The vacuum fluctuation theorem: Exact Schrödinger equation via nonequilibrium thermodynamics

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1. Three 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

2. Nothing 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

3. The 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

4. The 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

5. The 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

6. Grössing 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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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

Where it sits in the curriculum

• What the vacuum is