Showing papers in "Letters in Mathematical Physics in 1977"
TL;DR: In this paper, the mathematical properties of deformations of the Poisson Lie algebra and of the associative algebra of functions on a symplectic manifold are given, and the suggestion to develop quantum mechanics in terms of these deformations is confronted with the mathematical structure of the latter.
Abstract: Mathematical properties of deformations of the Poisson Lie algebra and of the associative algebra of functions on a symplectic manifold are given. The suggestion to develop quantum mechanics in terms of these deformations is confronted with the mathematical structure of the latter. As examples, spectral properties of the harmonic oscillator and of the hydrogen atom are derived within the new formulation. Further mathematical generalizations and physical applications are proposed.
238 citations
TL;DR: In this article, the authors give an algebraic condition in order that a completely positive dynamical semigroup of an N-level system has a unique (invariant) equilibrium state and that every initial state approaches this equilibrium state as t→∞.
Abstract: We give an algebraic condition in order that a completely positive dynamical semigroup of an N-level system has a unique (invariant) equilibrium state and that every initial state approaches this equilibrium state as t→∞. We apply our result to a semigroup arising in the weak coupling limit.
202 citations
TL;DR: In this paper, the quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered and it is proved that the operators introduced by Calogero et al. are integrals of motion and that they commute.
Abstract: The quantum version of the dynamical systems whose integrability is related to the root systems of semi-simple Lie algebras are considered. It is proved that the operators Ĵk introduced by Calogero et al. are integrals of motion and that they commute. The explicit form of another class of integrals of motion is given for systems with few degrees of freedom.
145 citations
TL;DR: For a quantum dynamical semigroup possessing a faithful normal stationary state, some conditions are discussed, which ensure the uniqueness of the equilibrium state and/or the approach to equilibrium for arbitrary initial condition as discussed by the authors.
Abstract: For a quantum dynamical semigroup possessing a faithful normal stationary state, some conditions are discussed, which ensure the uniqueness of the equilibrium state and/or the approach to equilibrium for arbitrary initial condition.
101 citations
TL;DR: In this article, the general form of the induced completely positive map of the C*-algebra of the canonical commutation relations is characterized given any operator on the test function space.
Abstract: Given any operator on the testfunction space, the general form of the induced completely positive map of the C*-algebra of the canonical commutation relations is characterized.
87 citations
TL;DR: In this paper, it was shown that the linear scattering problem for a non-linear evolution equation admits soliton solutions may be described in terms of a linear connection on a principal SL(2, ℝ), which is satisfied if and only if the curvature of this connection vanishes.
Abstract: It is pointed out that the linear scattering problem for a non-linear evolution equation which admits soliton solutions may be described in terms of a linear connection on a principal SL(2, ℝ). The equation in question is satisfied if and only if the curvature of this connection vanishes. Some other properties of the curvature are identified. The sine-Gordon, Korteweg-de Vries and modified Korteweg-de Vries equations are treated explicitly.
65 citations
TL;DR: In this paper, a generalization of the notion of coherent states is given, and one-to-one correspondences between covariant overcomplete systems of coherent state and a class of covariant semi-spectral measures are shown.
Abstract: A generalization of the notion of coherent states is given. The following one-to-one correspondences are pointed out: (1) between covariant overcomplete systems of coherent states and a class of covariant semi-spectral measures; (2) between covariant semispectral measures and unitary irreducible subrepresentations of induced representations of Lie groups; (3) between unitary irreducible representations of Lie groups with covariant overcomplete systems of coherent states and unitary irreducible subrepresentations of induced representations, whose representation spaces are reproducing kernel Hilbert spaces.
33 citations
TL;DR: In this paper, the connection between non-linear wave equations covariant under the action of a Lie group and the theory of nonlinear representations of the covariance group developed elsewhere is presented.
Abstract: A discussion about the connection between non-linear wave equations covariant under the action of a Lie group on one hand, and the theory of non-linear representations of the covariance group developed elsewhere [1] on the other hand, is presented here.
29 citations
TL;DR: In this article, coherent states on the dynamical group of the nonrelativistic Kepler (hydrogen atom) problem were introduced, and in the limit of high excitation these states are well concentrated wavepackets which move along classical trajectories.
Abstract: We introduce coherent states on the dynamical group of the nonrelativistic Kepler (hydrogen atom) problem. In the limit of high excitation these states are well concentrated wavepackets which move along classical trajectories.
25 citations
TL;DR: The existence of invariant twisted products on the cotangent bundles of classical groups and Stiefel manifolds is proved by explicit constructions as discussed by the authors, and all these products are positive.
