The Dirac operator for the pair of Ruelle and Koopman operators, and a generalized Boson formalism

Abstract

Denote by $\mu$ the maximal entropy measure for the shift map $\sigma$ acting on $\Omega =\{0,1{\}}^{\mathbb{N}}$, by $L$ the associated Ruelle operator and by $K=L^{†}$ the Koopman operator, both acting on ${\mathcal{L}}^2(\mu )$. The Ruelle-Koopman pair can determine a generalized boson system in the sense of \cite{Kuo}. Here $2^{-\frac{1}{2}}K$ plays the role of the creation operator and $2^{-\frac{1}{2}}L$ is the annihilation operator. We show that $[L,K]$ is the projection on the kernel of $L.$ In $C^{\ast }$-algebras the Dirac operator $\mathcal{D}$ represents derivative. Akin to this point of view we introduce a dynamically defined Dirac operator $\mathcal{D}$ associated with the Ruelle-Koopman pair and a representation $\pi$. Given a continuous function $f$, denote by $M_f$ the operator $g\to M_f(g)=fg.$ Among other dynamical relations we get $$\Vert \left[\mathcal{D},\pi (M_f)\right]\Vert =\underset{x\in \Omega }{\sup }\sqrt{\frac{|f(x)-f(0x){|}^2}{2}+\frac{|f(x)-f(1x){|}^2}{2}}={\left|\sqrt{L|Kf-f{|}^2}\right|}_{\infty }$$ which concerns a form of discrete-time mean backward derivative. We also derive an inequality for the discrete-time forward derivative $f\circ \sigma -f$: $$|f\circ \sigma -f{|}_{\infty }=|Kf-f{|}_{\infty }\ge \Vert \left[\mathcal{D},\pi (M_f)\right]\Vert \ge |f-Lf{|}_{\infty }.$$ Moreover, we get $\Vert \left[\mathcal{D},\pi (KL)\right]\Vert =1$. The Number operator is $\frac{1}{\sqrt{2}}K\frac{1}{\sqrt{2}}L.$ The Connes distance requires to ask when an operator $A$ satisfies the inequality $\Vert \left[\mathcal{D},\pi (A)\right]\Vert \le 1$; the Lipschtiz constant of $A$ smaller than $1$.