Dynamics and potentials
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Dynamics and Potentials1 Huzihiro Araki (Based on a joint work [2] with Hajime Moriya) Research Institute for Mathematical Sciences, Kyoto University email: [email protected]u.ac.jp Abstract—A dynamics (i.e. a oneparameter group of automorphisms) of a system described by a C*algebra with a local structure in terms of C*subalgebras Ꮽ(I) for local domains I of the physical space (a discrete lat tice) is normally constructed out of potentials P(I), each of which is a selfadjoint element of the subalgebra Ꮽ(I), such that the the first time derivative of the dynamical change of any local observable A is i times the convergent sum of the commutator [P(I), A] over all finite regions I. We will invert this relation under the assumption (obviously assumed in the usual approach) that local observables all have the first time derivative, i.e. we prove the existence of potentials for any given dynamics satisfying the abovestated condition. Further more, by imposing a further condition for the potential P(I) to be chosen for each I that it does not have a portion which can be shifted to potentials for any proper subset of I, we also show (1) the existence, (2) uniqueness, (3) an automatic convergence property for the sum over I, and (4) a quite convenient property for the chosen potential. The soobtained properties (3) and (4) are not assumed and are very useful, though they were never noticed nor used before. We consider a system of finite kinds of finite spins and fermions on a discrete lattice, local regions being all finite subsets of the lattice and all local algebras being full matrix algebras of finite dimensions. For all dynam ics for which all elements of any local algebra is once time differentiable, we prove that there exist a system of potentials which describe the time derivative of the given dynamics by a convergent sum stated above. The fundamental technique for finding such potentials is a noncommutative expectation which is defined on the basis of a product state of the algebra. For each choice of a product state, we obtain one expectation which produces one set of potentials, all of which satisfy what we call the standardness condition and the con vergence condition. We call this family of potentials standard potentials (corresponding to any specific choice of the product state). The standard potentials corresponding to different product states are different but pro duces the same time derivative, known as equivalent potentials. DOI: 10.1134/S1063779610060031
1.1 OUTLINE The System under consideration consists of Fermi ons and spins on a discrete lattice ⺜ (e.g. ⺪d), descrebed by the Algebra of Observables Ꮽ, which is a C*algebra with the following structure: (i) Grading: Θ ∈ Aut Ꮽ, Θ2 = id. Ꮽ = Ꮽ+ + Ꮽ–, Ꮽ± = {A ∈ Ꮽ; Θ(A) = ±A}. (Θ counts Fermion evenoddness.) (ii) Local Structure: For each finite subset I of ⺜ (denoted I ⺜), a C*subalgebra Ꮽ(I) of Ꮽ, ({A(I); I ⺜} satisfying natural mutual relations.) (iii) Graded Commutation Relations: If I1 ∩ I2 = ∅, Ai ∈ Ꮽ σi (Ii), σi =
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