Indecomposable representations of the algebra su(2,1)

Representations of the Lie algebra sl(3) with highest weight are analyzed. Invariant subspaces of indecomposable representations are determined. We study the decomposition of these representations with respect to the subalgebras su(2) and su(1,1) (in their obvious imbedding in su(2,1)).For special cases this decomposition gives indecomposable non multiplicity free representations (indecomposable pairs) with highest weight. These were discussed in [1] and appear also in the decomposition so(3,2) su(1,1) of the Rac representation, [7].     

Canonical quantization with indefinite inner product

In this paper an algebraic approach to canonical quantization on indefinite inner product space is presented. Concrete realization of such a quantization of a given classical system described by a symplectic space (M, σ) is obtained by means of a so-called J¹-representation of CCR algebra Δ(M, σ). So-called Fock-Krein representations of Δ(M, σ) determined by some class of complex structures on (M, σ) are studied in detail. It is shown that every Fock-Krein representation is unbounded. Starting with a fixed Fock-Krein representation of CCR sectors of non-Fock-Krein representations are constructed.     

Functional representations of conformally invariant free fields

Free fields in two-dimensional space-time are considered from the point of view of representation theory. Unitary representations of the conformal group are constructed in the representation space of equal-time commutation relations. The compatibility of the global conformal transformations with locality is discussed.     

Lax representations and zero-curvature representations by the Kronecker product

It is shown that the Kronecker product can be applied to construct not only new Lax representations, but also new zero-curvature representations of integrable models. A characteristic difference between continuous and discrete zero-curvature equations is pointed out.     

Indecomposable representations of Poincaré group and quantization of weak gravitational and electromagnetic fields

In this paper a quantization procedure of the electromagnetic and the weak gravitational fields, in an indefinite (but non-negative) metric space of states, is presented. We show that the covariance axiom can only be formulated with non-unitary representations (of the Poincaré group) whose restrictions to the one-particle states are indecomposable. In the gravitational case the corresponding indecomposable representation describes six states of helicity for gravitons. The quantized field appears as six interacting massless fields of helicities 2, ?2,1, ?1,0 and 0. These six values of helicity correspond to the six modes of polarization in the most general metric theory of gravity. For the electromagnetic field one obtains an indecomposable representation of the Poincaré group with helicities 1, ?1 and 0 corresponding to transverse and longitudinal photons.     

Spectral representations of Lorentz invariant distributions and scale transformation

An approach to the theory of Lorentz invariant distributions is developed in terms of covariant spectral representations. The behaviour of singular invariant distributions under a change of scale is analyzed. It is shown that the conventional extension of homogeneous singular functions into distributions inR ⁴, followed by a breakdown of homogeneity, is incomplete. Homogeneous extensions depending on an arbitrary scaling parameter are introduced, calculation techniques are developed and various formulae having applications in quantum field theory are derived.Supported by the Deutsche Forschungsgemeinschaft.     

An invariant formula for a star product with separation of variables

We give an invariant formula for a star product with separation of variables on a pseudo-Kähler manifold.     

Regular measures and inner product spaces

We present some results concerning the properties of regular finitely additive measures on the set of all splitting subspaces of a (real or complex) inner product spaceS and their relation to completeness ofS. These results are generalized for abstract quadratic spaces to be orthomodular. Moreover, some open problems are presented.     

Completeness of inner product spaces with respect to splitting subspaces

The set E(S) of all splitting subspaces, i.e., of all subspaces M of an inner product space S for which MM =S holds, is an orthocomplemented orthomodular orthoposet and it has already been shown that the ordering property on E(S) of being a complete lattice characterizes the completeness of inner product spaces. In this work this last result is generalized proving that S is a Hilbert space under the weaker request that E(S) is a -lattice. As a marginal result, we also prove that an inner product space is complete if and only if the complete lattice (S) of all subspaces is orthomodular.     

The Bogoliubov inner product in quantum statistics

A natural Riemannian geometry is defined on the state space of a finite quantum system by means of the Bogoliubov scalar product which is infinitesimally induced by the (nonsymmetric) relative entropy functional. The basic geometrical quantities, including sectional curvatures, are computed for a two-level quantum system. It is found that the real density matrices form a totally geodesic submanifold and the von Neumann entropy is a monotone function of the scalar curvature. Furthermore, we establish information inequalities extending the Cramér-Rao inequality of classical statistics. These are based on a very general new form of the logarithmic derivative.This work was supported by the Hungarian National Foundation for Scientific Research, grant No. 1900. Authors' e-mail addresses are: H1128PET@ella.hu and TOTH@zodiac.rutgers.edu.     

