joigus

Good point. +1 I've come to a point that I even accept the possibility that not only aspects of reality --sensory input organised by patterns-- could be empty of theoretical constructs as conceptual stand-ins for them, but also some theoretical constructs --especially mathematical ones, but not necessarily just those-- could have no sensory-input counterpart for them. My hope for this possible fly in the ointment of our intellectual satisfaction is that the map could be completed with different patches, very much in the way that cartographers map geographical reality with different patches, and consistency rules for leaping from one to another. A map that faithfully represents regions around the equator, presents major distorsions of Antarctica.

Yes, exactly. While not impossible to conceive, thinking that photons emitted 3 bya by remote quasars somehow code the decisions that primates will take here and now on a particularly rainy day in Vienna at, say, 8:00 AM, is next-to-inconceivable. That the information be so intimately interwoven as to produce this next-to-hallucinogenic coincidental effect is something I don't want to even start to consider. I think the solution to the perplexing results of the quantum theory of measurement are better tackled by some kind of "fiduciary internal determinism." Something very much in the vein of what John Bell proposed with his beable idea. It would involve the idea that a choice of wave function does not determine everything. Infinitely many gauge descriptions would hide the "hidden variables." The catch is: Hidden variables are hidden forever, they never show up completely. They simply can't be seen, ever. There's a logical --and experimental-- fundamental obstruction. I think that's the way out for the problem of measurement, and I think that's the natural logical continuation of John Bell's musings. But that's probably the topic for another thread. That seems to be the case. I cannot wait for her to give up on that, because she is very much the conscience of theoretical physics today.

Sure. I'm sure most everything can be approached by algorithms. The missing part here would be the rotation properties of vector fields, but I'm sure that can be done too. That you're confusing the map for the territory. A map of the world is not the world.

An example is worth a thousand rules. Can you give one?

No. And @studiot is giving you the answer: A rock is made of something. A mathematical object doesn't have to be, because it's defined by logic. It's not made of any ingredients.

Thanks for the clarifications. +1 The "that would mean" kind of gives it away. I've been worried by this for some days now... Is Zeilinger endorsing superdeterminism??? I will get hold of this book and read the paragraphs in their natural logical flow, if you know what I mean. I did understand that the quasar experiment seems to rule out superdeterminism, though. But at that point it seemed to me that Zeilinger was considering superdeterminism as the only way out to save locality. That would be a big "oh, no!!" by me.

Sometimes you can look at the same fact from different perspectives... Example: In order to find its path from A, a particle follows the clue of velocity and force or, In order to go from A to B, a particle tries to minimise its action Both are mathematically equivalent.

Not all fields are scalar fields. You're implying scalar fields. Fields can be scalar, real, complex, tensor, spinor, vector... They can code fibres, curvature, torsion, topological numbers... Fields are abstractions, they don't have to be made of anything. They must comply with certain rules though.

Exactly. It's clarified in the next paragraphs. Zeilinger does not endorse "spooky action at a distance": (quoting Zeilinger.) The part highlighted in red seems to be implying superdeterminism. I would postpone that, because you have a dangerous tendency to mix up different topics. Suffice it to say that superdeterminism is just one loophole for someone who wants to save some kind of determinism at all costs. People who are familiar with mathematical proofs know very well that finding a counterexample does not necessarily imply the counterexample is unique. The part highlighted in boldface black is the most important one, which ratifies what @Eise is saying. Good. If you understand what you've said, you could be on the verge of understanding your own confusion. It's not a minor point. It's very important. It's the essence of the projection postulate, which is deeply non-local, but has no non-local consequences. Only problem: It's just a convention. It's not an evolution law for the state, and certainly not a law of physics. And most importantly, it has no experimental consequences. It was theoretically designed to do exactly that. Go back to my comment on FAPP's Bell's comment. Most physicists that contributed to the formulation of QM had no problem in seeing the state vector as just an epistemic book-keeping device, reflecting our knowledge. That's why they had no problem in introducing a non-local, non-unitary mathematical convention of which no non-local consequences could be derived. And last, but not least, as Eise has said above: No. Non-locality does not imply non-realism. They are very different assumptions. Example: Plane waves propagate in a totally local way. Yet, by virtue of selecting the momentum, the position is totally undetermined. That's local non-realism at its simplest, even before we start talking about entanglement. @bangstrom. You also said this in answer to @MigL: The quote is from Bohr, in answer to Einstein's famous "God does not play dice" --a colourful way of saying he didn't accept indeterminism at that point. But, really, is that "the explanation" of everything we're talking about here?

