Deformation quantization of Heisenberg manifolds

ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.This work was supported in part by National Science Foundation grant DMS 8601900     

2-Cocycles and Twisting of Kac Algebras

We describe the twisting construction with the help of 2-cocycles on Hopf–von Neumann and George Kac algebras; we show that twisted Kac algebras are again Kac algebras. Using this construction, we give a wide class of new quantizations of the Heisenberg group and describe several series of non-trivial finite- dimensional -Hopf algebras (Kac algebras) of dimensions 4n and as twisting of finite groups. Received: Received: 21 March 1997 / Accepted: 2 June 1997     

The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra

The left spectrum of a wide class of the algebras of skew differential operators is described. As a sequence, we determine and classify all the algebraically irreducible representations of the quantum Heisenberg algebra over an arbitrary field.     

A general class of cut-off model field theories

We show that Heisenberg picture fields and their vacuum expectation values exist for a wide class of cut-off interactions among fermions and bosons.Alfred P. Sloan Foundation Fellow.On leave from Princeton University     

Regular Weyl-Systems and Smooth Structures on Heisenberg Groups

We study representation theory of the Weyl relations for infinitely many degrees of freedom. Differentiability of regular representations along rays in the parameter space E suggests to consider smooth structures on E. Switching from representations of CCR to group representations of the associated Heisenberg group over E we develop a framework for smooth representations of the Heisenberg group as an infinite dimensional Lie group. After careful inspection and translation of the necessary differential geometric input for Kirillov's orbit method we are able to construct a large class of smooth representations. These reproduce the Schr?dinger representation if E is finite dimensional. Received: 10 May 1996 / Accepted: 30 July 1996     

Projective representations of the inhomogeneous Hamilton group: Noninertial symmetry in quantum mechanics

Symmetries in quantum mechanics are realized by the projective representations of the Lie group as physical states are defined only up to a phase. A cornerstone theorem shows that these representations are equivalent to the unitary representations of the central extension of the group. The formulation of the inertial states of special relativistic quantum mechanics as the projective representations of the inhomogeneous Lorentz group, and its nonrelativistic limit in terms of the Galilei group, are fundamental examples. Interestingly, neither of these symmetries include the Weyl–Heisenberg group; the hermitian representations of its algebra are the Heisenberg commutation relations that are a foundation of quantum mechanics. The Weyl–Heisenberg group is a one dimensional central extension of the abelian group and its unitary representations are therefore a particular projective representation of the abelian group of translations on phase space. A theorem involving the automorphism group shows that the maximal symmetry that leaves the Heisenberg commutation relations invariant is essentially a projective representation of the inhomogeneous symplectic group. In the nonrelativistic domain, we must also have invariance of Newtonian time. This reduces the symmetry group to the inhomogeneous Hamilton group that is a local noninertial symmetry of the Hamilton equations. The projective representations of these groups are calculated using the Mackey theorems for the general case of a nonabelian normal subgroup.     

Quantum and Classical Brackets

We describe p-mechanical (Kisil, V. V. (1996). Journal of Natural Geometry 9(1), 1–14; Kisil, V. V. (1999). Advances in Mathematics 147(1), 35–73; Prezhdo, O. V. and Kisil, V. V. (1997). Physical Review A 56(1), 162–175) brackets that generate quantum (commutator) and classical (Poisson) brackets in corresponding representations of the Heisenberg group. We do not use any kind of semiclassical approximation or limiting procedure for 0     

Alternating-spin S = 3/2 and <Emphasis Type="Italic">σ</Emphasis> = 1/2 Heisenberg chain with isotropic three-body exchange interactions

The promotion of collinear classical spin configurations as well as the enhanced tendencytowards nearest-neighbor clustering of the quantum spins are typical features of thefrustrating isotropic three-body exchange interactions in Heisenberg spin systems. Basedon numerical density-matrix renormalization group calculations, we demonstrate that theseextra interactions in the Heisenberg chain constructed from alternating S = 3/2 and σ = 1/2 site spins can generate numerous specific quantum spinstates, including some partially-polarized ferrimagnetic states as well as adoubly-degenerate non-magnetic gapped phase. In the non-magnetic region of the phasediagram, the model describes a crossover between the spin-1 and spin-2 Haldane-typestates.     

Star product geometries

We consider noncommutative geometries obtained from a triangular Drinfeld twist. This allows us to construct and study a wide class of noncommutative manifolds and their deformed Lie algebras of infinitesimal diffeomorphisms. Principles of symmetry can be implemented in this way. Two examples are considered: (a) the general covariance in noncommutative spacetime, which leads to a noncommutative theory of gravity, and b) symplectomorphims of the algebra of observables associated with noncommutative configuration spaces, which leads to a geometric formulation of quantization on noncommutative spacetime (i.e., we establish a noncommutative correspondence principle from ⋆-Poisson brackets to ⋆-commutators).     

