On the Equivalence Principle and Relativistic Quantum Mechanics

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On the Equivalence Principle and Relativistic Quantum Mechanics Maciej Trzetrzelewski1  Received: 29 September 2019 / Accepted: 26 September 2020 © The Author(s) 2020

Abstract Einstein’s Equivalence Principle implies that the Lorentz force equation can be derived from a geodesic equation by imposing a certain (necessary) condition on the electromagnetic potential (Trzetrzelewski, EPL 120:4, 2018). We analyze the quantization of that constraint and find the corresponding differential equations for the phase of the wave function. We investigate these equations in the case of Coulomb potential and show that physically acceptable solutions do not exist. This result signals an inconsistency between Einstein’s Equivalence Principle and Relativistic Quantum Mechanics at an atomic level. Keywords  Equivalence principle · Electrodynamics · Relativity · Quantum mechanics

1 Introduction In our previous work [1] we considered a certain generalization of Einstein’s elevator experiment when the elevator is charged. Due to the screening effect, an observer inside the elevator cannot detect the electromagnetic field that surrounds it. This, together with Einstein Equivalence Principle, implies that the observer may identify his trajectory with a geodesic line in some curved space–time, even though the trajectory is given by the Lorentz force law. In the original formulation of Einstein’s Equivalence Principle [2] one assumes a complete physical equivalence of a reference frame K and K ′ where K corresponds to a uniform gravitational field and K ′ has no gravitational field but is uniformly accelerated. This equivalence is only to be understood locally i.e. in an arbitrary small neighbourhood of the observer in K (and K ′  ) since uniform gravitational

* Maciej Trzetrzelewski [email protected] 1



M. Smoluchowski Institute of Physics, Jagiellonian University, Łojasiewicza, St. 11, 30‑348 Kraków, Poland

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Foundations of Physics

fields are only idealisations therefore imperfections of real uniform fields could in principle be detected in terms of the tidal forces. The equivalence stated in this way is based on another equivalence between passive-gravitational mass mG and inertial mass mI of a body (also known as the Galilean Equivalence Principle) which has been confirmed by a number of experiments (see [3, 4] or [5] for a comprehensive review), most recently by the EötWash experiment [6] and the MICROSCOPE experiment [7, 8]. If mG and mI were not equal, one would be able to perform local experiments (such as dropping test bodies) in the accelerating frame K ′ which would produce differences between the outcomes of the same experiments performed in the reference K. The charged elevator thought experiment may be considered as a specific case of Einstein’s Equivalence Principle in which the source of the uniform acceleration of K ′ is given by the surrounding electromagnetic field. For that to work, it is also needful to assume that the observer is isolated from the charge. Finding the value o