Readers' Choice: The Crystallizing Universe

The axiom of choice (equivalent to Zorn's lemma and to the well order theorem) is often convenient for proofs, particularly when one wants to apply calculus and vector algebras to certain Hilbert spaces (limits have to be established); however, no well ordering procedure is required to support the _existence_ of Hilbert space, which is a generalization of the Euclidean space.

It's often said that the Banach-Tarski construction (usually called a paradox) depends on the axiom of choice. However, equidecomposable balls intrinsically allow construction of equal volume spheres (i.e., sets of equal cardinality). So B-T, yes, is supported by set theory, which is generally taken to be ZFC (Zermelo-Fraenkel plus the axiom of choice), but the space is always Euclidean. One could just as eaily prove that AC exists because of the Hilbert space.

None of this formality, though, worries physics. The usefulness of the mathematics to support physical results ends at the real geometry.

In regard to David Tong, one should point out the subtle difference between "simulation" and "emulation." Indeed, one could prove we are not living in a computer simulation. There is no way _in principle_ however, to distinguish an emulation from the original program. No physical principle obviates the universe acting like a computer program -- at least none that we could determine by objective measurement.

Eckard -- to speak of so-and-so "being wrong in their view of the world," is meaningless. _Everyone_ individually is wrong in their view of the world. The scientific view is an aggregate of theories, results and philosophies.

Tom

Eckard,

There is no controversy over the twin paradox. I am not sure why there is this "petition" or what the point of it is. This matter has been utterly beaten to death, and what are cited as "discrepancies" are probably different approaches to presentation. I have to implore people to avoid faux problems of this sort. Anyone who is caught up in these issues is really in some sort of cul-de-sac. I admonish people to not get into these traps.

The AC was fabricated in a sense, just as it might be argued that all of mathematics is a fabrication or model. Of course the math-realists or Platonists would object to this characterization. I might agree with them on Tuesday, Thursdays and Saturdays, while disagreeing on the other weekdays, Sundays are optional. I am not as I indicated out to rewrite mathematical foundations.

More continued in my response the TH Ray

Cheers LC

Tom,

Galileo Galilei correctly concluded by means of bijection: There are not more natural numbers 1, 2, 3, ... as compared with their squares 1, 4, 9, ... because the comparative relations are not valid for infinite quantities, only for finite ones. G. Cantor claimed having "proved" him wrong by arguing that there must be more irrational numbers than rational ones because something that is neither smaller nor equal to something must be larger." Accordingly Cantor introduced what he first called Maechtigkeit and later renamed cardinality.

Is this "another way of saying that a calculated result must be positive, negative or zero"? Well, explicit finite numerical results are rational numbers and therefore they obey this trichotomy. However, as I tried to explain in my last essay, Cantor's naive transfinite numbers have proven sterile. Already in 1922 Fraenkel admitted: Cantor's definition of sets, including infinite ones, is untenable.

"It is always assumed that a Hilbert space can have no more than a countable infinity of linearly independent state-vectors. This implies that there are no eigenstates of exact position, that the Dirac delta-function is illegitimate." [quoted from Gibbins, p. 90].

Regards,

Eckard

Tom,

The AC permits one to order up the eigenbasis of Hilbert space. In fact it is implicit in diagonalization or Schmidt orthogonalization, where the ordering of eigenvalues and eigenvectors. The connections with physics seem potentially to be with the matter of contextuality. The experimenter has the freedom to orient their apparatus to select a certain eigenbasis, where upon from there the ordering of the Hilbert eigen-vector space is determined. This is the classical piece of information that Alice must communicate to Bob in order to teleport a qubit, so Bob can convert an ancillary state into the teleported state. However, in the absence of such Hadamard transformations which demolish some entanglement the ordering of the Hilbert space is ambiguous. There is no ordering unless there is some selection process that takes place.

