Criteria for a good quantum interpretation

[vanhees71]

Do you reject that QM requires a classical context (observer/measurement device etc) to be properly formulated and corroborated?

This suprises me a bit, as I interpret you as defending the experimental/empirical side of thigs? I myself see myself as defend the conceptully consistent reasoning side of things, but even as such an some what armchair position, I take the "principle" of empirical and experimental "practical details" seriously. This is why it will not do IMO, to just "imagine" a reduction of the whole universe. Such scale of things can never be corroborated, and especially when you think of unification with gravity, these details are important. For me, it's clear the quantum theory and its principles are only corroborated for small subsystems taking place on very short time scales, monitoring from classical laboratory. To extrapolate those principles to cosmological times and cosmological scales is speculative.

/Fredrik

As I said before, I think the classical behavior of macroscopic systems is explained by quantum theory too as an emergent phenomenon. There is no contradiction between quantum theory and classical "context" in the sense you quote above. I also think that it doesn't make sense to talk about the state of the "whole universe", which is not even observable to begin with.

 

[Fra]

As I said before, I think the classical behavior of macroscopic systems is explained by quantum theory too as an emergent phenomenon.

Do you not consider this to be circular?

As I see it, both conceptually as well as historically, QM does not "replace" Classical mechanics, it is a special theory which explains the interaction of classical measurement devices with subatomic black boxes, and explains many phenomeman.

I get your point about that one can use QM to "explain the classical behaviour" of a larger system. But this is only from the perspective of another classical context, but does this circularity (or ifinite tower) not bother you`?

Also, I still insist that we do not KNOW if this extrapolation makes complete sense, suffucient to be accepted as a satisfacyory explanation.

/Fredrik

 

[vanhees71]

I don't see, where there is anything circular. Quantum theory as a mathematical theory is self-contained as is classical mechanics and field theory, and it's clear that the latter are approximations of the former.

 

[Fra]

Quantum theory as a mathematical theory is self-contained

This is the problem.

I argue that the mathematical theory of QM, is not univerally corroborated. Its has its domain of corroboration. Applying the mathematics of QM, to arbitrary systems, such as cosmology is IMO a fallacy.

"...In cosmology there is only a single history, so we loose the ability to do an experimentover and over again, while varying the initial conditions. So we have no operationalway to absolutely distinguish the influence of the choice of laws from the choice of initialconditions. When we attempt to impose the Newtonian paradigm on the interpretationof cosmological data, and ask questions that assume a strict separation between the role of law and the role of initial conditions, we end up asking confused questions that haveno clear answers. We call this running into thecosmological fallacy[1], which is the mistake of extendinga method that is designed to study small subsystems of the universe "
-- https://arxiv.org/pdf/1201.2632.pdf

/Fredrik

 

[Demystifier]

Sounds like an excuse.

No, I really can't. It's not you, it's me.

Now seriously. For many people, me included, it's hard and nonintuitive (hence nonsensical) to think of physical world without having in mind a more-or-less clear picture of something which is supposed to be real.

 

[PeterDonis]

For many people, me included, it's hard and nonintuitive (hence nonsensical) to think of physical world without having in mind a more-or-less clear picture of something which is supposed to be real.

Can what is real change with time? If so, thinking about the result of a coin flip that hasn't happened yet seems simple: you have two possible results, heads and tails, neither of which is real before the flip is made; but you can imagine what it would be like if either result happened. Which means you are imagining things that aren't real at the time you are imagining them.

After the flip, the result that happened is real, and the result that didn't happen is not. But you can still imagine what it would have been like if the result that didn't happen, had happened. Which means you are imagining something that not only isn't real at the time you are imagining it, but can never be real, because the coin flip has already happened and its result was what it was.

Do these cases qualify as imagining something which "is supposed to be real" (your words)? If not, why not?

 

[Demystifier]

Can what is real change with time? If so, thinking about the result of a coin flip that hasn't happened yet seems simple: you have two possible results, heads and tails, neither of which is real before the flip is made; but you can imagine what it would be like if either result happened. Which means you are imagining things that aren't real at the time you are imagining them.

After the flip, the result that happened is real, and the result that didn't happen is not. But you can still imagine what it would have been like if the result that didn't happen, had happened. Which means you are imagining something that not only isn't real at the time you are imagining it, but can never be real, because the coin flip has already happened and its result was what it was.

