This is an interesting question.

It is observed in the following paper that using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations:

Deformations of the Canonical Commutation Relations and Metric Structures

Article Deformations of the Canonical Commutation Relations and Metr...

This is an interesting question.

It is observed in the following paper that using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations:

Deformations of the Canonical Commutation Relations and Metric Structures

Article Deformations of the Canonical Commutation Relations and Metr...

Dear James,

Thanks for your message. This reference you suggested sounds really interesting. Thanks a lot for sending it to me. I was totally unaware of it.

Dear André,

The reference that the Prof. Peters has given you is interesting and there are a lot of different results in noncommutative algebras. But they are quite abstract and deep in the mathematical formalism.

Let me try to complement this with more physical and hystorically more basic background on this subject as can be see the symplectic manifolds. There is a classical book for this subject where you can find many physical examples to follow and see what is the meaning of the non commutativity, for instance through the Lie derivative in a vectorial flow, etc...

V.Arnold, Mathematical Methods of Classical Mechanics, 2nd Edition. Springer, 1989.

Dear Prof. Baldomir,

Thanks a lot for your message. I happen to have Arnold's book. I have been thinking about this question for some time. I looked into Lie derivatives and Lie dragging along vector fields, but I have not seem much of a direct connection with my question. I thought of symplectic manifolds in connection with quantum mechanics in phase space, that's when I looked into Arnold's book. Do you know any reference where my question is more or less addressed ?

Best Regards

Dear André,

Chapters 8 and 9 treat your question. There you have the definition of the Poisson bracket which is the macroscopic translation of the quantum commutator. In the case that you have this commutator different than zero you obtain a kind of uncertainty behaviour for the operators. There is a fundamental result related with your question and with quantum mechanics:

Every one-parameter group of canonical diffeomorphisms (i.e. preserving the exterior product of momentum by position) of a symplectic manifold preserves the symplectic structure if and only if it is a locally Hamiltonian phase flow.

Look at this book which is very clear but if you want more expecialised texts you can go to

https://people.math.ethz.ch/~acannas/Papers/lsg.pdf

Dear Prof. Baldomir,

Thanks a lot for your answer. I will look into the references.

Best Regards

Quantum Mechanics can be reformulated in a standalone and independent way in phase space --- see the excellent book by Zachos, Fairlie and Curtright "Quantum Mechanics in Phase Space: An Overview with Selected Papers" (2005). This is equivalent to deformation quantisation (see Bayen et al. papers in the above) and also to the Hilbert phase space formalism (see my paper Article Extended Hilbert Phase Space and Dissipative Quantum Systems

Now, although Heisenberg's Uncertainty Principle continues to hold for the pure state WFs, it does not always hold for the impure WF (despite the conjecture by Mori, Oppenheim and Ross to the contrary) and, in fact, the impurity of a state is not enough to break the uncertainty relations. What seems to be really essential here is that the Wigner function (WF) should dip deep enough into the negative values, so deep that even x- or p-projections no longer lead to the marginal probability densities, e.g. the spacelike shadow \int W(x,p) dp is NOT = |\psi(x)|^2 and likewise for p. At present such "exotic" solutions are encountered only during computer simulations, as far as I know.

To answer your question: a very rough geometric idea of the uncertainty relations is this: it is the area of a small rectangle in the phase space where density matrix (WF is a representation of the density matrix, as you probably know) behaves too wildly, but is tamed when restricted to the subset isomorphic to pure states only. I emphasised "very rough" because in the language of the 4-operator formalism (x,p,\theta,\lambda) you begin to appreciate that the quantum 'x' and 'p' are not at all the same as the quantum entities which in the classical limit correspond to the classical 'x' and 'p' and therefore comparing "area of rectangles" from one space to those of another space is not a very legitimate thing to do, but is good enough to give a rough idea...

UPDATE: Oh, I just saw from your project log that you are in fact a co-author with Dr Cabrera --- so you know ALL about Wigner functions and Hilbert phase space and those beautiful areas where the WF goes "BLUE" (as on the pictures you produced)

Forgive me for not checking whose question I am answering first! :)

Dear Tigran Aivazian, thanks for your message. Renan and I worked a lot together on many different subjects. I am quite familiar with the Wigner function. But with x and p being operators in Hilbert space, the geometrical meaning of their commutator becomes murky. The non conservation of positivity leads to non conservation of trajectories due to phase interferences. On the other hand, geometrically when the commutator is zero it means that p no longer lies on the tangential space, so at that point the trajectory ends abruptly. I was trying to have a geometrical picture for the operators that some intuition on these issues

I am currently reading the book by Maurice de Gosson "The Principles of Newtonian and Quantum Mechanics", 2001 and he describes there the result obtained in 1985 by M.Gromov called "no-squeezing theorem", namely that if you take a ball with the radius of R in the phase space and act on it with a Hamiltonian flow (classical of course) then no matter how it is deformed, the projection upon any plane of _conjugated_ coordinates (e.g. (x, px) but not, say, (x, py)) is such that its area cannot become less than its initial value of pi*R^2. I think this is relevant to your question in the sense that it shows both the geometrical interpretation of the principle of uncertainty as well as purely _classical_ equivalent thereof. Anyway, I recommend this book to you and I am also going to read other books by de Gosson, e.g. the most recent book "Born-Jordan Quantization" (2016) seems both interesting and very relevant to my research.

Dear Tigran,

Thanks for the suggestion. I will take a look at it.

Best

Another way to look at this is from a purely classical point of view. Namely, due to the fact that the class of distribution functions corresponding to the state of maximum entropy (with a fixed energy and unit normalisation) constitutes a one-parametric rather than two-parametric family (the parameter being the "inverse temperature" beta in exp(-beta H(x,p))) implies that the uncertainties of coordinate and momentum (i.e. their quadratic deviations from the mean equilibrium value) cannot be arbitrary: fixing one of them automatically places a constraint on the other. See formulae (50) and (51) and the text paragraph following them on my website http://quantuminfodynamics.com/ in the section Classical Infodynamics/Harmonic Oscillator. Actually, for the anharmonic case it becomes even more interesting (the classical energy and the sigma_x/sigma_p deviations are expressed with Bessel functions), but even the trivial case of harmonic oscillator, imho, illustrates the point I am making: i.e. the purely classical fundamental origin of the "uncertainty principle".