# Answering Mermin’s Challenge with Wilczek’s Challenge

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Nearly four decades ago, Mermin revealed the conundrum of quantum entanglement for a general audience [1] using his “simple device,” which I will refer to as the “Mermin device” (Figure 1). To understand the conundrum of the device required no knowledge of physics, just some simple probability theory, which made the presentation all the more remarkable. Concerning this paper Feynman wrote to Mermin, “One of the most beautiful papers in physics that I know of is yours in the American Journal of Physics’’ [2, p. 366-7]. In subsequent publications, he “revisited” [3] and “refined” [4] the mystery of quantum entanglement with similarly simple devices. In this Insight, I will focus on the original Mermin device as it relates to the mystery of entanglement via the Bell spin states.

###### Figure 1. The Mermin Device

The Mermin device functions according to two facts that are seemingly contradictory, thus the mystery. Mermin simply supplies these facts and shows the contradiction, which the “general reader” can easily understand. He then challenges the “physicist reader” to resolve the mystery in an equally accessible fashion for the “general reader.” Here is how the Mermin device works.

The Mermin device is based on the measurement of spin angular momentum (Figure 2). The spin measurements are carried out with Stern-Gerlach (SG) magnets and detectors (Figures 2 & 3). The Mermin device contains a source (middle box in Figure 1) that emits a pair of spin-entangled particles towards two detectors (boxes on the left and right in Figure 1) in each trial of the experiment. The settings (1, 2, or 3) on the left and right detectors are controlled by Alice and Bob, respectively, and each measurement at each detector produces either a result of R or G. The following two facts obtain:

1. When Alice and Bob’s settings are the same in a given trial (“case (a)”), their outcomes are always the same, ##\frac{1}{2}## of the time RR (Alice’s outcome is R and Bob’s outcome is R) and ##\frac{1}{2}## of the time GG (Alice’s outcome is G and Bob’s outcome is G).
2. When Alice and Bob’s settings are different (“case (b)”), the outcomes are the same ##\frac{1}{4}## of the time, ##\frac{1}{8}## RR and ##\frac{1}{8}## GG.

The two possible Mermin device outcomes R and G represent two possible spin measurement outcomes “up” and “down,” respectively (Figure 2), and the three possible Mermin device settings represent three different orientations of the SG magnets (Figures 3 & 4).

###### Figure 4. Three orientations of SG magnets in the plane of symmetry for Alice and Bob’s spin measurements corresponding to the three settings on the Mermin device.

Mermin writes, “Why do the detectors always flash the same colors when the switches are in the same positions? Since the two detectors are unconnected there is no way for one to ‘know’ that the switch on the other is set in the same position as its own.” This leads him to introduce “instruction sets” to account for the behavior of the device when the detectors have the same settings. He writes, “It cannot be proved that there is no other way, but I challenge the reader to suggest any.” Now look at all trials when Alice’s particle has instruction set RRG and Bob’s has instruction set RRG, for example.

That means Alice and Bob’s outcomes in setting 1 will both be R, in setting 2 they will both be R, and in setting 3 they will both be G. That is, the particles will produce an RR result when Alice and Bob both choose setting 1 (referred to as “11”), an RR result when both choose setting 2 (referred to as “22”), and a GG result when both choose setting 3 (referred to as “33”). That is how instruction sets guarantee Fact 1. For different settings, Alice and Bob will obtain the same outcomes when Alice chooses setting 1 and Bob chooses setting 2 (referred to as “12”), which gives an RR outcome. And, they will obtain the same outcomes when Alice chooses setting 2 and Bob chooses setting 1 (referred to as “21”), which also gives an RR outcome. That means we have the same outcomes for different settings in 2 of the 6 possible case (b) situations, i.e., in ##\frac{1}{3}## of case (b) trials for this instruction set. This ##\frac{1}{3}## ratio holds for any instruction set with two R(G) and one G(R).

The only other possible instruction sets are RRR or GGG where Alice and Bob’s outcomes will agree in ##\frac{9}{9}## of all trials. Thus, the “Bell inequality” for the Mermin device says that instruction sets must produce the same outcomes in more than ##\frac{1}{3}## of all case (b) trials. But, Fact 2 for the Mermin device says you only get the same outcomes in ##\frac{1}{4}## of all case (b) trials, thereby violating the Bell inequality. Thus, the conundrum of Mermin’s device is that the instruction sets needed for Fact 1 fail to yield the proper outcomes for Fact 2. That quantum mechanics accurately predicts the observed phenomenon without spelling out any means a la “instruction sets” for how it works prompted Smolin to write [6, 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.