Do these cases qualify as imagining something which "is supposed to be real" (your words)? If not, why not?

That's in fact an excellent example to explain what I mean by "real" and to compare with what other people may think by it.

In this example, the real thing is the coin in space, which always has both the heads side and the tails side. What we call "a definite outcome of coin flipping" is just a special case of that, the case in which the coin settled into a state in which one of the sides is up and the other down.

 

[PeterDonis]

In this example, the real thing is the coin in space, which always has both the heads side and the tails side.

Okay, but that doesn't mean each side always has to face the same way.

What we call "a definite outcome of coin flipping" is just a special case of that, the case in which the coin settled into a state in which one of the sides is up and the other down.

So if I take what you say here at face value, it means you think the coin is real, but which way the coin is facing is not, since you see "coin heads up" and "coin tails up" as just "special cases" of "coin with a heads side and a tails side".

Is that a correct understanding of your position? If not, why not?

 

[Demystifier]

So if I take what you say here at face value, it means you think the coin is real, but which way the coin is facing is not, since you see "coin heads up" and "coin tails up" as just "special cases" of "coin with a heads side and a tails side".

Is that a correct understanding of your position? If not, why not?

If we (not just you and me, but the whole community) cannot easily understand each other about something which is supposed to be obvious and trivial, it's no wonder that we cannot agree on quantum interpretations.

The coin is always facing somehow, so the facing is real. At each time, the facing is specified by the axis of facing and the orientation of facing. Do I need to draw pictures and write equations to explain what I mean by that?

 

[PeterDonis]

The coin is always facing somehow, so the facing is real.

Okay. That would seem to imply that you agree that what is real can change with time, since the coin's facing can change with time.

Now what about imagining the result of a coin flip that hasn't yet happened? Or imagining that the result of a coin flip that has happened was something other than the result that actually happened? Or, more generally, imagining the coin to be facing in a direction other than the one it is actually facing?

All of these seem to me to be cases of imagining something that is not real, but is "supposed to be real", to use your phrase again. Do you agree?

And finally, to get back to the original question of this subthread, do you see any difference in what is "supposed to be real" between the coin we have been discussing, and, say, an electron prepared in a z-spin up state when we measure its spin in the x-direction? If so, what is the difference? If not, why not?

 

[timmdeeg]

My 2 cents, in case of the coin we don't have to bother whether unitarity holds whereas in case of the electron we have to and then it turns out that the answer to the question what is "supposed to be real" is interpretation dependent.

 

[Fra]

If we (not just you and me, but the whole community) cannot easily understand each other about something which is supposed to be obvious and trivial, it's no wonder that we cannot agree on quantum interpretations.

The coin is always facing somehow, so the facing is real. At each time, the facing is specified by the axis of facing and the orientation of facing. Do I need to draw pictures and write equations to explain what I mean by that?

A control question to see if i am getting it:

Can what's "real" in your view, be "subjective" or "observer depdenente". One might first think you say no, but I am thinking about your solipsist HV.

Or is observer equivalence or "objectivity" implied in your notion of "real"?

/Fredrik

 

[Demystifier]

Okay. That would seem to imply that you agree that what is real can change with time, since the coin's facing can change with time.

Now what about imagining the result of a coin flip that hasn't yet happened? Or imagining that the result of a coin flip that has happened was something other than the result that actually happened? Or, more generally, imagining the coin to be facing in a direction other than the one it is actually facing?

All of these seem to me to be cases of imagining something that is not real, but is "supposed to be real", to use your phrase again. Do you agree?

I agree with all the above. Of course, the phrase "supposed to be real" is not very precise, but you understood correctly what I meant.

And finally, to get back to the original question of this subthread, do you see any difference in what is "supposed to be real" between the coin we have been discussing, and, say, an electron prepared in a z-spin up state when we measure its spin in the x-direction? If so, what is the difference? If not, why not?

I'd say there is no difference in the Bohmian interpretation. In the coin case, there is the coin and there are the forces that determine the motion of coin. In the spin case, analogously, there are the pointlike particles and there is the wave function that determines the motion of particles.

Are you now going to derive a paradox and/or contradiction from that?

 

[Demystifier]

Can what's "real" in your view, be "subjective" or "observer depdenente". One might first think you say no, but I am thinking about your solipsist HV.