The axiom of choice is not a decidable proposition, for all such orderings can be Cantor diagonalized and the register shifted diagonal can form another ordering which is not in the list. Consequently the AC is not proven from other axioms of set theory. This has a Turing machine analogue as well. Each orientation of the apparatus produces an independent ordering of the Hilbert eigenspace. However, the symbol strings corresponding to ordering is different and by the Chaitan Halting probability there is some measure of these which do not correspond to halting procedures. Consequently the ordering of the state space by these means is not decidable, or equivalently the AC is not provable.

This might suggest some foundational issue with the duality between the noncontextuality of quantum mechanics and the contextuality of the measurement procedure. The SLOCC entanglement states determine what sort of classical information may be communicated to teleport states in n-partite entanglements. These have correspondences with the moduli spaces for spacetime configurations, such as black holes. The entropy of these entanglements is computed by determinants, in the case of a 2x2 matrix, or hyperdeterminants for 2^n n > 2 matrices for n-partite entanglements. If quantum mechanics obtains on all levels these results are due to certain coarse graining which we impose on reality which prevent a complete characterization of the system. On the deepest level there is no ordering of the Hilbert space, this is something imposed on it from "outside." Hence the AC thought of as a "physical axiom" is turned on and off accordingly.

Cheers LC

Eckard,

Physics does not require, nor use, transfinite algebra. Every measured result has to be positive or zero. Negative and imaginary numbers are mathematical artifacts.

If one is disposed (I am) to argue for a continuum of mathematical results with physical phenomena, one has to be careful when speciifying domains. No physical domain that is not measure zero, is infinite. To use a simplistic analogy, though, of the uncountable molecules of water that go into making up a river at its source, we recognize finite phase transitions from vapor to moisture to puddle, etc. Extending that process to the origin of the universe is not a leap -- it is continuous.

You wrote: "Galileo Galilei correctly concluded by means of bijection: There are not more natural numbers 1, 2, 3, ... as compared with their squares 1, 4, 9, ... because the comparative relations are not valid for infinite quantities, only for finite ones. G. Cantor claimed having "proved" him wrong by arguing that there must be more irrational numbers than rational ones because something that is neither smaller nor equal to something must be larger." Accordingly Cantor introduced what he first called Maechtigkeit and later renamed cardinality.

Is this "another way of saying that a calculated result must be positive, negative or zero"?"

No. The idea of the cardinality of sets has nothing to do with numbers per se. It describes comparative relations, so it certainly is appropriate for infinite sets. Infinity is not a number.

"Well, explicit finite numerical results are rational numbers and therefore they obey this trichotomy."

We assume so. Intuitionists and some constructivists would disagree, allowing that without an explicit procedure to decide, one cannot know whether a result is positive, negative or zero. Again, though, this has nothing, at least directly, to do with physics.

You wrote, "However, as I tried to explain in my last essay, Cantor's naive transfinite numbers have proven sterile. Already in 1922 Fraenkel admitted: Cantor's definition of sets, including infinite ones, is untenable.

"It is always assumed that a Hilbert space can have no more than a countable infinity of linearly independent state-vectors. This implies that there are no eigenstates of exact position, that the Dirac delta-function is illegitimate." [quoted from Gibbins, p. 90].

Set theory (arithmetic) is useful to physics. Its usefulnes is limited, however, to the counting function. Physics doesn't address infinite sets.

Tom

The irrational numbers turn out to be countably infinite. The reason is that they are roots of polynomials and their rings, which have a countably infinite realization. However, this is a dense set in the reals, where transcendental numbers such as π that are not polynomial solutions fill in the gaps. The uncountably infinite number of reals is due to Cantor's diagonalization procedure, and the number of possible diagonal "slashes" one may perform for n numbers is 10^n. In the limit that n -- > ∞ the cardinality of the set of numbers formed by "slashes" is greater than countable infinity. So this get one into the continuum problem with C = 2^{X_0} > X_0, where X stands for aleph.

As Tom points out this is not terribly relevant for physics. We generally don't run around worried about levels of infinity. This is connected to Godel's theorem, and its algorithmic and information theoretic connection is with Turing's proof. At this point one can bring a computational aspect of this mathematics into physics. Things such as space represents relationships between physical objects, but in spite of all our advances with things such as curved spacetime it is still the case that spacetime is a model system. The assignment of degrees of freedom to spacetime results in problems, with over counting them and entropy measures that are too large. So there is no real worry with respect to the continuum of space. Where physics connects to this is with the algorithmic connections to set theory.

