Mathematics of noncanonical quantum theory

Giampiero V. M. Esposito

Chee‐Kong Lee, Jianshu Cao, Jiangbin Gong

Equilibrium canonical distribution in statistical mechanics assumes weak system-bath coupling (SBC). In real physical situations this assumption can be invalid, and equilibrium quantum statistics of the system may be noncanonical. By exploiting both polaron transformation and perturbation theory in a spin-boson model, an analytical treatment is advocated to study noncanonical statistics of a two-level system at arbitrary temperature and for arbitrary SBC strength, yielding theoretical results in agreement with exact Monte Carlo simulations. In particular, the eigen-representation of system's reduced density matrix is used to quantify noncanonical statistics as well as the quantumness of the open system. For example, it is found that irrespective of SBC strength, noncanonical statistics enhances as temperature decreases but vanishes at high temperature.

Claude D. George, Ilya Prigogine

We present a noncanonical transformation theory of large dynamical systems that leads to a formulation of dynamics in terms of processes, in which the concepts of well-defined “units” and “interactions” can be simultaneously accommodated.

P. A. M. Dirac

Mohammad Ali Gorji, K Nozari, B Vakili

Abstract We study a noncanonical Hilbert space representation of the polymer quantum mechanics. It is shown that the Heisenberg algebra gets some modifications in the constructed setup, from which a generalized uncertainty principle will naturally come out. Although the extracted physical results are the same as those obtained from the standard canonical representation, the noncanonical representation may be notable in view of its possible connection with the generalized uncertainty theories suggested by string theory. In this regard, by considering a Snyder-deformed Heisenberg algebra, we show that since the translation group is not deformed, it can be identified with a polymer-modified Heisenberg algebra. In the classical level, it is shown that the noncanonical Poisson brackets are related to their canonical counterparts by means of a Darboux transformation on the corresponding phase space.