From page 6, "In a local deterministic theory, each pair of entangled particles is described by a supplementary variable λ, often referred to as a hidden variable ... " — is this meant to be the definition of a "local deterministic theory" or is it a statement about the conventional wisdom of physicists concerning a "local deterministic theory"? Is it possible that time, space, energy, quantum...

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From page 6, "In a local deterministic theory, each pair of entangled particles is described by a supplementary variable λ, often referred to as a hidden variable ... " — is this meant to be the definition of a "local deterministic theory" or is it a statement about the conventional wisdom of physicists concerning a "local deterministic theory"? Is it possible that time, space, energy, quantum information, and the concept of “variable” are merely approximations that are not quite correct? Consider 2 questions: (1) Is Milgrom the Kepler of contemporary cosmology? (2) Are Riofrio, Sanejouand, and Pipino neglected geniuses instead of crackpots as Motl and others believe them to be? Consider Fredkin’s idea: Infinities, infinitesimals, perfectly continuous variables, and local sources of randomness do not occur in nature. I have presented various speculations: dark-matter-compensation-constant = (3.9±.5) * 10^-5, supersymmetry does not occur in nature, magnetic monopoles do not occur in free space, measurement is a natural process that separates the boundary of the multiverse from the interior of the multiverse, and so on. Motl thinks that I am crackpot, and so do Steven Weinberg, Sheldon Glashow, Frank Wilczek, and even Milgrom himself. They might be correct. There might be scientific proof that I am a crackpot: I claim that the Gravity Probe science team misinterpreted their own experiment. According to the Wikipedia article “Gravity Probe B” (as of 25 January 2020), “Analysis team realised that more error analysis was necessary (particularly around the polhode motion of the gyros) …”

Let us consider Clifford Will’s article:

”Viewpoint: Finally, results from Gravity Probe B” May 31, 2011, Physics, (physics.aps.org)

There is the statement: “Finally, during a planned 40-day, end-of-mission calibration phase, the team discovered that when the spacecraft was deliberately pointed away from the guide star by a large angle, the misalignment induced much large torques on the rotors than expected. From this they inferred that even the very small misalignments that occurred during the science phase of the mission induced torques that were probably several hundred times larger than the designers had estimated.” The Gravity Probe B science attribute the problem to “random patches of electrostatic potential fixed to the surface of each rotor, and similar patches on the inner surface of its spherical housing … The team was able to account for these effects in a parametrized model.” I attempted to convince the Gravity Probe B science team that Milgrom is the Kepler of contemporary cosmology, the 4 ultra-precise gyroscopes worked correctly, the gyroscopes indicated dark-matter-compensation-constant is nonzero (contrary to Newton and Einstein), and their parametrized model is a post-hoc rationalization for the expected outcome. The Gravity Probe B science team (or at least 3 of them) dismissed my opinion. My guess is that quantum information reduces to Fredkin-Wolfram information, and MOND is a direct consequence of string theory with the finite nature hypothesis. See the section “Abnormal Acceleration of the Pioneer 10, 11, Galileo and Ulysses Probes” in Pipino’s 2019 article “Evidences for Varying Speed of Light with Time”.

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Dear Tim,

congratulations on an eminently readable and engaging essay on a difficult topic! I will need some time to fully digest your arguments, but I wanted to leave a few preliminary comments---also because our two approaches have some overlap, in particular as regards undecidability and Bell/EPR.

I'll state my biases upfront: I'm skeptical of any sort of 'completion' of quantum...

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Dear Tim,

congratulations on an eminently readable and engaging essay on a difficult topic! I will need some time to fully digest your arguments, but I wanted to leave a few preliminary comments---also because our two approaches have some overlap, in particular as regards undecidability and Bell/EPR.

I'll state my biases upfront: I'm skeptical of any sort of 'completion' of quantum mechanics by hidden (or not-so-hidden) variables, that is, viewing quantum mechanics as a statistical theory of some deeper level of reality, and I'm in particular skeptical of superdeterminism.

That said, I'm always happy to see somebody giving an 'alternative' approach a strong outing---and that you certainly do. I do see now that I had dismissed these topics perhaps too quickly, so I've got to thank you for that. But on to some more detailed points.

