A general theoretical framework is developed for the treatment of a class of quantum field theories that are explicitly exactly solvable, but require the use of C*-algebraic techniques because time-dependent scattering theory cannot be constructed in any one natural representation of the observable algebra. The purpose is to exhibit mechanisms by which inequivalent representations of the observable algebra can arise in quantum field theory, in a setting free of other complications commonly associated with the specification of dynamics. One of two major results is the development of necessary and sufficient conditions for the concurrent unitary implementation of two automorphism groups in a class of quasifree representations of the algebra of the canonical commutation relations (CCR). The automorphism groups considered are induced by one-parameter groups of symplectic transformations on the classical phase space over which the Weyl algebra of the CCR is built; each symplectic group is conjugate by a fixed symplectic transformation to a one-parameter unitary group. The second result, an analog to the Birman–Belopol’skii theorem in two-Hilbert-space scattering theory, gives sufficient conditions for the existence of Mo/ller wave morphisms in theories with time-development automorphism groups of the above type. In a paper which follows, this framework is used to analyze a particular model system for which wave operators fail to exist in any natural representation of the observable algebra, but for which wave morphisms and an associated S matrix are easily constructed.
A general theoretical framework is developed for the treatment of a class of quantum field theories that are explicitly exactly solvable, but require the use of C*-algebraic techniques because time-dependent scattering theory cannot be constructed in any one natural representation of the observable algebra. The purpose is to exhibit mechanisms by which inequivalent representations of the observable algebra can arise in quantum field theory, in a setting free of other complications commonly associated with the specification of dynamics. One of two major results is the development of necessary and sufficient conditions for the concurrent unitary implementation of two automorphism groups in a class of quasifree representations of the algebra of the canonical commutation relations (CCR). The automorphism groups considered are induced by one-parameter groups of symplectic transformations on the classical phase space over which the Weyl algebra of the CCR is built; each symplectic group is conjugate by a fixed symplectic transformation to a one-parameter unitary group. The second result, an analog to the Birman–Belopol’skii theorem in two-Hilbert-space scattering theory, gives sufficient conditions for the existence of Mo/ller wave morphisms in theories with time-development automorphism groups of the above type. In a paper which follows, this framework is used to analyze a particular model system for which wave operators fail to exist in any natural representation of the observable algebra, but for which wave morphisms and an associated S matrix are easily constructed.
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.
The Hilbert space of the unitary irreducible representations of a Lie group that is a quantum dynamical group are identified with the quantum state space. Hermitian representation of the algebra are observables. The eigenvalue equations for the representation of the set of Casimir invariant operators define the field equations of the system. A general class of dynamical groups are semidirect products K *s N for which the representations are given by Mackey's theory. The homogeneous group K must be a subgroup of the automorphisms of the normal group N. The archetype dynamical group is the Poincare group. The field equations defined by the representations of the Casimir operators define the basic equations of physics; Klein-Gordon, Dirac, Maxwell and so forth. This paper explores a more general dynamical group candidate that is also a semi-direct product but where the 'translation' normal subgroup N is now the Heisenberg group. The relevant automorphisms of the Heisenberg group are the symplectic group. This together with the requirement for an orthogonal metric leads to the pseudo-unitary group for the homogeneous group K. The physical meaning and motivation of this group, called the quaplectic group, is presented and the Hermitian irreducible representations of the algebra are determined. As with the Poincare group, choice of the group defines the Hilbert space of representations that are identified with quantum particle states. The field equations that are the eigenvalue equations for the representation of the Casimir operators, are obtained and investigated. The theory embodies the Born reciprocity principle and a new relativity principle.
We investigate a class of nonrelativistic, self-interacting model field theories, assuming the existence of field and momentum operators that satisfy the canonical commutation relations (CCR's). The large symmetry that we assume permits us to determine explicitly all the relevant CCR representations and Hamiltonians. For each irreducible representation there exists only one Hamiltonian, which describes no interaction. Hence every model with interaction requires a reducible CCR representation. We expect a nontrivial S operator to exist for many of these models. We examine how close our models can come to relativistic theories.
We ask which kinematical groups allow local algebras of canonical quantum fields to commute at unequal times. The Galilei group acts transitively on pairs of non-simultaneous events with a given time separation, so a Galilean-invariant commutation relation between sharp-time algebras at unequal times also holds at coincident positions; at arbitrarily short separations it contradicts the canonical commutation relations, with no spectral assumption. Under unitary Bargmann covariance and positivity of the energy modulo the central mass, the vacuum is annihilated by the annihilation fields, hence not separating for local algebras, and the vacuum two-point function of a canonical field factorises into the free Schrödinger propagator and the characteristic function of an internal-energy distribution. Sharp-time algebras at unequal times then commute only at a closed null set of time separations, bounded below by the Mandelstam-Tamm and Margolus-Levitin times and empty for sharp or generic internal energy; any finite set of separations occurs. Algebras of open spacetime regions containing time-smeared fields never commute. Positivity cannot be dropped: without it, commutation can hold at all separations beyond any given one. Among spatially isotropic homogeneous spacetimes with absolute time, only the galilean ones are transitive at fixed time separation, up to discrete exceptions in the two oscillating cases, realised by a harmonic trap in the Newton-Hooke one. Non-commutation transfers from the vacuum to every representation of a simple C*-algebra on which the dynamics acts by automorphisms, including free Bose and Fermi gases at all temperatures and the Galilei-covariant cutoff dynamics of interacting fermions of Narnhofer and Thirring; without such a vacuum representation it can fail even in thermal equilibrium.
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 second in a series of four articles we create a mathematical formalism sufficient to represent nontrivial hamiltonian quantum dynamics, including resonances. Some parts of this construction are also mathematically necessary. The specific construction is the transforming of a pair of quantized free oscillators into a resonant system of coupled oscillators by analytic continuation which is performed algebraically by the group of complex symplectic transformations, thereby creating dynamical representations of numerous semi-groups. The quantum free oscillators are the quantum analogue of classical action angle variable solutions for the coupled oscillators and quantum resonances, including Breit-Wigner resonances. Among the exponentially decaying Breit-Wigner resonances represented by Gamow vectors are hamiltonian systems in which energy transfers from one oscillator to the other. There are significant mathematical constraints in order that complex spectra be accommodated in a well defined formalism which represents dynamics, and these may be met by using the commutative real algebra $\C (1,i)$ as the ring of scalars in place of the field of complex numbers. These mathematical constraints compel use of fundamental (spinor) representations rather than UIR's. By including distributional solutions to the Schr{ö}dinger equation, placing us in a rigged Hilbert space, and by using the Hamiltonian as a generator of canonical transformation of the space of states, the Schr{ö}dinger equation is the equation for parallel transport of generalized energy eigenvectors, explicitly establishing the Hamiltonian as the generator of dynamical time translations in this formalism.