A Principle Explanation of the “Mysteries” of Modern Physics

All undergraduate physics majors are shown how the counterintuitive aspects (“mysteries”) of time dilation and length contraction in special relativity (SR) follow from the light postulate, i.e., that everyone measures the same value for the speed of light c, regardless of their motion relative to the source (see this Insight, for example). And, we can understand the light postulate to follow from the principle of relativity, sometimes referred to as “no preferred reference frame” (NPRF). Simply put, if the speed of light from a source was only equal to $c=\frac{1}{\sqrt{\epsilon_o \mu_o}}$ (per Maxwell’s equations) for one particular velocity relative to the source, that would certainly constitute a preferred reference frame. Borrowing from Einstein [1], NPRF might be stated (see this Insight):

No one’s “sense experiences,” to include measurement outcomes, can provide a privileged perspective on the “real external world.”

While time dilation and length contraction follow “analytically” from the light postulate, there are those who do not consider the light postulate explanatory, since it does not provide “hypothetically constructed” mechanisms to “synthetically” account for time dilation and length contraction [2][3]. That is, the postulates of SR are “empirically discovered” principles offered without corresponding “constructive efforts.” In what follows, Einstein explains the difference between the two [4]:

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. Thus the kinetic theory of gases seeks to reduce mechanical, thermal, and diffusional processes to movements of molecules – i.e., to build them up out of the hypothesis of molecular motion. When we say that we have succeeded in understanding a group of natural processes, we invariably mean that a constructive theory has been found which covers the processes in question.

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. Thus the science of thermodynamics seeks by analytical means to deduce necessary conditions, which separate events have to satisfy, from the universally experienced fact that perpetual motion is impossible.

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. In order to grasp its nature, one needs first of all to become acquainted with the principles on which it is based.

Here is why Einstein formulated special relativity as a principle theory [5, pp. 51-52]:

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.

Despite the fact that “there is no mention in relativity of exactly how clocks slow, or why meter sticks shrink” (no “constructive efforts”), the “empirically discovered” principles of SR are so compelling that “physicists always seem so sure about the particular theory of Special Relativity, when so many others have been superseded in the meantime” [6].

As it turns out, we are in a similar position today with quantum mechanics (QM). For example, QM accurately predicts violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality all the way to the Tsirelson bound for Bell state entanglement without providing a corresponding constructive account (see this Insight, for example). This prompted Lee Smolin to write [7, p. 227]:

So, my conclusion is that we need to back off from our models, postpone conjectures about constituents, and begin again by talking about principles.

Other physicists are also calling for a principal account of QM. Chris Fuchs writes [8, p. 285]:

Compare [quantum mechanics] to one of our other great physical theories, special relativity. One could make the statement of it in terms of some very crisp and clear physical principles: The speed of light is constant in all inertial frames, and the laws of physics are the same in all inertial frames. And it struck me that if we couldn’t take the structure of quantum theory and change it from this very overt mathematical speak — something that didn’t look to have much physical content at all, in a way that anyone could identify with some kind of physical principle — if we couldn’t turn that into something like this, then the debate would go on forever and ever. And it seemed like a worthwhile exercise to try to reduce the mathematical structure of quantum mechanics to some crisp physical statements.

That QM accurately predicts the violation of the CHSH inequality to the Tsirelson bound without spelling out any corresponding constructive account prompted Smolin to write [7, p. xvii]:

I hope to convince you that the conceptual problems and raging disagreements that have bedeviled quantum mechanics since its inception are unsolved and unsolvable, for the simple reason that the theory is wrong. It is highly successful, but incomplete.

Of course, this is precisely the complaint leveled by Einstein, Podolsky, and Rosen (EPR) in their famous paper, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” [9]. In an equally famous paper (published after Einstein’s death), John Bell showed how any naïve “completion” of QM via hidden variables could conflict with the predictions of QM for two entangled particles (e.g., see this Insight) [10]. As long as the particles couldn’t communicate faster than light and didn’t have access to information about measurement settings until they were actually being measured, the outcomes for entangled particles with hidden variables would have to obey the “Bell inequality” (the CHSH inequality is a variation thereof). There are two-particle entangled states (“Bell states“) in QM that can violate the Bell inequality (and therefore, the CHSH inequality) and these violations are now experimentally well-confirmed. This “mystery” is in large part responsible for “the conceptual problems and raging disagreements that have bedeviled quantum mechanics.”

