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On correspondence between equations of motion for Dirac particle in curved and twisted space-times. (English) Zbl 0675.53055

Summary: In the general theory of relativity one has to assume the space-time to be endowed with a symmetric connection. Some generalizations of Einstein’s theory examine the most general metric space. Namely, these theories assume for the linear affine connection of space-time the expression: \(\Gamma^ k_{ij}= \left\{ \begin{matrix} k\\ ij\end{matrix} \right\}+ K^ k_{ij},\) where \(\left\{ \begin{matrix} k\\ ij\end{matrix} \right\}\) are Christoffel’s symbols and the \(K^ k_{ij}\) denote the components of the nonsymmetric torsion tensor.
V. I. Rodichev [Sov. Phys., JETP 13, 1029-1031 (1961); translation from Zh. Eksper. Teor. Fiz. 40, 1469-1472 (1961; Zbl 0104.445)], firstly, observed that in twisted space-time \((\Gamma^ k_{ij}=K^ k_{ij},\quad \left\{ \begin{matrix} k\\ ij\end{matrix} \right\}=0)\) the equation of motion for a Dirac particle, derived from the variational Einstein’s principle, is a nonlinear spinor equation of the Heisenberg-Pauli type. On the other hand, F. Hehl and B. K. Datta [J. Math. Phys. 12, 1334-1339 (1971)] have proved that the equation of motion for a Dirac particle in a curved space-time \((\Gamma^ k_{ij}=\left\{ \begin{matrix} k\\ ij\end{matrix} \right\},\quad K^ k_{ij}=0)\) obtained by Einstein’s principle is a well-known Dirac equation. In this paper we want to show the equivalence between the description of motion of a Dirac particle in a twisted space-time and the description of motion of the same particle in a curved space-time.

MSC:

53B50 Applications of local differential geometry to the sciences
83C10 Equations of motion in general relativity and gravitational theory
83C47 Methods of quantum field theory in general relativity and gravitational theory

Citations:

Zbl 0104.445
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