A Modification on the Ivanenko-Landau-K\"ahler Equation

TL;DR

This paper proposes a new modification to the Ivanenko-Landau-K"ahler equation in Minkowski space, focusing on restricting the tensor field space and altering the mass term to better represent physical states, with implications for symmetries.

Contribution

It introduces a novel approach to modify the Ivanenko-Landau-K"ahler equation by restricting tensor subalgebras and redefining the mass term, advancing towards general space-time applications.

Findings

Modified equation incorporates U(1) and Axial U(1) symmetries.

Restriction to tensor subalgebras refines the equation's structure.

Altered mass term frames the eigenvalue problem for physical states.

Abstract

In this paper, we illustrated one scenario to modify the Ivanenko-Landau-K\"ahler equation. Since Ivanenko and Landau introduced the equation in 1928, the equation has been regarded as having a certain role as a fermion in particular in the discrete Lattice. Also, although it correctly is formulated as an alternative classical field equation by the Ideal projection for the Dirac equation in the Minkowski space-time, so does it only in that flat geometry. I. M. Benn and R. W. Tucker in 1985 and Yu. N. Obukhov and S. N. Solodukhin in 1994 suggested two resolutions respectively. They modified the equation in order for it to make senses as an alternative for the Dirac equation in the general space-time. This paper advances a still another approach, however in the Minkowski space yet, as the first stage toward the generalization. Two ingredients for the modifications are essential. One is the restriction of the space of anti-symmetric tensor field to only its subalgebras, not to its Ideals. The other is the modification on the mass term for the equation to mean the eigenvalue problem generating physical states. The Vector U(1) and the Axial U(1) symmetries built in the modified equation are described.