Deformation quantization of the Heisenberg group

A *-product compatible with the comultiplication of the Hopf algebra of the functions on the Heisenberg group is determined by deforming a coboundary Lie-Poisson structure defined by a classicalr-matrix satisfying the modified Yang-Baxter equation. The corresponding quantum group is studied and itsR-matrix is explicitly calculated.     

Deformation quantization of fermi fields

Deformation quantization for any Grassmann scalar free field is described via the Weyl-Wigner-Moyal formalism. The Stratonovich-Weyl quantizer, the Moyal -product and the Wigner functional are obtained by extending the formalism proposed recently in [I. Galaviz, H. García-Compeán, M. Przanowski, F.J. Turrubiates, Weyl-Wigner-Moyal Formalism for Fermi Classical Systems, arXiv:hep-th/0612245] to the fermionic systems of infinite number of degrees of freedom. In particular, this formalism is applied to quantize the Dirac free field. It is observed that the use of suitable oscillator variables facilitates considerably the procedure. The Stratonovich-Weyl quantizer, the Moyal -product, the Wigner functional, the normal ordering operator, and finally, the Dirac propagator have been found with the use of these variables.     

On geometric quantization of compact,complex manifolds

A modified Kostant-Souriau geometric quantization procedure in a case of purely complex polarization is described. It is proved that each algebraic manifold admits such a quantization. The cases of a C-homogeneous manifold and a manifold covered with a bounded complex domain are studied.     

Deformation quantization and Nambu Mechanics

Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over ℝ of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products. Supported by the European Commission and the Japan Society for the Promotion of Science. NSF grant DMS-95-00557 This article was processed by the author using the L^AT_EX style filepljour1 from Springer-Verlag.     

Deformation quantization of the heisenberg group

After reviewing the way the quantization of Poisson Lie Groups naturally leads to Quantum Groups we use the existing quantum versionH(1)q of the Heisenberg algebra to give an explicit example of this quantization on the Heisenberg group.     

Path integral quantization corresponding to the deformed Heisenberg algebra

In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to fractional derivative terms in the corresponding quantum mechanical Hamiltonian, a formal meaning can be given to them by using the theory of harmonic extensions of function. Depending on this argument, the expression of the propagator of the path integral corresponding to the deformed Heisenberg algebra, has been obtained. In particular, the consistent expression of the one dimensional free particle propagator has been evaluated explicitly. With this propagator in hand, it has been shown that, even in free particle case, normal generalized uncertainty principle and doubly special relativity show very much different result.     

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.     

Deformation quantization, superintegrability and Nambu mechanics

Phase Space is the framework best suited for quantizing superintegrable systems—systems with more conserved quantities than degrees of freedom. In this quantization method, the symmetry algebras of the hamiltonian invariants are preserved most naturally. We illustrate the power and simplicity of the method through new applications to nonlinear σ-models, specifically for Chiral Models and de Sitter N-spheres, where the symmetric quantum hamiltonians amount to compact and elegant expressions, in accord with the Groenewold-van Hove theorem. Additional power and elegance is provided by the use of Nambu Brackets (linked to Dirac Brackets) involving the extra invariants of superintegrable models. The quantization of Nambu Brackets is then successfully compared to that of Moyal, validating Nambu’s original proposal, while invalidating other proposals.     

On the Fedosov deformation quantization beyond the regular Poisson manifolds

A simple iterative procedure is suggested for the deformation quantization of (irregular) Poisson brackets associated to the classical Yang–Baxter equation. The construction is shown to admit a pure algebraic reformulation giving the Universal Deformation Formula (UDF) for any triangular Lie bialgebra. A simple proof of classification theorem for inequivalent UDF's is given. As an example the explicit quantization formula is presented for the quasi-homogeneous Poisson brackets on two-plane.     

Asymptotic quantization for solution manifolds of some infinite dimensional Hamiltonian systems

The solution manifolds of some classes of Hamiltonian systems in Hilbert phase spaces are considered. Pseudodifferential operators with symbols on these manifolds are defined.     

Geometric quantization of symplectic manifolds with respect to reducible non-negative polarizations

The leafwise complex of a reducible non-negative polarization with values in the prequantum bundle on a prequantizable symplectic manifold is studied. The cohomology groups of this complex is shown to vanish in rank less than the rank of the real part of the non-negative polarization. The Bohr-Sommerfeld set for a reducible non-negative polarization is defined. A factorization theorem is proved for these reducible non-negative polarizations. For compact symplectic manifolds, it is shown that the above complex has finite dimensional cohomology groups, more-over a Lefschetz fixed point theorem and an index theorem for these non-elliptic complexes is proved. As a corollary of the index theorem, we deduce that the cardinality of the Bohr-Sommerfeld set for any reducible real polarization on a compact symplectic manifold is determined by the volume and the dimension of the manifold. Supported in part by NSF grant DMS-93-09653, while the author was visiting University of California Berkeley.     

Deformation theory applied to quantization and statistical mechanics

After a review of the deformation (star product) approach to quantization, treated in an autonomous manner as a deformation (with parameter ) of the algebraic composition law of classical observables on phase-space, we show how a further deformation (with parameter ) of that algebra is suitable for statistical mechanics. In this case, the phase-space is endowed with what we call a conformal symplectic (or conformal Poisson) structure, for which the bracket is the Poisson bracket modified by terms of order (1, 0) and (0, 1). As an application, one sees that the KMS states (classical or quantum) are those that vanish on the modified (Poisson or Moyal-Vey) bracket of any two observables, multiplied by a conformal factor.     

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.