Concerning his device Mermin wrote, “Although this device has not been built, there is no reason in principle why it could not be, and probably no insurmountable practical difficulties.” Sure enough, the experimental confirmation of the violation of Bell’s inequality per quantum entanglement is so common that it can now be carried out in the undergraduate physics laboratory [7]. Thus, there is no disputing that the conundrum of the Mermin device has been experimentally well verified, vindicating its prediction by quantum mechanics.

While the conundrum of the Mermin device is now a well-established fact, Mermin’s challenge to explain the device “in terms meaningful to a general reader struggling with the dilemma raised by the device” arguably remains unanswered. To answer this challenge, it is generally acknowledged that one needs a compelling model of physical reality or a compelling physical principle by which the conundrum of the Mermin device is resolved. Such a model needs to do more than the “Copenhagen interpretation” [8], which Mermin characterized as “shut up and calculate” [9]. In other words, while the formalism of quantum mechanics accurately predicts the conundrum, quantum mechanics does not provide a model of physical reality or underlying physical principle to resolve the conundrum. While there are many interpretations of quantum mechanics, even one published by Mermin [10], there is no consensus among physicists on any given interpretation.

Rather than offer yet another uncompelling interpretation of quantum mechanics, I will share and expand on an underlying physical principle [11] that explains the quantum correlations responsible for the conundrum of the Mermin device. While this explanation, conservation per no preferred reference frame (NPRF), may not be “in terms meaningful to a general reader,” it is pretty close. That is, all one needs to appreciate the explanation is a course in introductory physics, which probably represents the “general reader” interested in this topic.

Of course, entangled states like the Bell spin states result from conservation principles and quantum mechanics produces classical results on average, so this result is perhaps not surprising in a general sense. However, as we will see, the conservation principle represented by the Bell spin states holds only on average, not on a trial-by-trial basis. Conservation principles are usually underwritten by some dynamical mechanism and hold on a trial-by-trial basis. For example, conservation of momentum obtains when the sum of the forces is zero and Fermat’s principle of least time obtains due to refraction per Snell’s law, and both hold on a trial-by-trial basis. Thus, the “average-only” conservation represented by the Bell spin states is a constraint without any obvious deeper dynamical mechanism at work and I will motivate it via NPRF. Those who seek a dynamical explanation of the Mermin device might demand to know what makes the particles produce the specific outcomes in any given trial of the experiment. Therefore, while conservation per NPRF will likely strike the typical physicist as a sound principle, it only resolves the conundrum of the Mermin device if one is willing to accept the conservation principle as a constraint over the ensemble of measurement results without a corresponding ‘deeper mechanism’ at work to govern outcomes on a trial-by-trial basis. Thus, the solution to the mystery of the Mermin device per the average conservation principle (with or without NPRF) requires we move beyond dynamical (time-evolved, causal) explanation per the “ant’s-eye view” of physical reality to adynamical (constraint-based) explanation per the “4D view,” like that of Minkowski spacetime, per Wilczek’s challenge [12, p. 37]:

A recurring theme in natural philosophy is the tension between the God’s-eye [4D] view of reality comprehended as a whole and the ant’s-eye view of human consciousness, which senses a succession of events in time. Since the days of Isaac Newton, the ant’s-eye view has dominated fundamental physics. We divide our description of the world into dynamical laws that, paradoxically, exist outside of time according to some, and initial conditions on which those laws act. … The God’s-eye [4D] view seems, in the light of relativity theory, to be far more natural. Relativity teaches us to consider spacetime as an organic whole whose different aspects are related by symmetries that are awkward to express if we insist on carving experience into time slices. … To me, ascending from the ant’s-eye view to the God’s-eye [4D] view of physical reality is the most profound challenge for fundamental physics in the next 100 years.