Abstract: The existence of invariant twisted products (deformations of the associative algebra of C∞-functions) on the cotangent bundles of classical groups and Stiefel manifolds is proved by explicit constructions. All these products are positive.
17 citations
TL;DR: In this article, it was shown that any conservation law for differential equations is derived from the invariance properties of the equations with respect to a group of Lei-Backlund tangent transformations.
Abstract: The purpose of this paper is to establish a group theoretical foundation of conservation theorems for arbitrary systems of partial differential equations. It is shown that any conservation law for differential equations is derivable from the invariance properties of the equations with respect to a group of Lei-Backlund tangent transformations.
TL;DR: In this article, the authors summarize the results about the classical limit of relativistic quantum field models and discuss the validity of the loop expansion in the case of soliton calculations.
Abstract: Recently, there has been an increasing interest in computing quantum mechanical corrections to solutions of classical field equations. In this note, we want to proceed in the opposite way and we summarize theorems about the classical limit of relativistic quantum field models. These results are a byproduct of the so called ‘constructive’ approach to quantum field theory. After a section on generalities, we discuss in Section 2 the situation where no phase transitions occur in the limith→0 and in Section 3 we reformulate one result in the case where such a transition occurs (Glimmet al. [7]). We discuss the validity of the loop expansion. It seems however that the tools to show the rigorous validity of soliton calculations are not yet prepared.
TL;DR: A connection between deformation of Lie group representations and deformations of associated Lie algebra representations is established in this paper, where applications are given to the theory of analytic continuation of K-finite quasi-simple representations of semi-simple Lie groups.
Abstract: A connection between deformation of Lie group representations and deformations of associated Lie algebra representations is established. Applications are given to the theory of analytic continuation of K-finite quasi-simple representations of semi-simple Lie groups. A construction process of all TCI representations of SL(2,R) is obtained.
TL;DR: The connection between nonlinear autonomous dynamic systems, limit cycles, and one-parameter groups of transformations is described in this article, and an example is given to illustrate the approach.
Abstract: The connection between nonlinear autonomous dynamic systems, limit cycles, and one-parameter groups of transformations is described. To illustrate the approach, an example is given.
TL;DR: In this article, a method for finding non-linear spinor field equations which have stationary, localized solutions (solitons or droplets) of definite spin and parity is presented.
Abstract: A method is outlined for finding non-linear spinor field equations which have stationary, localized solutions (solitons or droplets) of definite spin and parity.
TL;DR: In this article, it was shown that these estimates are actually met for anyn ≥ 3 on an open set in IRinf+supn, and for any n ≥ 4 this open set is proper.
Abstract: Estimates on a minimal classification of relative equilibria in the planarn-body problem of celestial mechanics have been announced in [1], [2]. Our main theorem asserts that these estimates are actually met for anyn≧3 on an open set in IRinf+supn. For anyn≧4, this open set is proper.
TL;DR: In this article, it is pointed out that for the null-plane quantization a paradox arises in connection with Haag's theorem and a prescription is proposed to overcome this difficulty.
Abstract: It is pointed out that for the null-plane quantization a paradox arises in connection with Haag's theorem. A prescription is proposed to overcome this difficulty.
TL;DR: In this paper, the Schrodinger operator with magnetic vector potential and static scalar potential is considered and the existence of wave operators under considerations which allow strong oscillations of the potentials is shown.
Abstract: We consider the Schrodinger operator with magnetic vector potential and static scalar potential We show the existence of the wave operators under considerations which allow strong oscillations of the potentials
TL;DR: In this paper, the existence and uniqueness theorem of non-compact maximal space-like hypersurfaces for asymptotically flat metrics in a neighbourhood of the Minkowski's metric is proved.
Abstract: We study some properties of weighted Sobolev spaces with asymptotic conditions appropriate to physical conditions on non compact space times. An existence and uniqueness theorem of non compact maximal space-like hypersurfaces for asymptotically flat metrics in a neighbourhood of the Minkowski's metric is proved.
TL;DR: In this article, the authors considered operators H0 and V possessing the following properties:==================�€£££€�€€�£ £££ £€� ££€� £€£€£ £ £€�££$££/$££
Abstract: In this paper we consider operatorsH0 andV possessing the following properties:
(1)
H0 is a positive self-adjoint operator acting inL2(M, γ) with γ a probability measure, so that exp(−tH0) is a contraction onL1(M, γ) for eacht>0.
