joigus

No. It must have spin 2 if quantum field theory is to work for everything else. It's been tried with spin 0, 1/2, and so on. It doesn't work. It was understood in the '60s. And calling it a primal entity doesn't make it better, unfortunately. https://www.fnal.gov/pub/today/archive/archive_2012/today12-10-19_NutshellReadMore.html#:~:text=While the matter particles of,and momentum in the universe.

I think you got this absolutely right. V. Neumann tried to get a really robust proof that hidden variables were hopeless if QM is right. He thought he did. It's perhaps not widely well-known that Bell further elaborated on V. Neumann's argument and extended it to commuting observables, which resulted in what we know today as the Kochen-Specker theorem, which should be called Bell-Kochen-Specker. In a nutshell, what it says is that for some quantum systems, you cannot even assign reality to pairs of commuting variables. Analogously to the "regular" Bell theorem, this only happens for naggingly-difficult-to-spot pairs of variables. The thing about determinism is it blurs the boundary between causal and non-causal, as cause and effect are both co-determined by the deterministic law... I'm loosing my train of thought. I wanted to say more on @Markus Hanke's hopes that some version of superdeterminism coud be not so far-fetched --if not altogether plausible. They have to do with the possibility that the universe is actually holographic in nature. Maybe later.

Nietzsche is dead.

(My emphasis.) Not quite. I will only laugh at the matter simply because the matter is so insignificant.

Not always easy to do, but best advice IMO.

Logic by association, "just so" stories... Same old, same old... Why not DJL standing for Daniel Jay-Lewis? And why not a different lettering?, like:

Thanks! Yes, that's the source. Here's a reference to the recollection, In the forums below, G. 'tHooft himself clarifies these questions and some more. In a nutshell, and to the extent that I understand correctly, cellullar-automaton variables provide "onticity", but are affected by probabilities themselves, and produce the quantum states as something that very much looks "emergent". The space-time being essentially what we all know and love: https://physics.stackexchange.com/questions/34217/why-do-people-categorically-dismiss-some-simple-quantum-models As to non-locality of the Broglie-Bohm model, I'm aware that people claim it is. The claim is always kinda wrapped up in some obscure wording, never in the mathematics. I don't think relevant physicists have ever weighed in with the discussion, except to the effect of dismissing it from a distance --pun intended. <speculative> Of course, I'm sure the BB model cannot --at best-- be the whole story. It's got anthropocentrism written all over it. IMO, it must be some kind of toy-modelled approximation to some non-linear generalisation of field theory that exploits the (infinite dimensional) dynamical possibilities that gauge freedom affords. Throw in further assumptions on how this lumpiness of the gauge degrees of freedom correlates to the linear amplitude and there you are: your stand-in for realistic degrees of freedom, plus the reason why they cannot be ultimately determined: they can be changed by a simple gauge transformation, so they cannot ever be determined. </speculative>

OK. That's a historical point I cannot claim to be 100% sure about, so you may be right. I didn't mean he abhorred of the use of probabilities. I meant rather that Nature at the most fundamental level bespeaks probability. IMHO, Einstein's position towards QM, while it didn't significantly change, did experience a shift in emphasis perhaps. I know for a historical fact that at some point he's quoted as accepting probabilities appearing at the most fundamental level (whatever that means.) But his real qualms must have been not so much about probabilities as they were (must have been) about reality. What he did not accept until the bitter end was the possibility that there be no elements of reality below that level. He's been quoted as saying, https://www.scientificamerican.com/article/the-universe-is-not-locally-real-and-the-physics-nobel-prize-winners-proved-it/#:~:text=As Albert Einstein once bemoaned,regarded as a bad move. Correct me if I'm wrong, but I think this testimony came relatively late in his scientific career. It is a relevant matter to distinguish between something Einstein said in 1935 and, say, 1950. You know these things much better than I do. I think the concept of reality is the one that didn't let him sleep at night. He knew QM is deeply, unmistakably, irrecoverably at variance with the notion of a real world, existing independently of the observer. The problem with locality is that many people have used it for decades in several different ways, not all of them mutually overlapping. I'm not totally convinced that Bohm's theory is non-local in any meaningful sense that I can recognize. Of course I could be totally wrong and simply misunderstanding the finer points these experts are making. Strictly speaking, Copenhagen quantum mechanics is (mildly) non-local already, because of the fact that components of the wave function that now are light-years away, must go to zero just because I've measured something here on Earth. For better or worse, nobody can measure the consequences of something non-measurable disappearing out of existence[???]... But the user manual of QM does tell you to do that in your equations, which is a faux pas in local physics. Not that anybody has worried about that or (most) even noticed for about 50 plus years.

