Some speculative floating of loosely related ideas
(1) A Bayesian interpretation of quantum mechanics
Choice quotations from 'Quantum Mechanics as Quantum Information (and only a little more)', Christopher A. Fuchs :
The theory prescribes that no matter how much we know about a quantum system—even when we have maximal information about it—there will always be a statistical residue. There will always be questions that we can ask of a system for which we cannot predict the outcomes. In quantum theory, maximal information is simply not complete information. But neither can it be completed. (11)
...it turns out to be rather easy to think of quantum collapse as a noncommutative variant of Bayes’ rule. (35)
In this connection, it is interesting to note that the quantum de Finetti theorem and the conclusions just drawn from it work only within the framework of complex vector-space quantum mechanics. For quantum mechanics based on real Hilbert spaces, the connection between exchangeable density operators and unknown quantum states does not hold.(47)
One is left with a feeling—an almost salty feeling—that perhaps this is the whole point of the structure of quantum mechanics. Perhaps the missing ingredient for narrowing the structure ofBayesian probability down to quantum mechanics has been in front of us all along. It finds no better expression than in taking account of the challenges the physical world poses to our coming to agreement.(48)
The tensor-product rule for combining quantum systems can be thought of as secondary to the structure of local observables.(52) (See section 5)
(2) Monoidal categories as arenas for mathematics
Fuchs wants to deflate the mystery of quantum mechanics, including entanglement. Entanglement seems to have something to do with the (noncartesian) tensor product of Hilbert spaces, which for Fuchs is just a consequence of having local observables. Something I once posed to John Baez, who admittedly didn't look totally convinced, is whether one could see through some of the supposed strangeness of entanglement by looking away from the category of Hilbert spaces to another noncartesian monoidal category, Sets and Relations, whose objects are sets, and whose arrows from A to B are relations, i.e., subsets of the product A x B.
So, your sister goes to live in Australia. It is not surprising if someone observing her there knows about you by noting whether she marries (you're an in-law) or has a child (you're an aunt/uncle). What is different is that in the classical world a state of maximal information involves you knowing every property of an entity, in the quantum world this is no longer possible. If observables don't commute, then you can't know their values simultaneously.
(3) How much of the mathematics used in physics is describing our knowledge and ways of observing and intervening, and how much the physical world itself?
Fuchs sees the majority of the apparatus of quantum mechanics as representing how we can gamble wisely on how quantum measurements will turn out. Omnes was careful in 'Converging Realities' to identify mathematics and the laws of physics, rather than the physical world. Wigner in his 'On unitary representations of the inhomogeneous Lorentz group', Ann. Math. (2) 40, 149-204 (1939), gets far by thinking of invariances relative to different observers.
If Baez is right about QM and GR having something in common in that they both use symmetric monoidal categories with duals, do they share a common separation between world and knower?
(4) If for the Bayesian, probability theory is a generalised logic, what happens when you deform it?
Free probability theory forms noncommutative analogues of constructions from classical probability theory. E.g., Wigner's semicircle distribution is the analogue of the normal distribution.
Just as the tropical analogue of the fourier transform is the legendre transform, so the analogue of probability theory is optimisation theory.
Update: Abstracts from a conference studying Fuchs' ideas.
Choice quotations from 'Quantum Mechanics as Quantum Information (and only a little more)', Christopher A. Fuchs :
The theory prescribes that no matter how much we know about a quantum system—even when we have maximal information about it—there will always be a statistical residue. There will always be questions that we can ask of a system for which we cannot predict the outcomes. In quantum theory, maximal information is simply not complete information. But neither can it be completed. (11)
...it turns out to be rather easy to think of quantum collapse as a noncommutative variant of Bayes’ rule. (35)
In this connection, it is interesting to note that the quantum de Finetti theorem and the conclusions just drawn from it work only within the framework of complex vector-space quantum mechanics. For quantum mechanics based on real Hilbert spaces, the connection between exchangeable density operators and unknown quantum states does not hold.(47)
One is left with a feeling—an almost salty feeling—that perhaps this is the whole point of the structure of quantum mechanics. Perhaps the missing ingredient for narrowing the structure ofBayesian probability down to quantum mechanics has been in front of us all along. It finds no better expression than in taking account of the challenges the physical world poses to our coming to agreement.(48)
The tensor-product rule for combining quantum systems can be thought of as secondary to the structure of local observables.(52) (See section 5)
(2) Monoidal categories as arenas for mathematics
Fuchs wants to deflate the mystery of quantum mechanics, including entanglement. Entanglement seems to have something to do with the (noncartesian) tensor product of Hilbert spaces, which for Fuchs is just a consequence of having local observables. Something I once posed to John Baez, who admittedly didn't look totally convinced, is whether one could see through some of the supposed strangeness of entanglement by looking away from the category of Hilbert spaces to another noncartesian monoidal category, Sets and Relations, whose objects are sets, and whose arrows from A to B are relations, i.e., subsets of the product A x B.
