Talk on: Dynamics and Potential
 
Abstract: An infinite quantum lattice system is considered. The traditional way to define a dynamics of the system is first to give the "potential",namely, assign an operator for each subset of the lattice (the interaction energy between physical objects such as particles and/or spins in that subset), then the time derivative of an observable is defined as i times the sum of the commutator with potentials for all finite subsets of the lattice, and finally the dynamics is defined as the integral of the time derivatives up to time t, which is a one-parameter group of automorphisms of the algebra of all observable operators. Our approach is in the opposite direction. Given any such one parameter group of (time translation) automorphisms (called a C^*-dynamical system), satisfying a sole condition that all observable operators belonging to a finite subset of the lattice, has the time derivative at time 0, we prove the existence and uniqueness of the associated potential such that the time derivative is given as i times the sum of commutators with the potential for all finite subsets of the lattice. Furthermore, this unique potential satisfies automatically a natural convergence condition as well as a very useful standardness condition. As an illustration of its usefulness, we demonstrate a very elementary proof of the so-called energy estimate (for total interaction energy within any finite subset of the lattice), which uses only set theoretical computation apart from the standardness condition. This new approach has been used to give the proof of equivalence of a number of different characterizations of the equilibrium states for general potentials, which is a solution of an open problem given in the book of Bratteli and Robinson.