Deformation theory and quantization. I. Deformations of symplectic structures

We present a mathematical study of the differentiable deformations of the algebras associated with phase space. Deformations of the Lie algebra of C^∞ functions, defined by the Poisson bracket, generalize the well-known Moyal bracket. Deformations of the algebra of C^∞ functions, defined by ordinary multiplication, give rise to noncommutative, associative algebras, isomorphic to the operator algebras of quantum theory. In particular, we study deformations invariant under any Lie algebra of “distinguished observables”, thus generalizing the usual quantization scheme based on the Heisenberg algebra.     

Quantum corrections to ϕ 4 model solutions and applications to Heisenberg chain dynamics

The Heisenberg spin chain is considered in ? ⁴ model approximation. Quantum corrections to classical solutions of the one-dimensional ? ⁴ model within the correspondent physics are evaluated with account of rest d-1 dimensions of a d-dimensional theory. A quantization of the model is considered in terms of spacetime functional integral. The generalized zeta-function formalism is used to renormalize and evaluate the functional integral and quantum corrections to energy in a quasiclassical approximation. The results are applied to appropriate conditions of the spin chain model and its dynamics, for which elementary solutions, energy and the quantum corrections are calculated.     

A new model for the ground-state spin determination in ferromagnetics: The influence of many-electron exchange effects on the ground-state total spin in titanium and nickel

A new approximate exchange interaction operator, derived from the exact atomic second quantization formalism, has been applied to the study of the magnetic properties of atoms with spin $\text{S =}\text{1}\text{2}$ and S = 1 which obey Hund's rule. The new operator involves the effects of one-pair and two-pair electron exchanges. The one-pair exchange effects are compared with the Heisenberg model. It is shown that our model comprises the symmetry properties of the atomic wavefunction and the interaction of all electrons in the unfilled shells of two atoms, which are not considered in the Heisenberg theory. The conditions under which the new one-pair exchange effect may be used to improve the old one are pointed out. The significance of the two-pair exchange effects is discussed. It is demonstrated that, owing to the simultaneous exchange of two electron pairs, the total ground-state spin of two atoms may have intermediate values in addition to those predicted by the Heisenberg model. The importance of the interrelation between the first- and second-order effects, with respect to both sign and magnitude, is pointed out. The existence of the intermediate values, even in the case where the first order exchange integral is negative, is taken as a conjecture by which one may explain the fact that the mean magnetic moment per atom in magnetic 3d metals is not equal to the spin magnetic moment of an individual atom and accounts for the existence of magnetic ordering when the first order exchange integral is negative.     

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     

The Angular Momentum Dilemma and Born–Jordan Quantization

The rigorous equivalence of the Schrödinger and Heisenberg pictures requires that one uses Born–Jordan quantization in place of Weyl quantization. We confirm this by showing that the much discussed “ angular momentum dilemma” disappears if one uses Born–Jordan quantization. We argue that the latter is the only physically correct quantization procedure. We also briefly discuss a possible redefinition of phase space quantum mechanics, where the usual Wigner distribution has to be replaced with a new quasi-distribution associated with Born–Jordan quantization, and which has proven to be successful in time-frequency analysis.     

Fractional Heisenberg equation

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/?)[H,⋅], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this Letter, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/?)[H,⋅]. As a result, we obtain a fractional generalization of the Heisenberg equation. The fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. The suggested Heisenberg equation generalize a notion of quantum Hamiltonian systems to describe quantum dissipative processes.     

Approximation to the susceptibilities of three-dimensional antiferromagnets

Approximate susceptibilities of simple cubic S=∞ Heisenberg and $\text{S=}\text{1}\text{2}$ Ising antiferromagnets are calculated by using the new expansion method reported previously for the case of a square-plane S=∞ Heisenberg antiferromagnet. The present method gives a good approximation to the susceptibilities of both of the Heisenberg and Ising antiferromagnets above the Néel temperature.     

Quantum coadjoint orbits in gl(n)*

We present a doubleU _h(gl(n, ℂ))-equivariant quantization on semisimple coadjoint orbits of the group GL(n, ℂ) as a quotient of the extended reflection equation algebra by relations which are given explicitly. Such a quantization is a two-parameter family including an explicit GL(n)-equivariant quantization of the Kirillov-Kostant-Souriau Poisson bracket. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

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.     

Migdal renormalization group approach to quantum spin systems

The Migdal RG approximation is extended to quantum spin systems such as the Heisenberg and XY-models. This yields the non-existence of phase transition in the two-dimensional Heisenberg model. The phase transition of the two-dimensional XY-model is also studied.     

Nonstandard Poincaré and Heisenberg Algebras

Nonstandard deformations of the Poincaré group Fun(P(1+1)) and its dual enveloping algebra U (p(1+1)) are obtained as a contraction of the h-deformed (Jordanian) quantum group Fun( SL _h (2)) and its dual. A nonstandard quantization of the Heisenberg algebra U(h(1)) is also investigated.     

