On one-dimensional integrable quantum systems with infinitely many degrees of freedom

Yu-Ping Huo, Chi–Keung Ng

Michael Fowler

Fabio Benatti

Quantum systems with infinite degrees of freedom exhibit properties, like relaxation to equilibrium, phase-transitions and the existence of inequivalent representations of the CAR and CCR , that can satisfactorily be dealt with by means of the methods and techniques of algebraic quantum statistical mechanics [80, 81, 130]. The point of departure from standard quantum mechanics, is that in an infinite dimensional context one is usually provided with the algebraic properties of the relevant observables, but, in general, not with an a priori given representation on a Hilbert space; the latter rather depends on to the physical properties of the systems under consideration [324, 342, 343].

Sergio A Albeverio, Andrew Khrennikov

Gaussian measures on infinite-dimensional p-adic spaces are introduced and the corresponding L2-spaces of p-adic valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L2-spaces. p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L2-spaces with respect to a p-adic Gaussian measure.

Th.W. Ruijgrok, K. Sokalski