[Paper Review] Dirac-Kahler equation in curved space-time, relation between spinor and tensor formulations

В. М. Редьков|arXiv (Cornell University)|Sep 11, 2011

Algebraic and Geometric Analysis24 references3 citations

TL;DR

This paper establishes a general covariant formulation of the Dirac-Kähler equation in curved space-time by deriving tensor wave equations from spinor formalism, demonstrating equivalence between spinor and tensor descriptions. It shows that the Dirac-Kähler field decomposes into scalar, pseudoscalar, vector, and pseudovector components under general covariance, with constraints derived via tetrad or coordinate-based pseudo-tensor classification, yielding consistent wave equations for all four bosonic particle types in curved geometry.

ABSTRACT

A common view is that generalization of a wave equation on Riemannian space-time is substantially determined by what a particle is - boson or fermion. As a rule, they say that tensor equations for bosons are extended in a simpler way then spinor equations for fermions. In that context, a very interesting problem is of extension a wave equation for Dirac--Kähler field (Ivanenko--Landau field was historically first term, also the term a vector field of general type was used). The article relates a generally covariant tensor formalism to a spinor one when these both are applied to description of the Dirac-Kähler field in a Rimannian space-time. Both methods are taken to be equivalent and the tensor equations are derived from spinor ones. It is shown that, for characterization of Dirac-Kähler's tensor components, two alternative approaches are suitable: these are whether a tetrad-based pseudo tensor classification or a generally coordinate pseudo tensor one. By imposing definite restrictions on the the Dirac-Kähler function, we have produced the general covariant form of wave equations for scalar, pseudoscalar, vector, and pseudovector particles.

Motivation & Objective

• To establish a general covariant tensor formulation of the Dirac-Kähler equation in Riemannian space-time.
• To demonstrate equivalence between spinor and tensor formulations of the Dirac-Kähler field in curved geometry.
• To classify the field components into scalar, pseudoscalar, vector, and pseudovector particles using both tetrad-based and coordinate-based pseudo-tensor approaches.
• To derive consistent wave equations for each particle type (spin 0 and spin 1) under general covariance.
• To show that constraints separating the four bosonic fields in Minkowski space remain valid in curved space-time.

Proposed method

• Derives tensor components from a 16-component spinor wave function using the Dirac matrix expansion: $ U(x) = [-i\Psi + \gamma^l \Psi_l + i\sigma^{mn} \Psi_{mn} + \gamma^5 \tilde{\Psi} + i\gamma^l\gamma^5 \tilde{\Psi}_l] E^{-1} $.
• Applies general covariance by replacing partial derivatives with covariant derivatives $ \partial_a \to \nabla^\alpha $, preserving Lorentz structure in curved space-time.
• Uses both tetrad-based and coordinate-based pseudo-tensor classifications to characterize the field components and their intrinsic parities.
• Imposes constraints on the Dirac-Kähler function to isolate scalar, pseudoscalar, vector, and pseudovector fields.
• Derives four sets of wave equations for the respective fields: Proca-type for scalars and pseudoscalars, and Rarita-Schwinger-type for vectors and axial-vectors.
• Verifies that the constraints (e.g., $ \nabla^\alpha \Psi_\alpha = 0 $) hold identically in curved space-time using Riemann tensor terms and commutators of covariant derivatives.

Experimental results

Research questions

• RQ1 How can the Dirac-Kähler field be consistently extended to curved space-time using a tensor formulation?
• RQ2 What is the precise relationship between the spinor and tensor formulations of the Dirac-Kähler equation in a general Riemannian background?
• RQ3 Which classification scheme—tetrad-based or coordinate-based pseudo-tensor formalism—best preserves the physical interpretation of field components?
• RQ4 Do the constraints separating scalar, pseudoscalar, vector, and pseudovector fields in Minkowski space remain valid in curved space-time?
• RQ5 Can the wave equations for all four bosonic particle types (spin 0 and spin 1) be consistently derived from the general covariant Dirac-Kähler equation?

Key findings

• The tensor wave equations for the Dirac-Kähler field in curved space-time are derived from the spinor formulation using general covariance, with $ \partial_a \to \nabla^\alpha $.
• The scalar and pseudoscalar fields satisfy the Proca equations: $ \nabla^\alpha \Psi_\alpha + m\Psi = 0 $ and $ \nabla^\alpha \tilde{\Psi}_\alpha + m\tilde{\Psi} = 0 $, respectively.
• The vector and pseudovector fields are governed by equations involving covariant derivatives and Levi-Civita tensors: $ \nabla_\alpha \Psi + \nabla^\beta \Psi_{\alpha\beta} - m\Psi_\alpha = 0 $ and $ \tilde{\nabla}_\alpha \tilde{\Psi} - \frac{1}{2} \epsilon^{\beta\rho\sigma}_\alpha \nabla_\beta \Psi_{\rho\sigma} - m\tilde{\Psi}_\alpha = 0 $.
• Constraints such as $ \nabla^\alpha \Psi_\alpha = 0 $ and $ \nabla^\alpha \tilde{\Psi}_\alpha = 0 $ hold identically in curved space-time due to Riemann tensor symmetries.
• The decomposition into four bosonic fields (scalar, pseudoscalar, vector, pseudovector) is preserved under general covariance, with identical constraints as in Minkowski space.
• The use of tetrad formalism enables consistent treatment of discrete Lorentz symmetries, including intrinsic parity, in arbitrary curved space-time models.