Intensity–intensity correlations determined by dimension of quantum state in phase space: P-distribution

We use the P-distribution to show that the familiar values 1, 2 and 3 of the normalized second order correlation function at equal times $g^{\left(2\right)}\left(0\right)$ corresponding to a coherent state, a thermal state and a highly squeezed vacuum are a consequence of the number of dimensions these states take up in quantum phase space. Whereas the thermal state exhibits rotational symmetry and thus extends over two dimensions, the squeezed vacuum factorizes into two independent one-dimensional phase space variables, and in the limit of large squeezing is therefore a one-dimensional object. The coherent state is a point in the phase space of the P-distribution and thus has zero dimensions. The fact that for photon number states the P-distribution is even narrower than that of the zero-dimensional coherent state suggests the notion of 'negative' dimensions. (invited comment)