| Full text | |
| Author(s): |
Bru, J. -B.
;
Pedra, W. de Siqueira
Total Authors: 2
|
| Document type: | Journal article |
| Source: | Journal of Mathematical Physics; v. 63, n. 5, p. 33-pg., 2022-05-01. |
| Abstract | |
During the last three decades, Pavel Bona developed a non-linear generalization of quantum mechanics, which is based on symplectic structures for normal states. One important application of such a generalization is a general setting that is very convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bona's non-linear quantum mechanics. It is based on C-0-semigroup theory and has a domain of applicability that is much broader than Bona's original one. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. In this new mathematical approach, we build a Poisson bracket for the polynomial functions on the Hermitian weak*-continuous functionals on any C*-algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this C*-algebra by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative C*-algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate C-0-groups of contractions. As a matter of fact, in generic commutative C*-algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue. Some new mathematical concepts are introduced, which are possibly interesting by themselves: the convex weak* Gateaux derivative and the state-dependent C*-dynamical systems. Our recent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systems with long-range, or mean-field, interactions corroborate the relevance of the general approach we present here. Published under an exclusive license by AIP Publishing. (AU) | |
| FAPESP's process: | 17/22340-9 - Constructive methods for interacting fermions with applications to the microscopic theory of conductivity and superconductivity |
| Grantee: | Walter Alberto de Siqueira Pedra |
| Support Opportunities: | Research Grants - Visiting Researcher Grant - International |