Lorraine

Perhaps the most important thing to say in relation to my essay is that there is a difference between "superdeterminism" and "determinism". The former questions whether it is the case that, in a hidden variable model of the Bell experiment, the distribution of hidden variables are independent of the measurement settings. Without bizarre conspiracies, such distributions certainly are independent in classical models. However, in my essay I discuss a non-classical hidden-variable model which has properties of non-computability (in relation to state-space geometry) where this independence can be violated without conspiracy.

This has nothing to do with the Australian bush fires. In discussing the climate system (which is essentially a classical system) the concept of superdeterminism never arises explicitly. However, as a classical system it is implicitly not superdeterministic (we are not aware of any bizarre conspiracies in the climate system).

However, I think you are a little confused between the issues of determinism and superdeterminism. Somewhat perversely, given the name of the word, it is possible for a superdeterministic model to actually not be deterministic!

Instead, I think the question you are asking is about determinism, e.g. whether it was "predetermined" that the 2019/20 Australian bush fires would occur, ten million, or indeed ten billion years ago? Put another way, was the information that led to these fires somehow contained on spacelike hypersurfaces in the distant past. I sense that you feel it is somehow ridiculous to think that this is so, and I know colleagues who think like you. However, not everyone does and logically there is nothing to disprove the notion that the information was indeed contained on these hypersurfaces (albeit in a very inaccessible highly intertwined form).

However, this is not a discussion I would wish to have on these pages, not least because it rather deviates from the point of the essay which is that undecidability and non-computability provide a novel means to violate the Statistical Independence assumption in the Bell Theorem, without invoking conspiracy or violating the statistical equivalence of real-world sub-ensembles of particles.

Hope this helps.

Perhaps, since you mentioned the bush fires, I could tell you that I have a proposal for the Australian government if they want to reduce the risk of these fires in the future. My idea was published in the Sydney Morning Herald (and other Australian outlets) last week:

https://www.smh.com.au/environment/climate-change/we-should-be-turning-our-sunshine-into-jet-fuel-20200123-p53u09.html

I have a problem with the idea that the chaos is incompatible with relativistic invariance; I cant't give an example now, but a differential equation that is relativistic invariant and chaotic could be possible: I am thinking that in the solution set of the Einstein Field Equation there could be a solution that covers the space with a non-integer dimension, thus obtaining chaos for the metric tensors dynamics. I think that, for example, the Black Hole merger has an attractor (fixed point or almost limit cycle).

An Einstein field equation with weak field is a linearizable theory, so that there is an approximation nearly linear.

I don't understand: is a quantum non-locality the effect of the quantum field theory? The gauge boson interact between parts of the system, that transmit quantum information. So that to say that a system must satisfy bell's theorem is not equivalent to say: must a gauge boson exist?

    Tim Palmer,

    Thanks for your detailed reply. I think I WAS a little confused about the difference between determinism and superdeterminism: thanks for explaining. However, you are still in effect saying that every single koala death by fire was pre-determined.

    I will put the determinism issue another way, in terms of the problem of decidability: how we make decisions, and how we symbolically represent decision making. The issue is: what exactly do the symbols and numbers of physics represent, and what do the yellow blobs (Figure 3) represent, i.e. what is their deeper meaning? I have made various versions of the following case several times on the backreaction blogspot:

    According to physics there are no IF...THEN.... algorithmic steps in the laws of nature, there are only lawful relationships that are representable by equations. Try to do IF...THEN... with equations. You can't. So according to deterministic physics, you CAN'T make decisions, you CAN'T learn from your mistakes, and you CAN'T be responsible for your actions.

    Where are the models showing how IF...THEN... is done, using nothing but equations? IF...THEN... is about outcomes (the THEN... bit) that arise from logical analysis of situations (the IF... bit), but equations can't represent logical analysis. IF...THEN... is about non-deterministic outcomes, because logical analysis of situations is non-deterministic: there are no laws covering logical analysis. And you can't derive IF...THEN... from equations.

    The point of what I'm saying is this: If physicists need to use IF...THEN... logical analysis to represent the world (e.g. your Figure 1), then they are assuming that there exists a logical aspect of the world that is not representable by deterministic equations, and not derivable from deterministic equations. Your idea is that the world is deterministic, but the fact that you need to use symbolic representations of logical analysis and logical steps to represent the world contradicts the idea that the world is deterministic.

    (Please don't appeal to computer models. As a former computer programmer and analyst, I know that computers are 100% deterministic: they don't do IF...THEN... logical analysis. They deterministically process symbolic representations of IF...THEN... steps, which deterministically process symbolic representations of information.)

    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.

    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.

    Actually what I say is that chaos is only superficially incompatible with relativistic invariance. The key is to "geometrise" chaos and that can be done by considering the invariant sets of chaos. I then try to show that these invariant sets may in turn help make chaos compatible with quantum theory.

    My own view is that the resolution of the Bell Theorem is not through quantum field theory, since that is an extension of quantum theory. Rather my belief is that there is a deeper deterministic formalism based on non-computable fractal invariant sets which has quantum theory as a singular limit.

    I am currently working on an extension of these invariant set ideas to incorporate the formalism of relativistic quantum field theory.

    Tim,

    What do your diagrams and equations actually represent? Do they represent a world ruled by equations or do they represent a world that is taking logical steps and performing logical analysis? Where does the logical analysis and steps that you are personally taking end? Do the logical steps in Figure 1 represent logical steps you are taking (e.g. to solve a problem or equation) or do the logical steps in Figure 1 represent logical steps the world is taking?

