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In The Road to Reality by Roger Penrose, projective geometry as developed during the Renaissance is framed as (eventually) playing a pivotal role in quantum mechanics. (In fact, Penrose seems enamored with the idea that there is some connection between painting and physics, particularly where twistor theory is concerned. The book's epilogue is the most blatant example of this, though other examples abound.)

Not having studied quantum mechanics, I can't really imagine how projective spaces would be used to formalize what I know intuitively about the way QM works.

Can someone provide a very simple example?

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It's been a while since a looked into this paper. It deals with mutually unbiased bases in quantum computation and its connection to finite projective planes. There is more on the arXiv. Happy searching! – draks ... Feb 1 '12 at 21:58
What do you know intuitively about the way QM works?! – Qiaochu Yuan Feb 1 '12 at 22:24
@Qiaochu Yuan well, i've never been formally trained in mathematics but i've learned as much terminology as i possibly can... so, the intuition comes from knowing how the words and phrases mix together, which is what i'm naturally good at, without the sensation function (jung's term for that thing you guys use when you do math problems) kicking in. penrose's book is perfect for that, because he bothers to explain everything, down to the level of individual functions, variables, and constants, without throwing curtains over everything and trying to be poetic. which is why i love that book. – ixtmixilix Feb 3 '12 at 2:03
up vote 7 down vote accepted

In classical mechanics, the state of a particle on a smooth manifold $M$ is represented as a point of the cotangent bundle $T^{\ast}(M)$. In quantum mechanics, the state of a particle on a manifold $M$ is instead represented as a unit vector in the Hilbert space $H = L^2(M)$, except that two unit vectors differing in phase (that is, differing by multiplication by a scalar) are regarded as the same state because the results of all possible experiments performed on the two states are the same. So the state of a particle is represented, not really by a unit vector in $H$, but by a point in the projective space $\mathbb{P}(H)$. Symmetries of a quantum system are then identified with continuous homomorphisms $G \to \text{PGL}(H)$ of topological groups, or with projective representations of a (probably Lie) group $G$.

In quantum computing, we might consider smaller Hilbert spaces. For example, the Hilbert space $H = \mathbb{C}^2 = \text{span}(\langle 0|, \langle 1|)$ describes a qubit, and $\mathbb{P}(H)$ is the Riemann sphere, which in this context is known as the Bloch sphere.

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having constructed the best possible picture i could of what you just described, i can see that QM is incredibly beautiful. beautiful in a way that pop-science ideas about half-dead cats are not. thank you. – ixtmixilix Feb 3 '12 at 2:07
that also explains something i've always wondered about lie groups, i.e., why are they even necessary. they always seemed sort of redundant. but if they have to describe a system like the projective space you described... i can see how their flexibility (the part of them which has always seemed a bit boring) would be useful. i did once try to teach myself topological groups, but i had no idea it actually plugged in to QM. so thanks for that information, as well. – ixtmixilix Feb 3 '12 at 2:13
i also found this paper extremely useful in terms of understanding how projective spaces are used: -- in particular, page 13. – ixtmixilix Feb 28 '12 at 19:56
Question regarding the projective ray characteristic: "differing in phase, that is, differing by multiplication by a scalar". In QM the multiplicative factor has absolute value 1 (phase factor), however in PG the multiplicative factor is free, except 0 is not allowed. Correct? What does this difference imply? – Gerard Mar 18 '14 at 22:57
@Gerard: nothing. In quantum mechanics it's conventional to restrict attention to unit vectors, and "unit vectors modulo complex numbers of absolute value $1$" is the same space as "vectors modulo nonzero complex numbers." – Qiaochu Yuan Mar 30 '14 at 1:32

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