# No Preferred Reference Frame at the Foundation of Quantum Mechanics

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## Abstract

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## 1. Introduction

at the basis of information-theoretic reconstructions of QM already implies the relativity principle (aka “no preferred reference frame (NPRF)”) as it pertains to the invariant measurement of Planck’s constant h when applied to spin-$\frac{1}{2}$ measurements in spacetime. This is in total analogy to the Lorentz transformations of special relativity (SR) being based on the relativity principle as it pertains to the invariant measurement of the speed of light c (light postulate). Thus, the information-theoretic reconstructions of QM (hereafter, “reconstructions of QM”) provide a “principle” account of QM in total analogy to that of SR [23,25,26,27], revealing a deep unity between these pillars of modern physics where others have perceived tension [28,29,30,31].The total information of one bit is invariant under a continuous change between different complete sets of mutually complementary measurements.

The information-theoretic interpretation is the proposal to take Hilbert space as the kinematic framework for the physics of an indeterministic universe, just as Minkowski space provides the kinematic framework for the physics of a non-Newtonian, relativistic universe. In special relativity, the geometry of Minkowski space imposes spatio-temporal constraints on events to which the relativistic dynamics is required to conform. In quantum mechanics, the non-Boolean projective geometry of Hilbert space imposes objective kinematic (i.e., pre-dynamic) probabilistic constraints on correlations between events to which a quantum dynamics of matter and fields is required to conform.

Our goal here is to show how this key difference between classical and quantum probability theories per the principle of Information Invariance and Continuity relates directly to an application of NPRF in spacetime.We suggest that (continuous) reversibility may be the postulate which comes closest to being a candidate for a glimpse on the genuinely physical kernel of “quantum reality”. Even though Fuchs may want to set a higher threshold for a “glimpse of quantum reality”, this postulate is quite surprising from the point of view of classical physics: when we have a discrete system that can be in a finite number of perfectly distinguishable alternatives, then one would classically expect that reversible evolution must be discrete too. For example, a single bit can only ever be flipped, which is a discrete indivisible operation. Not so in quantum theory: the state $|0\rangle $ of a qubit can be continuously-reversibly “moved over” to the state $|1\rangle $. For people without knowledge of quantum theory (but of classical information theory), this may appear as surprising or “paradoxical” as Einstein’s light postulate sounds to people without knowledge of relativity.

Nearly every introductory physics textbook introduces SR via the relativity principle and light postulate without qualifying that introduction as somehow lacking an “interpretation”. With few exceptions, physicists have come to accept the principles of SR without worrying about a constructive counterpart. Thus, a principle account of QM based on NPRF as with SR certainly constitutes an important advance in our understanding of QM. Perhaps prophetically, Bell said ([51], p. 85):We can distinguish various kinds of theories in physics. Most of them are constructive. They attempt to build up a picture of the more complex phenomena out of the materials of a relatively simple formal scheme from which they start out. [The kinetic theory of gases is an example.] … Along with this most important class of theories there exists a second, which I will call “principle-theories.” These employ the analytic, not the synthetic, method. The elements which form their basis and starting point are not hypothetically constructed but empirically discovered ones, general characteristics of natural processes, principles that give rise to mathematically formulated criteria which the separate processes or the theoretical representations of them have to satisfy. [Thermodynamics is an example.] … The advantages of the constructive theory are completeness, adaptability, and clearness, those of the principle theory are logical perfection and security of the foundations. The theory of relativity belongs to the latter class.

