Fractional-dimensional Fock spaces, second quantization, and dynamical interaction of particles with Haldane’s exclusion statistics

We propose that the notion of Haldane's dimension and the corresponding statistics be treated in a probabilistic spirit. Motivated by the experience of dimensional regularization, the dimension of a space is defined as a trace of a diagonal ''unit operator'' where the diagonal matrix elements are not in unity in general but are probabilities to find a system in a given state. These probabilities are then uniquely defined by the rules of Haldane's statistics. The partition function for an ideal gas of the particles, a state-counting procedure, the entropy and a distribution function for this probabilistic definition are investigated and compared with previous works in the context of g particles. The corresponding creation and annihilation operators are introduced and Hamiltonians for interacting g particles are constructed. The stability of the thermodynamical properties of a gas of particles with exclusion statistics under the influence of hopping (or scattering) between states is investigated.