However, contra Smolin, we recently showed that Bell state entanglement does not render QM “wrong” or “incomplete” by extending NPRF to include the measurement of another fundamental constant of nature, Planck’s constant h [11][12][30]. As Steven Weinberg points out, measuring an electron’s spin via Stern-Gerlach (SG) magnets constitutes the measurement of “a universal constant of nature, Planck’s constant” [13, p. 3] (Figure 1). So if NPRF applies equally here, everyone must measure the same value for Planck’s constant h, regardless of their SG magnet orientations relative to the source, which like the light postulate is an “empirically discovered” fact. By “relative to the source” of a pair of spin-entangled particles, I mean relative “to the vertical in the [symmetry] plane perpendicular to the line of flight of the particles” [14, p. 943] (Figure 2). Thus, different SG magnet orientations relative to the source constitute different “reference frames” in QM just as different velocities relative to the source constitute different “reference frames” in SR.

Just as NPRF and the speed of light c produce the “mysteries” of time dilation and length contraction in SR, we showed that NPRF and Planck’s constant h produce the “mysteries” of Bell state entanglement and the Tsirelson bound for QM. Generally speaking, Bell spin state entanglement results from the conservation of spin angular momentum, and conservation principles and gauge invariance follow from Wheeler’s “boundary of a boundary” principle, $\partial\partial = 0$ [15][16][17] (Figure 3), which with NPRF underwrite all of physics [17][18].

Specifically, when Alice and Bob make their SG spin measurements at the same angle in the plane of symmetry (same reference frame), conservation of spin angular momentum dictates that they obtain the same result (both +1 or both -1) for the spin triplet states (opposite results for the spin singlet state). This suggests (naively) that if Bob makes his SG spin measurement at angle $\theta$ with respect to Alice (different reference frames), then he should obtain $\cos{(\theta)}$ when Alice obtains +1 in accord with the conservation of spin angular momentum (Figure 4). But, Bob can only ever measure $\pm 1$ per NPRF, just like Alice, so the conservation principle is constrained to hold only on average per NPRF (Figures 5 & 6). Thus, Bell state entanglement and the Tsirelson bound are “mysteries” precisely because of “average-only” conservation, which is conservation per NPRF (Figure 7). So we see that the principle of NPRF reveals an underlying coherence between non-relativistic QM and SR (Figure 8) where others have perceived tension.

For example, in his paper “On the Incompatibility of Special Relativity and Quantum Mechanics” Marco Mamone-Capria’s argument is based on “The EPR correlations … as a simple example of quantum mechanical macroscopic effects with spacelike separation from their causes” [19]. He points out that in 1972 Dirac wrote [20, p. 11]:

The only theory which we can formulate at the present is a non-local one, and of course one is not satisfied with such a theory. I think one ought to say that the problem of reconciling quantum theory and relativity is not solved.

And, Bell also voiced concerns about the compatibility of SR and QM based on quantum entanglement [21, p. 172]:

For me then this is the real problem with quantum theory: the apparently essential conflict between any sharp formulation and fundamental relativity. That is to say, we have an apparent incompatibility, at the deepest level, between the two fundamental pillars of contemporary theory.

Of course, we know QM is not Lorentz invariant and so it deviates trivially from SR in that fashion. In order to get QM from Lorentz invariant quantum field theory one needs to make low energy approximations [22, p. 173]. But, the charge of incompatibility based on QM entanglement actually carries serious consequences, because we have experimental evidence confirming the violation of the CHSH inequality per QM entanglement. So, if the violation of the CHSH inequality is in any way inconsistent with SR, then SR is being challenged empirically. By analogy, we know Newtonian mechanics deviates from SR because it is not Lorentz invariant. As a consequence, Newtonian mechanics predicts a very different velocity addition rule, so suppose we found experimentally that velocities do add as predicted by Newtonian mechanics. That would not merely mean that Newtonian mechanics and SR are incompatible, that would mean Newtonian mechanics has been empirically verified while SR has been empirically refuted. So, if one believes the violation of the CHSH inequality is in any way inconsistent with SR, and one believes the experimental evidence is accurate, then one believes SR has been empirically refuted. Clearly that is not the case, so their reconciliation as regards the violation of the CHSH inequality must certainly obtain in some fashion and here we see how the principle of NPRF does the job.

