Basic understanding of the Fock space of a quantized real scalar field

Question

The states in quantum mechanics belong to some Hilbert space while the states in quantum field theory belong to a Fock space. For simplicity, let me stick to the Fock space emerging after the quantization of a real scalar field.

A Fock space is defined as a direct sum, $$\mathcal{F}=\oplus_n\mathcal{H}_n$$ of Hilbert spaces $\mathcal{H}_n$, of physical $n$-particle states.

For a real scalar field, which after quantization (which lead to only one type of particle) the states in $\mathcal{H}_n$, are in general, linear combination of $n$-particle states $\{|p_1,p_2,...,p_n\rangle\}$ of all possible momenta satisfying $p^i_{\mu }p^{\mu i}=m^2$, and $p^0_i>0$.

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Questions

What is the physical interpretation of the Fock space being a direct sum of $\mathcal{H}_n$?

It looks like the Fock space has invariant subspaces of labels $n$ where $n\in \mathbb{Z}$. Does it mean that under Poincare transformation, the $n$-particle states, for a given $n$, represent an irreducible representation of the Poincare group i.e., under a Poincare transformation, the states within $\mathcal{H}_n$, for a given $n$, mix among themselves.

If the above interpretation is correct, is it also true that the states in different irreducible representations, for $n\neq m$, are labelled by different values of masses?

Does it also mean that the superposition of states belonging to two different irreducible representations (for example, superposition of a one-particle state with a two-particle state) is forbidden in nature?

Answer 1

Questions

What is the physical interpretation of the Fock space being a direct sum of $\mathcal{H}_n$?

I don't know what you are looking for here. This is the definition of a Fock space.

It looks like the Fock space has invariant subspaces of dimensions $n$ where $n\in \mathbb{Z}$. Does it mean that under Poincare transformation, the $n$-particle states, for a given $n$, represent an irreducible representation of the Poincare group i.e., under a Poincare transformation, the states within $\mathcal{H}_n$, for a given $n$, mix among themselves.

Yes, in a free (Gaussian) theory there exists a number operator $N = \sum_k a_k^\dagger a_k$ which can tell you exact number of particles in a state.

If the above interpretation is correct, is it also true that the states in different irreducible representations, for $n\neq m$, are labelled by different values of masses?

It depends on what you call the mass of a state. You could define it as $\sum_i p_i^2 = n m^2$. This mass is measured by the operator $P_1^2 \otimes P_2^2 \otimes \cdots \otimes P_n^2$ where $P_i^2$ is the momentum squared operator acting on the $i$th subspace in ${\cal H}_n$. This "mass" is of course different for $n_1 \neq n_2$.

A better definition is the invariant mass of the state, $P^2 = ( \sum P_i )^2$. The invariant mass can be the same even for $n_1 \neq n_2$.

Does it also mean that the superposition of states belonging to two different irreducible representations (for example, superposition of a one-particle state with a two-particle state) is forbidden in nature?

No, I don't see why it would be forbidden in nature. These would just not have the usual classical interpretation of a bunch of non-interacting moving particles, but in general is a perfectly good state in the quantum theory.