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From: Tim May <tcmay.domain.name.hidden>

Date: Mon, 30 Dec 2002 11:43:36 -0800

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
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*> 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

Received on Mon Dec 30 2002 - 14:47:49 PST

Date: Mon, 30 Dec 2002 11:43:36 -0800

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.)

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

Received on Mon Dec 30 2002 - 14:47:49 PST

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