If one can rise to Wilczek’s challenge in answering Mermin’s challenge, they will see that quantum mechanics is not only complete, contrary to Smolin, but it shares an underlying coherence with Einstein’s other revolution [5], special relativity. That is, the mysteries of both are grounded in the same principle, “no preferred reference frame.” This intuitive conservation principle resides in the correlation function, so I start there.

The correlation function between two outcomes over many trials is the average of the two values multiplied together. In this case, there are only two possible outcomes for any setting, +1 (up or R) or –1 (down or G), so the largest average possible is +1 (total correlation, RR or GG, as when the settings are the same) and the smallest average possible is –1 (total anti-correlation, RG or GR). One way to write the equation for the correlation function is
$$\langle \alpha,\beta \rangle = \sum (i \cdot j) \cdot p(i,j \mid \alpha,\beta) \label{average}$$
where ##p(i,j \mid \alpha,\beta)## is the probability that Alice measures ##i## and Bob measures ##j## when Alice’s SG magnet is at angle ##\alpha## and Bob’s SG magnet is at angle ##\beta##, and ##(i \cdot j)## is just the product of the outcomes ##i## and ##j##. The correlation function for instruction sets for case (a) is the same as that of the Mermin device for case (a), i.e., they’re both 1. Thus, we must explore the difference between the correlation function for instruction sets and the Mermin device for case (b).

To get the correlation function for instruction sets for different settings, we need the probabilities of measuring the same outcomes and different outcomes for case (b), so we can use Eq. (\ref{average}). We saw that when we had two R(G) and one G(R), the probability of getting the same outcomes for different settings was ##\frac{1}{3}## (this would break down to ##\frac{1}{6}## for each of RR and GG overall). Thus, the probability of getting different outcomes would be ##\frac{2}{3}## for these types of instruction sets (##\frac{1}{3}## for each of RG and GR). That gives a correlation function of

\langle \alpha,\beta \rangle = \left(+1\right)\left(+1\right)\left(\frac{1}{6}\right) + \left(-1\right)\left(-1\right)\left(\frac{1}{6}\right) +  \left(+1\right)\left(-1\right)\left(\frac{2}{6}\right) + \left(-1\right)\left(+1\right)\left(\frac{2}{6}\right)= -\frac{1}{3}

For the other type of instruction sets, RRR and GGG, we would have a correlation function of ##+1## for different settings, so overall the correlation function for instruction sets for different settings has to be larger than ##-\frac{1}{3}##. In fact, if all eight possible instruction sets are produced with equal frequency, then for any given pair of case (b) settings, e.g., 12 or 13 or 23, you will obtain RR, GG, RG, and GR in equal numbers giving a correlation function of zero. That means the results are uncorrelated as one would expect given that all possible instruction sets are produced randomly, i.e., with equal frequency. From this we would typically infer that there is nothing that needs to be explained.

Fact 2 for the Mermin device says the probability of getting the same results (RR or GG) for different settings is ##\frac{1}{4}## (##\frac{1}{8}## for each of RR and GG). Thus, the probability of getting different outcomes for case (b) must be ##\frac{3}{4}## (##\frac{3}{8}## for each of RG and GR). That gives a correlation function of
\langle \alpha,\beta \rangle = \left(+1\right)\left(+1\right)\left(\frac{1}{8}\right) + \left(-1\right)\left(-1\right)\left(\frac{1}{8}\right) +  \left(+1\right)\left(-1\right)\left(\frac{3}{8}\right) + \left(-1\right)\left(+1\right)\left(\frac{3}{8}\right)= -\frac{1}{2}

That means the Mermin device is more strongly anti-correlated for different settings than instruction sets. Indeed, if all possible instruction sets are produced with equal frequency, the Mermin device evidences something to explain (anti-correlated results) where instruction sets suggest there is nothing to explain (uncorrelated results). Thus, quantum mechanics predicts and we observe anti-correlated outcomes for different settings in need of explanation while its classical counterpart suggests there is nothing in need of explanation at all. Mermin’s challenge then amounts to explaining why that is true for the “general reader.”

At this point read my Insight Bell States and Conservation of Spin Angular Momentum.

