A Laplace operator and harmonics on the quantum complex vector space

The aim of this paper is to study the q-Laplace operator and q-harmonic polynomials on the quantum complex vector space generated by z_i,w_i, i=1,2,...,n, on which the quantum group GL_q(n) (or U_q(n)) acts. The q-harmonic polynomials are defined as solutions of the equation Delta_qp=0, where p is a polynomial in z_i,w_i, i=1,2,...,n, and the q-Laplace operator Delta_q is determined in terms of q-derivatives. The q-Laplace operator Delta_q commutes with the action of GL_q(n). The projector H_{m,m'}: A_{m,m'} --> H_{m,m'} is constructed, where A_{m,m'} and H_{m,m'} are the spaces of homogeneous (of degree m in z_i and of degree m' in w_i) polynomials and homogeneous q-harmonic polynomials, respectively. By using these projectors, a q-analogue of the classical zonal spherical and associated spherical harmonics are constructed. They constitute an orthogonal basis of H_{m,m'}. A q-analogue of separation of variables is given. The quantum algebra U_q(gl_n), acting on H_{m,m'}, determines an irreducible representation of U_q(gl_n). This action is explicitly constructed. The results of the paper lead to the dual pair (U_q(sl_2), U_q(gl_n)) of quantum algebras.Comment: 26 pages, LaTe

Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta

A noncommutative geometric generalisation of the quantum field theoretical framework is developed by generalising the Heisenberg commutation relations. There appear nonzero minimal uncertainties in positions and in momenta. As the main result it is shown with the example of a quadratically ultraviolet divergent graph in $\phi^4$ theory that nonzero minimal uncertainties in positions do have the power to regularise. These studies are motivated with the ansatz that nonzero minimal uncertainties in positions and in momenta arise from gravity. Algebraic techniques are used that have been developed in the field of quantum groups.Comment: 52 pages LATEX, DAMTP/93-33. Revised version now includes a chapter on the Poincare algebra and curvature as noncommutativity of momentum spac

Cartan Pairs

A new notion of Cartan pairs as a substitute of notion of vector fields in noncommutative geometry is proposed. The correspondence between Cartan pairs and differential calculi is established.Comment: 7 pages in LaTeX, to be published in Czechoslovak Journal of Physics, presented at the 5th Colloquium on Quantum Groups and Integrable Systems, Prague, June 199

Second law of thermodynamics for macroscopic mechanics coupled to thermodynamic degrees of freedom

Based only on classical Hamiltonian dynamics, we prove the maximum work principle in a system where macroscopic dynamical degrees of freedom are intrinsically coupled to microscopic degrees of freedom. Unlike recent identities between irreversible work and free energy, such as in the Jarzynski relation, the macroscopic dynamics is not governed by an external action but undergoes the back reaction of the microscopic degrees of freedom. Our theorems cover such physical situations as impact between macroscopic bodies, thermodynamic machines, and molecular motors.Comment: 4 pages, RevTe

Hamiltonian Quantization of Chern-Simons theory with SL(2,C) Group

We analyze the hamiltonian quantization of Chern-Simons theory associated to the universal covering of the Lorentz group SO(3,1). The algebra of observables is generated by finite dimensional spin networks drawn on a punctured topological surface. Our main result is a construction of a unitary representation of this algebra. For this purpose, we use the formalism of combinatorial quantization of Chern-Simons theory, i.e we quantize the algebra of polynomial functions on the space of flat SL(2,C)-connections on a topological surface with punctures. This algebra admits a unitary representation acting on an Hilbert space which consists in wave packets of spin-networks associated to principal unitary representations of the quantum Lorentz group. This representation is constructed using only Clebsch-Gordan decomposition of a tensor product of a finite dimensional representation with a principal unitary representation. The proof of unitarity of this representation is non trivial and is a consequence of properties of intertwiners which are studied in depth. We analyze the relationship between the insertion of a puncture colored with a principal representation and the presence of a world-line of a massive spinning particle in de Sitter space.Comment: 78 pages. Packages include

Kakutani Dichotomy on Free States

Two quasi-free states on a CAR or CCR algebra are shown to generate quasi-equivalent representations unless they are disjoint.Comment: 12 page

Generalized Fock Spaces, New Forms of Quantum Statistics and their Algebras

We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ``infinite'', Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot be mapped into single-indexed systems are studied. Our theory is based on a three-tiered structure consisting of Fock space, statistics and algebra. This general formalism not only unifies the various forms of statistics and algebras, but also allows us to construct many new forms of quantum statistics as well as many algebras of creation and destruction operators. Some of these are : new algebras for infinite statistics, q-statistics and its many avatars, a consistent algebra for fractional statistics, null statistics or statistics of frozen order, ``doubly-infinite'' statistics, many representations of orthostatistics, Hubbard statistics and its variations.Comment: This is a revised version of the earlier preprint: mp_arc 94-43. Published versio

Zero modes' fusion ring and braid group representations for the extended chiral su(2) WZNW model

The zero modes' Fock space for the extended chiral $su(2)$ WZNW model gives room to a realization of the Grothendieck fusion ring of representations of the restricted $U_q sl(2)$ quantum universal enveloping algebra (QUEA) at an even ($2h$-th) root of unity, and of its extension by the Lusztig operators. It is shown that expressing the Drinfeld images of canonical characters in terms of Chebyshev polynomials of the Casimir invariant $C$ allows a streamlined derivation of the characteristic equation of $C$ from the defining relations of the restricted QUEA. The properties of the fusion ring of the Lusztig's extension of the QUEA in the zero modes' Fock space are related to the braiding properties of correlation functions of primary fields of the extended $su(2)_{h-2}$ current algebra model.Comment: 36 pages, 1 figure; version 3 - improvements in Sec. 2 and 3: definitions of the double, as well as R- (and M-)matrix changed to fit the zero modes' one

Solutions of Klein--Gordon and Dirac equations on quantum Minkowski spaces

Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein--Gordon and Dirac equations are found. The Fock space construction is sketched.Comment: 21 pages, LaTeX file, minor change

Some remarks on the Gauss decomposition for quantum group GL_q(n)

In this letter some properties of the Gauss decomposition of quantum group $GL_q(n)$ with application to q-bosonization are considered.Comment: 11 page