So, your sister goes to live in Australia. It is not surprising if someone observing her there knows about you by noting whether she marries (you're an in-law) or has a child (you're an aunt/uncle). What is different is that in the classical world a state of maximal information involves you knowing every property of an entity, in the quantum world this is no longer possible. If observables don't commute, then you can't know their values simultaneously.
(3) How much of the mathematics used in physics is describing our knowledge and ways of observing and intervening, and how much the physical world itself?
Fuchs sees the majority of the apparatus of quantum mechanics as representing how we can gamble wisely on how quantum measurements will turn out. Omnes was careful in 'Converging Realities' to identify mathematics and the laws of physics, rather than the physical world. Wigner in his 'On unitary representations of the inhomogeneous Lorentz group', Ann. Math. (2) 40, 149-204 (1939), gets far by thinking of invariances relative to different observers.
If Baez is right about QM and GR having something in common in that they both use symmetric monoidal categories with duals, do they share a common separation between world and knower?
(4) If for the Bayesian, probability theory is a generalised logic, what happens when you deform it?
Free probability theory forms noncommutative analogues of constructions from classical probability theory. E.g., Wigner's semicircle distribution is the analogue of the normal distribution.
Just as the tropical analogue of the fourier transform is the legendre transform, so the analogue of probability theory is optimisation theory.
Update: Abstracts from a conference studying Fuchs' ideas.
3 Comments:
Regarding quantum mechanics and "monoidal categories as arenas for mathematics", try this:
Universal Algebra and Diagrammatic Reasoning
where I investigate the relation between categories with finite products (classical logic) and more general symmetric monoidal categories (quantum logic) by means of an adjunction between the two.
For example, I show how to get the definition of "Hopf algebra" by a perfectly systematic "quantization" of the definition of group... so systematic that one could simply turn the handle and crank out definitions of "quantum ring", "quantum commutative ring" and so on.
Near the end there's a speculative "philosophical postlude" which you might enjoy.
For Bayesian probability theory as a key to understanding quantum mechanics, you might like these rambling thoughts and references:
Bayesian Probability Theory and Quantum Mechanics
See how Bayesianism solves the puzzle of the free-falling Everettistas...
"Here is a sample conversation between two Everettistas, who have fallen from a plane and are hurtling towards the ground without parachutes:
Mike: What do you think our chances of survival are?
Ron: Don't worry, they're really good. In the vast majority of possible worlds, we didn't even take this plane trip."
This kind of Everettianism is no laughing matter. I have heard philosophers discussing the ethics of a situation where a quantum event will give you a million dollars 50% of the time, and kill you the other 50% of the time. If you're going to persist (rich) in one of these worlds, why not go for it? And if the outcome of a similar event is your death and my wealth or neither, should I be happy to go ahead, safe in the knowledge that if you die and I get rich I know you're living happily on another branch of the multiverse. I just hope no psychopath gets hold of these nutty ideas.
See page 27 of Cave's Bayesian interpretation of quantum mechanics for some acid criticism:
Many-worlds interpretations (Vaidman 1999a, Wallace 2001a). Find wave-function collapse distasteful, so banish it. Make the most naive realistic assumption: declare
the wave function to be objectively real, and then - damn the torpedos!- plow straight
ahead, undaunted by the mind-boggling consequences (well, actually, revel in them a
bit; they make good press and great science fiction). That's the spirit of many-worlds
interpretations. If you want to be perceived as a deep thinker without actually having
to do any thinking, this is your interpretation.
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