User:Bendesc/sandbox

Relativistically an observer can only interact locally with others inside his own light-cone. In 1972 Lieb and Robinson showed that such principal existed in discrete quantum mechanical systems, such as quantum spin systems. They showed that information can only propagate at a finite velocity, while the accessible amount of information outside this cone dies out exponentially.

Notation
Consider a $$\nu $$-dimensional lattice $$\mathbb{Z}^\nu $$, and associate with each point $$x$$of the lattice a finite N-dimensional Hilbert space $$\mathcal{H}_x^N $$. The Hilbert space of each finite subset $$\Lambda \subset \mathbb{Z}^\nu$$ is then $$\otimes_{x \in \Lambda}\mathcal{H}_x^N $$. A metric $$ d $$ is defined on the lattice. The local observables $$\mathcal{A}_\Lambda $$ of the subsystem is given by the algebra of the matrices acting on $$\mathcal{H}_\Lambda $$. The union over all the finite subsets is the *-algebra of local observables $$\mathcal{A}_{Loc} $$ and its completion is the quasi-local observables.

The interaction of the system is described by a self adjoint map $$\Phi $$ on each finite subset $$\Lambda $$  to $$\mathcal{A}_\Lambda $$. The Hamiltonian of the system is then
 * $$H = \sum\limits_{\Lambda \subset\mathbb{Z}^\nu} \Phi(\Lambda) $$

We then define a strongly continuous one paramter family of automorphism $$ \{\tau_t\}_{t \in \mathbb{R}} $$,
 * $$\tau_t: \mathcal{A}_{Loc} \to \mathcal{A}_{Loc}: A \to \exp{i t[H,.]}(A) $$

The Lieb-Robinson Bound
Given two subsets $$\Lambda_A, \Lambda_B \subset \mathbb{Z}^\nu$$, local observables $$ A \in \mathcal{A}_{\Lambda_A}$$ and $$ B \in \mathcal{B}_{\Lambda_B}$$. Let $$ t\in \mathbb{R} $$ and $$ \{\tau_t\}_{t \in \mathbb{R}} $$ be a strongly continuous one-parameter family of automorphisms then,
 * $$||[\tau_t(A),B]|| \leq C_1 \min{|A|,|B|} ||A|| ||B|| e^{\frac{d(\operatorname{supp}(A), \operatorname{supp}(B))-v t}{C_2}} $$

where $$ C_1,C_2, v $$ are constants depending on microscropic details of the system, such as the strength of and range the local interaction.