You note the similarity between the Liouville equation and von Neumann's equation; you probably know this, but that similarity can be made much more explicit by considering phase-space quantization. There, the Moyal equation emerges as a deformation of the Liouville equation (with deformation parameter hbar), and contains the same empirical content as von Neumann's (they are linked by the Wigner-Weyl transformation).

I'm of two minds whether this supports your contention, or not. On the one hand, you can explicitly link the quantum evolution to that of a stochastic system; on the other, the deformation by hbar essentially means that there's no finer grain to the phase space, you can't localize the state of the system any further.

I'd be interested in how your approach works with things like the Pusey-Barrett-Rudolph theorem, that's generally thought to exclude viewing quantum mechanics as a stochastic theory of some more fundamental variables---although, as usual, and as you seem to be adept at exploiting, there are various assumptions and caveats with all 'no-go' theorems. I think there's an assumption that successive preparations of a system can be made independently; I'm not sure, but maybe that fails, taking one out of the invariant set.

I'll also have to have a more thorough look at how your model system is supposed to yield Bell inequality violation. Of course, having a Hilbert space formulation is not sufficient---you can formulate classical mechanics in Hilbert space, too (the Koopmann-von Neumann formalism).

I've got to go now, I'll add more later when I have some time. In the meantime, congratulations on a very strong and interesting entry into this contest!

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We are going a bit off topic here. However, as I discuss in my essay, one can view free will as an absence of constraints that would otherwise prevent one from doing what one wants to do, a definition that is compatible with determinism. From this one could form a theory of how we make decisions based on maximising some objective function which somehow encodes our desires. This does allow one to...

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We are going a bit off topic here. However, as I discuss in my essay, one can view free will as an absence of constraints that would otherwise prevent one from doing what one wants to do, a definition that is compatible with determinism. From this one could form a theory of how we make decisions based on maximising some objective function which somehow encodes our desires. This does allow one to learn from previous bad decisions, since such previous experiences would provide us with data that a certain type of decision, if repeated, would lead to a reduction, not an increase, in that objective function.

However, we are veering into an area that has exercised philosophers for thousands of years and I suggest this is not the right place to discuss such matters. Of course, I respect your alternative point of view - there are many eminent philosophers and scientists who would agree with you.

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Hi Tim,

It is quite a revolutionary program you have embarked on, overthrowing the infinitesimal and subverting the continuum. Your standard of rationality includes its mathematical definition: that any rational quantity can be expressed as a ratio of whole numbers. The conviction that the infinite and the infinitesimal have no place in physics goes well with the idea that appropriate...

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Hi Tim,

It is quite a revolutionary program you have embarked on, overthrowing the infinitesimal and subverting the continuum. Your standard of rationality includes its mathematical definition: that any rational quantity can be expressed as a ratio of whole numbers. The conviction that the infinite and the infinitesimal have no place in physics goes well with the idea that appropriate mathematics ought to be involved. Nearly all of the infinities have been expelled from physics.

But there is still the century-old foundational problem with infinity in physics that appropriate mathematics might help resolve. It strikes me as ironic that Hilbert's admonishment about how "the infinite is nowhere to be found in reality" still stands today, considering his name is on the Einstein-Hilbert action which is involved in the unlimited gravitational energy required for inflation. This point is made clear by Paul Steinhardt who, when asked where the energy for inflation comes from, confirmed that it comes from a bottomless supply of gravitational energy. Since inflation requires infinite energy, the theory is inadmissible by Hilbert's standard of rationality, and so is the general theory of relativity which is supposed to deliver that energy.

On the presumption that it is essentially classical Newtonian gravitational potential energy which is apparently the source of the unlimited energy, it should be of interest that a relativistic version of gravitational potential energy can be constructed from a consideration of the composition of relativistic gravitational redshift due to a sphere.

For example, given a test particle of mass m, the classical element of potential energy due to a spherical shell of matter is du = -F(r) dr where F(r) is the force of gravity at radius r. The redshift due to the shell is given by dz = du / mc^2. The total redshift, z, from all shells can be composed relativistically as the product, 1 + z = PRODUCT[1 + dz] = exp[INTEGRAL dz], using Wikipedia's Pi notation (here "PRODUCT") for the Volterra product integral. The composite redshift due to a complete sphere of mass M, at radius R, is then z = exp(GM/Rc^2) - 1, not the conventional relativistic (1 - 2GM/Rc^2)^{-1/2} - 1, and not the first-order approximation, GMm/Rc^2. The corresponding relativistic gravitational potential energy must have similar exponential form to be consistent with the composition of relativistic gravitational redshift.

