Translation-invariant traces and fourier analysis on multiplier representation of vector groups

It is shown that for an arbitrary Borel multiplier b for the additive group of a finite-dimensional vector space V, there is an essentially unique trace on the enveloping von Neumann algebra of the regular b-representation which is translation invariant in a certain sense. An analogue of the Fourier-Plancherel transformation from the L² space of V to the non-commutative L² space defined by this trace is developed which generalizes the well-known canonical Fourier transformation associated with the Weyl commutation relations, and which essentially reduces to the classical Fourier-Plancherel transformation when b≡1.     

Classical and quantum statistical mechanics in a common Liouville space

It is shown that the extremal phase-space representations of quantum mechanics can be expressed in terms of wave-functions on L²-spaces which are embedded in L²(Γ). In L²(Γ) all these representations are restrictions of a globally defined representation of the canonical commutation relations. The master Liouville space $\text{B}$²(Γ) over L²(Γ) can accommodate representations of both classical and quantum statistical mechanics, and serves as a medium for their comparison. As a specific example, a Boltzmann-type equation on $\text{B}$²(Γ) is considered in the classical as well as quantum context.     

Ultralocal quantum field theory in terms of currents

The paper considers the possibility of constructing ultralocal theories, whose Hamiltonians contain no gradient terms and are therefore diagonal in position space, entirely in terms of currents with an equal time current algebra replacing the canonical commutation relations. It is shown that the free current theory can be defined in terms of a certain representation of the current algebra related to the group,S L(2,R). This representation is then constructed by using certain results of Araki and in the process a new infinitely divisible state on the universal covering group ofSL(2,R) is displayed. An ultralocal free theory can also be constructed for the canonical fields, and its relation to the free current theory is shown to involve a certain renormalization procedure reminiscent of the thermodynamic limit.Research sponsored by the Air Force Office of Scientific Research under Contract No. F 44620-71-C-0108 and Contract No. AF 49(638) 1545.     

An algebraic treatment of the MIC-Kepler problem on S 3 sphere

The quantum-mechanical problem of motion in a dual charged Coulomb field modified by a centrifugal term (MIC-Kepler problem) is considered in a three-dimensional space of constant positive curvature, S ³. Conserved operators are found, and their commutation relations are derived. It is shown that, in the MIC-Kepler problem in S ³ space, conserved operators form a cubic algebra similar to that of the Kepler problem in the same space. This symmetry algebra is used to obtain the energy spectrum of the problem.     

An effective method of investigation of positive maps on the set of positive definite operators

Let $\text{A}$₁ be the algebra of linear operators on the n-dimensional Hilbert space $\text{H}$₁, and let $\text{A}$₂ be the algebra of linear operators of the m-dimensional Hilbert space $\text{H}$₂. Let $\text{L}$($\text{A}$₁, $\text{A}$₂) denote the complex space of linear maps from $\text{A}$₁ to $\text{A}$₂. By a positive map we mean an element of the space $\text{L}$($\text{A}$₁, $\text{A}$₂) (superoperator with respect to $\text{H}$₁) which maps positive definite operators in $\text{A}$₁ into positive definite operators in $\text{A}$₂. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈$\text{L}$($\text{A}$₁, $\text{A}$₂) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.     

Locally covariant description of a π-meson field

The algebra of canonical commutation relations is constructed in the Weyl form and a locally covariant formulation of a charged scalar field is presented in terms of the covariant functor. The symplectic space of solutions of the Klein-Gordon-Fock equation is represented as a direct sum of symplectic spaces of positively and negatively charged ?? mesons. In each of the spaces, these fields satisfy the conditions of the locally covariant quantum field theory. Using natural transformation, the equivalence of two functors describing the ??⁺ and ??^? mesons is shown. From the physical viewpoint, the equivalence corresponds to the equality of the mesons?? masses.     

Non-commutative Euclidean and Minkowski structures

A noncommutative *-algebra that generalizes the canonical commutation relations and that is covariant under the quantum groups SO _q (3) or SO _q(1, 3) is introduced. The generating elements of this algebra are hermitean and can be identified with coordinates, momenta and angular momenta. In addition a unitary scaling operator is part of the algebra.     

Riesz Transforms on Deformed Fock Spaces

The Fock Von Neumann algebra , equipped with its canonical trace τ, is spanned by n hermitian operators acting on a Hilbert Fock space some commutation relations between and are defined by the n×n hermitian matrix A. We define a Riesz transform , where is the number operator, ∇^′ is aninner derivation (unbounded in general) and . Let 1<p<∞. We prove that is equivalent to for every with null trace, with constants which do not depend on n. Received: 24 November 1998 / Accepted: 2 March 1999     

Quantum holonomy theory

We present quantum holonomy theory, which is a non‐perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a ‐algebra that involves holonomy‐diffeo‐morphisms on a 3‐dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi‐classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi‐classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi‐classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost‐commutative algebra emerges from the holonomy‐diffeomorphism algebra in the same limit.     