    Your essay is interesting.

    Reading it made me think.

    For example if there was a chaotic state in general relativity, then is it possible that Hausdorff's measure of particle trajectory an relativistic invariant? If it were not so, then there would be an observer for whom the relativist trajectory is non-fractal, but this seems unlikely to me (it's like a change of topology, to change from a chaotic trajectory to a non-chaotic trajectory).

    Also for the Bell theorem (or the Einstein-Podolsky-Rosen paradox), it is possible to study the Feynmann diagram for the cross section in the scattering of two polarized Dirac particles (I read today the results in Greiner book) and to obtain the probability of the final state (with elicities). If there are interaction, so gauge bosons, then there is not an instantaneous effects; the collapse of Alice state communicate the state to Bob using the gauge bosons interaction, with the light speed.

    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

    "... the closed Hilbert Space of quantum mechanics only arises in the singular limit where my finite fractal parameter p is set equal to infinity, and this is an unphysical limit! Hence, rather than complete quantum mechanics, my own view is that, guided by quantum theory, we have to go back to basics ..."

    There may you meet the humble effort of an old engineer:

    Eckard Blumschein

    6 days later

    Hello again Tim,

    After reading Lawrence Crowell's paper; I have a greater appreciation for your work, and even moreso that you are able to write so lucidly about it for lay audiences. I am impressed. I will have more questions now, after all that fuel for thought.

    Would the correctness of your theory imply that the fabric of spacetime is fractal? This is a feature of several quantum gravity theories, in terms of the microstructure. Does that project onto the large scale structure of the cosmos in your view? Would it surprise you if I said it appears some of your starting assumptions would follow naturally, if my own theory pans out?

    Tip of the old iceberg for you.

    More later,

    Jonathan

      Thanks for these kind comments Jonathan.

      You ask a good question. However, to be honest, I am not 100% sure at present what my model implies about the structure of space-time, so I prefer to be agnostic about this for now. However, I am working on a generalisation of my model so that the properties of momentum/position commutators are (like spin commutators) describable by number theory. This will allow me to start reformulating relativistic quantum field theory in a more deterministic framework, and from there answers to your questions should emerge. However, I want to do this slowly and carefully, and not jump to conclusions that may at first sight seem reasonable, but will ultimately turn out to be wrong.

      That was a satisfying answer Tim...

      This speaks to the question of what sort of evidence of your theory would we see in the cosmos that might provide verification or refutation for its veracity. I asked a similar question of Gerard 't Hooft at one point and his answer was similar - that it was too early to tell what the cosmic evidence would be.

      The following year at FFP11; he elaborated in his talk about the desirability of and difficulties with obtaining Lorentz invariance in a CA based QG theory, but still no hard predictions about what we would observe (in black hole emissions perhaps) that would distinguish it from the standard.

      I've seen or heard several predictions from Loop Quantum Gravity folks about possible signature detections - such as Lorentz invariance violations, comb filtered emissions from black holes, and so on. But I see that each time such a prediction is made, folks will jump on it as excluding a theory if the exact signature predicted is not found. And String Theory folks seemingly refrain from making any hard predictions at all.

      All the Best,

      Jonathan

      Dear Tim,

      Great essay, congrats. Wish I had a background in physics to completely understand. Please indulge me if you have time.

      1) Are you essentially suggesting that mathematical incomputability/undecidability exists as space-time, emergent from quantum non-linearity (as that is what the wave function seems to suggest, which may in effect be the cause of macroscopic gravity?

      2) If something (anything that exists as part of detectable science) is incomplete as a matter of ontology (incomplete in the Godel sense) how could that ontology possibly verify determinism or superdeterminism?

      Best,

      Jack

      (Essay: Misalignment Problem - You may enjoy the amalgamated sleuths section)

        Yes I do think that relativistic space-time will be found to be emergent from this fractal state-space geometry. However, making this a precise notion, and not just an aspiration, is something that I am currently thinking hard about!

        I'm not sure I fully understand your second question. However, it triggers in my mind an important question: are there experimentally testable consequences of determinism? Again, this is something my collaborator Sabine Hossenfelder and I are currently thinking about.

        So, in short, I can't answer either question, but they both touch on important issues!

        Thanks Tim,

        I am very glad you are thinking (with the great tools of physics) about the same questions I am.

        Re Q2 I think you have grasped my question in your statement "are there experimentally testable consequences of determinism?" Because if Godel's incompleteness manifests physically (space-time & mass) then you could never test determinism because the physical system would have unknowable states that cannot be determined by the system itself. So you couldn't have a determinable system, could you?

        Best,

        Jack

        My view (which I tried to express in the essay) is that such undecidability only manifests itself in questions about the structure of state space, not in questions about the structure of, or processes in, space-time. Hence I do think there are experimentally testable consequences of determinism.

        Tim,

        What Bell had in mind (and explicitly expressed so in many interviews) is that, if particles are little machines, then his inequality must be respected. Now, as with any statement regarding the physical world, it tacitly assumes also `common sense'. One can bend this vague notion to an arbitrary extent, but there is a more direct attack on Bell's theorem, which has been staring us in the face for over a century: Particles (and chaotic systems and humans) are not machines! (no new-age stuff)

        You are invited to read my essay which is further relevant to your main area of expertise - predicting the behavior of chaotic systems. Ensemble average over initial conditions is probably not the right way to do so.

        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 1.29Г--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, qвџ© 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.