Re: projective geometry in theoretical physics

per.vognsen_at_gmail.com
Date: 12/10/04 

Date: Fri, 10 Dec 2004 11:02:28 +0000 (UTC)

rst wrote:
> I have read that Dirac used projective geometry in his derivations
...
> But however in all physics books I have seen there are no use of
> projective geometry ..

Most textbooks don't explicitly identify it as such. However, I'm sure
you've run into the idea that states are identified with rays (rather
than vectors) in a complex Hilbert space. That is, a pair of state
vectors u and v in the complex Hilbert space are considered physically
equivalent if and only if there exists a nonzero complex number lambda
such that u = lambda v. But this is just another way of saying that the
true state space is the projectivization of the Hilbert space,
resulting in a complex projective space.

We can do calculations in this complex projective space by doing
calculations in the underlying Hilbert space as long as we always keep
the identification between complex-parallel vectors in mind. What
physicists usually do is work with unit vectors and then identify unit
vectors that differ by a phase change, corresponding to a factor of a
unit complex number (which can be represented as exp(it) for some real
number t, this real number being interpreted as the phase difference).
Anyway, I'm sure you see how this restriction to unit vectors is
natural, given the usual probabilistic interpretation of the vectors as
probability amplitudes.

I am not a physicist and so if any of this is a misrepresentation, I'd
love to be corrected by someone more knowledgable.

Per

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