Now you understand how the correlation function for the Bell spin states results from average-only conservation (as a mathematical fact) resulting from the fact that Alice and Bob both always measure ##\pm 1 \left(\frac{\hbar}{2}\right)## (quantum), never a fraction of that amount (classical), as shown in Figure 2 (empirical fact). In other words, the mathematical facts (to include average-only conservation) map to the empirical facts (Facts 1 and 2 of the Mermin device). Indeed, many physicists are content with this as an explanation of Facts 1 and 2 for the Mermin device. But, stopping here would ignore what is clearly a conundrum for many others in the foundations community. Therefore, I now articulate for both the “physicist reader” and “general reader” why there is still a mystery and what can be done to resolve it.

The problem with the average conservation principle responsible for the quantum correlation function is that it holds only on average. Thus, it does not supply an explanation for outcomes on a trial-by-trial basis (Figure 5). This is quite unlike other constraints we have in classical physics. For example, conservation of momentum holds on a trial-by-trial basis because the sum of the forces equals zero and a light ray always takes the path of least time (Fermat’s principle) because of refraction at the interface per Snell’s law. Those constraints hold on average because they hold for each and every trial. In other words, constraints are typically explained dynamically and hold on a trial-by-trial basis. Therefore, we would like something other than a dynamical/causal mechanism to account for this “average-only” conservation.

###### Figure 5. A spatiotemporal (4D) ensemble of 8 experimental trials for the Bell spin states showing Bob’s outcomes corresponding to Alice‘s ##+1## outcomes when ##\theta = 60^\circ##. 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}##). It is impossible for angular momentum to be conserved explicitly in each trial since the measurement outcomes are binary (quantum) with values of ##+1## (up) or ##-1## (down) per no preferred reference frame. The conservation principle at work here assumes Alice and Bob’s measured values of angular momentum are not mere components of some hidden angular momentum with variable magnitude. That is, the measured values of angular momentum are the angular momenta contributing to this conservation, as I explained in my Insight Bell States and Conservation of Spin Angular Momentum.

I posit that the reason we have average-only conservation is ultimately due to “no preferred reference frame” (NPRF). To motivate NPRF for the Bell spin states, consider the empirical facts. First, Bob and Alice both measure ##\pm 1 \left(\frac{\hbar}{2}\right)## for all SG magnet orientations. In order to satisfy conservation of spin angular momentum for any given trial when Alice and Bob are making different measurements, i.e., when they are in different reference frames, it would be necessary for Bob or Alice to measure some fraction, ##\pm \cos(\theta)##, as I explained in Bell States and Conservation of Spin Angular Momentum. For example, if Alice measured ##+1## at ##\alpha = 0## for an ##S = 1## state and Bob made his measurement (in the plane of symmetry) at ##\beta = 60^\circ##, then Bob’s outcome would need to be ##\frac{1}{2}## (Figure 5). In that case, we would know that Alice measured the “true” angular momentum of her particle while Bob only measured a component of the “true” angular momentum for his particle. Thus, Alice’s SG magnet orientation would definitely constitute a “preferred reference frame.”

But, this is precisely what does not happen. Alice and Bob both always measure ##\pm 1 \left(\frac{\hbar}{2}\right)##, no fractions, in accord with NPRF. And, this fact alone distinguishes the quantum joint distribution from the classical joint distribution [4] (Figure 2). Therefore, the average-only conservation responsible for the correlation function for the Bell spin states leading to Facts 1 and 2 for the Mermin device is actually conservation resulting from NPRF. This is not the only mystery in modern physics resulting from NPRF, as I explained in Modern Physics Understood as an Unrecognized Kuhnian Revolution.

So, we see that the mathematical fact (average-only conservation) responsible for the mysterious empirical facts of the Mermin device can itself be understood to result from NPRF. The price for believing the ultimate explanation of Facts 1 and 2 resides in NPRF is no less than to replace time-evolved, causal mechanisms (dynamical explanation) as fundamental with 4D constraints (adynamical explanation). In fact, it means subscribing to the possibility that some phenomena are only explicable adynamically per Wilczek’s challenge [13]. This is a rather high price for some to pay, so I motivated NPRF using Einstein’s epistemology in Modern Physics Understood as an Unrecognized Kuhnian Revolution.

