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Matematičeskie struktury i modelirovanie

If Energy Is Not Preserved, Then Planck's Constant Is No Longer a Constant: A Theorem

Владик Крейнович, Andrés Felipe Ortiz

Published Jan 1, 2012• 1 citation
OpenAlex

Abstract

For any physical theory, to experimentally check its validity, we need to formulate an alternative theory and check whether the experimental results are consistent with the original theory or with an alternative theory. In particular, to check whether energy is preserved, it is necessary to formulate an alternative theory in which energy is not preserved. Formulating such a theory is not an easy task in quantum physics, where the usual Schroedinger equation implicitly assumes the existence of an energy (Hamiltonian) operator whose value is preserved. In this paper, we show that the only way to get a consistent quantum theory with energy non-conservation is to use Heisenberg representation in which operators representing physical quantities change in time. We prove that in this representation, energy is preserved if and only if Planck's constant remains a constant. Thus, an appropriate quantum analogue of a theory with non-preserved energy is a theory in which Planck's constant can change -- i.e., is no longer a constant, but a new field.

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For any physical theory, to experimentally check its validity, we need to formulate an alternative theory and check whether the experimental results are consistent with the original theory or with an alternative theory. In particular, to check whether energy is preserved, it is necessary to formulate an alternative theory in which energy is not preserved. Formulating such a theory is not an easy task in quantum physics, where the usual Schroedinger equation implicitly assumes the existence of an energy (Hamiltonian) operator whose value is preserved. In this paper, we show that the only way to get a consistent quantum theory with energy non-conservation is to use Heisenberg representation in which operators representing physical quantities change in time. We prove that in this representation, energy is preserved if and only if Planck's constant remains a constant. Thus, an appropriate quantum analogue of a theory with non-preserved energy is a theory in which Planck's constant can change -- i.e., is no longer a constant, but a new field.
Is the formalism of quantum mechanics complete? Examination of Max Karl Planck's original quantum work suggests the answer is "No".1-3 Retrospective analysis of Planck's foundational quantum derivation suggests that his action constant ("h") is actually the product of an energy constant ("h̃") and a time variable (tm"). Planck's ground-breaking quantum formula, "E = hν" (E is energy and ν is frequency) may more appropriately be denoted as, "E = h̃tmν". The energy constant represents the energy of a single oscillation or wave of electromagnetic (EM) energy, and is constant regardless of frequency, wavelength or photon energy. The time variable (as originally defined by Planck) is the measurement time of the electromagnetic energy. The invariance of the EM energy constant over a change in time or space suggests it represents a universal constant for light. Retrospective evaluation also reveals that Planck inadvertently fixed the EM measurement time (tm) at a value of one (1) second, thereby significantly limiting the time variable's degrees of freedom. Implications appear to be profound for restoring infinite degrees of freedom to the EM measurement time variable. Quantum re-interpretation using the energy constant and measurement time variable suggest conservation of energy, momentum and mass for a single oscillation of light. A deeper layer of foundations in physics may yet need to be built.
Planck's constant h is considered to be a fundamental Universal constant of Physics. And although we can experimentally determine its value to great precision, the reason for its existence and what it really means is still a mystery. Quantum Mechanics has adapted it in its mathematical formalism, as it also has the Quantum Hypothesis. But QM does not explain its meaning or prove its existence. Why does the Universe need h and energy quanta? Why does the mathematical formalism of QM so accurately reflect physical phenomena and predict these with great precision? Ask any physicists and uniformly the answer is that's how the Universe works. The units of h are in energy-time and the conventional interpretation of h is as a quantum of action. But in this brief note we take a different view. We interpret h as the minimal accumulation of energy that can be manifested in our measurements. Certainly the units of h agree with such interpretation. Based on this we provide a plausible explanation for the existence of Planck's constant, what it means and how it comes about. We show that the existence of Planck's constant is not so much dictated by the Universe but rather by Mathematics and the inner consistence and calibrations of Physics.
We formulate physically motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to quantum mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory, respectively. We show that even if the second group of axioms is dropped, there are no deformations of quantum mechanics which preserve the kinematic axioms. Thus, any theory going beyond quantum mechanics must represent a radical departure from the usual a priori assumptions about the laws of nature.
We formulate physically-motivated axioms for a physical theory which for systems with a finite number of degrees of freedom uniquely lead to Quantum Mechanics as the only nontrivial consistent theory. Complex numbers and the existence of the Planck constant common to all systems arise naturally in this approach. The axioms are divided into two groups covering kinematics and basic measurement theory respectively. We show that even if the second group of axioms is dropped, there are no deformations of Quantum Mechanics which preserve the kinematic axioms. Thus any theory going beyond Quantum Mechanics must represent a radical departure from the usual a priori assumptions about the laws of Nature.
We discuss the possibility that different realms of physics are described by distinct quantization constants. From the consistency of existing data, we infer limits on the differences between hypothetically distinct quantization constants for different elementary particles. Since the existence of multiple Planck constants implies violations of space-time symmetries, these limits may be viewed as precise tests of fundamental conservation laws, including the conservation of linear momentum and energy.
There is controversial evidence that Planck’s constant shows unexpected variations with altitude above the earth due to Kentosh and Mohageg, and yearly systematic changes with the orbit of the earth about the sun due to Hutchin. Many others have postulated that the fundamental constants of nature are not constant, either in locally flat reference frames, or on larger scales. This work is a mathematical study examining the impact of a position dependent Planck’s constant in the Schrödinger equation. With no modifications to the equation, the Hamiltonian becomes a non-Hermitian radial frequency operator. The frequency operator does not conserve normalization, time evolution is no longer unitary, and frequency eigenvalues can be complex. The wavefunction must continually be normalized at each time in order that operators commuting with the frequency operator produce constants of the motion. To eliminate these problems, the frequency operator is replaced with a symmetrizing anti-commutator so that it is once again Hermitian. It is found that particles statistically avoid regions of higher Planck’s constant in the absence of an external potential. Frequency is conserved, and the total frequency equals “kinetic frequency” plus “potential frequency”. No straightforward connection to classical mechanics is found, that is, the Ehrenfest’s theorems are more complicated, and the usual quantities related by them can be complex or imaginary. Energy is conserved only locally with small gradients in Planck’s constant. Two Lagrangian densities are investigated to determine whether they result in a classical field equation of motion resembling the frequency-conserving Schrödinger equation. The first Largrangian is the “energy squared” form, the second is a “frequency squared” form. Neither reproduces the target equation, and it is concluded that the frequency-conserving Schrödinger equation may defy deduction from field theory.