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

Abstract

Denote by $\mathbf{\mu}$ the maximal entropy measure for the shift map $\sigma$ acting on $\Omega = \{0, 1\}^\mathbb{N}$, by $L$ the associated Ruelle operator and by $K = L^{\dagger}$ the Koopman operator, both acting on $\mathscr{L}^2(\mathbf{\mu})$. 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 $C^*$-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)=f\, g.$ Among other dynamical relations we get $$\|\left[ \mathcal{D} , \pi (M_f) \right]\| = \sup_{x \in \Omega} \sqrt{\frac{|f(x) - f(0x)|^{2}}{2} + \frac{|f(x) - f(1x)|^2}{2}} = \left|\sqrt{L |K f - f|^{2}}\right|_{\infty}$$ which concerns a form of discrete-time mean backward derivative. We also derive an inequality for the discrete-time forward derivative $f \circ \sigma -f$: $$ |f \circ \sigma -f |_{\infty} = |K f - f|_{\infty} \geq \|\left[ \mathcal{D} , \pi (M_f) \right]\| \geq |f - L f|_{\infty}.$$ Moreover, we get $\|\, \left[\mathcal{D} ,\pi(K L)\right] \,\|=1$. 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 $\|\, \left[\mathcal{D} ,\pi(A)\right] \,\|\leq 1$; the Lipschtiz constant of $A$ smaller than $1$.