A dynamics (i.e. a one-parameter group of automorphisms) of a system described by a C *-algebra with a local structure in terms of C *-subalgebras A ( I ) for local domains I of the physical space (a discrete lattice) is normally constructed out of potentials P ( I ), each of which is a self-adjoint element of the subalgebra A ( 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 above-stated condition. Furthermore, 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 so-obtained 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 dynamics 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 non-commutative 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 convergence 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 produces the same time derivative, known as equivalent potentials.
A dynamics (i.e. a one-parameter group of automorphisms) of a system described by a C *-algebra with a local structure in terms of C *-subalgebras A ( I ) for local domains I of the physical space (a discrete lattice) is normally constructed out of potentials P ( I ), each of which is a self-adjoint element of the subalgebra A ( 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 above-stated condition. Furthermore, 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 so-obtained 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 dynamics 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 non-commutative 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 convergence 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 produces the same time derivative, known as equivalent potentials.
Introduction In this chapter, we shall discuss a unified axiomatic treatment of quantum lattice systems and quasi-free dynamics in Fermion field theory within the framework of C*-dynamical systems. In these systems, time evolution, equilibrium states (KMS states), ground states, stability under bounded perturbations and phase transitions are important physical notions. Here we shall present an abstract treatment of these notions within the theory of C*-dynamical systems. About time evolution, we shall emphasize the approximate innerness property for the corresponding time automorphism group in a given C*-dynamical system. In fact, the approximate innerness assures the existence of a ground state, and the existence of a KMS state at each inverse temperature under the assumption of ‘the existence of a tracial state’, which is always satisfied in quantum lattice systems and in canonical anticommutation relation algebras. We shall also discuss in detail one of the most important open problems (the Powers-Sakai conjecture) in C*-dynamical systems. In each section we shall explain relations between C*-dynamical systems and physical systems. For simplicity we shall assume that every C*-algebra has an identity (unless otherwise stated). Approximately inner C*-dynamics A quantum lattice system consists of a set of particles confined to a lattice and interacting at distance. There are two physical interpretations of these models. One is a lattice gas and the other is a spin system. The lattice gas views each point of the lattice as a possible site for a finite number of AT-particles, i.e. each point of the configuration space can be empty or occupied by 1, 2, …, N -particles.
An infinite quantum system with correctly defined dynamics τQ as an automorphism group of a C*-algebra 𝒞 of observables is determined by any continuous unitary representation U(G) of a connected Lie group G, as well as by an arbitrary differentiable real function Q on the dual space 𝔤* to the Lie algebra 𝔤 of G with the canonically defined Poisson flow φQ on 𝔤*. For specific choices of Q and G, the system can be obtained as the thermodynamic limit of a net of finite lattice systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)]. A simple nontrivial model of this type is the quasispin BCS model of superconductivity in the strong coupling limit, or a corresponding model of the Josephson junction. A peculiar feature of the considered models is τQ noninvariance of the usually considered C*-algebra 𝒜 of quasilocal observables, as well as an important role of classical dynamics of a set of macroscopic (intensive) observables Qφ in the description of τQ. The work is restricted here to norm-continuous representations U(G), in which case 𝒞 is isomorphic to the tensor product 𝒜⊗𝒩, where 𝒩 is the commutative algebra of classical (intensive) observables of the considered infinite quantum system.
Following the work of P. Bona [J. Math. Phys. 29, 2223 (1988)], a class of infinite mean-field quantum lattice systems is considered within the framework of algebraic quantum mechanics, and the following question is discussed: Which are the possible dynamical descriptions of the system, i.e., in which representations π of the system’s quasilocal C*-algebra 𝒜 do the local time evolutions converge in the thermodynamic limit to an automorphism group of a certain enlarged C*-algebra 𝒞π containing π(𝒜) and (some of) the classical observables of the system in π. To this end we associate a ‘‘classical state space’’ Eπ to any representation π (which is a subset of the phase-space E⊇RL defined by Bona for the largest possible representation), and show how it is characterized by the π-normal states, thereby obtaining a way to determine Eπ explicitly in many cases. The answer to the question posed then reads: The limiting (Heisenberg) dynamics exists in π on 𝒞π if and only if the set Eπ is invariant under the flow φQt on E which describes the time evolution of classical observables. Conversely, to any invariant, closed subset B of E, there exists a dynamical description of the quantum system with classical state space B.
In this chapter we set up a classical Lagrangian and Hamiltonian dynamics for matrix models. The fundamental idea is to set up an analog of classical dynamics in which the phase space variables are non-commutative, and the basic tool that allows one to accomplish this is cyclic invariance under a trace. Since no assumptions about commutativity of the phase space variables (such as canonical commutators/anticommutators) are made at this stage, the dynamics that we set up is not the same as standard quantum mechanics. Quantum mechanical behavior will be seen to emerge only when, in Chapters 4 and 5, we study the statistical mechanics of the classical matrix dynamics formulated here. In Section 1.1, we introduce our basic notation for bosonic and fermionic matrices, and give the cyclic identities that will be used repeatedly throughout the book. In Section 1.2, we define the derivative of a trace quantity with respect to an operator, and give the basic properties of this definition. In Section 1.3, we use the operator derivative to formulate a Lagrangian and Hamiltonian dynamics for matrix models. In Section 1.4, we introduce a generalized Poisson bracket appropriate to trace dynamics, constructed from the operator derivative defined in Section 1.2, and give its properties and some applications. Finally, in Section 1.5 we discuss the relation between the trace dynamics time evolution equations, and the usual unitary Heisenberg picture equations of motion obtained when one assumes standard canonical commutators/anticommutators. Bosonic and fermionic matrices and the cyclic trace identities We shall assume finite-dimensional matrices, although ultimately an extension to the infinite-dimensional case may be needed. The matrix elements of these matrices will be constructed from ordinary complex numbers, and from complex anticommuting Grassmann numbers.
Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].
Let M be a von Neumann algebra and let (Nt)t∈[0,T] be an increasing family of abelian von Neumann subalgebras encoding a (classical) information flow. Fix a faithful normal state φρ and a filtration of normal φρ-preserving conditional expectations Et:M→Nt satisfying the tower property. Using bounded functional-calculus cutoffs fn, we introduce a truncation-stable notion of localized (Nt,Et)-martingales for affiliated self-adjoint observables, and formulate a \emph{Local Informational Efficiency Principle} requiring symmetrically discounted traded prices to be martingales in this sense. Assuming a pricing state φ⋆ and a compatible family of normal φ⋆-preserving conditional expectations (Et⋆), we define for bounded terminal payoffs X∈MT the dynamic pricing operator Πt(X):=Bt1/2Et⋆(BT−1/2XBT−1/2)Bt1/2, where (Bt) is a strictly positive numéraire adapted to (Nt). We prove that (Πt) is normal, completely positive, Nt-bimodular, and time-consistent, and satisfies Πt(1)=Bt (equivalently, Πt(X):=Bt−1/2Πt(X)Bt−1/2 is unital). In the commutative reduction it agrees with risk-neutral valuation by conditional expectation. Finally, we develop an L2(M,φρ) prediction theory and introduce an operator-valued Fisher information relative to (Nt), obtaining a noncommutative Cramér--Rao lower bound for conditional mean-square prediction error; we compute the bound for compound Poisson lattice-jump models under ∑αγα(eαΔx−1)=r.