, correcting a fundamental flaw in the original DW formalism for gauge and gravitational fields. By reformulating CAM’s differential form-based canonical equations in tensor components, the authors show they match the corrected DW equations, and further present a modified Hamilton formalism that correctly derives the Dirac equations using only spinor fields as independent variables.

ABSTRACT

We show that the covariant analytic mechanics (CAM) is closely related to the De Donder-Weyl (DW) theory. To treat space and time on an equal footing, the DW theory introduces $D$ conjugate fields ($D$ is the dimension of space-time) for each field and the CAM regards the differential forms as the basic variables. The generalization of the canonical equations is called the DW equations. Although one of the DW equations is not correct for the gauge field and the gravitational field, we show the way to improve it. By rewriting the canonical equations of the CAM, which are manifestly general coordinate covariant and gauge covariant, using the components of the tensors, we show that these are equivalent to the improved DW equations. Additionally, we investigate the Dirac field. We present a modified Hamilton formalism which regards only the Dirac fields as the basic variables and show that it provides the Dirac equations correctly.

Motivation & Objective

• To resolve the inconsistency in the original De Donder-Weyl (DW) theory when applied to gauge and gravitational fields, where one of the canonical equations fails due to incorrect treatment of conjugate momenta symmetry.
• To demonstrate that the covariant analytic mechanics (CAM) framework, which uses differential forms as fundamental variables, is equivalent to a corrected version of the DW theory.
• To develop and validate a modified Hamilton formalism for the Dirac field that treats only the spinor fields as independent variables, avoiding spurious constraints and correctly reproducing the Dirac equations.
• To clarify the distinction between the Poisson brackets in CAM and the original DW theory, showing that CAM’s form-based brackets avoid the inconsistencies arising from tensor-component formulations.

Proposed method

• Reformulate the canonical equations of CAM—originally expressed in terms of differential forms—into component form using tensorial representations of the fields and conjugate momenta.
• Identify the flaw in the original DW equations: the assumption that polymomenta are independent across all spacetime indices fails for tensorial fields like gauge and gravitational fields.
• Introduce a constraint-based correction to the DW equations by enforcing complete antisymmetry of the conjugate momenta, leading to the improved DW equations.
• Show that the component-wise version of CAM’s canonical equations exactly matches the improved DW equations, proving their equivalence under general coordinate and gauge covariance.
• Apply the CAM formalism to the Dirac field by introducing Lagrange multipliers to handle constraints, then derive a modified Hamilton formalism that treats only the spinor fields as independent variables.
• Verify that the modified Hamilton formalism yields the correct Dirac equations by computing variations and confirming consistency with the Euler-Lagrange equations in the spinor sector.

Experimental results

Research questions

• RQ1 Why does the original De Donder-Weyl theory fail to correctly describe gauge and gravitational fields, and what is the precise nature of the error in its canonical equations?
• RQ2 Can the covariant analytic mechanics (CAM) framework be shown to be equivalent to a corrected version of the De Donder-Weyl theory through component-wise reformulation?
• RQ3 How can the Dirac field—known for its first-class constraints and fermionic nature—be consistently treated within a manifestly covariant Hamiltonian formalism?
• RQ4 What is the role of differential forms versus tensor components in defining Poisson brackets, and why does the CAM approach avoid the inconsistencies present in the original DW formulation?
• RQ5 Can a modified Hamilton formalism be constructed that treats only the Dirac spinor fields as independent variables and still reproduce the correct dynamics?

Key findings

• The original De Donder-Weyl theory contains an incorrect equation for gauge and gravitational fields due to the failure to enforce complete antisymmetry of the conjugate momenta across spacetime indices.
• By imposing the antisymmetry constraint on the polymomenta, the authors derive an improved set of DW equations that are consistent with the canonical structure of the theory.
• The component-wise form of the canonical equations in covariant analytic mechanics (CAM) is mathematically equivalent to the improved DW equations, establishing a rigorous equivalence between CAM and the corrected DW formalism.
• The Poisson bracket structure in CAM, defined on differential forms, avoids the inconsistencies present in the original DW theory, which uses tensor components and can lead to incorrect dynamics.
• A modified Hamilton formalism for the Dirac field—where only the spinor fields are treated as independent variables—correctly reproduces the Dirac equations, confirming the consistency of the approach.
• The derivation confirms that the Dirac equations emerge naturally from the Hamilton form when the variation is performed only over the independent spinor fields, with the conjugate forms being dependent variables.