The vacuum fluctuation theorem: Exact Schrödinger equation via nonequilibrium thermodynamics
Gerhard Grössing
Abstract and summary · read the original at the source
In one page
Gerhard Grössing at the Austrian Institute for Nonlinear Studies runs a thought experiment: suppose everything we know today about classical wave mechanics and about nonequilibrium thermodynamics — the physics of systems held away from equilibrium by a steady flow of heat — had been on the shelf a hundred years ago. What could have been derived? His answer is the Schrödinger equation, exactly, from three assumptions and no adjustable constants. The move is to stop treating a particle of energy h-bar times omega as an object that simply oscillates and start treating it as a dissipative system kept oscillating by a constant throughput of energy from its surroundings. Grössing then pushes the argument past the derivation into new territory, formulating what he calls the vacuum fluctuation theorem: a statement about how likely a given vacuum fluctuation is against its opposite, and about how large the fluctuations feeding a quantum system can in principle be.
Why it matters hereFor chapter 2 this is the strongest form of the claim that the vacuum is doing real work: the thermostat temperature the derivation needs comes out at exactly the zero-point energy per degree of freedom, so the heat bath in the equations is the zero-point field itself. For chapter 5 it is the vacuum-as-medium picture carried all the way to the Schrödinger equation, with the quantum potential turning out to be a force field written in terms of the flow of that heat.
What it claims
01Three assumptions suffice. First, the probability density of finding the particle equals the squared amplitude of a real classical wave. Second, the time evolution of that quantum probability density mirrors the evolution of the underlying sub-quantum distribution function. Third, the total energy is h-bar times omega plus a fluctuating kinetic term, the squared momentum fluctuation over twice the mass. From those three Grössing derives the exact Schrödinger equation.Chapter 3.1, Eqs. 3.1.1, 3.1.2 and 3.1.7; Chapter 3.2
Published and peer-reviewed02Nothing is fitted. Grössing points out that in Nelson’s stochastic-mechanics derivation the diffusion constant h-bar over twice the mass is put in by hand, whereas here no parameter adjustment, approximation or guessing of constants is made — which is the basis of his claim that this is the shortest derivation of the Schrödinger equation from modern classical physics in the literature, and the only exact one.Abstract; Chapter 1, paragraph 6; Conclusions and outlook, paragraph 1
Published and peer-reviewed03The temperature of the thermostat is not a free parameter either. Using the universal property that a sinusoidal oscillator carries half its total energy as kinetic energy, and requiring the internal and external temperatures of a steady-state system to match, Grössing gets kT over two equal to h-bar omega over two for each degree of freedom — and identifies that heat, in the simplest scenario, with the vacuum’s zero-point fluctuations, which steady-state systems absorb and release again.Chapter 3.2, Eq. 3.2.10 and the paragraph preceding it; Conclusions and outlook, paragraph 3
Published and peer-reviewed04The quantum potential is re-read as a dissipative force field, the negative gradient of that potential, which vanishes identically for conservative systems and is non-zero for nonconservative ones. Its velocity term can be written either from the gradient of the probability density or, equivalently, from the gradient of the heat flow — so the quantum potential carries an explicit dependence on how the heat moves through space.Conclusions and outlook, paragraph 2; Chapter 4, Eqs. 4.4 and 4.13
Published and peer-reviewed05The vacuum fluctuation theorem is Grössing’s extension of the transient fluctuation theorem to the sub-quantum domain: the probability of a fluctuation of a given size relative to the probability of its opposite is set by the average fluctuation of the quantum potential divided by h-bar omega. It follows that the fluctuation terms entering the total momentum can in principle be arbitrarily large, and that even where each probability is small on its own, the relative gradients still contribute significantly.Chapter 4, Eqs. 4.14 to 4.22
Published and peer-reviewed06Grössing reads the theorem as a route to nonlocality: in a delayed-choice experiment, where the walls of the configuration effectively move, minimal changes in amplitude over arbitrary distances inside the apparatus can produce significant momentum fluctuations — which he takes as a strong indication that the vacuum alone can serve as a resource for entanglement. He states that experimental tests of the theorem are conceivable which reach beyond the scope of present-day quantum theory.Chapter 4, final two paragraphs
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The way in
https://doi.org/10.1016/j.physleta.2008.05.007Published in Physics Letters A by Gerhard Grössing of the Austrian Institute for Nonlinear Studies, Akademiehof, Vienna. The manuscript is free to read on arXiv as 0711.4954, but that posting carries the arXiv default licence and the version of record carries the Elsevier text-and-data-mining user licence — neither is a Creative Commons licence — so this page holds the summary, the claims and the author’s own abstract and sends the reader to the source. Grössing’s later open-licence review of the same programme, which reproduces this derivation in full and adds the walking-droplet analogy, has its own sheet at /library/stm-0e3175348a. The claims are located against the manuscript’s numbered chapters and equations.
How to cite it
Gerhard Grössing (2008) The vacuum fluctuation theorem: Exact Schrödinger equation via nonequilibrium thermodynamics. doi:10.1016/j.physleta.2008.05.007
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