Readers' Choice: The Crystallizing Universe

It is tempting to identify the foliation of spaces with a "time." However, what ADM relativity describes are a set of spatial surfaces which are related to each other by diffeomorphism. What has not really been done, as far as I know, is to demonstrate explicitly that this is time.

Classically the block time does make sense. It does have to be realized this is a model system, it is not necessarily the universe at its foundations. When you bring quantum mechanics into the picture this model gets shaken to its core. Attempts to revise the block time picture with MWI or other state reduction schemes runs into the subtle questions I outline above.

Cheers LC

Event horizons are null congruencies and are invariants. Everything else that is timelike transforms in a covariant or frame dependent way.

Cheers LC

LC,

Lepp pointed me to Ranzan who commented on the difference between length contraction just in the direction of motion as imagined by Lorentz and the just imagined one introduced by Poincarè and Einstein. I already read the same distinction made by Tom Van Flandern. While the mathematics of the block universe seems to be flawless, I tried to show in my essay that it does not fit reality.

You wrote: "Attempts to revise the block time picture ... runs into the subtle questions I outline above." May I replace "the subtle questions" by a fundamental contradiction?

Regards,

Eckard

This could be an outright contradiction. A revision of the block world within a quantum mechanical setting needs to overcome the polarity between contextuality and noncontextuality. A world with a present time slice which emerges as it evolves requires some sort of context in which quantum basis vectors are selected. Without some external control this has to be done by quantum mechanics itself. However, quantum mechanics is noncontextual, and it is difficult to see how the emergence of a present space with all the field data on it according to some contextual meaning to quantum states can emerge.

There may be some subtle issue involved that escapes our attention. Quantum mechanics has a complementary logic, and contextuality and its logical complement noncontextuality have some quantum dualism. However, how to frame such a possibility is not apparent to me.

In a classical setting block time is the most direct model. The problem of setting up a relational block world in a quantum setting seems to have some connection to issues of quantum gravity and cosmology. It will be interesting to see if the block time can be revised, or whether block world is a classical structure that amounts to excess baggage that is abandoned in quantum gravity.

Cheers LC

LC,

I wonder why even physicists apparently dislike the clear distinction between (R) a road in reality and (M) the ideal line that represents it.

I see the block universe an obviously not totally appropriate model, and I also suspect that QM is too much mathematically idealized.

On the other hand, John Lucas considered it worth to illustrate the correct argument by Saint Augustinus that there is no extended time slice between past and future. Not just Einstein wrongly attributed the usual notion of "simultaneity" to simultaneous perception. What nonsense! The only reasonable idea of simultaneity can be explained with a fair duel where the bullets meet exactly in the middle.

While I do not hate anybody, I admire Karl Popper who not just called Einstein Parmenides, i.e. as wrong as Zeno but who also admitted hating Hegel, and I recall Marx speaking of "abstruse Hegelei".

Backed by Popper, I am pretty sure: The block universe is strictly speaking fundamentally wrong but might nonetheless be useful as a model to some extent.

What about noncontextuality, I admit being ignorant of the related and perhaps controversial theories. I merely guess, the real world could not be convincingly described without contextuality.

What about the logical complement, I would like to remind of what I consider flaws in the fundamentals of set theory.

Regards,

Eckard

Quantum mechanics is to physics almost what logic is to mathematics. It is a system which determines the relationships between qubits in a logico-algorithmic sense. The relationship to classical mechanics or macroscopic physics is what is odd. Of course the macro-world might just be some approximation that we observe in a coarse grained sense. Everything may in effect be quantum mechanical at all levels.

The block world and Einstein concept of time works fine in a classical setting. The thing which muddies the waters is quantum mechanics, and this is compounded by quantum gravity. Quantum mechanics is noncontextual, which means the eigenbasis the state vector is expanded in is not determined quantum mechanically. The context is determined by the observer, such as by the orientation of a Stern-Gerlach apparatus. So the context assigned to quantum outcomes of measurements is determined by the observer in a classical or macroscopic setting. The crystallizing block world in an MWI sense means the present that is materializing involves a set of eigenbranchings, where the "branches" are according to eigenbasis vectors with some contextuality. In the teleportation of states the classical content involves this contextual element, Alice telling Bob how she oriented her SG apparatus. How this happens in a purely quantum mechanical setting is unclear.

