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arXiv:2301.10718

Gauge is quantum?

Andrei T. Patrascu

Published Jan 5, 2023• 0 citations
arXivHTMLPDF

Abstract

An interesting phenomenon is happening in the construction of the Madelung equations from the Schrodinger equation. It seems like the Madelung equations require a rotational invariance symmetry to properly account for quantum vortices, and that Madelung equations are not fully determining the dynamics. The relation between Schrodinger's equations and Madelung equations are often debated with the observation that no clear understanding exists for why the additional rotational discretisation condition is required. Here I explain it as an additional gauge symmetry that speaks in favour of the recent idea that quantum is gauge (Q=G). Indeed, this additional symmetry seems to emerge as a gauge symmetry condition that needs to be incorporated in the Madelung equations in order to properly describe quantum mechanics. In that sense "Madelung Equation + Gauge symmetry = Quantum mechanics". Arguments in favour of understanding the quantum phase as a gauge symmetry component of the solution of Schrodinger's equations are also introduced.

DOI
10.48550/arxiv.2301.10718

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An interesting phenomenon is happening in the construction of the Madelung equations from the Schrodinger equation. It seems like the Madelung equations require a rotational invariance symmetry to properly account for quantum vortices, and that Madelung equations are not fully determining the dynamics. The relation between Schrodinger's equations and Madelung equations are often debated with the observation that no clear understanding exists for why the additional rotational discretisation condition is required. Here I explain it as an additional gauge symmetry that speaks in favour of the recent idea that quantum is gauge (Q=G). Indeed, this additional symmetry seems to emerge as a gauge symmetry condition that needs to be incorporated in the Madelung equations in order to properly describe quantum mechanics. In that sense "Madelung Equation + Gauge symmetry = Quantum mechanics". Arguments in favour of understanding the quantum phase as a gauge symmetry component of the solution of Schrodinger's equations are also introduced.
How to explain the Aharonov-Bohm (AB) effect remains deeply controversial, particularly regarding the tension between locality and gauge invariance. Recently Wallace argued that the AB effect can be explained in a local and gauge-invariant way by using the unitary gauge. In this paper, I present a critical analysis of Wallace's intriguing argument. First, I show that the unitary gauge transforms the Schroedinger equation into the Madelung equations, which are expressed entirely in terms of local and gauge-invariant quantities. Next, I point out that an additional quantization condition needs to be imposed in order that the Madelung equations are equivalent to the Schroedinger equation, while the quantization condition is inherently nonlocal. Finally, I argue that the Madelung equations with the quantization condition can hardly explain the the AB effect, even if in a nonlocal way. This analysis suggests that the unitary gauge does not resolve the tension between locality and gauge invariance in explaining the AB effect, but highlights again the profound conceptual challenges in reconciling the AB effect with a local and gauge-invariant framework.
It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.
It is known since Madelung that the Schrödinger equation can be thought of as governing the evolution of an incompressible fluid, but the current theory fails to mathematically express this incompressibility in terms of the wavefunction without facing problem. In this paper after showing that the current definition of quantum-mechanical momentum as a linear operator is neither the most general nor a necessary result of the de Broglie hypothesis, a new definition is proposed that can yield both a meaningful mathematical condition for the incompressibility of the Madelung fluid, and nonlinear generalisations of Schrödinger and Klein-Gordon equations. The derived equations satisfy all conditions that are expected from a proper generalisation: simplification to their linear counterparts by a well-defined dynamical condition; Galilean and Lorentz invariance (respectively); and signifying only rays in the Hilbert space.
Two unsolved problems in quantum mechanics are addressed: a source of randomness and an origin of entanglement. These problems being hidden in the Schrodinger equations became transparent in its Madelung version. Special attention is concentrated on equivalence between the Schrodinger and the Madelung equations. It has been demonstrated that randomness in quantum mechanics has the same mathematical source as that in turbulence and chaos, and the origin of entanglement is the global constraint imposed by the normalization constraint of the probability density that becomes an additional variable in the Madelung version of the Schrodinger equation.
Quantum mechanics is ‘emergent’ if a statistical treatment of large scale phenomena in a locally deterministic theory requires the use of quantum operators. These quantum operators may allow for symmetry transformations that are not present in the underlying deterministic system. Such theories allow for a natural explanation of the existence of gauge equivalence classes (gauge orbits), including the equivalence classes generated by general coordinate transformations. Thus, local gauge symmetries and general coordinate invariance could be emergent symmetries, and this might lead to new alleys towards understanding the flatness problem of the Universe.