Understanding Fock Spaces: Explanation and Examples

[SeventhSigma]

Can someone explain to me how Fock spaces work and a few examples of how they are used and what they tell us?

 

[tom.stoer]

Think about a harmonic oszillator state |n>. Instead of interpreting this as a state with one particle in the n-th state we interpret it as n particles in a state. Acting with a the creation operator on this state does not send one particle in the (n+1) state, instead it creates a new particle.

For one state this is boring, so let's think about a collection of harmonic oscillators (uncoupled, commuting operators) with a state |k,m,n,...>. Now we have k particles in one state (of one harmonic oscillator), m particles in another state and so on.

In quantum field theory the different states in the ket |...> belong to different momenta p. If these momenta are quantized (particles in a finite box) the interpretation of the fock state |k,m,n,...> means that we have k particles in the ground state, m particles is the first excited state etc. Acting with creation and annihilation operators (now equipped with a label p) on these states creates and annihilates these particles. Interaction terms will usually couple these different states and e.g. do something like sending a state |m,k,n> to |k-1,m+1,n>.

 

[SeventhSigma]

So is it safe to say that it's primarily just a convenient mathematical framework for addressing particular problems? Is it needed to actually explain anything?

What is physically meant by creation and annihilation operators?

 

[vanhees71]

In non-relativistic quantum mechanics, very many problems are posed, where the particle number is conserved. Take atomic physics as an example: The number of electrons running around a nucleus is conserved.

The reason for this however is that the energies involved here are small compared to the mass of the electron and thus there's not enough energy in the considered reactions of particles to create new particles (an exeption are photons which are always created, no matter how small the energies involved are since photons are massless, but that's not really part of non-relativistic physics since photons are always relativistic, again because they are massless).

Thus for many problems, in non-relativistic quantum mechanics you can work within a Hilbert space at fixed particle number. However, even then the use of annihilation and creation operators and thus the Fock space is convenient since the exchange-symmetry properties of many-body states (Bose or Fermi implying that all states must be superpositions of totally symmetrized or antisymmetrized tensor-product states, respectively) is automatically built in when using the corresponding field operators and thus the Fock-space formalism.

In relativistic quantum theory, you even have to use the Fock-space representation (sometimes somewhat misnamed as "second quantization") since there are no conserved particle numbers (except for the boring case of non-interacting particles) but you can always create particle-antiparticle pairs in reactions at relativistic energies, i.e., when the energies involved in the particle collisions are at the order of magnitude of the masses of particles.

Even in non-relativistic physics there are a plethora of particle-like excitations in many-body theory, where the number of these quasi-particles are not conserved (e.g., phonons, the quasi-particles (or field quanta) belonging the collective lattice vibrations of a solid body).

In that sense quantum field theory is the most versatile form of quantum theory and thus worth being studied not only in relativistic but also in non-relativistic physics.