The issues of simultaneity and the rest in relativity are pretty clear, and there is no controversy with respect to those results. I have also regarded the Zeno paradox as a funny thing that anyone should ever get wrapped up in. It always struck me as something overcome by calculus, and I think there was some proof to this effect using Robinson numbers derived a couple of decades ago.

As for Hegel, I suppose in keeping with his dialectics I both admire and dislike his work. In some sense what he said is true, and quantum mechanics is an example of Hegelian dialectics. The problem is that Hegel offered no method for using dialectics in some proper fashion. In fact I doubt anything like that could be founded. The founders of QM did not marinate themselves in Hegel before realizing the complementaritiy of QM. Hegel was also more concerned with social issues, which were in flux at his time with the French Revolution and Napoleon. That got taken up by Engels and Marx in a somewhat "loosy-goosy" fashion, but which motivated a lot of unfortunate political stuff in the 20th century.

Cheers LC

LC,

Hopefully you will agree with me that perception as well as measurement must be corrected for possibly different delays. I infer from this that Einstein's justification of Popincaré's (de)synchronization is not logical.

Hopefully you will not agree with G. Cantor on that something that is neither smaller nor equal to something must be larger. Logics allows a fourth possibility: incomparability.

Do we need a quantum logic? Having the textbook "Particles and Paradoxes - The limits of quantum logic" by Peter Gibbins at hand, I tend to deny that.

You are reiterating what is presently common opinion: "QM as well as SR are of eminent importance for modern physics. Therefore they must not be wrong."

-- Really?

You wrote; "Quantum mechanics is noncontextual, which means the eigenbasis the state vector is expanded in, is not determined quantum mechanically."

-- Isn't such expansion arbitrarily assumed?

You continued: "The context is determined by the observer, such as by the orientation of a Stern-Gerlach apparatus. So the context assigned to quantum outcomes of measurements is determined by the observer in a classical or macroscopic setting."

-- This reminds me of Heisenberg's likewise unacceptable interpretation: The path becomes reality if we measure it. Aren't state vectors and the like just abstract models? Could a point or a line become reality? I say no.

-- Maybe I am naive when I consider non-contextuality an simplifying idealization but context about the same as objective while possibly even sub-threshold influences. How do state vectors account for processes and for the embedding of any particle into reality?

You argued: "The crystallizing block world in an MWI sense means the present that is materializing involves a set of eigenbranchings, ... "

-- I desperately hope for understanding: Strictly speaking there is no present.

"... where the "branches" are according to eigenbasis vectors with some contextuality."

-- Isn't this a rather awkward and incomplete substitute for an admission of incalculable influences?

"In the teleportation of states the classical content involves this contextual element, Alice telling Bob how she oriented her SG apparatus. How this happens in a purely quantum mechanical setting is unclear."

-- At least to me the Alice/Bob metaphor is perhaps inappropriate.

Let me summarize: Zeno's flawed arguments indicate that Parmenides/Einstein (the block universe) is strictly speaking wrong. The putative necessity to use i in QM indicates a related and similar flaw in QM. In both cases, an abstract model is arbitrarily equated with reality.

At least laymen might be ready to understand: While time can be shifted, reversed, or otherwise manipulated in a record, a mathematical model, and the like, it cannot be influenced at all in reality. Why? Because pictures are abstractions that will never fully substitute the complete embedding into real life.

Regards,

Eckard

Eckard,

You wrote to Lawrence: "Hopefully you will not agree with G. Cantor on that something that is neither smaller nor equal to something must be larger."

This is another way of saying that a calculated result must be positive, negative or zero.

"Logics allows a fourth possibility: incomparability."

That's why quantum logic works. Logic in which value is assigned to undecidability allows a superposition of states. "Incomparable" quantities are undecidable -- a measured quantum result is always positive or zero.

