A solution of the non-uniqueness problem of the Dirac Hamiltonian and energy operators

TL;DR

This paper addresses the non-uniqueness of the Dirac Hamiltonian and energy operators in curved spacetime by proposing conditions on tetrad fields and their rotation rates, ensuring consistent physical descriptions across reference frames.

Contribution

It introduces a method to resolve the non-uniqueness problem of Dirac operators by aligning tetrad rotation rates with reference frame rotation, providing a clear criterion for unique Hamiltonian and energy operators.

Findings

Imposing that the tetrad rotation rate equals the frame's rotation rate solves the non-uniqueness problem.

Setting the tetrad rotation rate to zero also ensures uniqueness.

Other proposed solutions are analyzed and compared.

Abstract

In a general spacetime, the possible choices for the field of orthonormal tetrads lead (in standard conditions) to equivalent Dirac equations. However, the Hamiltonian operator is got from rewriting the Dirac equation in a form adapted to a particular reference frame, or class of coordinate systems. That rewriting does not commute with changing the tetrad field $(u_\alpha)$. The data of a reference frame F fixes a four-velocity field $v$, and also fixes a rotation-rate field $\Mat{\Omega}$. It is natural to impose that $u_0=v$. We show that then the spatial triad $(u_p)$ can only be rotating w.r.t. F, and that the title problem is solved if one imposes that the corresponding rotation rate $\Mat{\Xi}$ be equal to $\Mat{\Omega}$ - or also, if one imposes that $\Mat{\Xi}=\Mat{0}$. We also analyze other proposals which aimed at solving the title problem.