No. (Except when I am in the solipsistic HV mode of thinking, which is quite rare.)

Or is observer equivalence or "objectivity" implied in your notion of "real"?

Yes.

 

[Morbert]

Is thoroughgoing intelligibilty an obligation of the universe such that every aspect of reality no matter how far removed from our monkey existence must be representable in our models? Or is it wiser to limit our models to observables?

Ultimately this seems to be a subjective value call (and my loaded description probably betrays my own opinion). A more objective metric might be the success/progress/academic impact of the various competing realist and antirealist projects.

 

[Fra]

No. (Except when I am in the solipsistic HV mode of thinking, which is quite rare.)Yes.

Thanks, thant makes things a bit more clear. I am glad you answered yes on the last question, it makes sense.

In this case I would say that the reality you want/week, is expected to be emergent as a kind of equilibrium or attractor state in my view; but the explanatory path requires a walk in non-reality territory. This is why its hard to agree during intermediate discussions. I would expect that an agreement on the end result would be possible, but not on the research path as the thinking-tools are different. But OTOH, the thinking-tools aren't necessarily part of the final science.

/Fredrik

 

[PeterDonis]

I'd say there is no difference in the Bohmian interpretation.

What about other interpretations?

 

[Demystifier]

What about other interpretations?

It would take too much time to analyze each. Are you interested in one interpretation particularly?

 

[PeterDonis]

It would take too much time to analyze each.

Haven't you already done that? In your post that started this subthread, you said:

All realist interpretations are wrong, all non-realist interpretations are nonsensical. The realist interpretations (Bohm, GRW, many worlds, ...) are wrong because we don't know what is real and chances that our guesses are right are too slim. The non-realist interpretations (Copenhagen, QBism, statistical ensemble, relational, ...) are nonsensical because it does not make sense to think about things without imagining that they are real.

This indicates to me that you should already know the answer to my question; you shouldn't have to do any additional analysis.

 

[stevendaryl]

Haven't you already done that? In your post that started this subthread, you said:
This indicates to me that you should already know the answer to my question; you shouldn't have to do any additional analysis.

Demystifier was trying to guess what my opinion was. The lines you quoted were not his own opinion.

 

[PeterDonis]

Demystifier was trying to guess what my opinion was. The lines you quoted were not his own opinion.

In his subsequent exchange with @martinbn it looked like it was his opinion, not just a guess about yours. Perhaps I am misreading.

 

[Demystifier]

Haven't you already done that? In your post that started this subthread, you said:

This indicates to me that you should already know the answer to my question; you shouldn't have to do any additional analysis.

I know the answers implicitly, but if would take effort to make the answers explicit for each particular interpretation.

 

[Demystifier]

Demystifier was trying to guess what my opinion was. The lines you quoted were not his own opinion.

In his subsequent exchange with @martinbn it looked like it was his opinion, not just a guess about yours. Perhaps I am misreading.

It was both. I was trying to put the opinion by stevendaryl into a form that I could agree with.

 

[Sunil]

AFAIK there's no consistent non-local theory compatible with relativity.

Wrong. The non-local realist interpretations of QT are compatible with relativity interpreted as the Lorentz ether. More compatibility is not necessary, because this is sufficient for the agreement with all empirical predictions.

There is an incompatibility only with the fundamentalist spacetime interpretation of relativity. But this is only a metaphysical conflict with a particular interpretation of relativity, not a conflict with relativity as a theory of physics. Even if the spacetime interpretation is the most popular one, and for many people the only interpretation of relativity they have ever heard about.

 

[Sunil]

That illustrates the wrong versus nonsensical conundrum. Interpretations with a macro/micro distinction are (probably) wrong. Interpretations without such a distinction are nonsensical.

I disagree. Realist interpretations like dBB don't have a micro-macro distinction, they have a trajectory $q(t)\in Q$ for microscopic as well as macroscopic things, but this does not make them nonsensical.

 

[bhobba]

Wrong. The non-local realist interpretations of QT are compatible with relativity interpreted as the Lorentz ether.

