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.(adsbygoogle = window.adsbygoogle || []).push({});

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

Quantum Field Theory on Noncommutative Spaces, by

Richard J. Szabo

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. 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 [tex]\dot{x}^i, \dot{p}^j[/tex] which obey the Heisenberg commutation relations [tex][\dot{x}^j , \dot{p}^i] = i \hbar \delta^{ij}[/tex] . 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”, refering 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”, refering 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.

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?

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# Nocommutative space and QM

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