Abstract:
In 1925 Dirac observed that in the classical limit $\hbar \to 0$, the commutators of operators in a quantum algebra with Heisenberg multiplication rules reduce to the Poisson brackets of the corresponding classical observables. The quantum Heisenberg equation then reduces to the Hamiltonian equation in classical theory. In other words, the classical limit of a quantum algebra yields a commutative Poisson algebra of functions on the phase space. Quantum systems involving fermionic or spin degrees of freedom do not admit pure commutative limit as the quantisation parameter $\hbar \to 0$. Using the quantisation ideal approach, we discover a range of quantum integrable systems, in which the quantum algebra remains non-commutative for all specialisations of the quantisation parameter. All these examples can be regarded as quantum deformations of a noncommutative associative algebra $\mathcal A$.
In this talk, I will show that any deformation of a noncommutative algebra leads to a commutative Poisson algebra structure on
$$\Pi(\mathcal A) := Z(\mathcal A) \times (\mathcal A/Z(\mathcal A))$$
and endows $\mathcal A$ with the structure of a $\Pi(\mathcal A)$-Poisson module, where $Z(\mathcal A)$ denotes the center of $\mathcal A$. Commuting quantum Hamiltonians yield Poisson-commuting Hamiltonians defined on $\Pi(\mathcal A)$, and the Heisenberg equations reduce to Hamiltonian equations defined on $\mathcal A$. The theory of “hybrid systems” developed by A. Liashyk, N. Reshetikhin, and I. Sechin, “Quantum Integrable Systems on a Classical Integrable Background” Communications in Mathematical Physics, Volume 407, Issue 1 (December 2025), is closely related in spirit to our approach, although it is based on a different underlying philosophy and does not extend to constructing a Poisson algebra structure for the hybrid systems obtained.
The talk is based on the paper:
A.V. Mikhailov, P. Vanhaecke. Commutative Poisson algebras from deformations of noncommutative algebras. Lett. Math. Phys., 114(5), 1-51, 2024, arXiv:2402.16191v2.
