Re: extra dimensions

Posted by phobos on December 11, 19100 at 18:04:42:

In Reply to: Re: extra dimensions posted by ronron on December 11, 19100 at 14:05:41:

Some notes:

State vectors are not self-adjoint operators on L^2,
they are elements of L^2. Observables are the
self-adjoint operators on L^2.

I've treated the 11, 26, vs. infinity question below.
The Hilbert space is infinite dimensional, but the
domain of the functions in L^2 is an 11, 26, etc. dim
space, which is spacetime.

While Dirac delta functions are not always necessary in
standard QM, which deals mainly with bound states and
partial wave scattering states, the theory of
distributions is a perfectly viable mathematical
machinery. In a noncompact space, some bases of vectors
will only be normalizable in the sense of distributions,
but this is no less mathematically rigorous provided
the correct caveats are heeded. In fact, QFT is
based on the mathematical notion of operator-valued
fields on spacetime, which incorporates distributions.
(See Streater and Wightman for details.)

phobos

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