Talk:Projective space

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)

The lead section

The lead section of this article is problematic. I would like to be able to make wikiliks from more applied articles to this one but find that the general reader (not already familiar with the concept) will have a difficult time to make sense of it. There is too much formality, generality, and connections to concepts at a high level of abstraction. Here is a proposed alternative:

In mathematics a projective space is a set of elements constructed from a vector space in one of the following equivalent ways

• An distinct element of the projective space consists of all non-zero vectors which are equal up to a multiplication by a non-zero scalar.

• The projective space is the set of all lines through the origin of the vector space.

• The projective space is the quotient space of the vector space and the group of multiplications by the set of non-zero scalars.

Projective spaces can be studied as a separate field in mathematics, but it also used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation of elements in projective spaces and as a result, various relations between these objects can be described in simpler way than is possible using their basic representation.

Or something along these lines. The rest of the correct lead is correct, interesting and all that, but this information is not relevant for someone who is trying to understand what it is and how it can be of any use. It is not necessary to mention all aspects of the topic in the lead. This information can probably be worked into new or existing sections further down in the article if it is not already there. --KYN 22:27, 20 August 2007 (UTC)

Feel free to improve the article! However, the definitions you propose are (partly) there ("The basic construction, given a vector space V over a field K, is to form the set of equivalence classes of non-zero vectors in V under the relation of scalar proportionality: we consider v to be proportional to w if v = cw with c in K non-zero. This idea goes back to mathematical descriptions of perspective. If K is the real or complex numbers, and V has dimension n, then the projective space ℙ(V)—which we can talk about as the space of lines through the zero element 0 of V—carries a natural structure of a compact smooth manifold of real or complex dimension n − 1."). Also I guess it is not necessary to bring all the possible equivalent definitions to the lead, the most intuitive one should be enough. Finally, I think it is problematic to just cut down the lead such that no statements are made which may not interest every reader. But, you are right, the very first sentence could be far more comprehensible, so don't hesitate. Jakob.scholbach 00:11, 22 August 2007 (UTC)

disjoint union?

I'm no expert, but it seems to me misleading to say that the projective space is the disjoint union of R^k for k = 0 upto n. It is certainly not true in the topological sense (there's no discrete point, for example). Please elucidate (or fix?) this

Thanks, Amitushtush (talk) 06:55, 5 March 2008 (UTC)

Projective geometry

The article on Projective geometry broadly parallels this one. It does not seem to be getting much attention at the moment. Do we really want two separate articles (after all, there are no "parallels" in projective geometry _groan ), or should we merge them into a single article?

If we do want to keep both, I think we should move the axiomatic and analytical type content to the geometry and the visualisations type content to the space. -- Steelpillow (talk) 21:38, 9 May 2008 (UTC)

I asked generally about "geometry" vs. "space" articles on the Geometry talk page. Basically, "space" covers the 3d situation and "geometry" covers the general theory. As a result, I intend to move this page's section on Visualising the projective plane and merge it into the article on the projective plane. See also below on axiomatic projective space. -- Cheers, Steelpillow 13:46, 24 May 2008 (UTC)

Axiomatic projective space

The Projective geometry article also has a section on the Axioms which parallels the article on Axiomatic projective space. Since the axioms create the geometry theory and not the actual space, I would suggest merging these into a single article titled Axioms of projective geometry. -- Steelpillow (talk) 21:38, 9 May 2008 (UTC)

I think merging the two articles would decrease both articles' potential. As for merging the two sections: an article Axioms of projective geometry would be just a stub. You might want to check out Axiomatic_projective_space instead. But I agree, these three articles could use some improvement. Jakob.scholbach (talk) 17:43, 16 May 2008 (UTC)

Oops. It went quiet for so long I already did that - I created Axioms of projective geometry and merged it all there. Maybe that should be undone? But I am also confused about something else: these "axiomatic" projective spaces are what I would call "finite" projective spaces (and ISTR that Coxeter calls them this). Since the article does not mention any axioms distinguishing one from another, but only gives yet another variation on the general projective axioms, I wonder whether the title, presently given here, is correct? -- Cheers, Steelpillow 13:42, 17 May 2008 (UTC)

