Index of content:
Volume 37, Issue 11, November 1996

Isomorphisms between the Batalin–Vilkovisky antibracket and the Poisson bracket
View Description Hide DescriptionOne may introduce at least three different Lie algebras in any Lagrangianfield theory: (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin–Vilkovisky antibracket, which has attracted considerable interest recently; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number −1 and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number −1, while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from −1, a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin, and Vilkovisky.

On Green–Cusson Ansätze and deformed supersymmetric quantum mechanics
View Description Hide DescriptionSupersymmetric quantum mechanics cannot be deformed when the superposition of only a pair of usual bosons and fermions is considered, but it can if nontrivial parabosons and parafermions of the same order p of paraquantization are superposed. We take the simplest case p=2 and exhibit reducibility problems in that context by using Green–Cusson Ansätze following Macfarlane methods. Specific representations of the Lie superalgebra osp (22, R) play an interesting role in connection with possible deformations.

Becchi–Rouet–Stora cohomology of zero curvature systems. I. The complete ladder case
View Description Hide DescriptionWe present here the zero curvature formulation for a wide class of field theory models. This formalism, which relies on the existence of an operator δ which decomposes the exterior space–time derivative as a BRS commutator, turns out to be particularly useful in order to solve the Wess–Zumino consistency condition. The examples of the topological theories and of the B−C string ghost system are considered in detail.

Becchi–Rouet–Stora cohomology of zero curvature systems. II. The noncomplete ladder case
View Description Hide DescriptionThe Yang–Mills‐type theories and their BRS cohomologies are analyzed within the zero curvature formalism.

Density conditions for quantum propositions
View Description Hide DescriptionAs has already been pointed out by Birkhoff and von Neumann, quantum logic can be formulated in terms of projective geometry. In three‐dimensional Hilbert space, elementary logical propositions are associated with one‐dimensional subspaces, corresponding to points of the projective plane. It is shown that, starting with three such propositions corresponding to some basis {u,v,w}, successive application of the binary logical operation (x,y)⟼(x∨y)^{⊥} generates a set of elementary propositions which is countable infinite and dense in the projective plane if and only if no vector of the basis {u,v,w} is orthogonal to the other ones.

Geometrical interpretation of BRST symmetry in topological Yang–Mills–Higgs theory
View Description Hide DescriptionWe study the topological Yang–Mills–Higgs theories in two and three dimensions and the topological Yang–Mills theory in four dimensions in a unified framework of superconnections. In this framework, we first show that a classical action of topological Yang–Mills type can provide all three classical actions of these theories via appropriate projections. Then we obtain the Becchi–Rouet–Stora–Tyutin (BRST) and anti‐BRST transformation rules encompassing these three topological theories from an extended definition of curvature and a geometrical requirement of the Bianchi identity. This is an extension of Perry and Teo’s work in the topological Yang–Mills case. Finally, comparing this result with our previous treatment in which we used the ‘‘modified horizontality condition,’’ we provide a meaning of the Bianchi identity from the BRST symmetry viewpoint and thus interpret the BRST symmetry in a geometrical setting.

Reformulation of QCD in the language of general relativity
View Description Hide DescriptionIt is shown that there exists such a collection of variables that the standard QCDLagrangian can be represented as the sum of usual Palatini Lagrangian for Einstein general relativity and the Lagrangian of matter and some other fields where the tetrad fields and the metric are constructed from initial SU(3) Yang–Mills fields.

On the consistent effect histories approach to quantum mechanics
View Description Hide DescriptionA formulation of the consistent histories approach to quantum mechanics in terms of generalized observables (POV measures) and effect operators is provided. The usual notion of ‘‘history’’ is generalized to the notion of ‘‘effect history.’’ The space of effect histories carries the structure of a D‐poset. Recent results of J. D. Maitland Wright imply that every decoherence functional defined for ordinary histories can be uniquely extended to a bi‐additive decoherence functional on the space of effect histories. Omnès’ logical interpretation is generalized to the present context. The result of this work considerably generalizes and simplifies the earlier formulation of the consistent effect histories approach to quantum mechanics communicated in a previous work of this author.

Greechie diagrams, nonexistence of measures in quantum logics, and Kochen–Specker‐type constructions
View Description Hide DescriptionWe use Greechie diagrams to construct finite orthomodular lattices ‘‘realizable’’ in the orthomodular lattice of subspaces in a three‐dimensional Hilbert space such that the set of two‐valued states is not ‘‘large’’ (i.e., full, separating, unital, nonempty, resp.). We discuss the number of elements of such orthomodular lattices, of their sets of (ortho)generators and of their subsets that do not admit a ‘‘large’’ set of two‐valued states. We show connections with other results of this type.