On the representation theory of quantum Heisenberg group and algebra

We show that the quantum Heisenberg groupH _q (1) and its *-Hopf algebra structure can be obtained by means of contraction from quantumSU _q (2) group. Its dual Hopf algebra is the quantum Heisenberg algebraU _q (h(1)). We derive left and right regular representations forU _q (h(1)) as acting on its dualH _q (1). Imposing conditions on the right representation, the left representation is reduced to an irreducible holomorphic representation with an associated quantum coherent state. Realized in the Bargmann-Hilbert space of analytic functions the unitarity of regular representation is also shown. By duality, left and right regular representations for quantum Heisenberg group with the quantum Heisenberg algebra as representation module are also constructed. As before reduction of group left representations leads to finite dimensional irreducible ones for which the intertwinning operator is also investigated.     

Quantum deformations of the Heisenberg group obtained by geometric quantization

All multiplicative Poisson brackets on the Heisenberg group are classified and Manin groups [14] corresponding to a wide class of those brackets are constructed. A geometric quantization procedure is applied to the resulting symplectic pseudogroups yielding a wide class of pre-C^*-algebras with comultiplication, counit and coinverse, which provide quantum deformations of the Heisenberg group.     

Supersymmetric Dirac particles in Riemann-Cartan space-time

A natural extension of the supersymmetric model of Di Vecchia and Ravndal yields a nontrivial coupling of classical spinning particles to torsion in a Riemann-Cartan geometry. The equations of motion implied by this model coincide with a consistent classical limit of the Heisenberg equations derived from the minimally coupled Dirac equation. Conversely, the latter equation is shown to arise from canonical quantization of the classical system. The Heisenberg equations are obtained exact in all powers of and thus complete the partial results of previous WKB calculations. We touch also on such matters of principle as the mathematical realization of anticommuting variables, the physical interpretation of supersymmetry transformations, and the effective variability of rest mass.     

Analysis of the migdal transformation for models with Heisenberg spins on a d-dimensional lattice

We show for classical Heisenberg spins, with a general nearest neighbour interaction, that in the Migdal approximation the only low-temperature phase transitions are Ising ones (ferror antiferromagnetic). For d=2 neither the pure Heisenberg model nor the Lebwohl-Lasher model show a phase transition at a finite temperature. For d>2 transitions do exist at intermediate temperature and the complete flow diagram together with a two-parameter phase diagram is obtained numerically for d=3. Apart from critical temperatures and thermal exponents, also the magnetic exponents (for both Heisenberg and XY spins) are calculated. The latter are in very good agreement with exact results.     

A Particle-Picture Approach to Anomalies in Chiral Gauge Theories

The system of a chiral fermion field coupled to a background gauge field is considered. By taking what we call the particle picture and carefully defining the S-matrix in the Heisenberg picture, we investigate anomalous phenomena in this system. It is shown by explicit calculations that the gauge-field configuration with nonvanishing topological-charge causes anomalous production of particles that is directly responsible for the chiral U(1) anomaly. Unlike the chiral U(1) anomaly, the gauge anomaly, that is, gauge non-invariance of the S-matrix is a problem that arises in the phase of the S-matrix. It is shown that this phase is related to the freedom existing in the quantization method, and that a suitably chosen phase which of course is consistent with the equation of motion can remove the gauge anomaly. Finally, a modified form of path-integral quantization for this system is proposed.     

Quantization of reimannian gravity: interaction representation

A Riemann-covariant expression of Schwinger's procedure, leading from a Heisenberg to an interaction representation, completes here our quantization of the coupling of a massive graviton field and a spin-zero Kemmer field.     

Monte Carlo investigation of magnetic properties of a two-dimensional classical Heisenberg magnet

The equilibrium properties of a simple quadratic lattice of classical spins with nearest-neighbor Heisenberg interactions have been examined by a Monte Carlo Method. The susceptibility was found to have a singular temperature dependence χ ∝ exp (const/T²)/T above the Stanley-Kaplan transition temperature (T_SK). A plausible argument has been presented to explain peculiar properties of the 2?d Heisenberg magnet on the basis of the observed singular behavior of the susceptibility.     

BFV quantization on hermitian symmetric spaces

Gauge-invariant BFV approach to geometric quantization is applied to the case of hermitian symmetric spaces G/H. In particular, gauge invariant quantization on the Lobachevski plane and sphere is carried out. Due to the presence of symmetry, master equations for the first-class constraints, quantum observables and physical quantum states are exactly solvable. BFV-BRST operator defines a flat G-connection in the Fock bundle over G/H. Physical quantum states are covariantly constant sections with respect to this connection and are shown to coincide with the generalized coherent states for the group G. Vacuum expectation values of the quantum observables commuting with the quantum first-class constraints reduce to the covariant symbols of Berezin. The gauge-invariant approach to quantization on symplectic manifolds synthesizes geometric, deformation and Berezin quantization approaches.