Cheers LC

Thanks, Lawrence.

I accept the usefulness of AC, I just don't favor it as an axiom. "Contextuality" seems a slippery slope to me -- admittedly, I'm prejudiced by the hope that quantum mechanics derives in a most natural way from a Hilbert space of entire functions without having to invoke AC.

I suppose that just as Eistein wanted to believe "that the moon is there even when no one is looking," I want to believe that information and its mathematical processing is not different from the physics of the moon, even "when no one is looking."

Tom

TH Ray,

I hit send before pasting this in:

Before I continue onwards, I should make clear what I mean by contextual and noncontextual. Contextual means that reality assigned to a quantum system is only found through a measurement, "within the context of the eigenbasis established by the observer." Hidden variable theories often claim there is some reality which exists independent of this contextuality, and are often referred to as hidden variable theories with some reality that is noncontexual. What I mean by noncontextuality of QM is that there is no such reality that can be assigned in a noncontextual basis, or at least no reality as understood in a classical setting.

The contextuality of QM is what we observe through experiments, which may involve incommensurate and commensurate sets of detections. In other words the context of the apparatus (whether we measure momentum or position) or the orientation of a Stern-Gerlach apparatus is what determines the outcome, and a succession or ensemble of such measurements will demonstrate nonlocal properties and the incommensurability of measurements between conjugate observables. So the interpretation of quantum mechanics can only be inferred through the types of measurements we perform. These measurements are in the Copenhagen interpretation performed by perfectly classical systems and the outcomes determined by an infinite number of observations. This is an idealization. A perfectly macroscopic or classical system has an infinite mass, and a finite mass has some quantum wave spreading. The CI means context is made within a classical setting, where the world has a duality between quantum and classical physics. In the Many World Interpretation the contextual aspect of QM involves the basis of state vectors the world splits off into. The MWI entangles pointer states to a wave function super-position and the various pointer states that are observed correspond to a particular world-path. Decoherence indicates that wave function reduction is due to such entanglements which become spread into a vast reservoir of states, and the direct observation is a coarse grained result.

In all of these interpretations there is a ways that the context of the measurement produces its outcome: classical-quantum dichotomy, many world eigen-vectors splitting off, entanglement phase or pointer states that grows irreversibly in a coarse graining and so forth. With all of these there is a bit of "fuzz" that is introduced, where this fuzz is something imposed = the "machinery" behind the context. Quantum physics in its pure setting has no such context, and no reality hidden beneath any context = noncontextual quantum waves. The funny issue comes in if we do consider the world as fully quantum mechanical on all scales. The context from which quantum observables are detected must originate then from something which is ultimately noncontextual, and that "something" has no local reality.

The ordering of a Hilbert space is permitted by the AC, and any such ordering in a commutative basis of commensurate observables is what gives the context. However, if reality is quantum mechanical "all the way," then how do we get contextuality from nonlocal noncontextual waves that have no local or classical reality? This noncontextuality means there is nothing which orders eigen-vectors in the Hilbert space. So we may think of QM in this setting as having not-AC. As such we might have ultimately a duality which involves set theory and undecidable propositions underlying quantum mechanics.

I don't worry too much about the moon not being there if I am not looking at it. For one thing the separable states (eg thermal states) of photons from the sun interact with the moon and remove what little bit of coherent phase it might have. So this is a sort of Zeno measurement process where the wave function of the moon is constantly jolted away from being in any quantum superposition of position. So I am sure this holds for the entire macroscopic world. So I sure that 120 million years ago the photons reflecting off the moon formed an image in the retina of an Allosaurus, without any conscious awareness of what was being observed --- I doubt dinosaurs had any sense of the moon outside of being a "light."

