EP quantum mechanics

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In physics, EP quantum mechanics is a theory of motion of point particles, partly included in the framework of quantum trajectory representation theories of quantum mechanics, based upon an equivalence postulate (EP) similar in content to the equivalence principle of general relativity, rather than on the traditional Copenhagen interpretation of quantum mechanics. The equivalence postulate states that all one-particle systems can be connected by a non-degenerate coordinate transformation, more precisely by a map over the cotangent bundle of the position manifold, so that there exists a quantum action function [math]\displaystyle{ S(q) }[/math] that transforms as a scalar field. Here, the action defined as

[math]\displaystyle{ d S(q) = p_i(q) dq^i }[/math]

is the canonical one-form. This property lies at the heart of the EP formulation of quantum mechanics. An immediate consequence of the EP is the removal of the rest frame. The theory is based on symmetry properties of the Schwarzian derivative and on the quantum stationary Hamilton-Jacobi equation (QSHJE), which is a partial differential equation for the quantum action function [math]\displaystyle{ S(q) }[/math]. This quantum version of the Hamilton–Jacobi equations differs from the classical one by the presence of a quantum potential term

[math]\displaystyle{ Q(q) = \frac{\hbar^2}{4m}\{S(q),q\} }[/math]

where [math]\displaystyle{ \{,\} }[/math] denotes the Schwarzian derivative. The QSHJE can be demonstrated to imply the Schrödinger equation with square-summability of the wave function, and thus quantization of energy, due to continuity conditions of the quantum potential, without any assumption on the probabilistic interpretation of the wave function. The theory, which is a work in progress, may or may not include probabilistic interpretation as a consequence OR a hidden variable description of trajectories.

References

• Alon E. Faraggi, M. Matone (2000) "The Equivalence Postulate of Quantum Mechanics", International Journal of Modern Physics A, Volume 15, Issue 13, pp. 1869–2017. arXiv hep-th/9809127
• G. Bertoldi, Alon E. Faraggi, M. Matone (2000) "Equivalence principle, higher dimensional Mobius group and the hidden antisymmetric tensor of Quantum Mechanics", Class. Quantum Grav. 17 (2000) 3965–4005. arXiv hep-th/9909201

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