One of the answers in this MO post https://mathoverflow.net/questions/7917/non-commutative-algebraic-geometry draws a connection between localization in commutative algebra (or the failure thereof), and localization in the quantum mechanical sense (or the failure thereof, i.e. in the form of an uncertainty principle), in particular (emphasis mine):
At a higher level, suppose that $M$ and $N$ are finitely generated modules over a commutative ring $A$ such that $M\otimes_A N = 0$, then $Tor_i^A(M,N) = 0$ for all $i$. If $A$ is non-commutative, this is no longer true in general. This reflects the fact that $M$ and $N$ no longer have well-defined supports on some concrete spectrum of $A$. This is why localization is not possible (at least in any naive sense) in general in the non-commutative setting. It is the same phenomenon as the uncertainty principle in quantum mechanics, and manifests itself in the same way: objects cannot be localized at points in the non-commutative setting.
I will not describe the commutative algebra formalism of localization for this audience of mostly mathematicians, but I will paste some descriptions of the uncertainty principle and its relation to noncommutativity from various Wikipedia articles: https://en.wikipedia.org/wiki/Commutative_property#Non-commuting_operators_in_quantum_mechanics
According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely.
and also https://en.wikipedia.org/wiki/Uncertainty_principle#Introduction
In matrix mechanics, the mathematical formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue). For example, if a measurement of an observable $A$ is performed, then the system is in a particular eigenstate $\Psi$ of that observable. However, the particular eigenstate of the observable $A$ need not be an eigenstate of another observable $B$: If so, then it does not have a unique associated measurement for it, as the system is not in an eigenstate of that observable.
The aforementioned MO answer did mention that the analogous statement (of I guess this "no simultaneuous eigenstate" business for operators) was the non-vanishing of various "tensor products" $\text{Tor}_i^A(M,N)$, but it is unclear to me that these are connected at all.
My question: can anyone provide some more explanations, connecting the 2 notions of localization? Or even just an elaboration of the one sentence
"This reflects the fact
that M and N no longer have well-defined supports on some
concrete spectrum of A."
from the above MO answer.
P.S. then again there's also the Fourier analytic uncertainty principles which are obviously intuitively about localization (or the failure thereof), compared to the more opaque language of "no simultaneous eigenstates" that the quantum mechanical operator algebra, which requires the physical interpretation "An eigenstate of an observable represents the state of the wavefunction for a certain measurement value (the eigenvalue)" to be intuitive. Localization in commutative algebra/algebraic geometry also has a sort-of intuitive picture of looking at ALL our "functions" in $R$ "near" a point $p$, and being able to "divide" those that aren't $0$ "at" $p$. But still I don't see how it relates to Fourier analytic uncertainty principles, or operator algebraic uncertainty principles.
Perhaps also see https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8604559/, which mentions that
Noncommutative geometry, on our interpretation, does not have the resources to make meaningful claims about localisability below a certain magnitude.
In theories of noncommutative space, we again assume—but this time for reductio—that there is an ontic state corresponding to an arbitrarily precisely localised particle. We construct the analogue of an epistemic state: a density operator. We then attempt to localise this epistemic state to an arbitrarily small area and discover that this leads to ascriptions of negative probabilities. Since these measures are not elements of the state space, this signals a pathology. The only way to avoid this pathology, we argue, is to drop the assumption that there is an ontic state corresponding to an arbitrarily precisely located particle. Thus, even in principle, it is not possible, in a noncommutative space, to localise a particle below a certain area. Operationally, then, such areas—and a fortiori points—are undefinable.
We conclude that measurements attempting to localize particles to a linear scale smaller than $\sqrt \theta$ are unphysical; of course, this result is a reflection of the fact that there are no ontic states localizing particles below this scale. Then, from our tempered operationalism, we conclude that space in fact has no regions smaller than this scale—and in particular, is ‘pointless’.
To prove the last result is just an analytic calculation: taking some sort-of Fourier transform of some Gaussian (representing some perhaps localized state), gives rise to a matrix, for which we can analyze the eigenvalues. That's all well and good, but I doubt there's an abstract algebra analogue of this process. So although the MO answer says "It is the same phenomenon as the uncertainty principle in quantum mechanics", this Fourier analytic picture probably does not lead to an analogy with noncommutative rings.