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In the context of justifying the failure of modeling quantum observables in the 'more natural' way as real functions on the phase space (i.e. similar to the mathematical image modeling classical observables) and somehow justifying the use of the framework of Hilbert space and operators . In the following talk

https://www.youtube.com/watch?v=ODAngTW8deg (see 1:00 to 4:00)

Alain Connes argued that phase space and real functions on it cannot support the co-existence of continuous real observables (i.e. those whose image has the continuum cardinality) and discrete ones (i.e. whose image is finite or countable) by the following argument : Denote by $X$ a phase space and suppose the existence of a discrete observable $D$ and a continuous $C$ one on $X$. We then have necessarily that $X$ has the continuum cardinality. But then we will get at least a real value $\lambda$ in $\mathbb{R}$ which has an (even 'continuous') infinite number of pre-images by $D$. (i.e. a continuous infinite number of states which corresponds to this value..) so the values $\lambda$ will have an 'infinite multiplicity' and he said that this is impossible ! Can anyone help what is the impossible in this situation ? I mean do we have something in the classical paradigm which says that an observable value $\lambda$ (i.e. A measured value of an observable) cannot be hitten an infinite number of times ? (i.e. there cannot be an infinite number of states in the phase space having the value $\lambda$ as a measure of a fixed observable $A$). Is it a physical problem or a mathematical one ?

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  • $\begingroup$ The problem is strictly mathematical, but I'm not qualified to answer it, as I have had trouble understanding the speaker for over four decades. Indeed, naively, every point in phase space is a state, but in the phase-space formulation of QM this is not so, as in certain cases a real function observable must further satisfy a *-genvalue equation w.r.t. a single (Wigner) function describing the state, where these values are actually discrete. This comports (subtly!) with his identification of observables as operators in Hilbert space... $\endgroup$ Commented Dec 4, 2022 at 17:30
  • $\begingroup$ I see you have asked the very same question in MSE. The facile answer here is that observables are continuous real functions in phase space, but the conceit/ implicit assumption that states are points in phase-space is thoroughly unwarranted. $\endgroup$ Commented Dec 4, 2022 at 21:01
  • $\begingroup$ Sir @CosmasZachos, thank you very much for your help. $\endgroup$
    – user536450
    Commented Dec 4, 2022 at 21:15
  • $\begingroup$ my question was in the sens that 'is that impossible to find a mathematical image of QM I mean as a phase space and real function on it modeling the physical observables so that we go to the 'firstly appearing weird' formulation as linear spaces and operators and their spectra ? I mean we can consider real function (not necessarly contiunuous) as our QM observables. I m looking to understand why this generates I you said a Mathematical problem. $\endgroup$
    – user536450
    Commented Dec 4, 2022 at 21:19
  • $\begingroup$ It is impossible to map continuous phase-space points to a discrete spectrum, as AC says, but there is a monstrous fallacy in assuming such points amount to QM states, "as in classical mechanics". But nobody has seriously argued this is warranted, before AC. I can imagine mathematicians are stunned by a freakish straw man, but, really now... $\endgroup$ Commented Dec 4, 2022 at 21:29

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