Talk:Projective space/Archive 1

Archive 1

reminder to self

(or whoever else wants to edit this) I want to put something about notation. Introduce the notation P(V) (which is used in the text, but not defined. Then the notation KP^n. Also mention that P^n without qualification means complex projective. Though they have their own article, homogeneous coordinates need more explanation in this one. When you add that, add the useless fact that homogeneous coordinates are not actually coordinates, they are instead elements of the dual space V*. I'm getting on a plane in 20 minutes, but I should do this when I get home. See also the planetmath article. -Lethe | Talk 03:44, 27 July 2005 (UTC)

Remark

It's necessary that $T:V\to W$ let be one-to-one. —Preceding unsigned comment added by Molinagaray (talk • contribs)

What does "highly symmetric" mean? Please be more exact! 134.169.128.67 09:56, 22 September 2006 (UTC)

2 Questions

First, why say "vector space over a division ring" when this is improper, and further, the vector space page inconsistently (but correctly) builds on a field. More accurately, one should say module over a division ring, and of course, it would be more than acceptable to mention that this is as a vector space missing merely scalar commutativity. Second, why is that previous statement qualified by "in particular over a field"? Over a field, we have affine space, which is not isomorphic to the projective space, as even stated throughout the article (by mentioning that the projective plane is an affine plane unioned with a line at infinity.) 12.147.134.239 04:19, 4 May 2007 (UTC)

I also found the division ring showing up in the first place a bit distracting. I moved this to the main text and also replaced vector space over a division ring by module over a d.r. Jakob.scholbach 02:05, 5 May 2007 (UTC)

Merging with axiomatic projective space

There is already another article on projective spaces, poetntially they could be merged. For now i've just added the alternative approach section with a reference to the other definition.--Kmhkmh 05:14, 4 March 2007 (UTC)

That's the projective geometry article. It's definitely time to bring them together. — Preceding unsigned comment added by 64.132.207.253 (talk) 14:30, 16 June 2007 (UTC)

QM

Heh. I just learned something today, which I think is utterly fascinating. Turns out the quantum-mechanical Schroedinger equation is nothing more (and nothing less!) than a purely classical Hamiltonian flow on CP^n. Heh! Here, a quantum mechanical state is regarded as a point in CP^n, i.e. a point on the Bloch sphere. Hamiltonian flows can be defined only if there is a symplectic form on the manifold... but of course, CP^n has the Fubini-Study metric, and so has the requisite symplectic form! Golly! This is does have this forehead-slapping, "but of course, what else could it be" element to it, but .. well, I'm tickled. I'll have to work some sample problems in this new-found language. linas 04:21, 27 June 2006 (UTC)

Is that right? The Schrödinger equation controls the time dependence of the phase of a state vector, but points in projective space have had their phases modded out, so that time dependence is lost. In other words, a stationary state still has oscillating phase in Hilbert space, but is really static in projective space. Therefore it seems to me that the Schrödinger equation is something more than evolution in in projective space. A lot of kinematics happens in the phase! Am I missing something? -lethe ^{talk +} 05:41, 27 June 2006 (UTC)

Yes, you're missing that I stayed up past my bed time, and didn't know to keep my mouth shut. Seriously, I saw several interesting papers on the topic. Your argument about the phase is voided by the idea that the phase cannot be measured directly; it only shows up in interference effects (e.g. the Bohm-Aharonov effect); these are claimed to be modelled as projective lines (a projective line being a superposition of a pair of states). I have no clue what the AB effect would look like on projective space; will have to study this a good bit more. linas 05:42, 28 June 2006 (UTC)

Right, phase is not measurable, but you can't just do away with it. My gut tells me that classical mechanics on the sphere is not equivalent to quantum mechanics. I don't think the fact that phase is not directly measurable gives you the right to get rid of Hilbert space. Point me at some of these papers? -lethe ^{talk +} 05:54, 28 June 2006 (UTC)

By writing the wave function $\psi$ as a sum $(q^{n}+ip^{n})|n>$ as $n=0,1,2,...$, you'll also see that the Schroedinger equation reduces to Hamilton's equations with respect to the Hamiltonian $H(q,p)=<\psi |H|\psi >$. The Hilbert space inner product then gives you the symplectic form (its imaginary part) out of which the Poisson brackets come, and a configuration space metric (the real part), which is a Riemannian metric possessing a constant curvature that is directly related to the value of Planck's constant. It's this latter feature that may ultimately underlie the relation between the state space and projective space. — Preceding unsigned comment added by 64.132.207.253 (talk • contribs) 14:39, 16 June 2007 (UTC)