You should understand that "being at rest" are meaningless words in physics. Motion is relative. You should understand that what you are picturing as your "point of collision" is frame-dependent. You should understand that comparing how two different test masses accelerate towards a certain third mass which is source of a gravitational field acting upon those masses (context of the equivalence principle) is a different situation than the one you propose (two masses that collide under their mutual gravity). It's perhaps worth saying that gravity alone very rarely results in collisions in classical physics, because of centrifugal barriers, so the situation you propose is not nearly as general as it should be, especially considering the extraordinary character of your claim. You should have an understanding of relative motion*, vs centre of mass motion, vs motion described in a more general inertial frame. There are many important things that you misunderstand here, and so far you don't seem to be willing to understand, in spite on good efforts by several members. Please, be aware that you do not have a theory. You do seem to have several important misconceptions on basic Newtonian physics that lead you to believe a widely accepted theory, fastidiously checked, both experimentally and theoretically, forwards and backwards, is wrong. You should clarify your position, before anybody can prove you wrong. Nonsense cannot be proven wrong. * @Janus and @studiot in particular have emphasized this.

I have no idea what your thought experiment has to do with gravity, if anything. The principle of equivalence has to do with two different masses in the presence of a third mass (source of field). The fact that relative accelerations between mutual objects are equal and opposite is a trivial kinematical fact, and has nothing to do with gravity. Maybe I have misunderstood the whole thing. If that's the case, I do apologise.

I cannot be totally sure of what Bricmont's position is, but from what I can see, it seems compatible with mine, so here goes. At the point of publishing EPR, Einstein was not so much concerned with the possibility of hidden variables as he was with que question of whether uncertainty is just a consequence of our ignorance or goes deeper, as QM suggests. In order to do that, he tried to confront reality with relativistic causality. He might have devised an argument to confront it with the principle of relativistic frame independence or what may have you. But he was firmly convinced that both principles (relativistic causality and reality) must hold. As far as I can see, all the people who later worked on hidden-variables theorems --starting with V. Neumann-- were really working on theorems about reality (whether quantities A, B, C... can or cannot be said to have a value at the same time). The question of locality being in the back of everybody's mind, partly because Einstein invoked it, and partly because the projection postulate does invoke a non-local operation, even though it has no consequences in the way of signals, interactions, and the like. I also find the solution of superdeterminism unpalatable, even you there's always the possibility of saying that, at some point in the past, everything may have been causally connected. The De Broglie-Bohm solution I find the most reasonable, although it is extremely unappealing. In my mind, it could be but a very rough, very crude version of an idea that should be formulated in terms of field theory and gauge invariance. Both elements absent from the original formulation.

I wouldn't want to explain Lorentz transformations in terms of some alleged granular structure of space-time for the same reason that I don't need to appeal to a granular structure of space to explain rotations. Lorentz transformations are nothing but the analogue of rotations, but in a plane containing a spatial axis and a time axis. This is perfectly understood since like forever now and there is no need to explain it further.

If you don't mind my saying, it looks like you have passion for science, but tend to get too far ahead of yourself. Keep in mind that numbers have been studied for at least 6,000 years now. We don't know everything by any means, and some things we will never know; but chances are that you will gain insight much faster and efficiently if you study what's been conquered so far. Every step of the way you are allowed to take initiative, of course: Can I prove this, what about that? It's a beautiful voyage with far more unexpected vistas than you can even imagine.

Good insights, guys. (+1 to both) For @grayson,

The best approach to a tensor, I think, should be dealing first with Euclidean tensors. Ie, fixed entities that represent multilinear mappings acting on vectors on a flat space. On a flat space we can identify points with vectors quite directly by means of an affine structure. On a manifold OTOH, points are not vectors, nor can they be identified with such. We must introduce vector structures point to point (the so-called tangent space at point x TM(x) M standing for "manifold"). So I would introduce tensors in two steps. First: What they do at a point; then considering how what they do changes from point to point. That naturally leads to a parallel transport as a rule to take vectors at one point to vectors at a different point. Flat spaces have no connection (or parallel transport), even though they have tensors. Another thing that confuses people is the difference co-variant / contra-variant, which occurs long after one finds tensors and has to do with vectors "naturally" having two alternate bases when we are not in an orthonormal frame. Trying to bring all these aspects together into one pictorial explanation is, perhaps, misguided. I don't know. The root "tens-" in tensor doesn't help either. It seems to suggest something directly physically interpretable having to do with tension. They are multilinear operators; that's what they are. Angular velocity is a 2-tensor on a flat space, but in disguise. x-posted with @studiot

This is actually a question that's very close to my heart, so I'll be looking forward to derivations into both pure mathematics as well as physics, as Swanson and Hanke have suggested.