(2)
V is a semibounded multiplicative operator acting inL2(M, γ) {fx379-1}
TL;DR: The capabilities of the system are outlined and some of the physical problems it has considered as well as others it is examining at this time are described.
Abstract: We describe a new computational tool for physical calculations. It is the first computer system capable of performing indicial tensor calculus (as opposed to component tensor calculus). It is now operational on the symbolic manipulation system MACSYMA. We outline the capabilities of the system and describe some of the physical problems we have considered as well as others we are examining at this time.
TL;DR: In this article, a semigroup positivity preserving approach was used to prove asymptotic completeness of wave operators in many cases when they exist, where the wave operators are assumed to be wave operators.
Abstract: We use a semigroup positivity preserving to prove asymptotic completeness of the wave operators in many cases when they exist.
TL;DR: The Dirac theory of magnetic poles is equivalent to a Maxwell electrodynamics in which besides the point singularities, extended singularities (e.g. strings) occur as mentioned in this paper.
Abstract: Dirac theory of magnetic poles is equivalent to a Maxwell electrodynamics in which besides the point singularities (electric charges), extended singularities (e.g. strings) occur. The nature of singularities determine completely the theory, hence Betti numbers of space must occur as quantum numbers. Magnetic charge is one of the fundamental periods of the 2-formF.
TL;DR: The Backlund transformations for a physically interesting class of nonlinear partial differential equations can be interpreted as generalisations of the Cauchy Riemann equations or as nonlinear Dirac equations.
Abstract: It is pointed out that the Backlund transformations for a physically interesting class of nonlinear partial differential equations can be interpreted as generalisations of the Cauchy Riemann equations or as nonlinear Dirac equations. The generalisations are inhomogenisations of the Cauchy Riemann equations (or their hyperbolic analogue), whose condensed form makes the transformations easy to remember, which suggests ways to generalise to more than 2 dimensions, and which suggest that complex analysis techniques may be helpful in understanding the transformations.
TL;DR: In this paper, the coupled spin-2-spin-3/2 system (supergravity) is obtained in Cartan's geometrical language of differential forms, and the spin 3/2 field is introduced through a canonical spinor differential one-form.
Abstract: The coupled spin-2-spin-3/2 system (supergravity) is obtained in Cartan's geometrical language of differential forms. The spin-3/2 field is introduced through a canonical spinor differential one-form. Two points are discussed: the introduction of higher spin fields and the origin of local supersymmetry.
TL;DR: In this article, it was shown that the spectrum of the unitary partU of the polar decomposition of am-sectorial operator A=UK is contained in the closure of {(x|Ax):x∈D(A)}.
Abstract: It is shown that the spectrum of the unitary partU of the polar decomposition of am-sectorial operatorA=UK is contained in the closure of {(x|Ax):x∈D(A)}.
TL;DR: In this article, a constructive definition of parity, charge-conjugation and time-reversal operations for both a relativistic system of self-interacting classical Bose field and two massive Fermi fields with vector-axial vector interaction is given.
Abstract: A constructive definition of parity, charge-conjugation and time-reversal operations for both a relativistic system of self-interacting classical Bose field and a relativistic two massive Fermi fields with vector-axial vector interaction is given The ‘in’ and ‘out’ discrete operations are explicitely constructed and a simple dynamical mechanism of violation of these symmetries is suggested
TL;DR: In this paper, the decomposition of the tensor product of finite and infinite representations of a complex semigroup of a Lie group is examined by using the theory of characters of completely irreducible representations.
Abstract: The problem of the decomposition of the tensor product of finite and infinite representations of a complex semigroup of a Lie group is examined by using the theory of characters of completely irreducible representations. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite and elementary representation.
TL;DR: In this paper, the authors give a complete description of symmetric and phase invariant states on spin systems and construct explicitly the corresponding G.I.R.N. representations, and establish a correspondence between these states and their ergodic decomposition.
Abstract: We give a complete description of symmetric and phase invariant states on spin systems and construct explicitly the corresponding G.N.S. representations. We establish a correspondence between, on one hand, these states and their ergodic decomposition and, on the other hand, a class of unitary representations of E(2)×ℝ and their decomposition in U.I.R. We interpret the thermodynamic limit a contraction of representations of U(2) to representations of E(2)×ℝ.
TL;DR: In this paper, the authors emphasise the geometrical partnership of the vierbein and the spin3/2 field in the structure of the supergrvity Lagrangian.
Abstract: We emphasise the geometrical partnership of the vierbein and the spin3/2 field in the structure of the supergrvity Lagrangian. Both fields are introduced as components of the same matrix differential form. The only local symmetry of the theory is SL(2,C).