A quasi-free state over the algebra of the canonical anticommutations relations (CAR) is a state with respect to which the moments of the field (Segal-type) operators are calculated similarly to the expectation of a Gaussian random field (when one additionally takes into account the sign of a partition). An important subclass of quasi-free states is given by gauge-invariant states. For a given representation of the CAR, one formally defines its particle density as the product of the creation and annihilation operators at point, and again formally the smeared particle density is a family of commuting Hermitian operators. For a class of quasi-free states, we show that its particle density can be rigorously realised as a family of commuting self-adjoint operators and its joint spectral measure is a determinantal point process, i.e., a point process whose correlation functions are determinants built upon a correlation kernel $K(x,y)$. In the case of a gauge-invariant quasi-free state, the correlation kernel $K(x,y)$ is Hermitian. We also consider the particle-hole transformation in the continuum as a certain Bogoliubov transformation of a gauge-invariant quasi-free state, which leads to a non-gauge-invariant quasi-free state. For the corresponding particle density, the joint spectral measure is a determinantal point process with a correlation kernel $K(x,y)$ that is J-Hermitian. The latter means that the integral operator with integral kernel $K(x,y)$ is self-adjoint with respect to an indefinite inner product.