The exponential family of models is defined in a general setting, not relying on probability theory. Some results of information geometry are shown to remain valid. Exponential families both of classical and of quantum mechanical statistical physics fit into the new formalism. Other less obvious applications are predicted. For instance, quantum states can be modeled as points in a classical phase space and the resulting model belongs to the exponential family.
The first step in the counting operator analysis of the spectrum of any model Hamiltonian H is the choice of a Hermitean operator M in such a way that the third commutator with H is proportional to the first commutator. Next one calculates operators R and R † which share some of the properties of creation and annihilation operators, and such that M becomes a counting operator. The spectrum of H is then decomposed into multiplets, not determined by the symmetries of H, but by those of a reference Hamiltonian H ref , which is defined by H ref = H − R − R † , and which commutes with M . Finally, we introduce the notion of stable eigenstates. It is shown that under rather weak conditions one stable eigenstate can be used to construct another one.In the literature many attempts are found to generalise the notion of creation and annihilation operators. Some of these were introduced in the context of Bogoliubov's notion of quasi-particles -see for instance [1]. Others are related to the method developed by Darboux in the nineteenth century to find new solutions of non-linear equations (see for instance [2]). Lowering and raising operators [3] determine recurrence relations and generate a Lie algebra (see for instance [4]).
We prove three theorems about the use of a counting operator to study the spectrum of model Hamiltonians. We analytically calculate the eigenvalues of the Hubbard ring with four lattice positions and apply our theorems to describe the observed level crossings.
We introduce generalized notions of a divergence function and a Fisher information matrix. We propose to generalize the notion of an exponential family of models by reformulating it in terms of the Fisher information matrix. Our methods are those of information geometry. The context is general enough to include applications from outside statistics.
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