Is Fock space always separable?

[eoghan]

Hi!
During a lesson, the prof said that the Fock space could be non-separable. When can this happen?

 

[Bill_K]

Consider a system with N degrees of freedom: [a_α, a_β] = [a_α*, a_β*] = 0, [a_α, a_β*] = δ_αβ where α, β=1, 2,... N. The basis states of the Fock space are |n₁, n₂, n₃,... n_N>. That is, each possible set of occupation numbers corresponds to a linearly independent basis vector. As long as N is finite the number of basis vectors is countably infinite. But when N is infinite (as it will be for a quantized field) the number of basis vectors will be uncountably infinite, and therefore the Fock space is nonseparable.

 

[eoghan]

So, if for example I use the eigenstates of the position operator to label the state of a particle, then I have an infinite uncountable basis vectors(the particle can be in any |x> state with x real and non countable) and the space is non separable?

 

[strangerep]

So, if for example I use the eigenstates of the position operator to label the state of a particle, then I have an infinite uncountable basis vectors(the particle can be in any |x> state with x real and non countable) and the space is non separable?

Those eigenstates don't have a finite norm of the usual kind required in a Hilbert space. Some authors use a generalization, known as "rigged Hilbert space" or "Gel'fand triple" instead. Ballentine sect. 1.4 gives a gentle introduction.

 

[DarMM]

Fock Space is separable. Thermal Field theories have non-separable Hilbert spaces, but Fock space is separable.

 

[Bill_K]

Here's what Wikipedia has to say on the subject:

A Hilbert space is separable if and only if it admits a countable orthonormal basis... Even in quantum field theory, most of the Hilbert spaces are in fact separable, as stipulated by the Wightman axioms. However, it is sometimes argued that non-separable Hilbert spaces are also important in quantum field theory, roughly because the systems in the theory possesses an infinite number of degrees of freedom and any infinite Hilbert tensor product (of spaces of dimension greater than one) is non-separable. For instance, a bosonic field can be naturally thought of as an element of a tensor product whose factors represent harmonic oscillators at each point of space. From this perspective, the natural state space of a boson might seem to be a non-separable space. However, it is only a small separable subspace of the full tensor product that can contain physically meaningful fields (on which the observables can be defined).

 

[Bill_K]

So, if for example I use the eigenstates of the position operator to label the state of a particle, then I have an infinite uncountable basis vectors(the particle can be in any |x> state with x real and non countable) and the space is non separable?

This goes beyond what I described. I didn't say a continuum of modes like you'd get with eigenvectors of x. As strangerep points out, such states would not be normalizable.

What I said was, a discrete countable infinity of modes, normalized to a Kronecker delta.