Exploring the Connection Between Hilbert Space & Space-Time

[Gear300]

What is the relation between Hilbert Space and space-time? Are the two disjoint or is there something relating the two?

 

[Feldoh]

Well a Hilbert space is an infinite vector space that defines an inner product in a specific way.

Space-time is a way of describing mixed coordinates of space and time such that it's measure is relativistically invariant among other properties.

I don't really see the similarities...

 

[Demystifier]

They are quite unrelated. In particular, one of them is 4-dimensional, while the other is infinite dimensional.

 

[Gear300]

Oh...I see. Does the Hilbert space have any physical implementation, or is it primarily just a mathematical structure?

 

[martinbn]

Of course there are finite dimensional Hilbert spaces, in particular four dimensional.

 

[naima]

Can we say that a Quantum Field Theory assign an operator to each point of space-time and that these operators act on an Hilbert space?

 

[Fredrik]

Oh...I see. Does the Hilbert space have any physical implementation, or is it primarily just a mathematical structure?

Hilbert spaces are extremely important in quantum mechanics. The one-dimensional subspaces represent the pure states of a physical system. If a system is prepared in the state represented by a one-dimensional subspace R, the probability that a measurement will leave the system in a state represented by R' is given by |<u,u'>|², where u and u' are (any) normalized vectors that are members of R and R' respectively.

Martinbn is right that Hilbert spaces don't have to be infinite-dimensional. Minkowski spacetime can be defined as a vector space, but there's no inner product included in its definition (instead there's a symmetric, non-degenerate bilinear form that isn't positive definite like an inner product), so it doesn't satisfy the definition of a Hilbert space.

Can we say that a Quantum Field Theory assign an operator to each point of space-time and that these operators act on an Hilbert space?

That's pretty close to the truth (and is good enough for most purposes), but it's more accurate to say that it assigns an operator-valued distribution to each point.

 

[naima]

That's pretty close to the truth (and is good enough for most purposes), but it's more accurate to say that it assigns an operator-valued distribution to each point.

These distributions uses smearing functions defined on a limited zone of space-time . Is there anything physical in these functions. for example Rovelli wrote when a measure is done this function is null out of the laboratory! does it carry information?

 

[Demystifier]

Martinbn is right that Hilbert spaces don't have to be infinite-dimensional.

A long time ago, an expert in mathematical physics told me that Hilbert space is an infinite dimensional unitary space. Now I see that he was wrong. :shy:

 

[arkajad]

If you take the Hilbert space of square integrable functions over spacetime, if you take the algebra of bounded operators on this Hilbert space, then this algebra has an Abelian subalgebra and its spectrum is just spacetime. In general, if you have a vov Neumann algebra and a maximal Abelian sualgebra, you can interpret its spectrum as "spacetime". But it can also be the momentum space. From the Hilbert space alone you will not see a difference between space and momentum space, for instant. You need more structure, for instance some causal structure - usually described by a net of subalgebras organized by some special properties (like subalgebras corresponding to spacelike separated regions should commute with each other etc.) Such things are studied in algebraic quantum field theory.