Cheers LC

LC,

Perhaps the strongest argument against his SR was uttered by the late Einstein himself. He admitted that the Now worries him seriously, and he suspected something outside science. Well, I agree with Popper, Einstein's view of the world is the same as Zeno's. Has the Ritz matter really been "beaten to death"? I see the last three articles including The Crystallizing Universe tackling it with no convincing to me avail so far. You are not the only one who needs strong words like implore and even admonish instead of strong factual arguments. I guess: Those who signed the petition were aware of swimming against the ruling opinion.

What about me, I found out that it is a detour to include the future when analyzing measured, i.e. past data. In practice cosine transform actually is sufficient. Nonetheless, I was persistently lectured that one has always to use the Fourier transform because spacetime extends from minus infinity to plus infinity, full stop.

In order to find my mistake, I read a lot of original papers. Einstein's SR paper was a bit difficult to immediately understand because it did not give the due references to its sources, in particular Poincaré. However, Poincarè's method of synchronization is not at all convincing.

When Schroedinger succeeded calculating the hydrogen spectrum, he did not calculate relativistic.

Having no clue concerning GPS, I guess, Van Flandern was a good expert in this field, and he argued against SR.

Tom pointed me to a large Wiki collection of putative evidence for SR. Not a single one was obviously compelling to me.

A textbook "Elements of Non-Relativistic Quantum Mechanics" by Luis Sobrino did not even mention relativity except in the title.

Shouldn't I doubt?

Regards,

Eckard

LC and Tom,

David Tong wrote: "discreteness in the world is simply the Fourier transform of compactness". He did not answer my question whether this also holds for cosine transform.

I am voting for realism, i.e., strict distinction on the level of abstract notions.

This includes to not blur the distinction between positive and negative, past and future, countable in the sense of discrete and uncountable in the sense of attributed to a continuum. I am aware of many seeming exceptions. They do altogether belong to missing precision of the used definitions. While a river consists of discrete molecules, it does not have exact borders.

May I reiterate my suggestion to reinstall the Euclidean notion of number as a measure: Instead using f(x) I am suggesting to distinguish for a continuum between the two limits from below f(smaller than x toward x) and above f(greater than x toward x). See my essay.

|sign(0)|=1, not 0.

Regards,

Eckard

Lawrence,

My 2006 ICCS paper explains how to get a well ordered sequence from an n-dimension (Hilbert) space, without invoking AC.

As you point out, QM is fuzzy only in the classical context, when operators are aggregated (one tunes the dials of the "machinery"). In its own context, though, quantum mechanics is starkly real and focused one observation at a time.

You ask, " ... if reality is quantum mechanical 'all the way,' then how do we get contextuality from nonlocal noncontextual waves that have no local or classical reality?"

Exactly. I don't think we need contextuality at all -- if the wave function is the only reality. The reason for the "moon" analogy is to suggest that the wave function of the moon entangled with that of the observer (and therefore the observing apparatus) is independent of results that come from tuning dials on the "machinery." To use a weak analogy, as my glasses are independent of my eyes. My glasses do not create reality, though what I choose to see, may.

The moon has no such choice. I am reminded of a nursery rhyme my grandmother used to recite to me when a full moon shone through the window of my upstairs bedroom::

I see the moon

and the moon sees me;

The moon sees the one I'd like to see.

God bless the moon and God bless me,

God bless the one I'd like to see.

Hence no contextuality required. The moon doesn't have any classical dials to tune that make it fuzzy, and the Hilbert space is equally indifferent to the choice function.

Tom

Corrected link: ICCS 2006

Non relativistic QM is obvious for atomic physics. The gamma factor for the motion of an electron in an atom is γ = E/mc^2 = (mc^2 K)/mc^2, where K = kinetic energy of the electron. The mc^2 of an electron is .51Mev = 5.1x10^{5}ev, and the kinetic energy of an electron is on the order of 1-10ev. So for K = 5ev K/mc^2 ~ 10^{-5} and the gamma factor is γ =~ 1. 10^{-5} = 1.00001. Since γ = 1/sqrt{1 - (v/c)^2} the velocity of the electron is v = 0.00447c. Consequently relativity is not a major factor and amounts to a small perturbation on a nonrelativistic calculation. Research into atomic physics increasingly involves finding ever smaller effects which perturb atomic levels, so atomic physics calculations with the Dirac equation is frequently done to work on such small effects.

