5 ms·

archive link: http://web.archive.org/web/20170615010943/http://www.articlesbyaphysicist.com/quantummechanics1.html http://web.archive.org/web/20170615010943/htt

archive link: http://web.archive.org/web/20170615010943/http://www.articlesbyaphysicist.com/quantummechanics1.html http://web.archive.org/web/20170615010943/http://www.article... (w/o images) Skipping from the complex-valued wave function to its real-valued magnitude is a good simplification for an introductory article, but I think the need for complex numbers (the concept of phase) is very important and fundamental. That's what I would really liked explained? Why do cmplx numbers come up, and is there a QM-equivalent model using only real-valued functions?

evanb 9y ago
You may be interested in http://www.scottaaronson.com/democritus/lec9.html http://www.scottaaronson.com/democritus/lec9.html http://math.ucr.edu/home/baez/rch.pdf http://math.ucr.edu/home/baez/rch.pdf
n4r9 9y ago
Complex numbers aren't strictly necessary. You could even expand every complex dimension to a pair or real dimensions and get the same results (roughly speaking). However this is a bit unwieldy, so the question becomes "why do complex numbers provide such a convenient representation of quantum theory"? There is work being done on this sort of thing. If essentially all you care about are the probabilities of certain observations occurring, then you surely can represent any theory with real numbers. The field of "generalised probabilistic/physical theories" attempts to analyse quantum theory without assuming the baggage of Hilbert spaces etc... [0]. Chris Fuchs also has done a lot of work on representing quantum state spaces in terms of outcome distributions over special types of measurements called sic-povms[1,2]. 0. https://arxiv.org/abs/quant-ph/0508211 https://arxiv.org/abs/quant-ph/0508211 1. http://perimeterinstitute.ca/personal/cfuchs/ http://perimeterinstitute.ca/personal/cfuchs/ 2.https://en.m.wikipedia.org/wiki/SIC-POVM https://en.m.wikipedia.org/wiki/SIC-POVM
- akvadrako 9y ago
  This work (QBism) is certainly intriguing and the promise of deriving the Schrödinger equation from the Born rule using SIC-POVMs could really change how QM is understood. But I wonder why it gets so little attention...
  - n4r9 9y ago
    The "radical Bayesian" stance is very difficult to grasp, coming from an ordinary scientific background. At least it was for me - took months, maybe years of sinking-in to figure out what Fuchs was trying to say. I think a lot of people start reading about it, quickly get the (false) impression that it's solipsistic, and just lose interest.
    - akvadrako 9y ago
      To be honest, I still can't understand how it's not solipsistic. If you use QBism from two different perspectives, like Alice and Bob in a Bell test experiment, then you'll still either need many worlds or get incompatible results.
      - n4r9 9y ago
        Perhaps it would help to note that in QBism, reality is not denied. Rather, the objective reality of the quantum state is denied. Not sure what you mean by incompatible results. Certainly QBism is incompatible with Many Worlds! The latter states that the quantum state is real and objective, the former states the opposite. The QBist horsemen have devoted some attention to the question of compatible quantum state assignments. See: https://arxiv.org/abs/quant-ph/0206110 https://arxiv.org/abs/quant-ph/0206110 However, Chris Fuchs does not see any problem in someone assigning a probability-1 belief to a measurement outcome and being proven wrong. See section 2.2 of https://arxiv.org/pdf/1705.03483.pdf https://arxiv.org/pdf/1705.03483.pdf This last point in particular should bring home the point that Fuchs does think there is an ontological theory underlying physics, but that we are fooling ourselves about how much of it is objective/observer-independent.
jessriedel 9y ago
> is there a QM-equivalent model using only real-valued functions? Yes, the Wigner-Weyl transformation produces a completely equivalent way to formulate quantum mechanics without reference to complex numbers. The wavefunction over real space is replaced with a (generically not positive) real-valued pseudo-probability distribution on phase space called the Wigner function. https://en.m.wikipedia.org/wiki/Wigner–Weyl_transform https://en.m.wikipedia.org/wiki/Wigner–Weyl_transform

evanb 9y ago

You may be interested in http://www.scottaaronson.com/democritus/lec9.html http://www.scottaaronson.com/democritus/lec9.html http://math.ucr.edu/home/baez/rch.pdf http://math.ucr.edu/home/baez/rch.pdf

n4r9 9y ago

Complex numbers aren't strictly necessary. You could even expand every complex dimension to a pair or real dimensions and get the same results (roughly speaking). However this is a bit unwieldy, so the question becomes "why do complex numbers provide such a convenient representation of quantum theory"? There is work being done on this sort of thing. If essentially all you care about are the probabilities of certain observations occurring, then you surely can represent any theory with real numbers. The field of "generalised probabilistic/physical theories" attempts to analyse quantum theory without assuming the baggage of Hilbert spaces etc... [0]. Chris Fuchs also has done a lot of work on representing quantum state spaces in terms of outcome distributions over special types of measurements called sic-povms[1,2]. 0. https://arxiv.org/abs/quant-ph/0508211 https://arxiv.org/abs/quant-ph/0508211 1. http://perimeterinstitute.ca/personal/cfuchs/ http://perimeterinstitute.ca/personal/cfuchs/ 2.https://en.m.wikipedia.org/wiki/SIC-POVM https://en.m.wikipedia.org/wiki/SIC-POVM

akvadrako 9y ago
This work (QBism) is certainly intriguing and the promise of deriving the Schrödinger equation from the Born rule using SIC-POVMs could really change how QM is understood. But I wonder why it gets so little attention...
- n4r9 9y ago
  The "radical Bayesian" stance is very difficult to grasp, coming from an ordinary scientific background. At least it was for me - took months, maybe years of sinking-in to figure out what Fuchs was trying to say. I think a lot of people start reading about it, quickly get the (false) impression that it's solipsistic, and just lose interest.
  - akvadrako 9y ago
    To be honest, I still can't understand how it's not solipsistic. If you use QBism from two different perspectives, like Alice and Bob in a Bell test experiment, then you'll still either need many worlds or get incompatible results.
    - n4r9 9y ago
      Perhaps it would help to note that in QBism, reality is not denied. Rather, the objective reality of the quantum state is denied. Not sure what you mean by incompatible results. Certainly QBism is incompatible with Many Worlds! The latter states that the quantum state is real and objective, the former states the opposite. The QBist horsemen have devoted some attention to the question of compatible quantum state assignments. See: https://arxiv.org/abs/quant-ph/0206110 https://arxiv.org/abs/quant-ph/0206110 However, Chris Fuchs does not see any problem in someone assigning a probability-1 belief to a measurement outcome and being proven wrong. See section 2.2 of https://arxiv.org/pdf/1705.03483.pdf https://arxiv.org/pdf/1705.03483.pdf This last point in particular should bring home the point that Fuchs does think there is an ontological theory underlying physics, but that we are fooling ourselves about how much of it is objective/observer-independent.

jessriedel 9y ago

> is there a QM-equivalent model using only real-valued functions? Yes, the Wigner-Weyl transformation produces a completely equivalent way to formulate quantum mechanics without reference to complex numbers. The wavefunction over real space is replaced with a (generically not positive) real-valued pseudo-probability distribution on phase space called the Wigner function. https://en.m.wikipedia.org/wiki/Wigner–Weyl_transform https://en.m.wikipedia.org/wiki/Wigner–Weyl_transform