On 10/22/2014 5:35 PM, Russell Standish wrote:
On Wed, Oct 22, 2014 at 05:23:38PM +0200, Bruno Marchal wrote:
On 22 Oct 2014, at 11:37, Richard Ruquist wrote:
Brent,
That is certainly true for Schrodinger's equations,
but is it also true for matrix theory?
Re: real and complex numbers.
Why would it be different for the matrix. In non relativistic QM,
the position observable in a continuous matrix of complex (and thus
couple of real numbers), same for momentum.
In a quantized space-time, that might be different. But we don't
find good quantization for space-time, I think. Loop gravity seems
to be refuted on this point.
Note also that if Brent is right that QM assume real numbers, it
does not imply that nature (whatever that is) needs them. All what
we can measure are rational numbers. Is there a circle in nature. I
think plausible that circle exists only in the mind of machine in
arithmetic, or they exists as infinite collection of natural numbers
with some relations, etc. Well, it has to like that if we assume
computationalism, and don't eliminate consciousness to save a
primary matter that nobody has seen or even can defined ...
To reiterate on Bruno's point, observables corresponding to x or d/dx
do not exist in reality. Every measurement made is done to some finite
precision - the number of digits of a numerical readout, or the needle of
an analogue meter lying between one graduation and the next.
Consequently, the actual observables have eigenvalues and eigenvectors
drawn from the rational complex numbers. Reals do not exist except as
an approximation that is convenient for doing calculations. And even then,
countable models of the reals' axioms exist, by virtue of the
Löwenheim–Skolem theorem. These countable models exist in Bruno's
ontology, and suffice for any practical purpose QM is put to.
But by the same kind of positivist attitude there's no reason to think that every integer
has a successor. It's just a convenient assumption for doing proofs and calculations.
Brent
Cheers
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