Seriously, there is no real debate over special relativity. It has been tested to death and it forms the global symmetries of spacetime. General relativity uses these symmetries as local symmetries on frames that are patched together with connection coefficients. There simply is no scientific controversy over the validity of special relativity in this appropriate domain of applicability.

Cheers LC

The wave function is not random and discrete. It is perfectly continuous and is a C^∞ solution to a partial differential equation. The randomness and discrete aspect of QM emerges with the measurement or decoherence of wave functions as observed with coarse grained coherent sets. Further, the wave function is not "real" in a standard sense, or equivalently it has no ontology, but is rather a set of existential potentialities or an epistemology. This means that any ordering occurs with the measurement of a wave function and is imposed on an ensemble of quantum systems, not on a quantum system by itself. The ordering of the Hilbert space is then derived by this imposition.

The GHZ state does demonstrate that the nonlocal aspects of quantum mechanics can be inferred without an ensemble of measurements, but rather with one measurement. However, even with one measurement the teleportation of this 4-partite state requires the SLOCC state ~ C^4/[sl(2,C)]^4, with a root system that imposes an ordering.

Cheers LC

No, the wave function is not real in the standard sense (of observer-dependent and measurement-verified) reality.

Applying Einstein's definition* of "physically real," however: " ... independent in its physical properties, having a physical effect but not itself influenced by physical conditions ..." we can successfully argue for _either_ the ontology of the wave function or the ontology of spacetime (which is where Einstein applied his definition).

Suppose that spacetime and the wave function are identical, in the sense of a Hilbert space plus time, rather than a complete Euclidean 3-space plus time as in general relativity. One would then have an n-dimension extension of general relativity, as well as access to the hyperbolic space where string theory and holography originate -- that's what my "time barrier" preprint is all about.

While Einstein maintained,** in the spirit of classical mechanics, particularly the mechanics of Mach, that "From the standpoint of epistemology it is more satisfying to have the mechanical properties of space completely determined by matter . . ." I doubt that Einstein would object to a spacetime ontology and a matter epistemology. The advantage would be to lay a foundation toward explaining the origin of mechanics, and hence the origin of inertia.

I continue to maintain that science has no use for the assumptioon of any particular "reality." We can as easily live with the general relativity result of a reality finite in time and unbounded in space, as with the reality I propose: one that is finite in space and unbounded in time.

Tom

*Einstein, _The Meaning of Relativity_ Princeton 1956

**ibid

LC,

"The gamma factor for the motion of an electron in an atom is ..." [near to 1]. Hm, who measured that motion. Being not a physicist, I recall an argument that electrons should have a speed far in excess of c as to overcome the electrostatic attraction of the positively charged nucleus. Do not get me wrong. I merely guess, both models might be premature.

If I recall correctly, the result by Schroedinger differs from relativistically calculated one by a factor two. Remarkably, the relativistic Doppler effect may also differ from ordinary one by a factor of the same order.

I agree, SR "forms the global symmetries of spacetime": symmetries between past and future, world and anti-world, matter and anti-matter, and the like. Exactly these putatively natural but actually artificial symmetries gave rise to me looking for the flaw I am claiming to have found in a view of the world that goes back at least to Parmenides. If there was not yet a debate on this, we should start it now, no matter where we will begin. The topic includes the so called arrow of time, real time physics, the nature of time, emergence, why did von Neumann in 1935 admit he did no longer believe in (his own creation) Hilbert space, all failed efforts to unite for instance the block universe with reality, etc., etc.

Regards,

Eckard

Correcting link again. :-(

[link:home.comcast.net/~thomasray1209/timebarrier80108.pdf]"time barrier"

I am not sure what the departure by a "factor of 2" refers to. However, I must implore there is nowhere the sort of problem you think there is with special relativity. To fret over this is almost as silly as Putnam who thinks that Newton's F = ma is wrong. I can't argue these points endlessly, but frankly people who think relativity is all wrong are barking up the wrong tree.

LC