You wouldn't happen to have a paper linking LET and QFT? It's not a particular interest of mine, but books on QFT I have read mentioned attempts at such have not been successful. Modern SR requires no interpretation - it is simply the geometry of Space-Time implied by the symmetries of an inertial frame. It is like saying Euclidian Geometry requires an interpretation. It's simply the geometry that results from certain symmetries:
https://en.wikipedia.org/wiki/Euclidean_group

If it is true or not, just like SR, it is an experimental, not interpretational matter.

With my mentor's hat on, we do not discuss LET here except in the context of a peer-reviewed paper or in a historical context. If you wish to pursue it further, a peer-reviewed source, textbook or similar is needed. Why is the reason I stated - the aether is simply redundant. If I push an object, one will naturally say its movement is from the force I applied. I could say it wasn't - my push agitated an angel to move it. There is no way to tell the difference. But the angel is obviously redundant unless you have independent experimental evidence for it. So such theories are not part of mainstream physics.

Thanks
Bil

 

[Sunil]

You wouldn't happen to have a paper linking LET and QFT? It's not a particular interest of mine, but books on QFT I have read mentioned attempts at such have not been successful.

? What could be the problem with introducing a hidden preferred frame into standard QFT which could lead to some "not successful"?

Modern SR requires no interpretation - it is simply the geometry of Space-Time implied by the symmetries of an inertial frame. It is like saying Euclidian Geometry requires an interpretation. It's simply the geometry that results from certain symmetries:
https://en.wikipedia.org/wiki/Euclidean_group

The question of the nature of those "certain symmetries" - fundamental or emergent - requires interpretation. LET claims the symmetry is emergent, not fundamental, the spacetime interpretation assumes that it is fundamental.

With my mentor's hat on, we do not discuss LET here except in the context of a peer-reviewed paper or in a historical context. If you wish to pursue it further, a peer-reviewed source, textbook or similar is needed.

Ok, the standard reference to Lorentz ether in the context of modern QFT should do the job:

Schmelzer, I. (2009). A Condensed Matter Interpretation of SM Fermions and Gauge Fields, Found. Phys. 39(1) 73-107, resp. arxiv:0908.0591

I have already heard that it has been discussed a lot (even if I was unable to find much of a discussion here), but have not heard that some fatal error has been found that would invalidate that paper. So, for the purpose of this particular argument it should be sufficient.

Just to clarify, this is a reaction to your request, not an attempt to start a discussion about this paper (I have recognized that Schmelzer is anathema here). And my reply had simply the aim to correct a wrong statement, namely

there's no consistent non-local theory compatible with relativity.

It remains wrong, and my argument remains valid.

Why is the reason I stated - the aether is simply redundant. If I push an object, one will naturally say its movement is from the force I applied. I could say it wasn't - my push agitated an angel to move it. There is no way to tell the difference. But the angel is obviously redundant unless you have independent experimental evidence for it. So such theories are not part of mainstream physics.

First, even if there would be only an angle interpretation of relativity compatible with a consistent non-local theory, the claim I have questioned would be wrong too, and would be falsified by the angel interpretation. Second, Schmelzer computes a lot of things out of his model, like the SM gauge group and all three generations of fermions. And for gravity he derives the EEP. All this the mainstream theories have to postulate. So, to compare it with your angle example does not look fair.

 

[PeterDonis]

AFAIK there's no consistent non-local theory compatible with relativity.

@Demystifier has some papers based on a version of Bohmian mechanics in which Lorentz invariance is an emergent, approximate symmetry, not an exact one. This kind of theory would qualify as a non-local theory compatible with relativity, since experimentally we can't say that Lorentz invariance is an exact symmetry, we can only say it appears to be valid down to the smallest distance scale we can currently probe.

 

[martinbn]

@Demystifier has some papers based on a version of Bohmian mechanics in which Lorentz invariance is an emergent, approximate symmetry, not an exact one. This kind of theory would qualify as a non-local theory compatible with relativity, since experimentally we can't say that Lorentz invariance is an exact symmetry, we can only say it appears to be valid down to the smallest distance scale we can currently probe.

May be a link.

 

[AnssiH]

I'm sorry Bhobba but you are holding a complete misconception about special relativity and its intepretation.

Modern SR requires no interpretation - it is simply the geometry of Space-Time implied by the symmetries of an inertial frame.

That is an interpretation. It is not required nor was presented in the original paper, linked here:
http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

It was specifically put forward as an interpretation by Herman Minkowski.

