Talk:Projective Hilbert space

CP(1)

Well, isn't CP^1 a one-dimensional projective space...?

Yes, one-complex-dimensional, two real-dimensional, i.e. it is a sphere. 67.198.37.16 (talk) 19:00, 6 May 2019 (UTC)

lambda

Because $\lambda$ is complex, it is misleading to say that $\lambda$ has to have absolute value of 1. Rather, the modulus has to be one, which implies $\lambda =e^{i\theta },\theta$ real. Thus we can alter the phase of the element of the Hilbert space. —Preceding unsigned comment added by Nardis miles (talk • contribs) 07:42, 29 October 2007 (UTC)

but this is equivalent, is it not? --- User:ibotty (not logged in) —Preceding unsigned comment added by 91.67.59.57 (talk) 10:57, 17 December 2008 (UTC)

This article needs attention

I hate to write such a negative talk page section, but I think we need to make a decision about this article.

1. This article has existed since 2004 and cites no sources.
2. It introduces jargon (C*-algebras, convex linear combinations, Segre mappings, density matrices) that serves no purpose.
3. Is $\lambda$ meant to represent a phase factor of modulus 1, or any arbitrary complex scalar? The symbol seems to be doing double-duty throughout.
4. What exactly do categorical products have to do with projective Hilbert space? Are we referring to the category of projective Hilbert spaces with morphisms continuous functions between them, or some other morphisms? Is the article trying to say that the categorical product of the projectivizations of two Hilbert spaces is isomorphic to the projectivization of the tensor product of the Hilbert spaces? This is highly technical and of questionable relevance. If it is true, it needs a source.
5. This article's definition of a ray is different from the definition at Wigner's theorem, which has two citations. (Specifically, the Wigner's theorem article says that two vectors are in the same ray when one is a multiple of the other by a complex number of unit norm.)
6. What makes a phase a "global phase"? While this page links to the article phase factor, the string "global" does not occur anywhere there.
7. What is meant by the articles' statement that a finite-dimensional projective Hilbert space is a homegeneous space for the unitary or orthogonal group? Is this a property or an alternative definition?

The most alarming issue with this article is the absence of citations. As far as I can tell from the talk page, this article hasn't included a reference since it was created 13 years ago. Why should a reader believe even that two vectors differing by a complex scalar multiple represent the same quantum state?

I don't have the physics background to fix this article on my own, but I suggest that a deletion or complete rewrite is justified. Thoughts or ideas?

Norbornene (talk) 15:36, 2 September 2017 (UTC)

It's unfortunate that the article is neglected. It could be made much better. To answer some of your questions:

2. C-star mappings and Segre mappings is kind-of the whole point of this article: C-star is the backbone operator algebra of quantum mechanics, and Segre mappings are one of the stock-in-trade of algebraic geometry, where projective spaces are studied. It is presumed that the reader is interested in quantum, and/or in algebraic geometry.

3. lambda is an arbitrary complex number, which can be restricted to a phase when the length is normalized to one. That's what the article says. This is a common decomposition in complex projetive spaces, which sometimes leads to Hopf fibrations. Again, this is part of the standard tradecraft.

4. Categorial products are ... universal. That's why categorical-anything is good-to-know. The Segre mapping is categorical, as many of the main-stream, heavy-hitter textbooks on algebraic geometry will happily point out.

6. The term "global phase" is being used to contrast it to the Berry phase, which is a local phase (it's a holonomy), and causes the Bohm-Aharonov effect, for example.

7. Homogeneous spaces are quotients of Lie groups. The group of rotations is a Lie group. So, yes, it's a property, and its also an alternative defintion. They're all one-and-the-same thing.

I can answer these questions because I "know these things". I tried to answer them in a way that indicates that this is "common knowledge" to others, and specifically, that the reason the article is saying what it is saying is because that really is the topic of the article. You can define a projective hilbert space in about one or two sentences (its a projective space, duhh), but such a short definition does not explain why its an interesting topic to study (it geometrizes both quantum, and some of these spaces even show up in general relativity, e.g. the n=2 fubini-study gravitational instanton.) Yes, this article is stubby; much much more can be said. It does require a somewhat high level of understanding though -- it requires being at least an upperclassman in mathematics. 67.198.37.16 (talk) 18:55, 6 May 2019 (UTC)

As for 5., there is no conflict because “symmetry transformations” preserve the norm. In fact, all the Wigner's theorem stuff can be made on the unit sphere because of it. All Hilbert space (except for 0) over ℂ^× is the same as its unit sphere over U(1). Incnis Mrsi (talk) 20:03, 6 May 2019 (UTC)