Applications of methods of random differential geometry to quantum statistical systems

1 Abstract We apply concepts of random differential geometry connected to the random matrix ensembles of the random linear operators acting on finite dimensional Hilbert spaces. The values taken by random linear operators belong to the Liouville space. This Liouville space is endowed with topological and geometrical random structure. The considered random eigen-problems for the operators are applied to the quantum statistical systems. In the case of random quantum Hamiltonians we study both hermitean (self-adjoint) and non-hermitean (non-self-adjoint) operators leading to Gaussian and Ginibre ensembles Refs. [1], [2], [3]. [1] M. M. Duras, " Finite-difference distributions for the Ginibre ensemble, " J. Opt. B: 2 Introduction We study generic quantum statistical systems with energy dissipation. Let us consider Hilbert's space V with some basis {|Ψ i }. The space of the linear bounded operatorsˆX acting on Hilbert space V is called Liouville space and is denoted L(V). The Liouville space is again Hilbert's space with the scalar product: ˆ X|ˆY = Tr(ˆ X † ˆ Y). Hence, it is a Banach space with norm: ||ˆX|| = (ˆ X|ˆX) 1/2 , and it is metric space with distance: ρ(ˆ X, ˆ Y) = ||ˆX − ˆ Y ||. Finally, it is topological space with the balls B(ˆ X, r) = { ˆ Y |ρ(ˆ Y , ˆ X) < r}. The Liouville space is also a differentiable manifold. The exist a tangent space T ˆ X L(V), tangent bundle TL(V), cotangent space T ⋆ ˆ X L(V), and cotangent bundle T ⋆ L(V). The quantum operatorˆX ∈ L(V) is in the given basis a matrix with elements X ij. We are allowed to define random operator variablê X : Ω ∋ ω → ˆ X (ω) ∈ L(V), where Ω is sample space, and ω is sample point. In the basis it is reduced to random matrix variable