Unitary representations of semi-direct product groups with infinite dimensional Abelian normal subgroup

We generalize Mackey's theory of induced representations to groups which are semidirect products of a locally compact group H by an infinite dimensional Abelian group A. A theorem on inducing in stages is proved and conditions ensuring conservation of the irreducibility of the representation through the inducing process are given. When A is a nuclear space or the Hilbert space A_O of a Gelfand triplet A_O ? A ? A′_O, a cylindrical ergodic measure on the dual space A′ of A (respectively A′_O) is associated to each irreducible representation of G, and it is proved that the representation is induced when the measure is concentrated on an orbit. In the last part, A is supposed to be a Hilbert space and sufficient conditions are given for the preceding ergodic measures to be concentrated on an orbit.     

Semi-indefinite inner product and generalized Minkowski spaces

In this paper we develop the theories of normed linear spaces and of linear spaces with indefinite metric, for finite dimensions both of which are also called Minkowski spaces in the literature.     

The inverse scattering method in an invariant form with respect to internal symmetry algebra representations

By the example of the sine—Gordon equation, we develop a technique for obtaining soliton solutions. This procedure is not connected with a concrete matrix realization of the Lax pair operators but it directly appeals to the properties of internal symmetry spectrum of the corresponding equations.     

Indecomposable fusion products

We analyse the fusion products of certain representations of the Virasoro algebra for c = −2 and c = −7 which are not completely reducible. We introduce a new algorithm which allows us to study the fusion product level by level, and we use this algorithm to analyse the indecomposable components of these fusion products. They form novel representations of the Virasoro algebra which we describe in detail.We also show that a suitably extended set of representations closes under fusion, and indicate how our results generalise to all (1, q) models.     

Indecomposable BRS modules

Explicit structures of indecomposable BRS modules have been constructed with suitable illustrations.     

Gleason's theorem and completeness of inner product spaces

We generalize the result of Hamhalter and Pták showing that an inner product space whose dimension is either a nonmeasurable cardinal or an arbitrary cardinal is complete iff its lattice of strongly closed subspaces possesses at least one state or one completely additive state, respectively. Moreover, we show that this lattice of any separable space possesses many-finite measures, and we give the Gleason analogue for them.     

Solving general gauge theories on inner product spaces

By means of a generalized quartet mechanism we show in a model independent way that a BRST quantization on an inner product space leads to physical states of the form ph〉 = exp [Q, ψ]ph〉₀ where Q is the nilpotent BRST operator, ψ a hermitian fermionic gauge-fixing operator, and ph〉_o BRST invariant states determined by a hermitian set of BRST doublets in involution. ph〉₀ does not belong to an inner product space although ph〉 does. Since the BRST quartets are split into two sets of hermitian BRST doublets there are two choices for ph〉₀ and the corresponding ψ. When applied to general, both irreducible and reducible, gauge theories of arbitrary rank within the BFV formulation we find that ph〉₀ are trivial BRST invariant states which only depend on the matter variables for one set of solutions, and for the other set ph〉₀ are solutions of a Dirac quantization. This generalizes previous Lie group solutions obtained by means of a bigrading.     

Translation invariant tensor product states in a finite lattice system

We show that the matrix (or more generally tensor) product states in a finite translation invariant system can be accurately constructed from a same set of local matrices (or tensors) that are determined from an infinite lattice system in one or higher dimensions. This provides an efficient approach for studying translation invariant tensor product states in finite lattice systems. Two methods are introduced to determine the size-independent local tensors.     

On the tensor product of representations of semisimple Lie groups

All topologically irreducible representations involved in the tensor product of finite and infinite dimensional representations of the Principal nonunitary series (including the unitary series) of a semisimple Lie group are defined.     

On the reduction ofn-fold tensor product representations of noncompact groups

The reduction ofn-fold tensor products of induced unitary representations of noncompact groups into irreducible constituents is shown. Clebsch-Gordon coefficients are then calculated. The technique is applied to then-fold tensor products of the positive mass representations of the Poincaré group.