Further, contrary to Smolin and EPR, Bell state entanglement does not mean that QM is “incomplete” or “wrong.” Rather, QM is as complete as possible per the principles of $\partial\partial = 0$ and NPRF.

Given this result, one immediately wonders if general relativity (GR) can be brought into the mix via NPRF and the gravitational constant G. Of course it can and the associated counterintuitve aspect (“mystery”) in GR is the contextuality of mass. We already showed how this might resolve the missing mass problem without having to invoke non-baryonic dark matter [23][24].

Specifically, I am pointing out the well-known result per GR that matter can simultaneously possess different values of mass when it is responsible for different combined spatiotemporal geometries. Here “reference frame” refers to each of the different spatiotemporal geometries associated with one and the same matter source. Tacitly assumed in this result is of course that G has the same value in each reference frame, which is consistent with NPRF as applied to c and h above. This spatiotemporal contextuality of mass is not present in Newtonian gravity where mass is an intrinsic property of matter. For example, when a Schwarzschild vacuum surrounds a spherical matter distribution the “proper mass” $M_{p}$ of the matter, as measured locally in the matter, can be different than the “dynamic mass” $M$ in the Schwarzschild metric responsible for orbital kinematics about the matter [25, p. 126]. This difference is attributed to binding energy and goes as $dM_p = \left(1-\frac{2GM(r)}{c^2r}\right)^{-1/2} \: dM$. In another example, suppose a Schwarzschild vacuum surrounds a sphere of Friedmann-Lemaitre-Robertson-Walker (FLRW) dust, as used originally to model stellar collapse [15, pp. 851-853]. The dynamic mass $M$ of the surrounding Schwarzschild metric is related to the proper mass $M_{p}$ of the FLRW dust, as joined at FLRW radial coordinate $\chi_o$, by

\begin{equation}
\frac{M_p}{M} = \frac{3(2\chi_o -\sin(2\chi_o))}{4 \sin ^3(\chi_o)} \label{massratio}
\end{equation}

where

\begin{equation}
ds^2 = -c^2d\tau^2 + a^2(\tau)\left(d\chi^2 + \sin^2\chi d\Omega^2 \right) \label{FLRWmetric}
\end{equation}

is the closed FLRW metric [26]. I should quickly point out that this may prima facie seem to constitute a violation of the equivalence principle, as understood to mean inertial mass equals gravitational mass, since inertial mass can’t be equal to two different values of gravitational mass. But, the equivalence principle says simply that spacetime is locally flat [27, pp. 68-69] and that is certainly not being violated here nor with any solution to Einstein’s equations.

Thus, contrary to what some believe about SR, QM, and GR collectively, these theories are comprehensive (not “incomplete” as claimed in [7] and [9]) and coherent (not “in conflict” as claimed in [19] and [20]). In order to appreciate the beauty of these theories collectively, one need only view them per the principles of $\partial\partial = 0$ and NPRF with their associated “mysteries” corresponding to c, h, and G, respectively. I close with this quote from Wolfgang Pauli [28, p. 33]:

‘Understanding’ nature surely means taking a close look at its connections, being certain of its inner workings. Such knowledge cannot be gained by understanding an isolated phenomenon or a single group of phenomena, even if one discovers some order in them. It comes from the recognition that a wealth of experiential facts are interconnected and can therefore be reduced to a common principle. In that case, certainty rests precisely on this wealth of facts. The danger of making mistakes is the smaller, the richer and more complex the phenomena are, and the simpler is the common principle to which they can all be brought back. … ‘Understanding’ probably means nothing more than having whatever ideas and concepts are needed to recognize that a great many different phenomena are part of a coherent whole.

