The Dirac operator for the pair of Ruelle and Koopman operators, and a generalized Boson formalism
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2024
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| _version_ | 1866909639837220864 |
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| author | Braucks, William M. M. Lopes, Artur O. |
| author_facet | Braucks, William M. M. Lopes, Artur O. |
| contents | Denote by $\mathbfμ$ the maximal entropy measure for the shift map $σ$ acting on $Ω= \{0, 1\}^\mathbb{N}$, by $L$ the associated Ruelle operator and by $K = L^{\dagger}$ the Koopman operator, both acting on $\mathscr{L}^2(\mathbfμ)$. 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 $π$. Given a continuous function $f$, denote by $M_f$ the operator $ g \to M_f(g)=f\, g.$ Among other dynamical relations we get $$\|\left[ \mathcal{D} , π(M_f) \right]\| = \sup_{x \in Ω} \sqrt{\frac{|f(x) - f(0x)|^{2}}{2} + \frac{|f(x) - f(1x)|^2}{2}} = \left|\sqrt{L |K f - f|^{2}}\right|_{\infty}$$ which concerns a form of discrete-time mean backward derivative. We also derive an inequality for the discrete-time forward derivative $f \circ σ-f$: $$ |f \circ σ-f |_{\infty} = |K f - f|_{\infty} \geq \|\left[ \mathcal{D} , π(M_f) \right]\| \geq |f - L f|_{\infty}.$$ Moreover, we get $\|\, \left[\mathcal{D} ,π(K L)\right] \,\|=1$. 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 $\|\, \left[\mathcal{D} ,π(A)\right] \,\|\leq 1$; the Lipschtiz constant of $A$ smaller than $1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_18133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Dirac operator for the pair of Ruelle and Koopman operators, and a generalized Boson formalism Braucks, William M. M. Lopes, Artur O. Functional Analysis Mathematical Physics Dynamical Systems 37D35, 37N20, 81P16, 46C05, 46L05, 60K40 Denote by $\mathbfμ$ the maximal entropy measure for the shift map $σ$ acting on $Ω= \{0, 1\}^\mathbb{N}$, by $L$ the associated Ruelle operator and by $K = L^{\dagger}$ the Koopman operator, both acting on $\mathscr{L}^2(\mathbfμ)$. 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 $π$. Given a continuous function $f$, denote by $M_f$ the operator $ g \to M_f(g)=f\, g.$ Among other dynamical relations we get $$\|\left[ \mathcal{D} , π(M_f) \right]\| = \sup_{x \in Ω} \sqrt{\frac{|f(x) - f(0x)|^{2}}{2} + \frac{|f(x) - f(1x)|^2}{2}} = \left|\sqrt{L |K f - f|^{2}}\right|_{\infty}$$ which concerns a form of discrete-time mean backward derivative. We also derive an inequality for the discrete-time forward derivative $f \circ σ-f$: $$ |f \circ σ-f |_{\infty} = |K f - f|_{\infty} \geq \|\left[ \mathcal{D} , π(M_f) \right]\| \geq |f - L f|_{\infty}.$$ Moreover, we get $\|\, \left[\mathcal{D} ,π(K L)\right] \,\|=1$. 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 $\|\, \left[\mathcal{D} ,π(A)\right] \,\|\leq 1$; the Lipschtiz constant of $A$ smaller than $1$. |
| title | The Dirac operator for the pair of Ruelle and Koopman operators, and a generalized Boson formalism |
| topic | Functional Analysis Mathematical Physics Dynamical Systems 37D35, 37N20, 81P16, 46C05, 46L05, 60K40 |
| url | https://arxiv.org/abs/2409.18133 |