How does vacuum state look in first quantization? Wikipedia says that the vacuum state is the unit of tensor product. In my understanding then, a first-quantized wavefunction for the vacuum state would be just constant in the each particle's configuration space. But this is a state where all the particles are present and have uniform probability density to be found anywhere.
If all particles are present, then how does this define a $0$-particle state?
 A: The vacuum state is not part of the $n$-particle sectors of the Fock space, apart when $n=0$. To be more precise, given a single-particle Hilbert space $\mathscr{H}$, define the $n$-particle space, $n\in \mathbb{N}^*$, by
$$\mathscr{H}_n=\underbrace{\mathscr{H}\otimes_{s/a}\dotsm\otimes_{s/a}\mathscr{H}_n}_n$$
where $s/a$ stand for either symmetrized or antisymmetrized tensor product. Then define the $0$-particle state as $\mathscr{H}_0=\mathbb{C}$. The Fock space is the direct sum
$$\Gamma_{s/a}(\mathscr{H})=\bigoplus_{n=0}^\infty\mathscr{H}_n$$
As a matter of fact we can represent a vector of the Fock space as
$\psi=(\psi_0,\psi_1,\dotsc,\psi_n,\dotsc)$,
where each $\psi_n\in\mathscr{H}_n$, the $n$-particle space.
The vacuum $\Omega$ is, a normalized vector with only non-zero component in the $0$-particle space (the customary (unimportant) choice is to have it to be $1$):
$$\Omega= (1,0,\dotsc,0,\dotsc)$$
As a matter of fact, the vacuum is also orthogonal, in the Fock space, to any other $n$-particle vector (with $n>0$).
edit (to incorporate the comments): Suppose you have two states $\psi,\phi\in\Gamma_{s/a}(\mathscr{H})$; then they sum componentwise (i.e. sum each n-particle sector components) $\psi+\phi=(\psi_0+\phi_0,\psi_1+\phi_1,\dotsc,\psi_n+\phi_n,\dotsc)$. So, given the vacuum state $\Omega$, and a 1-particle state $\psi=(0,\psi_1(x),0,\dotsc,0,\dotsc)$, their superposition is
$$\psi+\Omega=(1,\psi_1(x),0,\dotsc,0,\dotsc)$$
that has components on the vacuum and the 1-particle space.
Now this has to be normalized, if we want a meaningful superposition, suppose we choose a symmetric $\psi+\omega=(1/\sqrt{2},\psi_1(x)/\sqrt{2},0,\dotsc,0,\dotsc)$. This is normalized, because the norm of the Fock space is
$$\lVert\psi\rVert^2=\sum_{n=0}^\infty\lVert\psi_n\rVert^2$$
The interpretation is that the state $\psi+\Omega$ has the probability 1/2 of being the vacuum (no particles), and the probability 1/2 of being a one particle state $\psi_1(x)$; as the usual superposition principle predicts.
Simply in the Fock space we have (infinite) vectors of states, that represent all the possible configurations of $n$-particles, and that in principle may coexist on the same state in the sense that we may have a certain probability of having either 0, 1, 2, or $n$ particles (with any $n$).