 

Figure 1. A Stern-Gerlach (SG) spin measurement showing the two possible outcomes, up ($+\frac{\hbar}{2}$) and down ($-\frac{\hbar}{2}$) or +1 and -1, for short. The important point to note here is that the classical analysis predicts all possible deflections, not just the two that are observed. The difference between the classical prediction and the quantum reality uniquely distinguishes the quantum joint distribution from the classical joint distribution for the Bell spin states [29].

 

Figure 2. Alice and Bob making spin measurements on a pair of spin-entangled particles with their Stern-Gerlach (SG) magnets and detectors in the xz-plane. Here Alice and Bob’s SG magnets are not aligned so these measurements represent different reference frames.

 

Figure 3. The boundary of a boundary is zero. The oriented plaquettes bound the cube and the directed edges bound the plaquettes. As you can see from the picture, every edge has oppositely oriented directions that cancel out. Thus, the boundaries of the plaguettes (the edges), which bound the cube, sum to zero.

 

 

Figure 4. The angular momentum of Bob’s particle $\vec{S}_B = \vec{S}_A$ projected along his measurement direction $\hat{b}$. This does not happen with spin angular momentum because Bob must always measure $\pm 1$, no fractions, in accord with NPRF.

 

Figure 5. A spatiotemporal ensemble of 8 experimental trials for the spin triplet states showing Bob’s outcomes corresponding to Alice’s +1 outcome when $\theta = 60^\circ$. Blue arrows depict SG magnet orientations and yellow dots depict the measurement outcomes. Spin angular momentum is not conserved in any given trial, because there are two different measurements being made, i.e., outcomes are in two different reference frames, but it is conserved on average for all 8 trials (six up outcomes and two down outcomes average to $\cos{(60^\circ)}=\frac{1}{2}$).

 

Figure 6. Average View for the Spin Triplet States. Reading from left to right, as Bob rotates his SG magnets relative to Alice’s SG magnets for her +1 outcome, the average value of his outcome varies from +1 (totally up, arrow tip) to 0 to -1 (totally down, arrow bottom). This obtains per conservation of spin angular momentum on average in accord with no preferred reference frame. Bob can say exactly the same about Alice’s outcomes as she rotates her SG magnets relative to his SG magnets for his +1 outcome. That is, their outcomes can only satisfy conservation of spin angular momentum on average in different reference frames, because they only measure $\pm 1$, never a fractional result. Again, just as with the light postulate of special relativity, we see that no preferred reference frame leads to a counterintuitive result. Here it requires quantum outcomes $\pm 1 \left(\frac{\hbar}{2}\right)$ for all measurements leading to the “mystery” of “average-only” conservation.

 

Figure 7. Bub’s version of Wheeler’s question “Why the quantum?” is “Why the Tsirelson bound?” The “constraint” is the principle of “conservation per no preferred reference frame”.

 

Figure 8. Comparing SR with QM according to no preferred reference frame (NPRF). Because Alice and Bob both measure the same speed of light c, regardless of their motion relative to the source per NPRF, Alice(Bob) may claim that Bob’s(Alice’s) length and time measurements are erroneous and need to be corrected (length contraction and time dilation). Likewise, because Alice and Bob both measure the same values for spin angular momentum $\pm 1$ $\left(\frac{\hbar}{2}\right)$, regardless of their SG magnet orientation relative to the source per NPRF, Alice(Bob) may claim that Bob’s(Alice’s) individual $\pm 1$ values are erroneous and need to be corrected (averaged, Figures 5 & 6). In both cases, NPRF resolves the “mystery” it creates. In SR, the apparently inconsistent results can be reconciled via the relativity of simultaneity. That is, Alice and Bob each partition spacetime per their own equivalence relations (per their own reference frames), so that equivalence classes are their own surfaces of simultaneity and these partitions are equally valid per NPRF. This is completely analogous to QM, where the apparently inconsistent results per the Bell spin states arising because of NPRF can be reconciled by NPRF via the “relativity of data partition.” That is, Alice and Bob each partition the data per their own equivalence relations (per their own reference frames), so that equivalence classes are their own +1 and -1 data events and these partitions are equally valid.

Comment Thread

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