Thermal States on Mittag-Leffler Fock Space of the Slitted Plane
Natanael Alpay, Tiju Cherian John
TL;DR
This work extends the theory of number and thermal states from the usual Bosonic Fock space to the Mittag-Leffler Fock space on the slitted plane. It defines ML-number states as eigenvectors of the ML number operator with eigenvalues $n_q$, and introduces thermal states via $\rho_s=\frac{e^{-s a^\dagger a}}{\mathrm{Tr}(e^{-s a^\dagger}a)}$, proving that $e^{-s a^\dagger a}$ is trace class for all $s>0$ by analyzing the Dirichlet-series with eigenvalues $n_q=\frac{\Gamma(qn+1)}{\Gamma(q(n-1)+1)}$ and establishing an abscissa of convergence $\sigma_c=0$. This ensures well-defined ML-thermal states and recovers the standard Fock-space results in the limit $q=1$. The results lay a foundation for ML analogues of quantum gaussian states and suggest further directions, such as ML Weyl/Bogoliubov representations and ML coherent and squeezed states, broadening the interface between operator theory, special functions, and quantum probability on generalized Fock spaces.
Abstract
Number states and thermal states form an important class of physical states in quantum theory. A mathematical framework for studying these states is that of a Fock space over an appropriate Hilbert space. Several generalizations of the usual Bosonic Fock space have appeared recently due to their importance in many areas of mathematics and other scientific domains. One of the most prominent generalization of Fock spaces is the Mittag-Leffler (ML) Fock space of the slitted plane. Natural generalizations of the basic operators of quantum theory can be obtained on ML Fock spaces. Following the construction of the creation and annihilation operators in the Mittag-Leffler Fock space of the slitted plane by Rosenfeld, Russo, and Dixon, (J. Math. Anal. Appl. 463, 2, 2018). We construct and study the number states and thermal states on the ML Fock space of the slitted plane. Thermal states on usual Fock space form an important subclass of the so called quantum gaussian states, an analogous theory of more general quantum states (like squeezed states and Bell states) on ML Fock spaces is an area open for further exploration.