Thursday, December 30, 2021

Quantum Mechanics needs Complex Numbers.

LiveScience reports:
Now, two studies, published Dec. 15 in the journals Nature and Physical Review Letters, have proved Schrödinger wrong. By a relatively simple experiment, they show that if quantum mechanics is correct, imaginary numbers are a necessary part of the mathematics of our universe.
Sounds big, right?

One source sends an entangled pair of photons into nodes A and B, while another sends a pair into B and C. Experiment showed that the photons in A and C were uncorrelated.

No surprise here. I am sure that no one expected correlations from light from different and unrelated sources.

Somehow this shows that some hypothetical real-number variant of quantum mechanics is wrong.

I did not follow the details, but apparently their real-number variant is a nonlocal theory. No one has discovered any experiment with this sort of nonlocal properties. Why did they bother doing any experiment? A nonlocality result like this would be one of the most important in the history of science.

Maybe they should have tested a real-number quantum with locality similar to quantum mechanics.

Quantum mechanics does use complex numbers. You could do all the calculations with real numbers if you wanted to, but there would be no point.


  1. Dear Roger,

    1. Why must the wavefunction be complex valued?

    The best answer I've ever ran into came, IIRC, from Dr. Lubos Motl.

    However, unfortunately, I cannot locate it now! But I am sure that the piece I have in mind is *not* his answer here: . I am almost certain that it was at his blog, though I can't recall which post. (It's *not* the post he mentions in his preceding StackExchange reply; it's probably a later post.)

    2. Putting the logic (I learnt from Motl) in my own words:

    QM has to have complex-valued wavefunction because:

    1. The de Broglie relations (i.e. Planck's relation E = h \nu and Einstein's relation p = \hbar k) are based on experimental observations. To challenge them is to challenge reality.

    2. The de Broglie hypothesis (viz., that matter has
    wave-like properties) also springs from experimental observations. Wave nature of electrons comes out in experiments (involving large flux situations).

    3. If you try to build a wave equation for \Psi while also satisfying both the energy and momentum relations used in de Broglie's hypothesis, then, the time derivative of \Psi has to be of the first order, even as the space derivative remains of the second order.

    A second-order time derivative (as in the usual real-valued wave equation) wouldn't satisfy de Broglie's relations. A first-order time derivative wouldn't give oscillatory solution. (It would be the diffusion equation.)

    Now, the only way an oscillatory solution can be had even while using the *first-order* time derivative, requires that \Psi be complex-valued. (Or, alternatively, there be two coupled real-valued equations, as indicated by you in the last line.)

    3. I am just a slight bit surprised that the paper got published in Nature. The postulatory basis of QM is too tightly integrated, that's why.

    Nothing wrong whatever in trying to probe if there is still some gap left (or a crack can be made) in what otherwise appears as a monolithic layer of those postulates. But as the above logic shows, there are certain things which get introduced into QM because (and only because) of experimental findings. Mother Nature forces our hand, so to speak.

    So, IMO, challenging the complex-valued nature of QMcal solutions wouldn't be a very good way to try and see if there can be any gaps.

    4. I haven't yet read the paper, but I am earmarking it, for an entirely different reason (not at all related to the complex- vs. real-valued issue). ... They are talking of photons, right? Photons. Not electrons. That's interesting.

    Would one expect correlations if the A and the C were electrons? If the answer is a "no", then what happens to the idea of the universal wavefunction --- to the very phenomenon of entanglement across arbitrary distances? And if the answer is a "yes", then why wouldn't correlations exist also for photons? .. I do think that the last is a valid question (given the theory and the state of the mainstream QM, anyway!)

    ...Anyway, thanks for pointing out the paper and the coverage. I had missed it.

    ...Happy New Year!


  2. ... On the second thoughts...

    It's about the difference between the real-valued (negative) exponentials (the exponential decay) vs. the complex-valued exponentials vs. the (co)sinusoidals.

    [Easy to figure out once some many ones already did!]


  3. The McGucken Nonlocality Principle: All quantum nonlocality begins in locality.

    Locality becomes nonlocality via the expansion of the fourth dimension at the rate of c as given by dx4/dt=ic, which naturally gives rise to Huygens’ Principle, time and all its arrows, all of relativity via the spacetime metric x4=ict, and quantum nonlocality, etanglement, and probability.

    In the grand endeavor of physics, a physicist observes nature and then composes principles, postulates, and equations reflecting physical reality. Other physicists are invited to disprove the principles, postulates, and equations via mathematics and thought experiments, and/or conduct experiments and make observations that demonstrate the said principles, postulates, and equations to be false.
    The McGucken Principle: Quantum nonlocality begins in locality.
    The world’s top scientists (and philosophers/quantum computing mavericks) have yet to share how they would go about entangling two unentangled electrons in NY and LA. Please, if they (or anyone) could tell me how they would entangle the two distant electrons, without bringing them in direct, local contact, or by using a system or systems of particles that originated in a local manner with local contact, then I would consider myself defeated in my argument, and I would retract my principle: “ALL QUANTUM NONLOCALITY BEGINS IN LOCALITY.” This provides a direct physical test of the expansion of the fourth dimension given by dx4/dt=ic.
    But, if they are unable to explain how they would go about entangling the two electrons in NY and LA without using some form of nonlocality which begins in locality, then SCIENCE and PHILOSOPHY must declare
    The McGucken Principle of Nonlocality the victor here.
    The McGucken Principle: Quantum nonlocality begins in locality.