Tom

Eckard & Tom,

I suppose you are referring to the Banach-Tarski paradox or the addition of transfinite numbers. The axiom of choice (AC) does involve the well ordering of a set. Hilbert space exists because of the AC. The Schmidt orthogonalization procedure employed in quantum mechanics and the theory of Banach-Hilbert spaces is an algorithm which works because the space is well ordered. This is a sort of choice function.

I have been working on the integer partition theorem. Given an integer n, there exists a set of integers (n1,n2,...) which add up to n, and then there are other sets as well, and the number of these sets is the partition. This has physical implications for how microstates of a black hole can be arranged amongst n Planck areas on the event horizon. The partition number grows approximately exponentially for the number of integer sets which sum to n as n --> ∞. Consequently, the set of all possible integer partitions for the integers is a power set, which is an C = 2^{X_0} result. X = aleph

The AC is an undecidable proposition as well. Some research was done on this, which I know about but I don't have references available, and it was found that the AC is not a consequence of the other axioms in ZF. The axiom of replacement I think has a similar property.

The AC does result in some quirky results though. A sphere can be decomposed in a certain way, group rotations (eg SO(3)) performed, and the pieces further decomposed and rotated, and ... infinitely onwards. The pieces may then be reassembled to construct two spheres identical to the first. This is the result of Banach and Tarski.

Some mathematicians consider the Banach-Tarski theorem to be a reducto-absurdum argument against the AC. That's one of the reasons for considering other axioms. The Perfect Set hypothesis "Every uncountable subset of the real line has a non-empty perfect subset." is inconsistent with the AC and seems just as intuitive. This gets into the subject of Polish spaces. However, in what I do it is not my interest to rewrite the foundations of mathematics.

Special relativity and quantum mechanics are "true" in the same way that Newtonian mechanics is "true." They both work in a broad domain of observation and have been extensively tested. There really are no controversies over the issue of simultaneity and clock synchronization. I worked on problems related to the question of synchronizing clocks for GPS and various other satellites. That gets more complicated with general relativity, but there are no serious controversies with the basic issues. Much the same holds with QM, and recently an experimental version of the Kochen-Specker theorem on quantum nonlocality was performed and the KS result supported.

My take on the issue of space and time is they are configuration variable representations which have a type of complementarity. This being the case there is no physical axiom which can tell us which of these is "real." Fundamentally there is no such axiom, and the observer imposes the context upon which is real, or how the two are related to each other. Classical spacetime does not share this feature, which is why in the classical domain you can talk about a block time.

The complementarity of space and time and the quantum mechanics of black holes could have had a much earlier start. At the 1930 Solvay conferences Neils Bohr and Albert Einstein debated the nature of quantum mechanics. Einstein was convinced of reality and locality and argued staunchly for an incompleteness of quantum mechanics. Quantum theory could only be made complete if there are some hidden variables that underlay the probabilistic, nonlocal quirky aspects of quantum mechanics. Einstein proposed an interesting thought experiment. Einstein considered a device which consisted of a box with a door in one of its walls controlled by a clock. The box contains radiation, similar to a high-Q cavity in laser optics. The door opens for some brief period of time t, which is known to the experimenter. The loss of one photon with energy E = ħω reduces the mass of the box-clock system by m = E/c^2, which is on a scale. Einstein argued that knowledge of t and the change in weight provides an arbitrarily accurate measurement of both energy and time which may violate the Heisenberg uncertainty principle ΔEΔt~ħ

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Bohr realized that the weight of the device is made by the displacement of a scale in spacetime. The clock's new position in the gravity field of the Earth, or any other mass, will change the clock rate by gravitational time dilation as measured from some distant point the experimenter is located. The temporal metric term for a spherical gravity field is 1 - 2GM/rc^2, where a displacement by some δr means the change in the metric term is \simeq~(GM/c^2r^2)δr. Hence the clock's time interval T is measured to change by a factor

T-- >T sqrt{(1 - 2GM/c^2)δr/r^2} ~ T(1 - GMδr/r^2c^2),

so the clock appears to tick slower. This changes the time span the clock keeps the door on the box open to release a photon. Assume that the uncertainty in the momentum is given by the Δ p ~ ħ/Δr \lt TgΔm, where g = GM/r^2. Similarly the uncertainty in time is found as ΔT = (Tg/c^2)δr. From this ΔT > ħ/Δmc^2 is obtained and the Heisenberg uncertainty relation ΔTΔE > ħ. This demands a Fourier transformation between position and momentum, as well as time and energy.