LOL. You've made a good point before, and something deep inside of me tells me you're about to understand an important part of this problem. Maybe tomorrow. Hint: You said "minor point." It's not minor at all. It's the key to all the confusion.

Here's a riddle for you: If there's no way to know which "action" came first --in a given frame of reference--, how do you know the time interval between the "actions" can be tagged as FTL?

Non-realism is just the character of QM. So it's not "we don't need it." It's what it's like. |electron> = |electron here> + |electron there> |vacuum> = |particle> + |antiparticle> Etc. It's what it is. A non-realistic theory out and out. OK? It's not "we don't need it" or "maybe tomorrow" or "I don't like it" or "we can do without." Well, maybe you don't need it, but it's what it is. That's factual. The double-slit experiment has no entanglement, and already tells you that much. I know you don't like my objections, but I'm here again.

(boldness by you, boldfacedness by me.) Exactly! Thank you.

No, it's not. It's actually a very quick and surprisingly deep understanding of the question, IMO, from someone that declares not to be all that familiar with the mathematical formalism. @Eise has understood this very clearly IMO. He's a very careful and attentive, and deep reader, and he has corrected me when I (wrongly) quoted Bell, when it really was Einstein quoted by Bell. It's you who, for some reason, keep saying that non-locality has been proven. After the last entry on my part, you've decided not to respond. You do this again and again. Mind you, the most dangerous thing concerning this topic is taking at face value what many people in a half-arsed way think they understand about it. You may end up wasting millions of SGD (Singapore Dollars) in declaring such stupid things as "oh, wow, my tardigrade got entangled with my qubit, how about that?" Silly, silly, stupid, mind-numbing nonsense "scientoids" --rather than science facts--, hyped to the max, totally void of content, expressed in a deliberately confusing and ambiguous language, and responsible in a big way for the bad image of science in the minds of many intelligent laypeople in this revolting, repugnant, post-truth age!

Disclaimer: I don't mean I endorse this view. In fact, it's shocking to me that someone like Einstein held any hopes that a classical theory of EM and gravity could some day explain spin, radiation spectra, particle spectra, etc. The very first time you see and understand commutation rules in QM, and the utterly fundamental role they play in it, it dawns on you: There's something here that no classical way of thinking can reproduce. I once read --it may have been on one of James Gleick's books-- that Einstein was presented with Feynman's brand-new path-integral formulation of QM by Oppenheimer, so as to intimidate him with the beauty of it. His answer was something to the effect of "Oh, I think I've earned my right to be wrong by now." If the story is true, Einstein was very much aware that he must have been left something out. Something essential. So no, in case there's any doubt, I do not endorse Einstein's view. But his argument about particle trajectories. Oh, be in no doubt; that still holds. Dear @bangstrom. Here, again:

Sure. Because Hoola said "to the degree I understand it", I felt I needed to avoid words like "a metrical property" which are more adecuate. This metrical property goes with the mod squared. Here. Again: There are many questions here that can be invoked in the same context: Completeness (where do "classical" data come from?) Indefiniteness (where were those "classical" data stored before the measurement?) Locality (do we need to introduce some kind of superluminal updating to implement "classical" data?) Counterfactuals (why are there measurements without interaction?) You could throw in more (contextuality, delayed-choice, etc.), but they're all related to each other. The quantum state holds the key to most of the "paradoxes," but you need to keep the character of QM in all of them. I told you, it's like a house of cards. If you remove randomness in a particular way --print something in the state, so to speak, and still use its built-in correlations-- you would be able to send FTL signals..., etc. Why did I talk about gloves? Was I endorsing Einstein's view? No, I wasn't. I introduced it the same reason that John Bell introduced Bertlmann's socks example: To show that in order to have perfect anti-correlation at a distance, you don't need any non-local mechanism going on. But no pair of gloves, boots, coins, astronauts, etc. can give you the other quantum correlations for spin: statistical variance of (A) = 1/2 statistical variance of (not A) = 1/2 statistical variance of (A+not A) = 0 statistical variance of (A+C) = statistical variance of (not A+C) = 1/2 for other C's incompatible with A, because classical observables don't have this character of incompatibility. A, B, and C are spin projections. Incompatibility comes from non-commutativity. If you listen carefully to Sidney Coleman's talk --or read the transcript--, this particular point becomes very clear, as well as how John V. Neumann understood this very clearly, and never got entangled in silly discussions about non-locality. How that endorses Einstein's view of realism, when Einstein spent the last 25-odd years of his life thinking about other things and not accepting QM, is beyond me. Einstein was concerned with trajectories of particles. Go back to the paper and tell me if he ever mentions spin. He doesn't. He never did. He didn't even accept lambda hyperons, and other elementary particles, because he thought somehow all these quantised features would some day be deduced from a unified theory of gravity and electromagnetism. Einstein was thinking about particle trajectories, nothing else. The lesser-known paper with Tolman and Podolsky, Knowledge of Past and Future in Quantum Mechanics, and others, prove that very clearly.

