The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z, z  =  0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z  =  0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ .     

Statistics of Poincaré recurrences for maps with integrable and ergodic components

Recurrence gives powerful tools to investigate the statistical properties of dynamical systems. We present in this paper some applications of the statistics of first return times to characterize the mixed behavior of dynamical systems in which chaotic and regular motion coexist. Our analysis is local: we take a neighborhood A of a point x and consider the conditional distribution of the points leaving A and for which the first return to A, suitably normalized, is bigger than t. When the measure of A shrinks to zero the distribution converges to the exponential e(-t) for almost any point x, if the system is mixing and the set A is a ball or a cylinder. We consider instead a system, a skew integrable map of the cylinder, which is not ergodic and has zero entropy. This map describes a shear flow and has a local mixing property. We rigorously prove that the statistics of first return is of polynomial type around the fixed points and we generalize around other points with numerical computations. The result could be extended to quasi-integrable area preserving maps such as the standard map for small coupling. We then analyze the distribution of return times in a region which is composed by two invariants subdomains: one with a mixing dynamics and the other with an integrable dynamics given by our shear flow. We show that the statistics of first return in this mixed region is asymptotically given by the exponential law, but this limit is attained by an intermediate regime where exponential and polynomial laws are linearly superposed and weighted by some factors which are proportional to the relative sizes of the chaotic and regular regions. The result on the statistics of first return times for mixed regions in the phase space can provide a basis to analyze such a property for area preserving maps in mixed regions even when a rigorous result is not available. To this end we present numerical investigations on the standard map which confirm the results of the model.     

Poincaré Algebra and Space-Time Critical Dimensions for Paraspinnings Strings

In this paper, we paraquantize the spinning string theory in the Neveu-Shwarz model. Unlike the Ardalan and Mansouri work [Phys. Rev. D, Vol. 9, (1974) 3341], the paraquantum system is such that both the center of mass variables and the excitation modes of the string verify paracommutation relations. The commutators of the Poincaré algebra are satisfied, except the [p ,p ] one, since one can only write [p ,p ]= 0, for Q1. Because of the relation [x ,x ] =,0 and with the sole use of the trilinear relations, we find existence possibilities of spinning strings defined in a noncommutative space-time at space-time dimensions other than D=10.     

Lie symmetries, perturbation to symmetries and adiabatic invariants of Poincaré equations

Based on the invariance of differential equations under infinitesimal transformations, Lie symmetry, laws of conservations, perturbation to the symmetries and adiabatic invariants of Poincaré equations are presented.The concepts of Lie symmetry and higher order adiabatic invariants of Poincar\'{e} equations are proposed. The conditions for existence of the exact invariants and adiabatic invariants are proved, and their forms are also given. In addition, an example is presented to illustrate these results.     

Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces,Orbit Invariants and Applications

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi, μCH and μDP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.     

The End of a Classical Ontology for Quantum Mechanics?

In this paper, I argue that the Shrapnel–Costa no-go theorem undermines the last remaining viability of the view that the fundamental ontology of quantum mechanics is essentially classical: that is, the view that physical reality is underpinned by objectively real, counterfactually definite, uniquely spatiotemporally defined, local, dynamical entities with determinate valued properties, and where typically ‘quantum’ behaviour emerges as a function of our own in-principle ignorance of such entities. Call this view Einstein–Bell realism. One can show that the causally symmetric local hidden variable approach to interpreting quantum theory is the most natural interpretation that follows from Einstein–Bell realism, where causal symmetry plays a significant role in circumventing the nonclassical consequences of the traditional no-go theorems. However, Shrapnel and Costa argue that exotic causal structures, such as causal symmetry, are incapable of explaining quantum behaviour as arising as a result of noncontextual ontological properties of the world. This is particularly worrying for Einstein–Bell realism and classical ontology. In the first instance, the obvious consequence of the theorem is a straightforward rejection of Einstein–Bell realism. However, more than this, I argue that, even where there looks to be a possibility of accounting for contextual ontic variables within a causally symmetric framework, the cost of such an account undermines a key advantage of causal symmetry: that accepting causal symmetry is more economical than rejecting a classical ontology. Either way, it looks like we should give up on classical ontology.     

Clebsch-Gordan Coefficients for Two-Parameter Deformed Quantum Algebra SU(1,1)q,s(Ⅱ)

Abstract The Bargmann represen tations in the tensor product space of the irreducible representations for the two-parameter deformed quantum algebra SU(1,1)_q,s corresponding to the negative discrete series (b) are introduced, and the corresponding Bargmann expressions for the bases of irreps, the coherent state and the operators are also derived. The Clebsch-Gordan coeficients (CGC) for the two-parameter deformed quantum algebra SU(1,1)_q,s corresponding to the negative discrete series (b) are obtained.     

