joigus

That's the rho for the scatterers!! Form factors measure the spatial shape of scatterers. OMG. Please don't ask artificial intelligence again. It's almost indistinguishable form natural stupidity. Can you set me free now? Yeah, let's keep this short, please, oh please. where ρ(r) is the spatial density of the scatterer about its center of mass (r=0), and Q is the momentum transfer. (quote from https://en.wikipedia.org/wiki/Atomic_form_factor). If you've done some physics it takes you about half a second to figure out that's what they mean. Even if you don't remember the whole context. For Pete's sake.

Yes, you have mixed them up. Scattering theory is about an incoming state that comes from \( t=-\infty \) and evolves towards an outgoing state that evolves towards \( t=+\infty \) . Quite different from \[ \left|\psi\right|^{2} \]. Born's rule is about one and the same state. Scattering is about an incoming and outgoing states, both close to plane waves, and infinitely distant in time. Do you really know about quantum mechanics? Doesn't sound like you do. Oh, come on, drop the attitude, will you? I know quite a bunch of details about quantum mechanics, and I don't need prosthetics for my intelligence. AI has failed to answer some of my deepest questions. Miserably so. No wonder, really. AI works on the logical span of what humans have already thought. It's clueless about what's next. If you could paraphrase what it's trying to do (sometimes to astonishing perfection, I'll give you that), it is: How could I convince myself this is what human interlocutors would want to hear?

I think you're conflating here Born's rule with the first Born approximation (scattering theory). Very different things, even if both bear the name "Born". Born's rule is the general assumption that probabilities are bilinears of amplitudes. Born's approximation, OTOH, is the simplification that scatterers can be considered as given, and unaffected by the scattered. Further, the Fourier transform of any x-dependent quantum distribution is yet another thing. Namely, the momentum representation of said local distribution. So you're conflating three different things here. Two of them bear the name 'Born". The other one was born at about the same time. None of them is born out by what I know about quantum mechanics.

What I mean --hopefully-- more precisely is that, as you imply, wave functions are determinations from a range of physical conditions (examples: hydrogen atom's ground state, free particles, etc) whose only undetermined co-factor are global or even local (in cases where the dynamical theory is a gauge theory) phases that can be chosen locally at will (not measured or indeed measurable at all). Summarising: Wave funcions have, 1) A factor that is completely determined 2) Another co-factor that is a huge arbitrary Mind you, "huge" here is, if anything, an understatement. It is futile to keep discussing here if there isn't a least common denominator of what is known to be the case for quanta. Having said that, I sympathise with attempts to "better understand" what possible sub-reality[?] quantum mechanics is telling us about. But variables half of which merit the qualifier of "completely determinable" and the other half "completely arbitrary", to me at least, cannot qualify as random, as @Killtech seems to purport. Sorry for my tardiness in answering, but life-changing events are taking place for me lately. [?] Less-stringent reality than what our intuitive criterion would have it?

Random variables have probability distributions. As such, they must be observable. Otherwise what does it even mean for a hidden variable to have a certain probability to adopt a particular value, if that value cannot be observed? AFAIK, in HMM, probabilities are assigned to the Y's (the measured variables), not to the X's (the hidden states). The X's are there to provide conditional probabilities P(Y|X). Someone might have called these X's "random". If that's the case, I think it's a misnomer. Anyway, the wave function is not a random variable, except for mixture states, in which it is. But its status as such is a little bit flaky (there is a huge arbitrariness in the choice of wave function).

Not really. The wave function is not considered to be a random variable, for good reasons. It's a representative of infinitely many valid wave functions that all embody the totality of statistical properties of the system. All these wave functions differ from each other in a constant phase factor. You said yourself the wave function cannot be observed. Random variables can be measured. Otherwise it doesn't make much sense to call them random variables, does it?

No. The momentum operator is an endomorphism (a function from square-integrable functions to square-integrable functions). It is no random function. It's the measured values are random.

I didn't know whom it was addressed to.

Please learn to use the quote function. We don't know what particular argument you're answering to. Yes. Can you give us an outline of what it is?

Aha! So deterministic is a special case of non-deterministic... They do. QM is non-Markovian. Markovian systems don't keep memory of their previous history. Quantum mechanical systems do. Except when you measure. When you measure you do erase a big chunk of the past history of the system. So you have to be very specific and very precise about when this whole Markovian thing comes into play. It's not impossible. I didn't say it's nonsense. It's just you haven't convinced me. Others don't seem convinced either. So far, I'm afraid I can't follow your 'aspect'.

A little bit along the lines of what Phi and Swansont said, tell your friend: Try to start by being your own critic. Be objective. Try to think, 'how could this idea be wrong?' instead of so much on 'why don't people immediately see how beautiful this idea is?' The history of science sometimes focuses too much on the epiphany, the eureka moment, and glosses over the bouts of painful self-criticism the authors themselves had to apply. Also by others. The history of sience is full of simply elegant ideas that had to be hammered out into useful ones.