OK. I have a feeling I haven't been very helpful. Let me pick up on @MigL's observation that, Exactly right. These waves, among many other interesting properties, have one particular funny property: They have some kind of a "volume" that must be the same at all times. We call this law "unitarity." This "volume" is actually not ordinary volume. It has no dimensions, it must always add up to one, and we understand it as probability. The probability of all the alternatives of an experiment must add up to one. It makes sense. You cannot have a quantum wave become zero, even for a split second, because this "zero wave" would have "zero volume." So you can never have any physical process make the quantum wave shrink out of "existence," so to speak, even for the shortest period of time. Let me tell you something. Believe me, there's no "laughable" idea that you can think of which there isn't a similarly "laughable" idea I haven't thought of before, or MigL, or anyone of us. It's not laughable at all what you say. We know it doesn't work for the reasons I tried, to the best of my abilities, to explain. Very eminent physicists and mathematicians have ventured down really deep rabbit holes in order to try and explain these things. So bad it is that humanity lost very valuable people down those holes. Einstein is perhaps the best example. I think David Bohm was another one. But that's another story.

Another interesting application of complex numbers inspired by physics: Materials have a refractive index. The reflected wave is given in terms of the real part, the imaginary part conveniently embodies the property of absorbance. So a metal has a considerable imaginary part in its complex refractive index. This is a nice example of what @Markus Hanke said about complex numbers as exponents. I see nothing "imaginary" in the ability of a material to absorb light. Why wouldn't one want to use such a handy book-keeping device?

You cannot add wavefunctions corresponding to two particles. Superposition is only valid for states of the same system. The wave function of a two-particle quantum state cannot be arranged to add up to zero. In the formalism, this is reflected in that they are functions of different variables. Furthermore, you can never totally cancel a quantum state. The zero vector in the state space has no physical significance. The vacuum that we use in quantum field theory \( \left|0\right\rangle \), for example, is not the zero state vector. There is no zero state vector in quantum mechanics.

As I never claimed that, there is no way that I might still claim that. Einstein's view of realism is untenable today. However, Einstein's original argument was about space variables. But spin cannot be understood in terms of space variables. In fact, spin variables cannot even be consistently understood as coming from any internal reality based on commuting variables, or "just parameters." Period. Why don't you read what people say, draw conclusions carefully, are rigorous about what you say yourself, and stop offending? If you did, and were, it would be possible to talk about the interesting problem of why QM --when you hold a certain interpretation of it-- produces this illusion of non-locality that's not really there. There are interesting parallels, possibly, in how biological evolution --when you hold certain interpretation of it-- leads to an illusion of design that's not really there. But you don't and aren't, so this discussion is leading nowhere.

Last paragraph in response to: Realism is violated by QM in the most glaringly obvious way: QM uses states for which, x-projection of spin is up x-projection of spin is down x-projection of spin is 40% up / 60% down, 70% up / 30% down, 50% up / 50% down, etc. In fact, for those states for which x-projection of spin is definitely "up," y-projection of spin is 50%up/50% down, and so is z-projection of spin. So, if you want to make one variable determined, all incompatible variables are undetermined. It has all these tradeoffs built in. IOW, QM violates realism in a way in which it implements an inherent, irreducible, totally from-the-ground-up indeterminism. Deeply-rooted indeterminism. Indeterministic to the bone and marrow. It does it in such a way that, for Hilbert spaces of dimension 3 upwards, you can even build 3 mutually commuting observables for which attributing hidden variables to determine the 3 corresponding eigenvalues is impossible (Kochen-Specker theorem.) With space-time not playing even the remotest part in the argument. Furthermore: For entangled states with maximal entanglement entropy (Bell, GHZ, etc.) not even the individual quantum states are determined, let alone the underlying "reality" of eigenvalues. Furthermore: Identity of a particle doesn't mean anything --anything measurable, that is-- in QM.