Lie Algebras of Derivations and Resolvent Algebras

This paper analyzes the action δ of a Lie algebra X by derivations on a C*–algebra ${\mathcal{A}}$ . This action satisfies an “almost inner” property which ensures affiliation of the generators of the derivations δ with ${\mathcal{A}}$ , and is expressed in terms of corresponding pseudo–resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*–algebra ${\mathcal{A}}$ , it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo–resolvents. If the Lie action δ is ergodic, i.e. the only elements of ${\mathcal{A}}$ on which all the derivations in δ _X vanish are multiples of the identity, then this extension is given by a (non–degenerate) symplectic form σ on X. Moreover, the algebra generated by the pseudo–resolvents coincides with the resolvent algebra based on the symplectic space (X, σ). Thus the resolvent algebra of the canonical commutation relations, which was recently introduced in physically motivated analyses of quantum systems, appears also naturally in the representation theory of Lie algebras of derivations acting on C*–algebras.     

Hilbert space representation of an algebra of observables forq-deformed relativistic quantum mechanics

Using a representation of theq-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the massp ² is diagonal.     

Bogolyubov transformations for regular representations of canonical commutation relations in space with an indefinite metric

Bogolyubov transformations for regular representations of the algebra of canonical commutation relations (CCRs) are considered in space with an indefinite metric.     

Representations of the CCR in the (φ4)3 model: Independence of space cutoff

The algebra of observables for the renormalized ⁴ interaction in 3-dimensional space-time is constructed. It is shown that the von Neumann algebras associated with observables in a bounded regionB are independent of the space cutoff which is used in the construction of a Hamiltonian for this interaction. This result is shown to be useful in the construction of a translation invariant ⁴ theory in three dimensions. It also gives a physical criterion for the equivalence of non-Fock representations of the canonical commutation relations.     

State-Vector Space and Canonical Coherent States in Noncommutative Plane

The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose-Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.     

Infinite-parameter symmetries and the Higgs effect in gravity-coupled sigma models in D+4 dimensions

We compute the mass spectra of small fluctuations of four-dimensional fields for Kaluza-Klein models in which the compactification from D+4 to 4 (flat) dimensions is induced by the scalar fields of a nonlinear sigma model defined on an S^N or CP^N manifold. The compactifications are stable for all values of N. The fact that the spectra contain no massless vector fields is traced to the absence of a local gauge invariance for the sigma-model action. We introduce a complete basis for the infinite-parameter symmetries that arise from the harmonic analysis of the higher-dimensional dynamical invariances. The spectrum of spin-one and spin-two fields is consistent with the Higgs effect associated with the breaking of the local symmetries corresponding to these generators. The commutation relations of the infinite parameter algebra for the case of CP¹ are also given. The algebra includes the spectrum-generating algebra SO(1,3) of Salam and Strathdee.     

Analogue of the Weyl representation of algebra of canonical commutation relations in the case of nonphysical particles

An analogue of the Weyl representation of the algebra of canonical commutation relations is proved to exist in the anti-Fock case achieved in Krein space.     

Kac–Moody theories for colored phase space (quantum Hall) droplets

We derive the canonical structure and Hamiltonian for arbitrary deformations of a higher-dimensional (quantum Hall) droplet of fermions with spin or color on a general phase space manifold. Gauge fields are introduced via a Kaluza–Klein construction on the phase space. The emerging theory is a nonlinear higher-dimensional generalization of the gauged Kac–Moody algebra. To leading order in ℏ this reproduces the edge state chiral Wess–Zumino–Witten action of the droplets.     

Coherent states for quantum compact groups

Coherent states are introduced and their properties are discussed for simple quantum compact groupsA _l, B_l, C_l andD _l. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general quantum dressing orbit. The coherent state is interpreted as a holomorphic function on this orbit with values in the carrier Hilbert space of an irreducible representation of the corresponding quantized enveloping algebra. Using Gauss decomposition, the commutation relations for the holomorphic coordinates on the dressing orbit are derived explicitly and given in a compactR-matrix formulation (generalizing this way theq-deformed Grassmann and flag manifolds). The antiholomorphic realization of the irreducible representations of a compact quantum group (the analogue of the Borel-Weil construction) is described using the concept of coherent state. The relation between representation theory and non-commutative differential geometry is suggested. Dedicated to Professor L.D. Faddeev on his 60th birthday