##### References
1. Mermin, N.D.: Bringing home the atomic world: Quantum mysteries for anybody. American Journal of Physics 49, 940-943 (1981).
2. Feynman, M.: Perfectly Reasonable Deviations from the Beaten Track. Basic Books, New York (2005).
3. Mermin, N.D.: Quantum mysteries revisited. American Journal of Physics 58, 731-734 (Aug 1990).
4. Mermin, N.D.: Quantum mysteries refined. American Journal of Physics 62, 880-887 (Aug 1994).
5. Garg, A., and Mermin, N.D.: Bell Inequalities with a Range of Violation that Does Not Diminish as the Spin Becomes Arbitrarily Large, Physical Review Letters 49(13), 901–904 (1982).
6. Smolin, L.: Einstein’s Unfinished Revolution: The Search for What Lies Beyond the Quantum. Penguin Press, New York (2019).
7. Dehlinger, D., and Mitchell, M.W.: Entangled photons, nonlocality, and Bell inequalities in the undergraduate laboratory, American Journal of Physics 70(9), 903–910 (2002).
8. Becker, A.: What is Real? The Unfinished Quest for the Meaning of Quantum Physics. Basic Books, New York (2018).
9. Mermin, N.D.: Could Feynman Have Said This? Physics Today 57(5), 10 (Apr 2004).
10. Mermin, N.D.: What is quantum mechanics trying to tell us? American Journal of Physics 66(9), 753-767 (1998).
11. Unnikrishnan, C.S.: Correlation functions, Bell’s inequalities and the fundamental conservation laws, Europhysics Letters 69, 489–495 (2005).
12. Wilczek, F.: Physics in 100 Years, Physics Today 69(4), 32–39 (2016).
13. Silberstein, M. and Stuckey, W.M. and McDevitt, T.: Beyond the Dynamical Universe, Oxford University Press, Oxford (2018).

88 replies
1. RUTA says:
My first impression: This "adynamical explanation" is simply a euphemism for giving up classical explanation, in situations where one does not like the consequences of the necessity of explanation.

In the case of the violation of the Bell inequalities, the requirement of a classical causal (dynamical) explanation can be fulfilled, and in an easy way, by accepting a preferred frame. The formulas are, then, given by dBB theory and other realistic and causal interpretations. Essentially everything is simple here, there are even the formulas how a particular Bohmian trajectory dynamically influences the trajectory of some other particle far away. So, there is, first of all, no lack of a classical dynamical explanation.

The only problem is that a preferred frame is anathema. Those who risk proposing it can be banned. And you can silently ignore that there are well-known simple dynamical explanations and write things like "it means subscribing to the possibility that some phenomena are only explicable adynamically".

Dynamical explanation also gets you into trouble in GR and constraint-based explanation comes to the rescue there, too (I have Insights on that). It then leads to entirely new approaches to dark matter, dark energy, unification, and quantum gravity (see Chapter 6 of our book). If it was only in QM that adynamical explanation bailed you out, maybe people would consider giving up NPRF in SR and using a preferred frame in QM. I choose constraint-based explanation motivated by NPRF as fundamental to time-evolved, causal explanation for the reasons articulated here and in Chapters 7 and 8 of our book (having to do with the hard problem of consciousness). It gives me coherence and integrity in my worldview as a whole. It’s just a personal preference, though.

2. bhobba says:
For those interested here is Mermin’s original paper:

Feynman in a letter to Mermin said ‘One of the most beautiful papers in physics that I know of is yours in the American Journal of Physics.’

I personally am finding my view of QM evolving a bit. Feynman said the essential mystery of QM was in the double slit experiment. I never actually thought so myself, but was impressed with it as an introduction to the mysteries of QM at the beginning level. I am now starting to think entanglement may be the essential mystery.

Thanks
Bill

3. RUTA says:
For those interested here is Mermin’s original paper:

Feynman in a letter to Mermin said ‘One of the most beautiful papers in physics that I know of is yours in the American Journal of Physics.’

Would you please send me a reference for that? I’ll add it to this Insight and a paper we’re writing. Thnx

4. bhobba says:
Would you please send me a reference for that? I’ll add it to this Insight and a paper we’re writing. Thnx

Got it from here:

‘Richard P. Feynman in a letter to N. David Mermin, related to his AJP paper Bringing home the atomic world: Quantum mysteries for anybody, American Journal of Physics, Volume 49, Issue 10, pp. 940-943 (1981), as quoted in Michelle Feynman (2005). Perfectly Reasonable Deviations from the Beaten Track. Basic Books. p. 367. ISBN 0-7382-0636-9.’

