# What is the zero-particle subspace in the direct sum of Fock space?

Fock space is defined as $$\mathcal{F}_s(\mathcal{H})=\bigoplus_{k=0}^\infty \overset{k}{\bigotimes}_s\mathcal{H},$$ and we can write it as $$\mathcal{F}_s(\mathcal{H})=H_0\oplus H_1 \oplus H_2 \oplus \dots,$$ where $$H_i$$ means the Hilbert space of $$i$$ particles.

I understand the spaces with nonzero number of particles. But what is the Hilbert space of zero number of particles, $$H_0$$? It seems like the only state should be $$\vert 0 \rangle$$. Is that still a legitimate Hilbert space (that follows the axioms)?

The complex numbers $$\mathbb{C}$$ form a one-dimensional Hilbert space. The axioms of a Hilbert space are that it be a complete metric space with a metric $$d(x,y)=\sqrt{\langle x-y,x-y\rangle}$$ that is defined in terms of an inner product $$\langle x,y\rangle$$. The complex numbers are endowed with a sesquilinear inner product $$\langle x,y\rangle=x^{*}y$$, which generates the usual distance measure $$d(x,y)=|x-y|^{2}$$, under which $$\mathbb{C}$$ is, of course, complete.
The one-dimensional Hilbert space $$H_{0}$$ with the bare vacuum $$|0\rangle$$ as its only basis state is just isomorphic to $$\mathbb{C}$$. You may be getting confused by the fact that the physical state of a quantum-mechanical system is not specified by an element of a Hilbert space, but by a ray in Hilbert space (or, equivalently, an element of a projective Hilbert space). The norm of a state $$|0\rangle$$, versus say $$2|0\rangle$$, is not physically meaningful. They both lie along the same ray (as does every element of $$H_{0}$$), but they are different elements of the Hilbert space.