Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral

Version 1 : Received: 28 January 2021 / Approved: 29 January 2021 / Online: 29 January 2021 (11:48:01 CET)

How to cite: Dannenberg, R. Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral. Preprints 2021, 2021010612 (doi: 10.20944/preprints202101.0612.v1). Dannenberg, R. Variable Planck’s Constant: Treated As A Dynamical Field And Path Integral. Preprints 2021, 2021010612 (doi: 10.20944/preprints202101.0612.v1).

Abstract

The constant ħ is elevated to a dynamical field, coupling to other fields, and itself, through the Lagrangian density derivative terms. The spatial and temporal dependence of ħ falls directly out of the field equations themselves. Three solutions are found: a free field with a tadpole term; a standing-wave non-propagating mode; a non-oscillating non-propagating mode. The first two could be quantized. The third corresponds to a zero-momentum classical field that naturally decays spatially to a constant with no ad-hoc terms added to the Lagrangian. An attempt is made to calibrate the constants in the third solution based on experimental data. The three fields are referred to as actons. It is tentatively concluded that the acton origin coincides with a massive body, or point of infinite density, though is not mass dependent. An expression for the positional dependence of Planck’s constant is derived from a field theory in this work that matches in functional form that of one derived from considerations of Local Position Invariance violation in GR in another paper by this author. Astrophysical and Cosmological interpretations are provided. A derivation is shown for how the integrand in the path integral exponent becomes Lc/ħ(r), where Lc is the classical action. The path that makes stationary the integral in the exponent is termed the “dominant” path, and deviates from the classical path systematically due to the position dependence of ħ. The meaning of variable ħ is seen to be related to the rate of time passage along each path increment. The changes resulting in the Euler-Lagrange equation, Newton’s first and second laws, Newtonian gravity, Friedmann equation with a Cosmological Constant, and the impact on gravitational radiation for the dominant path are shown and discussed.

Subject Areas

Modified Gravity; variable constants; fundamental constants

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