I, for one, don't feel any need to visualize a tensor. Never have, I must say. To me, it's an algebraically motivated concept. In some cases it might be useful to picture something geometric going on (example: the energy momentum tensor in GR). As an example of the converse, the Einstein tensor is very notorious for being essentially the only second-order tensor you can form that's covariantly constant. I'm sure pure mathematicians will tell you that there is no known role that this tensor plays in pure geometry --unless you invoke GR for some reason.

Oh, OK. I was confused by your words "you are given a space of unknown geometry". You must be given something. You must depart from some assumption. I think that may be at the root of why some of us have thought you were talking about physics. I've been known to do that...

Can you actually provide the example? Otherwise it's like... "I'm thinking of a space... you know. It's fractal here and non-fractal there, non orientable...", and so on. You see what I mean? Which one is it? Meanwhile, in coordinate land, say... f1(x,y,z)= x+y2-1=0 f2(x,y,z)=2x+y-cos(xyz)=0 The implicit function theorem guarantees there are ranges of x, y and z that make this a 2-dim analytic manifold, at least for certain values of x, y, z. Then you can worry about metric, angles, and parallel transport. Thus we would have a metric space with parallel transport, so we can talk about infinitesimal distance, angles, and distance along a path. But dimension is there at the very beginning.

Skimming through this very interesting topic, with special emphasis on Genady's quote from MTW. The first thing that came to my mind is that @Genady's question is, I think, equivalent to, Is there any way to define dimension --in pure mathematics-- that can be considered more primitive* than counting coordinates? A connection requires a differentiable structure. For that you have to have to be given your space in terms of equations, whether implicit or explicit, parametric, etc. From there, derivatives allow you to give a sense to the concept of "moving" (vectors). The so-called tangent space. Metric and parallel transport (connection) can be introduced independently. In spaces given in this way, you can start talking about dimension long before you have a metric or parallel transport. Eg, in thermo you have systems defined by eq. of state like f(p,V,T)=0. Even though you don't have any meaningful metric, or parallel transport (although you could talk if you want about a tangent space consisting in the different thermodynamic coefficients); you do have a dimension, which in the example is = 2. MTW's criterion is, I think, based on the topological notion of space. A topological space is basically a set with an inclusion operator on which you can define interior, exterior, and boundary. This I have found in https://u.math.biu.ac.il/~megereli/final_topology.pdf, which in turn I've found by googling for "dimension of a topological space". It seems that this criterion is somewhat different from MTW's, but they agree in that they're both topological. I must confess I'm a tad out of my depth with these "coverings" and "refinements of a covering" in topology. I must also say I'm always baffled by these questions when no analitic example (ie, using coordinates) is allowed. How are you even given a space when no coordinates are allowed? * Relying on fewer assumptions, that is.

You don't make sense. More examples: (My emphasis.) This is like saying that Einstein, rather than being a physicist, was German. Thus, whether something is a field, or a high-dimensional state --of what, BTW?-- belong in different categories. And more: (Again, my emphasis.) Blend QFT with ST?! You seem to forget that when people say "string theory" that's just short for "supersymmetric quantum field theory of strings". So string theory is but one kind of quantum field theory. Again using analogy, what you're saying here is very much like saying "we should blend calculus and mathematics". It's obvious to most everybody here that you're not making any sense. You've found a narrative that pleases you in terms of these characters "entropy", "mass", and so on. That's not science.

Can you "bunk" it first? It hasn't escaped my attention that you've used the word 'therefore' incorrectly three times since you've been here.

Sounds like an appealing approach to physicists and engineers. Thank you, Studiot. Although I've been much less active lately, I was also wondering about Markus. I found I'd missed a brief message announcing he was to be away for a while. I hope he's OK too.

There's a reason why science fiction is called fiction. The property you want to circumvent is called "confinement" of the strong nuclear force. It would be interesting to try and see if people have thought about this. I bet it's impossible, but that's never stopped people from trying.