On Monday, December 30, 2002, at 11:18 AM, Tim May wrote:

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
operates

in the space of Integers (Z), is this correct?

Any 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.

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.)

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.

--Tim May

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