On non-singular generalised dynamical formalisms

This work shows that a certain class of classical dynamical formalisms, characterised by non-singular Lie structures more general than the usual (Poisson) one, are derivable from ordinary constrained dynamical formalisms. As a consequence, the Lie brackets considered are special cases of suitably chosen Dirac brackets. Both unconstrained and constrained generalised dynamical formalisms are considered. The relations of our results with the problem of constructing classical analogues of generalised quantum systems are stressed.     

NAMBU BRACKET FORMULATION OF NONLINEAR BIOCHEMICAL REACTIONS BEYOND ELEMENTARY MASS ACTION KINETICS

We develop a Nambu bracket formulation for a wide class of nonlinear biochemical reactions by exploiting previous work that focused on elementary biochemical mass action reactions. To this end, we consider general reaction mechanisms including for example enzyme kinetics. Furthermore, we go beyond elementary reactions and account for reactions involving stoichiometric coefficients different from unity. In particular, we show that the stoichiometric matrix of biochemical reactions can be expressed in terms of Nambu brackets. Finally, we solve the sign problem that arises in the context of coupled biochemical reactions.     

Entanglement conditions for four-mode states via the uncertainty relations in conjunction with partial transposition

From the Heisenberg uncertainty relation in conjunction with partial transposition, we derive a class of inequalities for detecting entanglements in four-mode states. The sufficient conditions for bipartite entangled states are presented. We also discuss the generalization of the entanglement conditions via the Schrödinger-Robertson indeterminacy relation, which are in general stronger than those based on the Heisenberg uncertainty relation.     

Equivariant quantization on quotients of simple Lie groups by reductive subgroups of maximal rank

We consider a class of homogeneous manifolds including all semisimple coadjoint orbits. We describe manifolds of that class admitting deformation quantizations equivariant under the action ofG and the corresponding quantum group. We also classify Poisson brackets relating to such quantizations. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

On BRST quantization of second class constraint algebras

A BRST quantization of second-class constraint algebras that avoids Dirac brackets is constructed, and the BRST operator is shown to be related to the BRST operator of first class algebra by a nonunitary canonical transformation. The transformation converts the second class algebra into an effective first class algebra with the help of an auxiliary second class algebra constructed from the dynamical Lagrange multipliers of the Dirac approach. The BRST invariant path integral for second class algebras is related to the path integral of the pertinent Dirac brackets, using the Parisi-Sourlas mechaism. As an application the possibility of string theories in subcritical dimensions is considered.     

The fermionic Heisenberg group and its Q-representations

A nonstandard way of representing canonical anticommutation relations is presented. It is connected with a generalization of the Heisenberg group to the case of graded phase space. We show how Grassmann harmonic analysis can be performed and what are the Q-representations of such a generalized Heisenberg group. As in the conventional case, the Schrödinger and Bargmann-Fock realizations are shown to exist. The Grassmann-Hermite polynomials via the generalized Bargmann transform are presented and new Grassmann-Laguerre polynomials are obtained.     

Correlation inequalities for some classical spin vector models

Correlation inequalities are derived for a class of lattice systems including some classical anisotropic X-Y and Heisenberg ferromagnets. In particular, a comparison is estab- lished between some correlation functions of the X-Yand Heisenberg models.     

Ising-type critical exponents of the fully frustrated spin-1/2 Heisenberg FM/AF square bilayer at a critical magnetic field

The fully frustrated spin-1/2 Heisenberg FM/AF square bilayer in a magnetic field with the ferromagnetic inter-dimer interaction and the antiferromagnetic intra-dimer interaction is explored by the use of localized many-magnon approach, which allows to connect the original purely quantum Heisenberg spin model on a square bilayer with the effective ferromagnetic Ising model on a simple square lattice. Magnetization and specific heat are investigated exactly at a field-driven phase transition from the singlet-dimer phase towards the fully saturated ferromagnetic phase, which changes from a discontinuous phase transition to a continuous one at a certain critical temperature. The mapping correspondence between the spin-1/2 Heisenberg FM/AF square bilayer and the ferromagnetic Ising square lattice suggests for this special critical point of the spin-1/2 Heisenberg FM/AF square bilayer critical exponents from the standard two-dimensional Ising universality class.     

Proof of the Wehrl-type Entropy Conjecture for Symmetric {SU(N)} Coherent States

The Wehrl entropy conjecture for coherent (highest weight) states in representations of the Heisenberg group, which was proved in 1978 and recently extended by us to the group \({SU(2)}\), is further extended here to symmetric representations of the groups \({SU(N)}\) for all N. This result gives further evidence for our conjecture that highest weight states minimize group integrals of certain concave functions for a large class of Lie groups and their representations.