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

Abstract

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 proof consists in the analysis of three expressions for Hamiltonians, which are derived from the Dirac equation and describe the dynamics of spin 1/2 particles in the gravitational field of the Kerr solution. The Hamiltonians correspond to different choices of tetrad vectors and differ from each other. The differences between the Hamiltonians confirm the conclusion known from many studies that the Hamiltonians derived from the Dirac equation are non-unique. Application of standard pseudo-Hermitian quantum mechanics rules to each of these Hamiltonians produces the same Hermitian Hamiltonian. The eigenvalue spectrum of the resulting Hamiltonian is the same as that of the Hamiltonians derived from the Dirac equation with any chosen system of tetrad vectors. For description of the dynamics of spin 1/2 particles in stationary gravitational fields can be used not only the formalism of pseudo-Hermitian Hamiltonians, but also an alternative approach, which employs the Parker scalar product. The authors show that the alternative approach is equivalent to the formalism of pseudo-Hermitian Hamiltonians.

20 pages

References in corpus (3)

• Spin dynamics in gravitational fields of rotating bodies and the equivalence principle
• Hermiticity of the Dirac Hamiltonian in Curved Spacetime
• Post-Newtonian equation for the energy levels of a Dirac particle in a static metric

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• Quantization of spinor field in the Schwarzschild spacetime and spin sums for solutions of the Dirac equation
• Stationary solutions of the second-order equation for fermions in Kerr-Newman space-time
• Quantum mechanical equivalence of the metrics of a centrally symmetric gravitational field
• Interacting Dirac fields in an expanding universe: dynamical condensates and particle production
• Motion of half-spin particles in the axially symmetric field of naked singularities of static q-metric
• Pauli equation and charged spin-1/2 particle in a weak gravitational field