On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

Kristian Camilleri: Heisenberg and the Interpretation of Quantum Mechanics—The Physicist as Philosopher

The book Heisenberg and the Interpretation of Quantum Mechanics—The Physicist as Philosopher, by Kristian Camilleri is critically reviewed. The work details Heisenberg’s philosophical development from an early positivist commitment towards a later philosophy of language. It is of interest to researchers and graduate students in the history and philosophy of quantum mechanics.     

Automorphisms and Derivations of the Insertion–Elimination Algebra and Related Graded Lie Algebras

This paper addresses several structural aspects of the insertion–elimination algebra \({\mathfrak{g}}\), a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of \({\mathfrak{g}}\), the automorphism group of \({\mathfrak{g}}\), the derivation Lie algebra of \({\mathfrak{g}}\), and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.     

A Possible Way for Constructing Generators of the Poincaré Group in Quantum Field Theory

Starting from the instant form of relativistic quantum dynamics for a system of interacting fields, where amongst the ten generators of the Poincaré group only the Hamiltonian and the boost operators carry interactions, we offer an algebraic method to satisfy the Poincaré commutators.We do not need to employ the Lagrangian formalism for local fields with the N?ether representation of the generators. Our approach is based on an opportunity to separate in the primary interaction density a part which is the Lorentz scalar. It makes possible apply the recursive relations obtained in this work to construct the boosts in case of both local field models (for instance with derivative couplings and spins ≥ 1) and their nonlocal extensions. Such models are typical of the meson theory of nuclear forces, where one has to take into account vector meson exchanges and introduce meson-nucleon vertices with cutoffs in momentum space. Considerable attention is paid to finding analytic expressions for the generators in the clothed-particle representation, in which the so-called bad terms are simultaneously removed from the Hamiltonian and the boosts. Moreover, the mass renormalization terms introduced in the Hamiltonian at the very beginning turn out to be related to certain covariant integrals that are convergent in the field models with appropriate cutoff factors.     

Lieb–Schultz–Mattis Theorem with a Local Twist for General One-Dimensional Quantum Systems

We formulate and prove the local twist version of the Yamanaka–Oshikawa–Affleck theorem, an extension of the Lieb–Schultz–Mattis theorem, for one-dimensional systems of quantum particles or spins. We can treat almost any translationally invariant system with global U(1) symmetry. Time-reversal or inversion symmetry is not assumed. It is proved that, when the “filling factor” is not an integer, a ground state without any long-range order must be accompanied by low-lying excitations whose number grows indefinitely as the system size is increased. The result is closely related to the absence of topological order in one-dimension. The present paper is written in a self-contained manner, and does not require any knowledge of the Lieb–Schultz–Mattis and related theorems.     

Statistical characteristics of the Poincaré return times for a one-dimensional nonhyperbolic map

Characteristics of the Poincaré return times are considered in a one-dimensional cubic map with a chaotic nonhyperbolic attractor. Two approaches, local one (Kac’s theorem) and global one related with the AP-dimension estimation of return times, are used. The return times characteristics are studied in the presence of external noise. The characteristics of Poincaré recurrences are compared with the form of probability measure and the complete correspondence of the obtained results with the mathematical theory is shown. The influence of the attractor crisis on the return time characteristics is also analyzed. The obtained results have a methodical and educational significance and can be used for solving a number of applied tasks.     

Microcanonical Analysis of the Curie–Weiss Anisotropic Quantum Heisenberg Model in a Magnetic Field

The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy \(s(e,\varvec{m})\) is obtained as a function of the energy \(e\) and the magnetization vector \({\varvec{m}}\) in the thermodynamic limit. Since, for this model, \(e\) is uniquely determined by \({\varvec{m}}\) , the same information can be encoded either in \(s(\varvec{m})\) or \(s(e,m_1,m_2)\) . Although these two entropies correspond to the same physical setting of fixed \(e\) and \({\varvec{m}}\) , their concavity properties differ. The entropy \(s_{{\varvec{h}}}(u)\) , describing the model at fixed total energy \(u\) and in a homogeneous external magnetic field \({\varvec{h}}\) of arbitrary direction, is obtained by reduction from the nonconcave entropy \(s(e,m_1,m_2)\) . In doing so, concavity, and therefore equivalence of ensembles, is restored. \(s_{{\varvec{h}}}(u)\) has nonanalyticities on surfaces of co-dimension 1 in the \((u,\varvec{h})\) -space. Projecting these surfaces into lower-dimensional phase diagrams, we observe that the resulting phase transition lines are situated in the positive-temperature region for some parameter values, and in the negative-temperature region for others. In the canonical setting of a system coupled to a heat bath of positive temperatures, the nonanalyticities in the microcanonical negative-temperature region cannot be observed, and this leads to a situation of effective nonequivalence even when formal equivalence holds.     

Discrete Lagrangian Reduction,Discrete Euler–Poincaré Equations,and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincaré equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.