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hendrikboom at 08:34pm on 25/08/2008
hendrikboom at 08:34pm on 25/08/2008Let me quote Hermann Weyl:
The continuum of real numbers has retained its ancient prerogative in physics for the expression of physical measurements, but it can justly be maintained that the essence of the new Heisenberg-Schrödinger-Dirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense, the elements of which are the physical quantities themselves. (1928)
Now as far as I can tell, this is the idea behind the noncommutative geometries the mathematicians among quantum-mechanical researchers have been excited about recently.
What puzzles me is that it seems to have taken seven or eight decades to get to the point that people are taking this seriously. Could it be that operators like -i hbar d/dx don't fit physicists' physical intuition, but real numbers too?
I've had an idea like that myself, too, about ten years ago, but have not the technical facility and background knowledge to develop it -- that the operator representing momentum that is the real physical quantity, and that the real-numbers eigenvalues that we "measure" are just momentary glimpses into a deeper reality.
Isn't the idea that real numbers are the results of measurements also an experimental falsehood? All the experimental results I've ever seen have been rational numbers. Why assume that the actual physical values we can only measure approximately are real? Why not assume they're members of a noncommutative algebra?
The continuum of real numbers has retained its ancient prerogative in physics for the expression of physical measurements, but it can justly be maintained that the essence of the new Heisenberg-Schrödinger-Dirac quantum mechanics is to be found in the fact that there is associated with each physical system a set of quantities, constituting a non-commutative algebra in the technical mathematical sense, the elements of which are the physical quantities themselves. (1928)
Now as far as I can tell, this is the idea behind the noncommutative geometries the mathematicians among quantum-mechanical researchers have been excited about recently.
What puzzles me is that it seems to have taken seven or eight decades to get to the point that people are taking this seriously. Could it be that operators like -i hbar d/dx don't fit physicists' physical intuition, but real numbers too?
I've had an idea like that myself, too, about ten years ago, but have not the technical facility and background knowledge to develop it -- that the operator representing momentum that is the real physical quantity, and that the real-numbers eigenvalues that we "measure" are just momentary glimpses into a deeper reality.
Isn't the idea that real numbers are the results of measurements also an experimental falsehood? All the experimental results I've ever seen have been rational numbers. Why assume that the actual physical values we can only measure approximately are real? Why not assume they're members of a noncommutative algebra?
(no subject)