I'm not an expert in these axiomatic things. Perhaps you could look at a reference and see what it's usually called? Then a {{main}} tag would be nice in the section here, pointing to Axiomatic_projective_space. Jakob.scholbach (talk) 16:52, 17 May 2008 (UTC)

I'm not an expert either. I read several books not long ago (Coxeter, Hilbert & Cohn-Vossen, Greenberg), and I came away with the impression that they were called finite spaces or finite geometries. Googling "axiomatic projective space" just now yielded 10 hits, many of which are scrapings from Wikipedia. Googling "Axiomatic projective geometry" yields endless references to titles of that name, one or two hints that the terms is used in the same sense as here, and eventually a decent number of links implying that it refers to the general axiomatisation of PG. "Finite projective space" gets around 500 hits, while "finite projective geometry" gets over 3,000 - in both cases dealing with the subject matter in question. Anyway, I would suggest that the present discussion of "axiomatic projective space" be renamed "finite projective geometry". But I'm not sure whether it deserves its own page, or just a section within Projective geometry; that may depend on what we do with the "geometry vs. space" issue. -- Cheers, Steelpillow 20:21, 18 May 2008 (UTC)

By "finite projective space" I would rather understand a space which has only finitely many points? How would "finite" point to some axiomatic charactersiation? Jakob.scholbach (talk) 21:22, 18 May 2008 (UTC)

The PG[m,n] notation defines the number of dimensions m and the number n of lines incident with some line at a given point (i.e. Number of lines through any point = n+1). So I assume that axiomatising these numbers, in addition to the general axioms of PG, would axiomatise the space. But I do not recall this finite axiomatisation being spelled out in the few summaries that I have read - ISTR they only covered the general projective axioms, which is perhaps why I remember them talking of "finite" space rather than "axiomatic" ones. -- Cheers, Steelpillow 10:52, 19 May 2008 (UTC)

So I intend to rename the page as "Finite projective geometry", and to move the general axiomatisation of PG back to projective geometry, where I originally moved it from. -- Cheers, Steelpillow 13:46, 24 May 2008 (UTC)

Sure, go ahead. Jakob.scholbach (talk) 20:43, 24 May 2008 (UTC)

R²

Can someone confirm that R in this article is The set of all real numbers, often written as $\mathbb {R}$ or R, as stated in R (disambiguation)? MacStep (talk) 10:30, 25 June 2011 (UTC)

Yes, R is definitely the set of real numbers. Thank you for pointing out that this notation is not explained! Mgnbar (talk) 12:22, 25 June 2011 (UTC)

PG(n, q)

This notation is used 5 times, I think, in the section Axioms for Projective Geometry, but is never defined anywhere in the entire article. — Preceding unsigned comment added by StatisticsMan (talk • contribs) 18:34, 10 September 2012 (UTC)

Sorry about that, definition of this alternate notation now appears in the definition section. Bill Cherowitzo (talk) 20:33, 10 September 2012 (UTC)

Morphisms

Hi! - Is L(V,W) the space of all linear maps? In this case, its projective cannot be indentified with the space of morphisms from P(V) to P(W), you need to take only injective maps. Quantrillo (talk) 14:35, 10 November 2012 (UTC)

Perhaps the right statment is: the morphisms from P(V) to P(W) form an open and dense set in P(L(V,W))... I'm not sure.Quantrillo (talk) 14:41, 10 November 2012 (UTC)

This section needs to be completely rewritten and renamed Projective map. Basically, like for algebraic varieties, there are two kinds of morphisms, the regular maps, which, here, are all injective, and the rational maps, that, in our case, are called projective maps. Clearly, the injectivity constraint is too restrictive and make, in our case, the regular maps of few interest. On the other hand, projective maps include projections, and this is the origin of the use of the word "projective" for naming "projective geometry". The (small) difficulty with projective maps is that, as functions, their are not defined everywhere but only outside some linear subspace. In other words, the right statement is: "The projective maps may be identified with the elements of P(L(V,W))". --D.Lazard (talk) 16:08, 10 November 2012 (UTC)

There are several problems with this section, for instance, the mention of [. . .] which is not included in the definitions. There is also a problem with notation; is the underlying field called K or k? And we do not define K*, which might be the center of Aut(V), or might not be. Sorry if I am sort of dense (in the material sense) here. But I want to make this more clear.