It's well known that Lorentz ether theory is only philosophically different from Special Relativity - Special Relativity did not push any specific interpretation forward strongly. But Minkowski did, and it is harmful that there are so many people professionals alike, who are not clear on the difference between Minkowski and Special Relativity.

Minkowski requires static spacetime, in which non-locality leads into inconsistency. But that is only so if you literally believe reality is fundamentally static (that there is fundamentally no specific momentary state to reality). The only difference for Lorentz version is that there is a specific state to reality - we just can't probe what it is.

But if that limit really is just an observation limit (as oppose to ontological limit), then in principle there would be no inconsistency issues from non-locality. Being able to measure superluminal signals would just mean we establish a new limit.

I would also not call a static universe interpretation elegant, because it forces you to detach consciousness from the mechanisms of reality. That's pretty terrible complication in my opinion :P

So overall, Sunil's assertion is actually completely valid.

-Anssi

 

[PeterDonis]

Special Relativity did not push any specific interpretation forward strongly. But Minkowski did

By "Special Relativity" you appear to mean "what Einstein originally published in 1905". But @bhobba used the phrase "Modern SR", which does not mean that; it means SR as it is taught and practiced today, which does include spacetime geometry as first formulated by Minkowski. Indeed, the particular flat spacetime geometry used in SR today is called "Minkowski spacetime".

Minkowski requires static spacetime

More precisely, it requires a specific static spacetime geometry, that of flat Minkowski spacetime. (There are other static spacetime geometries which are not flat, but curved. These are solutions of the Einstein Field Equation in GR, but are not part of SR.)

in which non-locality leads into inconsistency

No, it doesn't. It's perfectly possible to have a set of measurement results in Minkowski spacetime (or any static spacetime) that violate the Bell inequalities.

What you cannot have in classical relativity in general (not just Minkowski spacetime or even any static spacetime, but any classical spacetime whatever) is more than one measurement result occurring at a single event (point) in the spacetime.

The only difference for Lorentz version is that there is a specific state to reality

If by "state" you mean "preferred frame" or "preferred set of surfaces of simultaneity", this is correct.

Being able to measure superluminal signals

Is not the same thing as "non-locality". Allowing superluminal signaling is actually a stronger condition than just non-locality. Note that standard QFT allows non-locality (violation of the Bell inequalities) but not superluminal signaling.

a static universe interpretation

"Static universe" is not an interpretation. It's a property that a spacetime geometry can either have or not have.

 

[bhobba]

? What could be the problem with introducing a hidden preferred frame into standard QFT which could lead to some "not successful"?

We discuss standard physics here that, by our rules, you agreed to when you signed up, is peer-reviewed papers, textbooks, talks by respected scientists etc. If ever there is any worry about a relevant source, you can write to a mentor. Instead of simply posting such a source to discuss it, as I requested, it indicates these are just ideas you have. Personal theories (unless published in peer-reviewed journals or equivalent) are not allowed. As I said, this aether thing is not an interest of mine, but as you mentioned, Glet would do:
https://www.ilja-schmelzer.de/glet/

I asked for the link between the aether and QFT. Yes GLET is a legit theory about that the aether, now you need to link it to QFT. Dymysifyer, as mentioned by Peter, has papers on that, so they exist. Quantum fields, of which there are many, are not similar to the old idea of light as classical undulations of the aether. QFT obeys Lorentz symmetry, meaning it has the same properties regardless of the inertial reference frame. In contrast, the aether has Galilean symmetry which means if you move relative to the fixed frame of the ether, the fields will be different. LET explains this using the Lorentz hypothesis of length contraction. Internal length contractions of clock components changed time. In modern times we have atomic clocks, so such an explanation will not work. Electric fields are a dielectric shift in the aether - totally at odds with what they are in QFT. These issues and others all need to be explained by LET.

It is not well known that LET is only philosophically different to modern SR any more than my angel theory is only philosophically different to Newton's laws. It makes a specific claim, a claim that is not just philosophy. For it to be a legit physical theory, you need evidence for the existence of the angels. We have none, so it is rejected. The same with the aether.

All you have to do is post up a source so we can discuss your ideas. It is not hard. There are issues here, such as the quantum vacuum is the aether (it isn't), dark matter, or the CBMR etc., is the aether (again, they are not), but to discuss it, you need a source. Or you could ask why the CBMR is not considered the aether, but that would require a new thread - here you are assuming such exists.