I'm with Gauss on this one. +1 Remove all nonsense propping-up-to-mystical nuances from anything intended to clarify our understanding. Vocabulary is a good place to start. I share your taste. I'd rather multiply complex numbers than add them. Adding complex numbers is messy; multiplying them is nice. I also agree that complex numbers are an indispensible part of QM. Unfortunately QM forces us, not only to add them, but normalise the result, after having added them, which is even more awkward than just adding them. I think the problem of what mathematical representation makes our description least awkward will always be an open question. The parametrisation of the sphere is, in this respect, a cautionary note that I never forget.

Disclaimer: It wouldn't be all of 2x2 matrices. It would only be those of the form, \[ z=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \] Those, and not other 2x2 real matrices, are complex numbers. These rings have fascinated me for decades. You could get an approximation as close as desired to the real numbers by means of rational combinations of 1 and the square root of any integer number that's not a perfect square. The p and q would have to be rational, instead of integer. Would that be a way in which the continuum can be approximated by a discrete mapping that frees physics from singularities and other similar "diseases"? Absolutely spot on. Just a name. Nothing "spooky" in imaginary. Agree. Non-commutativity: totally peculiarly quantum, but not due to complex nature.

Well, one single ocurrence doesn't tell you anything. That's a very important point. It's easily dismissed as a coincidence. OTOH, if every single time you saw a particular cat doing that, you would have to conclude it's likely not a coincidence. If you saw that happen with the same cat every single time, maybe a natural hypothesis would be that the owner of both the cat, the TV set, and being in control of the sequences shown to that cat, has arranged things in such a way that the cat has been trained to recognise the situation and produce the desired effect. Conclusion: Coincidence is not such. There's been an antecedent common cause to give rise to observed correlations. If, OTOH, every single cat you try this with displayed same behaviour... Well, that would be a puzzle. Every time the correlation becomes more and more puzzling, because you widen the sampling, and it becomes less and less likely that it be a coincidence. You would have both to widen your statistical scope --perhaps invoke Bayesian methods--, have a theory to explain why this would happen, further test your theory by sampling more and more of the parameter space that your theory suggests, etc.

Sorry, I meant addition. A false friend tricked me. A little bit more on this fascinating --at least to me-- topic: Suppose that, for some reason, you are repulsed by numbers which are the square root of a negative real number. You can always obtain a numeric structure that's totally equivalent to complex numbers by means of the following trick: Complex numbers "secretly" are 2x2 real matrices. Now, 2x2 real matrices can always be uniquely expanded into a symmetric part and an antisymmetric part. Here's how you do it. Introduce the special matrices that are going to be respective stand-ins for 1 and i: \[ E=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) \] \[ I=\left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right) \] and define a complex number \( z \) with real part \( x \) and imaginary part \( y \) --and its conjugate-- as "secretly," \[ z=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right) \] \[ z^{*}=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \] Then, the absolute value (squared) of \( z \) is, \[ z^{*}z=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right)\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right)=\left(\begin{array}{cc} x^{2}+y^{2} & 0\\ 0 & x^{2}+y^{2} \end{array}\right)=\left(x^{2}+y^{2}\right)E \] The product of \( z \) and \( z' \) --another complex number-- is \[ zz'=\left(xx'-yy'\right)E+\left(xy'+x'y\right)I \] etc. Now, whether complex numbers are "secretly" 2x2 real matrices, or conversely 2x2 real matrices "secretly" are complex numbers is, of course, totally immaterial from a purely mathematical POV. Errata It should be: \[ z=\left(\begin{array}{cc} x & y\\ -y & x \end{array}\right) \] \[ z^{*}=\left(\begin{array}{cc} x & -y\\ y & x \end{array}\right) \] So I guess my answer is: Yes, we do need complex numbers. We can dress them as 2x2 real matrices if we want, but we need them is some disguise or another.