Hilbert space vs. Projective Hilbert space

Question

Hilbert space and rays:
In a very general sense, we say that quantum states of a quantum mechanical system correspond to rays in the Hilbert space

H , such that for any

c∈C the state

ψ and

cψ map to the same ray and hence are taken as equivalent states.

How should one interpret the above in order to understand why ψ and cψ are the same states? Clearly for c=0 it doesn't hold, and for c=1 it is trivial, but why should this equivalence hold for any other c ?
Knowing Hilbert space is a complex vector space with inner product, is ray just another way of saying vectors?
In the case that c just corresponds to a phase factor of type e iϕ with ϕ∈R, then obviously |ψ|=|e iϕ ψ|, i.e. the norms didn't change, then what is the influence of e iϕ at all? In other words what does the added phase of ϕ to the default phase of ψ change in terms of the state of the system?

Projective Hilbert space:
Furthermore, through a process of projectivization of the Hilbert space

H, it is possible to obtain a finite dimensional projective Hilbert space

P(H). In the Projective Hilbert space, every point corresponds to a distinct state and one cannot talk in terms of rays anymore.

What does such projectivization entail in a conceptual sense? I guess in other words, how are rays projected to single points in the process? and what implies the distinctness? Is such process in any way analogous to the Gram–Schmidt process used to orthonormalise a set of vectors in linear algebra?
When one limits the Hilbert space to that of a certain observable of the system at hand, e.g. momentum or spin space (in order to measure the momentum and spin of a system respectively), does that mean we're talking about projective spaces already? (e.g. is the spin space spanned by up |↑⟩ and down |↓⟩ spins states of a system referred to as projective spin Hilbert space?)

The aim is to develop a better and clearer understanding of such fundamental concepts in quantum mechanics.

score 7 · Accepted Answer

Why are states rays?

^{(Answer to OP's 1. and 2.)}
One of the fundamental tenets of quantum mechanics is that states of a physical system correspond (not necessarily uniquely - this is what projective spaces in QM are all about!) to vectors in a Hilbert space

H , and that the Born rule gives the probability for a system in state

∣ψ⟩ to be in state

∣ϕ⟩ by

P (ψ, ϕ) = ∣ ⟨ ψ | ϕ ⟩ ∣ 2 ∣ ⟨ ψ | ψ ⟩ ⟨ ϕ | ϕ ⟩ ∣

^{(Note that the habit of talking about normalised state vectors is because then the denominator of the Born rule is simply unity, and the formula is simpler to evaluate. This is all there is to normalisation.)}
Now, for any

c∈C−{0} ,

P(cψ,ϕ)=P(ψ,cϕ)=P(ψ,ϕ) , as may be easily checked. Therefore, especially

P(ψ,ψ)=P(ψ,cψ)=1 holds, and hence

c∣ψ⟩ is the same states as

∣ψ⟩ , since that is what having probability 1 to be in a state means.
A ray is now the set of all vectors describing the same state by this logic - it is just the one-dimensional subspace spanned by any of them: For

∣ψ⟩ , the associated ray is the set

R ψ : = {∣ ϕ ⟩ \in H | \exists c \in C : ∣ ϕ ⟩ = c ∣ ψ ⟩}

Any member of this set will yield the same results when we use it in the Born rule, hence they are physically indistiguishable.

Why are phases still relevant?

^{(Answer to OP's 3.)}
For a single state, a phase

e iα ,α∈R has therefore no effect on the system, it stays the same. Observe, though, that "phases" are essentially the dynamics of the system, since the Schrödinger equation tells you that every energy eigenstate

∣E i ⟩ evolves with the phase

e iE i t .
Obviously, this means energy eigenstates don't change, which is why they are called stationary states. The picture changes when we have sums of such states, though:

∣E 1 ⟩+∣E 2 ⟩ will, if

E 1 ≠E 2 , evolve differently from an overall multiplication with a complex phase (or even number), and hence leave its ray in the course of the dynamics! It is worthwhile to convince yourself that the evolution does not depend on the representant of the ray we chose: For any non-zero complex

c ,

c⋅(∣E 1 ⟩+∣E 2 ⟩) will visit exactly the same rays at exactly the same times as any other multiple, again showing that rays are the proper notion of state.

The projective space is the space of rays

^{(Answer to OP's 4. and 5. as well as some further remarks)}
After noting, again and again, that the physically relevant entities are the rays, and not the vectors themselves, one is naturally led to the idea of considering the space of rays. Fortunately, it is easy to construct: "Belonging to a ray" is an equivalence relation on the Hilbert space, and hence can be divided out in the sense that we simply say two vectors are the same object in the space of rays if they lie in the same ray - the rays are the equivalence classes. Formally, we set up the relation

ψ \sim ϕ \Leftrightarrow ψ \in R ϕ

and define the space of rays or projective Hilbert space to be

P (H) : = (H - {0}) / \sim

This has nothing to do with the Gram-Schmidt way of finding a new basis for a vector space! This isn't even a vector space anymore! (Note that, in particular, it has no zero) The nice thing is, though, that we can now be sure that every element of this space represents a distinct state, since every element is actually a different ray.¹
(Side note (see also orbifold's answer): A direct, and important, consequence is that we need to revisit our notion of what kinds of representations we seek for symmetry groups - initially, on the Hilbert space, we would have sought unitary representations, since we want to preserve the vector structure of the space as well as the inner product structure (since the Born rule relies on it). Now, we know it is enough to seek projective representations, which are, for many Lie groups, in bijection to the linear representations of their universal cover, which is how, quantumly,

SU(2) as the "spin group" arises from the classical rotation group

SO(3) .)
OP's fifth question

When one limits the Hilbert space to that of a certain observable of the system at hand, e.g. momentum or spin space (in order to measure the momentum and spin of a system respectively), does that mean we're talking about projective spaces already? (e.g. is the spin space spanned by up |↑⟩ and down |↓⟩ spins states of a system referred to as projective spin Hilbert space?)

is not very well posed, but strikes at the heart of what the projectivization does for us: When we talk of "momentum space"

H p and "spin space"

H s , it is implicitly understood that the "total space" is the tensor product

H p ⊗H s . That the total/combined space is the tensor product and not the ordinary product follows from the fact that the categorial notion of a product (let's call it

× cat ) for projective spaces is

P (H 1) \times cat P (H 2) = P (H 1 \otimes H 2)

For motivations why this is a sensible notion of product to consider, see some other questions/answers (e.g. this answer of mine or this question and its answers).
Let us stress again that the projective space is not a vector space, and hence not "spanned" by anything, as the fifth question seems to think.

phymath999

Sunday, October 18, 2015

一个物理系统于时间点的状态可以由希尔伯特空间中的一个归一化矢量 hilbert space vector ray

量子力學基礎 - 第 32 頁 - Google 圖書結果

Hilbert space vs. Projective Hilbert space

5 Answers 5

Why are states rays?

Why are phases still relevant?

The projective space is the space of rays

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