5. RUTA says:
Feel free to share these insights.

I understand your aim to sell your book. But such claims sound far too freaky to me to pay something to see them, my immediate reaction is "why not new approaches to astrology", sorry.

Do not pretend to understand my motives for doing this work. And you should read the work before making any comments pertaining thereto. Here are three published papers for the DM and DE results where we fit galactic rotation curves (THINGS data), the mass profiles of X-ray clusters (ROSAT and ASCA data), the angular power spectrum of the cosmic microwave background (CMB, Planck 2015 data), and the supernova type Ia curve (SCP data) all without DM or DE meeting or exceeding other fits, e.g., metric skew-tensor gravity (MSTG), core-modified NFW DM, scalar-tensor-vector gravity (STVG), ΛCDM, MOND, and Burkett DM. Are those the kinds of analyses you do in astrology? I don’t study astrology, so I wouldn’t know.

References to consciousness add suspicion. The google books version is too limited to get any information out of it. But you can put at least the relevant parts for this discussion somewhere with open access.

Again, you should not criticize an idea out of ignorance. Read the relevant material and then render informed feedback. Here is a paper relating the physics and consciousness to appear in an edited volume.

6. bhobba says:
The only problem is that a preferred frame is anathema. Those who risk proposing it can be banned.

I think I have explained before that what is banned is, except in the area of a historical discussion, LET – ie a theory that involves not only a preferred frame, but a medium that light is supposed to undulate in, have physical effects that shorten objects when they move in it etc. The reason is it is unobservable and superseded by a theory based on simpler testable symmetry assumptions. You can discuss a preferred frame as part of discussions of peer reviewed papers, textbooks, lectures by reputable scientists etc. But discussing it as part of personal theories you may have is not allowed. GLET you mentioned before is borderline because I do not think it ever got past the peer review process. However from my relativity newsgroup days many knowledgeable people did think it was of publishable quality, as do I. If that is what you wanted to discuss, then the mentors would need to approve it.

Thanks
Bill

7. bhobba says:

Just an observation with my mentors hat on – is such a comment really productive? You probably do not know this but Ruta and I have different interests in physics – mine is more mathematical. For example he is quite interested in the the Blockworld view of physics – but its not something I am into. Ruta knows this and actually counselled me that getting his book may not suit me. I did buy it because Ruta is a knowledgeable member of our community here and I was interested in his view. Again as a mentor if anyone started actually forcibly touting books they wrote, or chided anyone for not buying such etc, they would be warned.

Thanks
Bill

8. PeterDonis says:
Once you travel around a loop, and have memory of the past

If you’re truly traveling around a loop (a closed timelike curve), then each time you reach a particular event, your memory state must be the same as all the previous times you reached that event. Otherwise the physical state at that particular event would not be fixed, and it must be.

9. bhobba says:
So, there is a conflict between the two rules. So, a risk to be banned certainly exists.

Why this worry about being banned? You would not get immediately banned for discussing a work that is borderline in meeting our standards. The mentors will discuss it and advise if its ok to continue to discuss. And no I do not agree that the preferred frame in GLET is the same as the aether in LET, but that is for a discussion on it, not this thread.

Thanks
Bill

10. RUTA says:
This does not get rid of the singularity itself, it remains a singularity. And so it explains nothing, the BB remains a point where the equations of the theory fail.

There are two mysteries about the Big Bang, both resulting from dynamical explanation via time-evolved causal mechanisms from initial conditios and both resolved by adynamical explanation via constraints per 4D self consistency. The initial conditions in dynamical explanation are independent of the causal mechanisms and for cosmology, they are therefore a mystery. That is resolved in the self-consistency approach because initial conditions are just as explanatory as any other point on the spacetime manifold. As for the initial singularity, that is also avoidable in at least two ways via adynamical means. For example, one may simply choose the scaling factor to be something other than zero at t = 0. The second-order differential equation for the time evolution of a(t) does not demand a(0) = 0. That is a result of dynamical thinking. The second is the “stop-point problem” of Regge calculus, where you can pick your fundamental lattice spacing based on whatever you like.