Nocommutative space and QM

[selfAdjoint]

I am reading up on the application of noncommutative coordinates to quantum mechanics, and I found this paragraph which I think many here will find interesting.


From http://arxiv.org/PS_cache/hep-th/pdf/0109/0109162.pdf

Quantum Field Theory on Noncommutative Spaces, by
Richard J. Szabo

The idea behind spacetime noncommutativity is very much inspired by quantum mechanics.
A quantum phase space is defined by replacing canonical position and momentum variables xi, pj with Hermitian operators $$\dot{x}^i, \dot{p}^j$$ which obey the Heisenberg commutation relations $$[\dot{x}^j , \dot{p}^i] = i \hbar \delta^{ij}$$ . The phase space becomes smeared out and the notion of a point is replaced with that of a Planck cell. In the classical limit ¯h → 0, one recovers an ordinary space. It was von Neumann who first attempted to rigorously describe such a quantum “space” and he dubbed this study “pointless geometry”, referring to the fact that the notion of a point in a quantum phase space is meaningless because of the Heisenberg
uncertainty principle of quantum mechanics. This led to the theory of von Neumann algebras and was essentially the birth of “noncommutative geometry”, referring to the study of topological spaces whose commutative C*-algebras of functions are replaced by noncommutative algebras [2]. In this setting, the study of the properties of “spaces” is
done in purely algebraic terms (abandoning the notion of a “point”) and thereby allows for rich generalizations.

Of course a phase space is not spacetime: by definition it's the space spanned by the canonical variables in the Hamiltonian: the Canonical Coordinates and the Canonical Momenta. Nevertheless the coordinates are convertible to spacetime coordinates and the momenta to the observed kind of momenta. So his point about spacetime being non-commutative at short distances is well taken.

Now this raises a question in my mind. The difference between the quantum world and the macroscopic one is not always one of scale, but rather of coherence. Quantum effects involving noncommutative operators over distances that can be seen with the naked eye have been demonstrated. So does spacetime noncommutativity extend to those visible cases too? Could it be experimentally demostrated?

 

[eljose79]

As a scientist studying noncommutative geometry and its applications to quantum mechanics, I find this paragraph very interesting as well. The concept of noncommutative coordinates and their relation to quantum mechanics has been a topic of much research and discussion in recent years.

The idea of replacing canonical position and momentum variables with noncommutative operators is a key aspect of noncommutative geometry. This allows for the concept of a point to be replaced with a Planck cell, which is a fundamental unit of spacetime. This is in line with the Heisenberg uncertainty principle, which states that the more precisely we know the position of a particle, the less precisely we can know its momentum and vice versa.

It is important to note that a phase space is not the same as spacetime, but rather a mathematical construct used in quantum mechanics. However, the conversion of these noncommutative coordinates to spacetime coordinates and momenta is crucial in understanding the implications of noncommutative geometry in the quantum world.

The question raised about the potential extension of spacetime noncommutativity to visible cases is an interesting one. While quantum effects have been demonstrated at the macroscopic level, there is still much research to be done in this area. It is possible that noncommutative coordinates could play a role in these visible cases, but further experimentation and research is needed to confirm this. The study of noncommutative geometry and its applications to quantum mechanics is a constantly evolving field, and I look forward to seeing how it develops in the future.

 

[denian]

I find this topic fascinating and full of potential for further research. It is true that the concept of noncommutative spaces is inspired by quantum mechanics and has led to the development of noncommutative geometry. This approach allows for a deeper understanding of the nature of space and its properties.

The idea of spacetime being noncommutative at a fundamental level raises interesting questions about the nature of our universe and the role of quantum mechanics in shaping it. It also highlights the need for further exploration and experimentation in this area to fully understand the implications of noncommutative spaces in quantum mechanics.

Regarding the question about the extension of spacetime noncommutativity to the macroscopic world, it is worth noting that quantum effects have been observed at larger scales, as mentioned in the paragraph. However, the coherence of these effects may still be limited, and it is an open question whether noncommutative spacetime plays a role in these cases. This is an exciting area for future research and experimentation.

Overall, the application of noncommutative coordinates to quantum mechanics is a fascinating topic that has the potential to greatly enhance our understanding of the universe. Further studies in this area will undoubtedly lead to new insights and advancements in our knowledge of physics.