Thanks.

Fcy (talk) 02:23, 16 January 2020 (UTC)

Problem with Definition

Am I correct in thinking that the Definition section should read:

"with the equivalence relation (x₀, ..., x_n+1) ~ (λx₀, ..., λx_n+1), where λ is an arbitrary non-zero real number."

instead of

"with the equivalence relation (x₀, ..., x_n) ~ (λx₀, ..., λx_n), where λ is an arbitrary non-zero real number."

Monsterman222 (talk) 00:40, 27 March 2013 (UTC)

No. However you label them, there should be exactly n + 1 variables. You can label them from 1 to n + 1, or from 0 to n. But labeling them from 0 to n + 1 results in n + 2 variables. Mgnbar (talk) 02:17, 27 March 2013 (UTC)

Uncommon set notation

In the section "Definition of projective space", the following notation is used:

Pⁿ(R) := (Rⁿ⁺¹ \ {0}) / ~,

I cannot find any hint to what the slash and tilde symbols mean. I figure that because the backslash commonly denotes set difference, or exclusion, the slash is intended to symbolize inclusion, but that is merely speculation. Also, the use of the tilde is completely unknown to me in this context. I also could not find any explanation in the following articles: Real projective space, Set theory, Set (mathematics), and i would not know where else to look.

Could someone please introduce the notation in the article? Thanks. --Doubaer (talk) 11:48, 17 September 2013 (UTC)

It is not set notation: the tilde ~ denotes the equivalence relation that is defined next line (in the article). The slash / is the usual notation for quotient sets. I have just edited this link and quotient space for having a proper target to be linked to in the article. D.Lazard (talk) 12:50, 17 September 2013 (UTC)

Ah, now I see. Thank you very much. I had overlooked the tilde in the following line, now it comes out clearer. --Doubaer (talk) 08:46, 8 October 2013 (UTC)

Ray

Does the term (projective) ray from projective Hilbert space apply to general (or at least complex) projective spaces? Petr Matas 02:57, 30 April 2016 (UTC)

What do you mean? A ray in a vector space may be thought of as a point in the corresponding projective space (of rays). Boris Tsirelson (talk) 05:21, 30 April 2016 (UTC)

Though, I prefer to say that a one-dimensional subspace (of a vector space) may be thought of as a point in the corresponding projective space. As for me, a ray is rather a half of a (straight) line. But I know that (especially in the quantum theory context) punctured one-dimensional subspaces are often called rays. Boris Tsirelson (talk) 06:01, 30 April 2016 (UTC)

It seems that the use of "ray" to denote the set of nonzero scalar multiples of a vector (equivalence class in the definition of a projective space) is specific to quantum theory. It is sourced in Ray (quantum theory). As far as I know, "ray" is not used in this sense in pure mathematics, where "vector line" is commonly used, although, formally, a vector line contains 0. However, I have not found a definition of "vector line" in Wikipedia, although it is used, at least, in Homography § Projective frame and coordinates. D.Lazard (talk) 06:29, 30 April 2016 (UTC)

"Vector line", really? I did not face it. I've just google it, unsuccessfully. Where did you see it? Boris Tsirelson (talk) 07:23, 30 April 2016 (UTC)

Searching on Google Scholar with "vector line" "vector space" provides many example of this use. (Searching only for "vector line" provides too many non-mathematical use of "vector line" for finding easily the mathematical meaning). D.Lazard (talk) 08:06, 30 April 2016 (UTC)

I see, thanks. Boris Tsirelson (talk) 08:25, 30 April 2016 (UTC)

Thanks from me as well. Petr Matas 21:23, 30 April 2016 (UTC)