Ok, let's stop right here and do a sanity check. In what sense do you see evolution of the quantum state (never mind pure states evolving via Schrödinger or density matrices evolving via V. Neumann eq.) as a Markovian process? I fail to see how they are related. Schrödinger eq. allows for evolution of superpositions. V. Neumann too, if only perhaps more obscurely, because they are integrated over in the p's (the statistical weights). You must mean something like results of measurements somehow occur "internally" as Markovian processes that must be implemented in the evolution of the quantum state, completing it, quite unpredictably. For some reason to be ascertained later, they comply with Born's rule that the odds follow the square of the wave function, or bilinears of it. Otherwise, I don't know what you mean.

Sure, but you said, It is that what I argued not to make much sense. For starters, typical quantum mechanical space states are infinite-dimensional. But even in the case of relevant sub-spaces (eg, spin), the space of pure states has dimension 2S+1, while the space of density matrices has dimension (2S+1)2. So, again, what do you mean a density matrix is "much smaller"?

The key to the concept of a density matrix is that, whenever your quantum state is not maximally determined, the most natural thing to assume mathematically is that the collectivity you're handling is a statistical mixture of different so-called "pure" vectors with statistical weights p1 , p2, etc. It is a mathematical convenience to define it as a matrix: p1|1><1|+p2|2><2|+... Nothing more. Anything you can define with a matrix you could define equally well with a series of scalars (bilinears, in the case of QM). This doesn't make much sense, as a basis of the Hilbert space spans the whole set of possible states. The total is the span of the basis states... Or I'm missing your point completely. Quantum mechanics is non-Markovian. Present states depend on their past histories... Errr... more stuff... Non-linearity could be relevant at many levels. Evolution could be non-linear, observables could be non-linear functionals defined on elements of a linear vector space, etc. These things have been tried to death, of course. And sub-quantum Markovian processes have and are being tried. Cellular automata, for example. I'm almost sure of that, although I'm no expert. Look up Gerard 't Hooft interpretation of quantum mechanics...

Ok. So let's leave it alone, as long as you agree that Bell's theorem experimental violation implies there can be no local hidden variables (a local reality that underlies quantum mechanics). I agree with @studiot that you don't need Markovian probabilities to accomodate quantum mechanics. You only need a concept of probability in the terms that he defined. What's different is the existence of these "potentialities" if you will, that we call "probability amplitudes".

Again, no. There is a restriction on local realism, because there is a restriction on realism, because three perfectly sensible experimental questions, namely "Is A true?", "Is B true?", and "Is C true?" cannot have yes / no as answers simultaneously. In fact, the CHSH state was prepared in one and only one causally connected patch of space-time, so nothing non-local is going on. Many people misunderstand this. Some of them write books. What can you do... Then why bring it up, especially as an incorrect statement? QM's concept of probability differs from classical probability only in that classical probability doesn't have anything in the way of quantum amplitudes, which as we know give rise to interference phenomena. This gives way for interesting correlations that do not appear classically. Otherwise it's more or less the same concept.

Not really. What QM and CHSH experimental results display is incompatibility with a somewhat involved hypothesis called "local realism". If I had an alternative life to repeat this discussion again, I think I would be able to convince you that the downfall of local realism is due to the "realism" part of it, not to the "local" part of it. The Schrödinger equation is perfectly local. Field theory is perfectly local. This should be enough of a clue that nothing non-local is going on in QM. What is formally non-local is the projection postulate (far-away and long-gone subamplitudes of the wave function must "die" immediately). But it has no discernible consequences that would allow to --eg-- send signals, transport energy, etc. Which only reinforces the idea that nothing non-local is actually going on.

That would probably be considered hijacking. Maybe @AbstractDreamer could expand on their statement in a thread of their own, I suppose. Not for me to decide anyway.

I'd put it even more simply: GR was valid 5 minutes ago, but not 5 minutes from now? Quantum mechanics will be valid only after I finish my ice cream? There is no such thing in physics as "the present" (the all-pervading separator between past and future as far as human experience is concerned). What is the present, according to physics? If you have an answer, any answer, let me know. Thereby my McEnroe point.

I've had a John McEnroe moment: You cannot be serious!

I'd say it's homeopathic in the blend. Rigour of physics being the substance to be dilute to zero, and wildness of intuition being the diluent. And no mathematics...

This I remember from previous readings referred-to as "discontinuity of the fossil record", and if I remember correctly, Darwin already was very much aware of it.

It's long been known that the Dirac algebra is isomorphic to quaternions. Unfortunately, it's also been known for a long time that quaternions are insufficient to represent the properties of elementary particles. Models based on octonions have been tried with more success than those based on quaternions (Hestenes et al.). See, eg, https://pirsa.org/c21001 And ripples in space? Space is not a substance. It's more like a format.

I am. I wish I were.

There is a fine line between a numerator and a denominator. Only a fraction can make sense of this.