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On a generalization of quantum mechanics of biquaternions

E. Conte

Published Jun 1, 1993• 0 citations
OpenAlex

Abstract

A generalization of the quantum mechanical formalism by biquaternions is proposed. By a direct example, it is evidenced that such a generalization explains in an algebraic manner the {open_quotes}wave packet reduction{close_quotes} of quantum mechanics. 8 refs.

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•Jan 1, 2002•0 citations
This paper begins with the introduction of biquaternion algebra and its properties in compact, profound and comprehensible approach. The Schrödinger equation including both the scalar Aharonov-Bohm(sAB) and Aharonov-Casher(AC) effects, that are currently popular in particle physics and condensed-matter physics, has been re-written by biquaternions. The newly biquaternionic Schrödinger equation, which is defined with various assumptions and approaches, also includes the biquaternionic generalized momentum in Hamiltonian.
A "minimal" generalization of Quantum Mechanics is proposed, where the Lagrangian or the action functional is a mapping from the (classical) states of a system to the Lie algebra of a general compact Lie group, and the wave function takes values in the corresponding group algebra. This formalism admits a probability interpretation and a suitable dynamics, but has no obvious classical correspondence. Allowing the Lagrangian or the action functional to take values in a general Lie algebra instead of only the real number field (actually the u(1) algebra) enlarges the extent of possible physical laws that can describe the real world. The generalized quantum dynamics of a point particle in a background gauge field is given as an example, which realizes the gauge invariance by a Wilson line structure and shows that some Schrodinger-like equation can be deduced within this formalism. Some possible developments of this formalism are also discussed.
It is now well established that the presence of an opaque classical object can be unambiguously determined by an {open_quotes}interaction-free{close_quotes} measurement (IFM), where the object and the probe never directly interact. For quantum objects, we examine {open_quotes}interaction-free{close_quotes} measurement (the object wavefunction is unchanged) and {open_quotes}interaction-free{close_quotes} preparation (the object wavefunction is changed without physical interaction) and find that in general, neither is possible. We propose using high efficiency IFM{close_quote}s as a quantum information bus to connect disjoint quantum systems. {copyright} {ital 1999 American Institute of Physics.}
The possibility of generalizing quantum mechanics in such a way as to retain its predictive results, while comprehending additional solutions, is examined. It is found that this can be done through a perfected formal correspondence with Hamilton-Jacobi mechanics, by which one is led to consider generalizations of the Heisenberg postulate of the form ${p}_{k}{q}_{j}\ensuremath{-}{q}_{j}{p}_{k}=S{\ensuremath{\delta}}_{\mathrm{jk}}$, where SSS is a quantum analog of Hamilton's principal function. The formalism is shown to be equivalent to a simple change in Hamiltonian, with transformed momentum operators satisfying conventional commutation relations, and with an additional relationship involving formal analogs of the classical "initial constants" adjoined. A particular choice of $S(=\frac{\ensuremath{\hbar}}{i})$ leads to a theory identical with wave mechanics a part from a constant (unobservable) phase factor on the wave function. The fact that SSS may possess other, nonconstant values, demonstrated by a specific example, suggests the ability of the mechanical equations to describe a broader class of physical states than has hitherto been investigated.