By revealing the relativity principle’s role at the foundation of QM, information-theoretic reconstructions of QM have revealed what QM is telling us about Nature to no less an extent than SR. And, SR’s principle explanation of Nature certainly constituted a “big new development” for physics in 1905. As emphasized by Fuchs, “Where present-day quantum-foundation studies have stagnated in the stream of history is not so unlike where the physics of length contraction and time dilation stood before Einstein’s 1905 paper on special relativity” [5]. At that time, “Maxwellian physicists were ready to abandon the relativity of motion principle” [45] and even “Einstein was willing to sacrifice the greatest success of 19th century physics, Maxwell’s theory, seeking to replace it by one conforming to an emission theory of light, as the classical, Galilean kinematics demanded” before realizing that such an emission theory would not work [43]. Thus, concerning his decision to produce a principle explanation instead of a constructive explanation for time dilation and length contraction, Einstein writes [52]:I think the problems and puzzles we are dealing with here will be cleared up, and … our descendants will look back on us with the same kind of superiority as we now are tempted to feel when we look at people in the late nineteenth century who worried about the aether. And Michelson-Morley …, the puzzles seemed insoluble to them. And came Einstein in nineteen five, and now every schoolboy learns it and feels … superior to those old guys. Now, it’s my feeling that all this action at a distance and no action at a distance business will go the same way. But someone will come up with the answer, with a reasonable way of looking at these things. If we are lucky it will be to some big new development like the theory of relativity.

Therefore, being in a similar situation today with QM, it is not unreasonable to seek a compelling principle account of QM along the lines of SR. Again, a principle account of QM that maps to NPRF applied to h at its foundation would be as valuable to understanding QM as NPRF applied to c is to understanding SR and, as we will show, the information-theoretic reconstructions of QM entail exactly that understanding.By and by I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more despairingly I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results.

## 2. The Qubit and NPRF

## 3. Planck’s Constant and Spin

**Figure 4.**The classical constructive model of the Stern-Gerlach (SG) experiment. If the atoms enter with random orientations of their “intrinsic” magnetic moments (due to their “intrinsic” angular momenta), the SG magnets should produce all possible deflections, not just the two that are observed [59,66].

**Figure 5.**The “intrinsic” angular momentum of Bob’s particle $\overrightarrow{S}$ projected along his measurement direction $\widehat{b}$. This does not happen with spin angular momentum due to NPRF.

**Figure 6.**State space for a qubit showing two reference frames of mutually complementary SG spin measurements [55].

## 4. Implication for Entanglement via the Bell States

## 5. Conclusions

he nonetheless acknowledged ([74], p. 230):Einstein simply postulates what we have deduced, with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field.

We have shown how the principle of Information Invariance and Continuity at the basis of axiomatic reconstructions of QM provides an understanding of the qubit and Bell state entanglement that is every bit the equal of Einstein’s postulates of SR for understanding time dilation and length contraction. Thus, it is no longer true that “nobody understands quantum mechanics” unless it is also true that nobody understands special relativity. Very few physicists would make that claim.By doing so, [Einstein] may certainly take credit for making us see in the negative result of experiments like those of Michelson, Rayleigh, and Brace, not a fortuitous compensation of opposing effects but the manifestation of a general and fundamental principle.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Probability state space for the qubit $|u\rangle $ in the z basis. Since this state space is isomorphic to 3-dimensional real space, the Bloch sphere is shown in a real space reference frame with its related Stern-Gerlach (SG) magnet orientations (see Knight ([59], p. 1307) for an explanation of the SG experiment). The probability is given for a $+1$ outcome at the measurement direction shown [55]. Compare this with Figure 2.

**Figure 2.**In this set up, the first SG magnets (oriented at $\widehat{z}$) are being used to produce an initial state $|\psi \rangle =|u\rangle $ for measurement by the second SG magnets (oriented at $\widehat{b}$). Compare this with Figure 1.

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Stuckey, W.; McDevitt, T.; Silberstein, M. No Preferred Reference Frame at the Foundation of Quantum Mechanics. *Entropy* **2022**, *24*, 12.
https://doi.org/10.3390/e24010012

**AMA Style**

Stuckey W, McDevitt T, Silberstein M. No Preferred Reference Frame at the Foundation of Quantum Mechanics. *Entropy*. 2022; 24(1):12.
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Stuckey, William, Timothy McDevitt, and Michael Silberstein. 2022. "No Preferred Reference Frame at the Foundation of Quantum Mechanics" *Entropy* 24, no. 1: 12.
https://doi.org/10.3390/e24010012