The canonical form of the Rabi Hamiltonian
View Description Hide DescriptionThe Rabi Hamiltonian, describing the coupling of a two‐level system to a single quantized boson mode, is studied in the Bargmann–Fock representation. The corresponding system of differential equations is transformed into a canonical form in which all regular singularities between zero and infinity have been removed. The canonical or Birkhoff‐transformed equations give rise to a two‐dimensional eigenvalue problem, involving the energy and a transformational parameter which affects the coupling strength. The known isolated exact solutions of the Rabi Hamiltonian are found to correspond to the uncoupled form of the canonical system.

Boson–fermion model with two‐body interactions: Exact results
View Description Hide DescriptionA one‐dimensional boson–fermion model with two‐body interactions between the two types of chiral fermions is considered. It is shown that the model is exactly soluble and the general Bethe eigenstates are constructed. On the basis of the Bethe ansatz equations, the ground state, the low lying elementary excitations, and the thermodynamics are also given in some closed integral equations.

Exact results on a Dirac‐like Lee model
View Description Hide DescriptionA one‐dimensional massive Lee model is studied via the Bethe ansatz method. The exact engenstates and the energy spectrum are obtained. The general picture of the excitations is discussed.

Microcanonical ensemble and algebra of conserved generators for generalized quantum dynamics
View Description Hide DescriptionIt has recently been shown, by application of statistical mechanical methods to determine the canonical ensemble governing the equilibrium distribution of operator initial values, that complex quantum field theory can emerge as a statistical approximation to an underlying generalized quantum dynamics. This result was obtained by an argument based on a Ward identity analogous to the equipartition theorem of classical statistical mechanics. We construct here a microcanonical ensemble which forms the basis of this canonical ensemble. This construction enables us to define the microcanonical entropy and free energy of the field configuration of the equilibrium distribution and to study the stability of the canonical ensemble. We also study the algebraic structure of the conserved generators from which the microcanonical and canonical ensembles are constructed, and the flows they induce on the phase space.

Decay of correlations and uniqueness of Gibbs lattice systems with nonquadratic interaction
View Description Hide DescriptionThe aim of this paper is to develop the classical lattice models with unbounded spin to the case of nonquadratic polynomial interaction. We demonstrate that the distinct relation between the growths of potentials leads to the uniqueness and the fast decay of correlations for Gibbs measure.

Integration of thermodynamic identities for a relativistic general equation of state
View Description Hide DescriptionIn a recent paper Mason and Kgathi have integrated some well‐known thermodynamic identities for an equation of statep=nkT and thereby have obtained some explicit expressions for the thermodynamic variables concerned. In the present paper we integrate the same identities for a more general equation of statep=p(n,T) and thereby obtain more general expressions for the same thermodynamic variables.

The low activity phase of some Dirichlet series
View Description Hide DescriptionWe show that a rigorous statistical mechanics description of some Dirichlet series is possible. Using the abstract polymermodel language of statistical mechanics and the polymer expansion theory we characterize the low activity phase by the suitable exponential decay of the truncated correlation functions.

Criteria for inverted temperatures in evaporation–condensation processes
View Description Hide DescriptionA recent criterion for shock waves which predicts the behaviors of the internal energy across the shock is extended to a vapor between two interfaces. We look at the predictions of monotonic or not behavior and at the possible inversion of the internal energy inside the gas. We consider different discrete Boltzmann models, construct new classes of exact solutions (models with a rest particle and models leading to 3×3 Riccati systems), and verify the criteria predictions.

Extension of Giacomini’s results concerning invariants for one‐dimensional time‐dependent potentials
View Description Hide DescriptionOne‐dimensional time‐dependent potentials are considered for which an invariant can be expressed in terms of the potential and the momentum according to the formulation of Giacomini. New solutions of Giacomini’s equations are derived. In addition, possibilities are discussed for extending Giacomini’s approach to more general systems.

A second invariant for one‐degree‐of‐freedom, time‐dependent Hamiltonians given a first invariant
View Description Hide DescriptionAn explicit formula for a second invariant of a one‐degree‐of‐freedom time‐dependent Hamiltonian is derived in terms of the Hamiltonian and an assumed first invariant. If the first invariant is expressed as a function of two canonical functions, a transformation to an autonomous Hamiltonian system is possible.

Inhibition of chaotic escape from a potential well using small parametric modulations
View Description Hide DescriptionIt is shown theoretically for the first time that, depending on its period, amplitude, and initial phase, a periodic parametric modulation can suppress a chaotic escape from a potential well. The instance of the Helmholtz oscillator is used to demonstrate, by means of Melnikov’s method, that parametric modulations of the linear or quadratic potential terms inhibit chaotic escape when certain resonance conditions are met.