On Monday, December 30, 2002, at 11:18 AM, Tim May wrote:
Lest there be no confusion, I meant that all actual systems can be computed with finite-dimensional vector spaces which have inner products. Or in Von Neumann's more precise language, "complete complex inner product spaces." (Since all Hilbert spaces with an infinite number of dimensions are isomorphic, this gives rise to just saying "Hilbert space" in the singular.)On Monday, December 30, 2002, at 10:44 AM, Stephen Paul King wrote:QM comp seems to operate in the space of the Reals (R) and TM operatesAny finite system, which of course all systems are, can have all of its quantum mechanics calculations done with finite-dimensional vector spaces. The "full-blown machinery" of an infinite-dimensional Hilbert space is nice to have, in the same way that Fourier analysis is more elegantly done with all possible frequencies even though no actual system (including the universe!) needs all frequencies.
in the space of Integers (Z), is this correct?
The point is that the arbitrary-dimension elegance of a full-blown Hilbert space is nice to have, especially for theorem-proving, but not essential.
More speculatively, postulating that a quantum state in the real world (in a quantum computer, or atom cage, etc.) is "actually" a vector with an infinite degree of positional accuracy, is akin to saying that it computes with the reals, which touches on the Blum-Shub-Smale issue I talked about earlier this morning.
As Hal says, the world is not actually Newtonian. And neither is it actually quantum-mechanical in the ideal, limiting, infinite-dimensional case.