Unlike Newtonian potential energy which is negative, relativistic gravitational potential energy is positive, and equal to mc^2 exp(-GM/Rc^2). In the absence of a gravitational field, it is equal to rest energy. Gravitational potential energy is taken from that rest energy, and thus has a finite limit. Newtonian potential energy -GMm/R, is an approximation to mc^2 [exp(-GM/Rc^2) -1] for weak fields. Relativistic gravitational potential energy is an exponential map of the classical potential energy.

Relativistic gravitational potential energy gives an escape velocity sensibly limited to the speed of light, as might be expected from a relativistic theory, whereas this condition is violated in both the classical theory and general relativity. The singularity-free metric corresponding to the escape velocity is the same as Brans-Dicke. In that theory, inertial and gravitational mass differ slightly, by a presently undetectable amount. I suspect this discrepancy could arise from failing to account properly for the exponential nature of gravitational energy.

The original work can be found at the link in the file shells2010dec29.pdf. It has some simple examples to demonstrate the essential concepts. I was not aware of product integrals when it was written in 2010. This derivation of the product integral addresses the issue of normalization, which can be inferred from the physics of the problem. I don't have an essay for this contest, but here is a link to an essay from the last contest that shows some radical consequences of accepting the composite relativistic gravitational redshift.

It seems to me that there might be a way to incorporate these relativistic compositions for gravity into general relativity via the product integral and arrive at the Brans-Dicke metric. I wonder, what would be your intuition on this possibility?

Colin Walker

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I finally got to reading your paper. I have been working to get a piece of instrumentation developed meant to go to another planet. In reading this I think what you say is maybe not that different from what I develop.

Your paper drives home the point on using the Blum, Shub, and Smale (BSS) concept of computability. This is an odd concept for it involves complete computation of the reals...

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I finally got to reading your paper. I have been working to get a piece of instrumentation developed meant to go to another planet. In reading this I think what you say is maybe not that different from what I develop.

Your paper drives home the point on using the Blum, Shub, and Smale (BSS) concept of computability. This is an odd concept for it involves complete computation of the reals to infinite precision and where our usual idea of close approximations are not real computations. This is a certain definition of incomputability. Since these I_U fractal subsets for an underlying fractal system are forms of Cantor sets the p-adic number or metric system is used to describe them. As fractal sets are recursively enumerable their complements are what are incomputable in a standard Church-Turing sense. Since this fractal is really defined in a set of such, there is a set of p-adic numbers or metrics and by Matiyasevich this is not globally computable. By this there is no principal ideal for the entire set or equivalently a single algorithm for all possible Diophantine equations. This is the approach I take with my FQXi paper. As a result, at this time I am relatively disposed to your concept here.

This meaning to incomputability in the BBS system is different, but not that out of line with the standard Church-Godel-Turing understanding. We can see that determinism is not always computable. The Busy Beaver algorithm of Rado has the first five numbers 0, 1, 4, 6, 13, but beyond that things become tough. The 6th is thought to be 4098, though not proven as yet. The 7th is a number greater than 1.29×10^{865}. It is not possible to compute higher Busy Beaver numbers. The failure to do so is a form of the Berry paradox or undecidability. The Busy Beaver is then a sort of model idea of a strange attractor with the exponential separation of differing initial conditions for two systems.

We have for coherent states, a general form of laser states of light, the occurrence of states of the form |p, q⟩ that have both symplectic and Riemannian geometry. My mind is pondering what connection this concept of incomputability has to coherent states. The occurrence of Riemannian geometry for spacetime, particularly if spacetime is a large N entanglement or condensate of states, and an underlying quantum geometry may be ordered as such. Einstein in his Annus Mirabilus proposed that states of light have blackbody or Boltzmann thermal distributions with a coherent set of states in his coefficients. This may really describe quantum gravitation as well.

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