Quaternionic Aharonov–Bohm Effect

Abstract

A quaternionic analog of the Aharonov-Bohm effect is developed without the usual anti-hermitian operators in quaternionic quantum mechanics (QQM). A quaternionic phase links the solutions obtained to ordinary complex wave functions, and new theoretical studies and experimental tests are possible for them.

Virial theorem and generalized momentum in quaternionic quantum mechanics

“…In this article we have revisited and emended the mathematical machinery of HQM in real Hilbert space [30][31][32]. The previous results demonstrate that the theory is provided with wave equation, momentum operator, conservation of probability, expectation values, classical limit and spectral decomposition.…”

“…This quaternic vector product is not identical to the usual real or complex vector product, one immediately sees that X × Y = −Y × X. Next, using (24), (30) and (31), we get…”

“…However, a recent series of papers has indicated an anternative to anti-hermitian HQM [30][31][32][33][34]. First of all, it is realized that non-anti-hermitian Hamiltonian may generate probability conserving solutions.…”

“…In this article, we continue grounding HQM without the anti-hermitian assumption. The general momentum operator obtained in [30] has been used to obtain the quaternic Virial theorem in Section III and a quaterninionic quantum Lorentz force in Section IV. As a by-product, we were oblidged to redefine the expectation value in Section V. However, before considering these results, the next section provides a summary of the results obtained in the three previous articles.…”

Quaternionic Quantum Particles

“…One may replace usual QQM wave functions with regular quaternionic functions [20], and maintain anti-hermitian formalism. A more radical approach emerged after the discovery of quaternionic solutions obtained throughout non-anti-hermitian (NAH) Hamiltonians in the study of the quaternionic Aharonov-Bohm (AB) effect [21]. This discovery has enabled a formal expression of an NAH-QQM, where the probability current and the expectation value are redefined [22].…”