What is Geometry of Quantum Mechanics?

[cartuz]

I read about Projective Hilbert space in quant/ph9906086. It is the space of rays in Hilbert space. Than Projective Hilbert space is non-linear with curves and described by Riemann’s metrics. But it is a mathematic only. Is here a physical sense? Is it the way of cooperation of Quantum Mechanic and General Relativity Theory?

 

[Perturbation]

Are you talking about removing parrallelism in (rigged) Hilbert spaces of QM and introducing a (pseudo-)Riemannian metric?

If we were to include some form of non-flat metric in Hilbert space then we'd have to modify the inner product as follows, I think,

Hilbert space inner product of QM: $$\langle \psi |\phi\rangle =\int^{\infty}_{-\infty}\psi^{*}\phi d^3x$$

General Hilbert Space inner product: $$\langle \psi |\phi\rangle =\int^{\infty}_{-\infty}\psi^{*}\phi \rho (\mathbf{x})d^3x$$

Where $\rho$ is a weighting function.

However, general relativity is based in space-time. Hilbert space is an infinite dimensional abstract vector space, not actual space-time itself. If we were to unify QFT and general relativity we'd need to change the geometry of the space-time we do QFT in, not Hilbert space.

I may be wrong...

 

[cartuz]

Are you talking about removing parrallelism in (rigged) Hilbert spaces of QM and introducing a (pseudo-)Riemannian metric?

If we were to include some form of non-flat metric in Hilbert space then we'd have to modify the inner product as follows, I think,

Hilbert space inner product of QM: $$\langle \psi |\phi\rangle =\int^{\infty}_{-\infty}\psi^{*}\phi d^3x$$

General Hilbert Space inner product: $$\langle \psi |\phi\rangle =\int^{\infty}_{-\infty}\psi^{*}\phi \rho (\mathbf{x})d^3x$$

Where $\rho$ is a weighting function.

However, general relativity is based in space-time. Hilbert space is an infinite dimensional abstract vector space, not actual space-time itself. If we were to unify QFT and general relativity we'd need to change the geometry of the space-time we do QFT in, not Hilbert space.

I may be wrong...

May be you are right..
Here $\rho$ is sense of metric of Projective Hilbert space.

 

[Hurkyl]

Quantum mechanically, there is no physical difference between a state $\psi$, and any complex multiples $k \psi$. (More precisely, multiplying the state by k doesn't change the expectation of any observable)

So, this gives a reason why one might want to take an entire ray and treat it as a single object -- if all of the states along that ray correspond to the same physical entity, why not make them all the same thing mathematically?


This construction is actually a common one -- for finite-dimensional spaces, anyways. It's the usual way of constructing, say, the projetive plane.


I doubt that this has anything to do with general relativity -- it looks like their thesis is just that homogenized QM is a really nice theory that one ought to learn. (Just like anyone serious about geometry ought to be comfortable with the projective plane)

 

[Perturbation]

Could I have a link to that thesis?

 

[Hurkyl]

http://arxiv.org/pdf/quant-ph/9906086

 

[cartuz]

Quantum mechanically, there is no physical difference between a state $\psi$, and any complex multiples $k \psi$. (More precisely, multiplying the state by k doesn't change the expectation of any observable)
QUOTE]
Yes,
But in Projective Hilbert space it is different.
Of cause, it is more general case. Is it to complete the description of quantum states?

 

[Hurkyl]

Let me try this again.

All of the states $k \psi$ (for any complex k) all correspond to the same physical reality.

So, the ordinary Hilbert space contains a lot of irrelevant information.

So, what one might want to do is to get rid of that extra information -- all of those $k \psi$ from the original Hilbert space have been collapsed into a single point of the projective Hilbert space.

 

[cartuz]

Let me try this again.

All of the states $k \psi$ (for any complex k) all correspond to the same physical reality.

So, the ordinary Hilbert space contains a lot of irrelevant information.

So, what one might want to do is to get rid of that extra information -- all of those $k \psi$ from the original Hilbert space have been collapsed into a single point of the projective Hilbert space.

Is it an analog to 2-dimentinal space and 3-dimentinal space?
Is 2-dimentinal space have extra information about position of the point in the space?

 

[Hurkyl]

If you take 3-dimensional space and perform this construction, the result can be interpreted as a 2-dimensional space with some "points at infinity". And this is true in general.


For example, consider a photon. It has two states: spin up and spin down, which I'll write as |+> and |->.

The corresponding Hilbert space has two complex dimensions. However, this vector space is too big, because |+>, 2|+>, i|+>, (-3)|+>, et cetera, are all the same physical state.

So, we apply the construction described in the paper to produce a projective Hilbert space. Each point of this projective Hilbert space corresponds to an entire ray of the original Hilbert space; i.e. to a single physical state.

Since our original Hilbert space had two complex dimensions, the corresponding projective Hilbert space looks like one complex dimension, plus points at infinity. (Just one point, actually)

Topologically, the result is homeomorphic to an ordinary sphere. (It's the one-point compactification of the set of complex numbers)

This is called the Bloch sphere.

 

[cartuz]

If you take 3-dimensional space and perform this construction, the result can be interpreted as a 2-dimensional space with some "points at infinity". And this is true in general.


For example, consider a photon. It has two states: spin up and spin down, which I'll write as |+> and |->.

The corresponding Hilbert space has two complex dimensions. However, this vector space is too big, because |+>, 2|+>, i|+>, (-3)|+>, et cetera, are all the same physical state.

So, we apply the construction described in the paper to produce a projective Hilbert space. Each point of this projective Hilbert space corresponds to an entire ray of the original Hilbert space; i.e. to a single physical state.

Since our original Hilbert space had two complex dimensions, the corresponding projective Hilbert space looks like one complex dimension, plus points at infinity. (Just one point, actually)

Topologically, the result is homeomorphic to an ordinary sphere. (It's the one-point compactification of the set of complex numbers)

This is called the Bloch sphere.

Thank you very much for your help.

 

[dextercioby]

I've never seen the merge of projective Hilbert space and rigged Hilbert space as to provide a unique & complete description of the mathematical foundation for quantum mechanics...

Daniel.