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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. ...

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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

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