# Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations

@article{Arminjon2013EquivalentFO, title={Equivalent Forms of Dirac Equations in Curved Space-times and Generalized de Broglie Relations}, author={Mayeul Arminjon and Frank Reifler}, journal={Brazilian Journal of Physics}, year={2013}, volume={43}, pages={64-77} }

One may ask whether the relations between energy and frequency and between momentum and wave vector, introduced for matter waves by de Broglie, are rigorously valid in the presence of gravity. In this paper, we show this to be true for Dirac equations in a background of gravitational and electromagnetic fields. We first transform any Dirac equation into an equivalent canonical form, sometimes used in particular cases to solve Dirac equations in a curved space-time. This canonical form is needed…

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## References

SHOWING 1-10 OF 45 REFERENCES

Dirac-Type Equations in a Gravitational Field, with Vector Wave Function

- Mathematics, Physics
- 2008

An analysis of the classical-quantum correspondence shows that it needs to identify a preferred class of coordinate systems, which defines a torsionless connection. One such class is that of the…

Dirac Equation from the Hamiltonian and the Case With a Gravitational Field

- Physics
- 2006

Starting from an interpretation of the classical-quantum correspondence, we derive the Dirac equation by factorizing the algebraic relation satisfied by the classical Hamiltonian, before applying the…

Basic quantum mechanics for three Dirac equations in a curved spacetime

- Physics, Mathematics
- 2010

We study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation:…

Four-vector vs. four-scalar representation of the Dirac wave function

- Mathematics, Physics
- 2012

In a Minkowski spacetime, one may transform the Dirac wave function under the spin group, as one transforms coordinates under the Poincare group. This is not an option in a curved spacetime.…

A non-uniqueness problem of the Dirac theory in a curved spacetime

- Physics, Mathematics
- 2009

The Dirac equation in a curved spacetime depends on a field of coefficients (essentially the Dirac matrices), for which a continuum of different choices are possible. We study the conditions under…

Uniqueness and Self-Conjugacy of Dirac Hamiltonians in arbitrary Gravitational Fields

- Physics
- 2011

Proofs of two statements are provided in this paper. First, the authors prove that the formalism of the pseudo-Hermitian quantum mechanics allows for describing the Dirac particles motion in…

On the relation Hamiltonian-wave equation, and on non-spreading solutions of Schrödinger's equation

- Physics
- 1999

According to Schrödinger’s ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the…

Dirac equation: representation independence and tensor transformation

- Physics
- 2008

We define and study the probability current and the Hamiltonian operator for a fully general set of Dirac matrices in a flat spacetime with affine coordinates, by using the Bargmann-Pauli hermitizing…

Solution of the problem of uniqueness and hermiticity of hamiltonians for Dirac particles in gravitational fields

- Physics
- 2010

The authors prove that the dynamics of spin 1/2 particles in stationary gravitational fields can be described using an approach, which builds upon the formalism of pseudo-Hermitian Hamiltonians. The…

Trajectories and spin motion of massive spin-1/2 particles in gravitational fields

- Physics
- 1981

From covariant Dirac theory in curved space-time, dynamical equations for the motion of the spin and the spin-induced non-geodesic behaviour of the particle trajectories are deduced. This is done for…