This holds in some part to the quantum level with gravity, even if we do not fully understand quantum gravity. Consider the clock in Einstein's box as a black hole with mass m. The quantum periodicity of this black hole is given by some multiple of Planck masses. For a black hole of integer number n of Planck masses the time it takes a photon to travel across the event horizon is t ~ Gm/c^3 ~ nT_p, which are considered as the time intervals of the clock. The uncertainty in time the door to the box remains open is

ΔT ~ Tg/c(δr - GM/c^2),

as measured by a distant observer. Similarly the change in the energy is given by E_2/E_1 ~= sqrt{(1 - 2M/r_1)(1 - 2M/r_2)}, which gives an energy uncertainty of

ΔE ~ (ħ/T_1)g/c^2(δr - GM/c^2)^{-1}.

Consequently the Heisenberg uncertainty principle still holds Δ EΔT ~ ħ. Thus general relativity beyond the Newtonian limit preserves the Heisenberg uncertainty principle. It is interesting to note in the Newtonian limit this leads to a spread of frequencies Δω ~ sqrt{c^5/Għ}, which is the Planck frequency.

The uncertainty ΔE ~ ħ/Δt larger than the Planck mass gives an event horizon. The horizon has a radius R ~ 2GΔE/c^4, which is the uncertainty in the radial position ΔR associated with the energy fluctuation. Putting this together with the Planckian uncertainty in the Einstein box we then have

ΔrΔt ~ 2Għ/c^4 = L^2_{Planck}/c.

So this argument can be pushed to understand the nature of noncommutative coordinates in quantum gravity.

This complementarity means that fundamentally with quantum mechanics there is no meaning to space and time outside of the context of a measurement, or the choice of observation. QM has no contextuality of its own, and so any statement made about the spatial and temporal nature of the world is something which is determined by the choice of basis by the observer.

Quantum logic is interesting in some ways, but I don't think it really buys us that much. It is sort of a formal set theory way of doing what we already understand.

Cheers LC

"The fundamental process of nature lies outside space-time but generates events that can be located in space-time."

H.Stapp

Anatomy of quantum superposition

(3- bit Universe)

We are studying the simplest model of a finite deterministic world. Here, we have attempted to answer the following question: what our artificial world would look like from the point of view of an observer (subject), placed in our modeled finite world? To do this, it's necessary to formulate abstract model of the observer. Only in that way is it possible to answer this complicated question. Herein, we have endeavored to show the quantum-similar character of the laws, discovering by the objective observer. As a consequence of this exercise, we can now assume that physical laws of our real world have a similar origin

Digital Physics

LC,

"Some mathematicians consider the Banach-Tarski theorem to be a reducto-absurdum argument against the AC." You wrote: "...is not my interest to rewrite the foundations of mathematics." You meant the putative ones of mathematics, i.e. the foundations of contemporary mathematics, which are based on Cantor's belief that there are more than infinitely many numbers (ueberabzaehlbar means than countable).

To those who are not familiar with history: The AC was arbitrarily fabricated by Zermelo in 1904 in order to rescue Cantor's well ordering of uncountable sets.

Well, those mathematicians who provided most useful contributions to mathematics used the irrational numbers as if they were rational ones. However, I do not see any compelling reason to ascribe trichotomy to them. If "Hilbert space exists because of the AC" then it might be questionable. I am anyway wondering why Tong meant "no one knows how to write down a discrete version of the Standard Model". Maybe, his essay is not just the usual antithesis to my essay. At least I agree with his last sentence: "We are not living inside a computer simulation".

You repeatedly declared SR correct: "There really are no controversies over the issue of simultaneity and clock synchronization." Don't some hundred petitors consider the twin paradox an reductio-ad-absurdum argument against SR? What about Van Flandern? What about Popper? Weren't the Pythagoreans, Parmenides, Zeno, G. Cantor, Einstein, and Hilbert most likely wrong altogether in their view of the world?

Regards,

Eckard

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,