> The real mystery is about the wave/particle duality of reality

Does the Hilbert space where the quantum state lives automatically fall out of this? That the state of a combined system is the tensor product of the states of its components? That is more mystifying to me than wave-particle duality.

There are all kinds of simple models like lattice gases, the Ising model, etc. where you can get various classical equations like Navier-Stokes out by making the mesh fine enough. It’s been bugging me whether anything like that ends up having complex probability amplitudes as a continuum limit, like in quantum mechanics. But maybe they can’t, because in those models, the dimension stays the same as the system evolves, which means you can simulate it in polynomial time, but we believe BQP>P.

So where does the proliferation of dimensions come from in QM?

]]>*“preserving, for example, that no information about position is knowable for a definite momentum.”*

My understanding is that any wave “packet” has that property, e.g. even a super localized spike-like wave packet (exact position) has to have an infinite frequency content (Fourier transform) so momentum/energy is unknown. And a wave packet with a pure frequency (known momentum/energy) has to be infinite in space (unknown position).

And the actual wavelength of such waves comes from the mystery of quantization – if continuous spaces (position, energy, …) get turned into discrete spaces, an absolute scale appears.

E.g. if electrons have stable orbits, their waves has to be such that they interfere positively with itself along the orbits, so it ties the wave number and the energy.

The connection of complex numbers to phase certainly is of physical importance, not least because it explains an important instance of gauge freedom in the equations. And those freedoms of the complex numbers (like that replacing every instance of i by -i would give identical predictions, or like that gauge freedom) is one reason to ask whether one can get rid of them. As long as they are there, the freedom they bring is most likely there too, and hence prevents a unique “best” formulation.

Now you may think that this discussion is all academic, because there is no reasonable natural formulation of QM without complex numbers anyway. But things are not necessarily so clear cut. See my (rather long) comment on the formulation of QM in section “2.1 The Ehrenfest picture of quantum mechanics” via (6), (7), and (8) in A. Neumaier’s Foundations of quantum physics II. The thermal interpretation:

]]>… appreciate the beauty and depth of (6), (7), and (8). That beauty goes deeper than my parenthetical remark. Note for example that those equations don’t contain i. (How could they, given that they unify classical and quantum mechanics?)

…

After appreciating the beauty, the difficult next step would be to understand why this is still not enough. Even so i no longer occurs explicitly in the equations, and all beables are real valued, complex numbers will continue to play a key role behind the scenes. How physical are those real valued beables? I once wrote: “I admit that it is often easier to compute with the vector potential instead of the actual fields. But the actual fields are measurable, at least in principle, while the vector potential is not.” Are those real valued beables more like the actual fields than like the vector potential? They still share properties with the vector potential, i.e. some gauge freedom is still left. Only the complex phase which explained the gauge freedom is more hidden now.

It seems the Renou paper is saying complex numbers are (part of) the point.

In classical mechanics, yes, complex numbers are “just a tool”.

But try to explain the SG experiment and complex numbers become indispensable and inevitable for describing (two-state) systems with more than two incompatible observables.

And it is not just that there is a wave structure but it is that it is a different kind of wave structure than classical because the complex exponential has a wavelength AND a constant absolute value – preserving, for example, that no information about position is knowable for a definite momentum.

What is remarkable about the Renou et al result is that complex numbers (the structures that we call as such) are a verifiable part of those “real mysteries” – and not just a tool we use to work with those mysteries.

I guess to me the fact that (the structure of the things we call) complex numbers is right at the heart of QM is something like the mystery of the fine structure constant. Complex numbers have all of these “perfect” qualities that are “just right” for what we need in a working model of the universe (see Scott’s paper on this). ]]>

My only memory about Gisin is that I froze in my chair when he said it’s only a matter of time that quantum computers will break any post-quantum crypto algorithm. (Sure, I wish he was right, of course, but still…)

]]>Scott, may I ask why you don’t treat the Renou et al. paper like you did with other papers in the past that “proved” their fancy points by basically stating that the predictions of quantum mechanics are correct?

One reason may be that Nicolas Gisin is a coauthor, and that Marc-Olivier Renou did his PhD in Gisin’s group. Scott may list as many good properties of this paper as he wants, it would still not explain why he did put in the time to evaluate this paper.

So who is Gisin that his presence as a coauthor gives weight to a paper proposing an experimental test for a question related to the foundations of quantum mechanics? In 1997, he (and his group) could show violation of Bell’s inequality outside of highly controlled Laboratory condition (using standard optical fiber over a distance of more than 10 km). In 2003 he achieved something similar for quantum teleportation. But those achievements don’t properly explain his weight. His talk delivered at the first John Stewart Bell prize award ceremony does a better job: He showed how applications of quantum weirdness (like quantum key distribution) could be decoupled from the validity of quantum mechanics (or the abscence of backdoors in the equipment) and be based solely on the presence of the observable effects.

But even this last part somehow doesn’t capture my impression (of him) while reading Nicolas Gisin’s short book Quantum Chance. He somehow goes beyond the notion of true randomness or “bit strings with proven randomness” and tries to capture the experimentally observable nature of quantum chance itself, independent of any interpretation of quantum mechanics (or even the validity of quantum mechanics). Something like that it is a nonlocal randomness, and because the nonlocal correlations of quantum physics are nonsignalling, it has to be random, because otherwise it would allow faster than light communication.

]]>Complex numbers are just a tool to represent rotations, i.e. something that has a phase in it.

i is a 2d rotation of 90 degrees, and two rotations of 90 degrees is a rotation of 180 degrees, aka i*i = -1.

Similarly you can represent 3D rotations with quaternions.

The real mystery is about the wave/particle duality of reality: things have a built-in wave length in them, i.e. there’s a phase that depends on the traveled distance. And when different possibilities/paths exist and are indistinguishable, such possible paths interfere, until there’s some sort of interaction ruining the “indistinguishability” and the wave-like behavior is lost (the phases become noise-like).

]]>So, does the imaginary unit i “exist”? […] If we took the standard quantum formalism and restricted the Hilbert spaces to be real, possibly of larger dimensions, could we still explain the same phenomena? […] For years this was considered the final answer to our question: in quantum theory complex numbers are only convenient, but not necessary. Here we prove this conclusion wrong.

First of all, the “conclusion” is obviously right (we don’t need complex numbers and Hilbert spaces to describe quantum mechanics, if we are masochistic enough).

Second, complex numbers aren’t more mysterious than negative numbers (one can use Gauss’s “lateral unit” idea to argue that). So in the 21st century, why should anyone come to the idea to restrict complex amplitudes to just real ones in the first place? Go immediately for natural numbers, those are the ones almost everyone believes to “exist”.

So my problem is that I feel this paper is trying to sell itself on the wrong philosophical grounds.

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