Hamilton-Jacobi and Klein-Gordon-Fock equations for a charged test particle in space-time with simply transitive four-parameter groups of motions

Abstract

Metric components of potentials of admissible electromagnetic fields in spaces with simply transitive motion group G4 are found. The components of vector tetrads corresponding to the components of the metric tensors found by Petrov are given. The results obtained complement the coordinate-free classification given in Magazev et al. [Theor. Math. Phys. 156, 1127–1141 (2008)]. Previously, admissible electromagnetic fields were found for the case when three- and four-parameter groups of motions act on hypersurfaces of spacetime. Thus, non-equivalent sets of potentials for all electromagnetic fields that admit three- and four-parameter groups of motions are known now.

Noether Symmetry Analysis of the Klein–Gordon and Wave Equations in Bianchi I Spacetime

“…In a general program of research into the problem of integrating the classical and quantum equations of motion of a test particle in external fields of different nature in spaces with symmetry following the sets of Killing fields, Obukhov found all admissible electromagnetic fields for the case, when the groups of motions G 3 act simply transitively on the hypersurfaces of spacetime V 4 [32][33][34][35]. In [32], he found all external electromagnetic fields in which the Klein-Gordon-Fock equation admits the first-order symmetry operators and completed the classification of admissible electromagnetic fields in which the Hamilton-Jacobi and Klein-Gordon-Fock equations admit algebras of motion integrals that are isomorphic to the algebras of operators of the r-parametric groups of motions, G r , of spacetime manifolds if r ≤ 4 [33].…”

“…In the paper [34], the case when the groups G 4 act on V 3 was considered. The remaining case in the latter article, when the groups G 4 act simply transitively on the space V 4 , was studied in the paper [35].…”

“…where f 0 = t L+p+q ψ, m is a non-zero constant parameter, and K 4 is the HKV given in (35). Thus, one can write the conserved flow vectors for the Noether symmetries X 5 4 , ..., X 5 as…”