# Understanding Fock space

Let's start with a single particle Hilbert space $$H$$ with basis $$\{|\alpha\rangle\}$$, where $$\alpha$$ represents a complete set of quantum numbers. If we now take two particles of the same type, we must consider the space $$H_2=H_1\otimes H_1,$$ and more generally $$n$$ particles live in $$H^{\otimes n}=H_1\otimes...\otimes H_1.$$ The (free) Fock space is now defined by taking the direct sum $$\mathcal{F(H)}:=\bigoplus_{n\ge 0} H^{\otimes n}.$$ I understand why we take the tensor products: we wish to describe the whole system of $$n$$ particles at the same time. However, it is not at all clear to me why we also need to take the direct sum. What is its physical interpretation?

Consider identical particles. Consider also first just a single particle in your system. This "1-particle sector" is described by Hilbert space $$H_1$$. This is the first term, $$n=1$$, in the direct sum you wrote. ($$n=0$$ can be probably ignored.) Consider now two particles. This system's Hilbert space is $$H_1 \otimes H_1$$. This is the second term in the direct sum. And so on. The sectors with different numbers of particles are independent and thus you take the direct sum over them.
• This is all nice and well, but what is the need to actually consider such "$m$-particles sectors" in the first place? – Mr. Mister Sep 29 at 10:10