The issue of non-locality in QM is different. I was even a bit confused by it until I came across a paper at CERN. The CERN server is not working for me right now, so I can't give a link, but here is the gist. Bell showed, assuming the Kolmogorov axioms, showed QM is incompatible with counterfactual definiteness. If we relax the Kolmogorov axioms requirement, i.e. assume from the start QM is a Generalised Probability Theory, then the whole 'issue' is bypassed. Technically outcome and parameter independence is assumed. See section 3.1.2 of the following:
https://plato.stanford.edu/entries/bell-theorem/

The generalised probability view of QM is fascinating in its own right:
https://en.wikipedia.org/wiki/Generalized_probabilistic_theory

It shows such theories, as a class, allow for many features of QM, with QM perhaps the simplest. This has been my view for a long time. We also have discussed many times on this forum its compatibility with the cluster decomposition property as expressed by Wienberg. But that requires its own thread.

Thanks
Bill

 

[bobob]

"Interpretation" invariably means some scheme to make quantum mechanics behave like classical mechanics even at the expense of introducing very weird potentials and equally weird fictitious entities along with some "explanation" of how these things "really" don't violate other well known physics, like relativity. How about something more simple for starters, like what quantum mechanics is telling you is that you only get to define so much about nature, but you have a choice between mutually exclusive "pieces of information" (to avoid the word "observable") and that even (or obviously, really) nature has nothing else to offer because that's all that actually exists. I think that idea has been around from the very beginning (starting with the earliest ideas of the copenhagen interpretation).

I think the choices come down to accepting that nature can't divulge what doesn't exist (as in simultaneous values for mutually exclusive observables) or believing nature is engaged in some conspiracy to hide that from us and make it only appear as it does for no particular physical reason.

 

[bhobba]

"Interpretation" invariably means some scheme to make quantum mechanics behave like classical mechanics

That is not quite what is going on as well. QM is a theory about observations that occur here in the classical macro world. What is going on between observations is not addressed. This is unsatisfactory to some. Specifically:

1. What is going on when not observed.
2. QM itself is supposed to explain the classical world yet assumes it from the beginning.

Interpretations, for some, are meant to explain those issues. Some use 'classical' ideas, some bizarre stuff like many worlds. Others are minimalist and seek to give meaning to their probabilistic nature, e.g. the Ensemble interpretation. John Baez thinks many of the issues are just a continuation of those in what probability means:
https://math.ucr.edu/home/baez/bayes.html

Thanks
Bill

 

[bobob]

That is not quite what is going on as well. QM is a theory about observations that occur here in the classical macro world. What is going on between observations is not addressed. This is unsatisfactory to some. Specifically:

1. What is going on when not observed.

This is a complete misrepresentation of quantum mechanics which has been made popular in magazines like New Scientist, etc. When you prepare a state, you have choices of the states you can prepare with some properties being mutually exclusive. That exclusivity applies to nature as well and it has nothing to do with an "observer" in the colloquial meaning of that word. If there is one notion that is constantly being repeated, despite being wrong, it's anthropomophizing the word "observer."

2. QM itself is supposed to explain the classical world yet assumes it from the beginning.

You have it backwards. Quantum mechanics from the beginning was modeled on classical mechanics. Why would you not expect quantum mechanics to reduce to classical mechanics as a limiting case? If anything, it's classical mechanics that would be very puzzling as any sort of fundamental explanation of anything. Try explaining the collision between two billiard balls classically at the level of the collision without any approximationslike are usually just assumed (like finite propagation speed through the billiard balls or parameratizations to avoid those infinities like an elastic modulus, which explains nothing). Why is everyone so gung ho to try to make quantum theory into some bastardized version of classical theory when classical theory cannot explain anything without approximating away or parameterizing things which are quantum in origin?

Interpretations, for some, are meant to explain those issues.

Except that none of those interpretations explain anything. They just reflect the interpreter's bias of where they can shift what they perceive as an issue to some place they are comfortable ignoring it. Whatever they perceived as an issue did not go away. They just found a way to make it more palatable while wiping away any physics that there is to be found in whatever they saw as an issue. Here's a simple question. Assume nature allows you to position something with infinite precision. (Nature obeys the same laws of physics as we do.) How much information could we store at a single (in fact mathematical) point?