Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories

The super-Hamiltonian and supermomentum in canonical geometrodynamics or in a parametried field theory on a given Riemannian background have Poisson brackets which obey the Dirac relations. By smearing the supermomentum with vector fields V^→∈ L Diff Σ on the space manifold Σ, the Lie algebra L Diff Σ of the spatial diffeomorphism group Diff Σ can be mapped antihomomorphically into the Poisson bracket algebra on the phase space of the system. The explicit dependence of the Poisson brackets between two super-Hamiltonians on canonical coordinates (spatial metrics in geometrodynamics and embedding variables in parametrized theories) is usually regarded as an indication that the Dirac relations cannot be connected with a representation of the complete Lie algebra L Diff M of spacetime diffeomorphisms.We show how this difficulty may be overcome and construct a homomorphic mapping of spacetime vector fields V ∈ L Diff M into the Poisson bracket algebra on the phase space of the system. In the present paper, I, we explain how the technique works in the case of a parametrized field theory and in the following paper, II, we generalize it to canonical geometrodynamics. In a parametrized theory, the phase space of the system is the ordinary phase space of the field augmented by the embedding variables X: Σ →Mand their conjugate momenta. The dynamical variable H(V) which represents V ∈ L Diff M generates a deformation of the embedding along the flow lines of V accompanied by the correct dynamical evolution of the field data and preserves the constraints in the extended phase space of the system. We also establish the relation between the representations of Diff Σ and DiffM.     

Representations of D-posets

A generalization of an orthoalgebra, which includes the set of all effects (i.e., s.a. operators between 0 and 1 on a Hilbert space) is a D-poset or an effect algebra, equivalently. Two generalizations of test spaces the logics of which are D-posets are investigated and their equivalence is shown.     

Gauge theories as theories of spontaneous breakdown

Any gauge theory is proved to arise from spontaneous breakdown of symmetry under certain infinite parameter group, the corresponding gauge field being the Goldstone field by which this breakdown is accompanied.     

Representations of BRS algebra

In the canonical formulation of gauge theories the BRS transformation plays a fundamental rôle. The generator of this transformation along with the ghost number forms an algebra called the BRS algebra. Certain properties of this algebra are essential to the proof of unitarity of the S matrix in the physical sector and also to the discussion of color confinement.In the present paper we present all the possible representations of the BRS algebra in the light of indefinite metric.     

Representations of conformal supersymmetry

The superalgebras of (generalized) conformal supersymmetry have some very interesting unitarizable representations that contain only massless representations of the conformal subalgebra, in spite of contrary claims that have recently been made , .

---

     

Topologizations of Chiral Representations

We analyze and compare two families of topologies that have been proposed for representation spaces of chiral algebras by Huang and Gaberdiel & Goddard respectively. We show, in particular, that for suitable pairs the topology of Gaberdiel & Goddard is coarser. We also give a new proof that the chiral two-point blocks are continuous in the topology of Huang.     

Representations of twisted Yangians

We study highest weight representations of certain Yangian-type quantum algebras connected with the series B, C, D of complex classical Lie algebras. In the symplectic case, we obtain a complete parametrization of irreducible finite-dimensional representations in terms of their highest weights. We apply these results to the well-known missing label problem in the reduction sp(2n)sp(2n–2).     

Residual Representations of Spacetime

Spacetime is modelled by binary relations—by the classes of the automorphisms of a complex two-dimensional vector space with respect to the definite unitary subgroup U(2). In extension of Feynman propagators for particle quantum fields representing only the tangent spacetime structure, global spacetime representations are given, formulated as residues using energy–momentum distributions with the invariants as singularities. The associated quantum fields are characterized by two invariant masses—for time and position—supplementing the one mass for the definite unitary particle sector with another mass for the indefinite unitary interaction sector without asymptotic particle interpretation.     

Superfields as Representations

We study irreducible and reducible representations of the generalized Lie algebra of Wess and Zumino. The algebra is integrated to a group with the help of Grassmann algebras and the representations of the algebra are made into representations of the group. We construct invariant sesquilinear forms that are positive definite for the Wess-Zumino algebra over the complex field. We define irreducible superfields for any spin J as specific realizations of such representations. All superfields appearing in the literature are either equivalent to one of these or built up out of these superfields.     

Completely integrable theories as theories of spontaneous breakdown

It is shown that the field theory described by a completely integrable equation (in particular, the theory of any free field) is the theory of spontaneous breakdown of symmetry under certain infinite dynamical group, the corresponding field being the Goldstone field by which this breakdown is accompanied.     

All free string theories are theories of BRST cohomology

The fully gauge-invariant theory of the free open (super) strings is generalized to treat closed (super) strings in the BRST formalism. It is shown that the actions for all theories can be written in the form (Φ,Q_BΦ) using the BRST charge Q_B and the string functional Φ, if the inner product is interpreted appropriately with necessary ghost insertions. Also is shown that consistent truncations of our actions recover all known covariant formulations of (super) strings.     

Representations of Super Yang-Mills Algebras

We study in this article the representation theory of a family of super algebras, called the super Yang-Mills algebras, by exploiting the Kirillov orbit method à la Dixmier for nilpotent super Lie algebras. These super algebras are an extension of the so-called Yang-Mills algebras, introduced by A. Connes and M. Dubois-Violette in (Lett Math Phys 61(2):149–158, 2002), and in fact they appear as a “background independent” formulation of supersymmetric gauge theory considered in physics, in a similar way as Yang-Mills algebras do the same for the usual gauge theory. Our main result states that, under certain hypotheses, all Clifford-Weyl super algebras ${{\rm {Cliff}}_{q}(k) \otimes A_{p}(k)}$ , for p ≥ 3, or p = 2 and q ≥ 2, appear as a quotient of all super Yang-Mills algebras, for n ≥ 3 and s ≥ 1. This provides thus a family of representations of the super Yang-Mills algebras.     

Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras U_q,q(gl(n)) and U_q,q(sl(n)) which are in duality with the multiparameter quantum groups GL_qq(n), SL_qq(n), respectively. These objects depend on n(n − 1)/2+ 1 deformation parameters q, q_ij (1 ≤ i< j ≤ n) which is the maximal possible number in the case of GL(n). The representations are labelled by n − 1 complex numbers r_i and are acting in the space of formal power series of n(n − 1)/2 non-commuting variables. These variables generate quantum flag manifolds of GL_qq(n), SL_qq(n). The case n = 3 is treated in more detail.

     

Representations of aq-deformed Heisenberg algebra

We extend the symmetric operators of theq-deformed Heisenberg algebra to essentially self-adjoint operators. On the extended domains the product of the operators is not defined. To represent the algebra we had to enlarge the representation and we find a Hilbert space representation of the deformed Heisenberg algebra in terms of essentially self-adjoint operators. The respective diagonalization can be achieved by aq-deformed Fourier transformation.     

Representations of generalized oscillator algebra

The representations of the oscillator algebra introduced by Brzeziski et al. [Phys. Lett. B 311 (1993) 202] are classified.     

Matter as Spectrum of Spacetime Representations

Bound and scattering state Schrödinger functions of nonrelativistic quantum mechanics as representation matrix elements of space and time are embedded into residual representations of spacetime as generalizations of Feynman propagators. The representation invariants arise as singularities of rational representation functions in the complex energy and complex momentum plane. The homogeneous space GL(²)U(2) with rank 2, the orientation manifold of the unitary hypercharge-isospin group, is taken as model of nonlinear spacetime. Its representations are characterized by two continuous invariants whose ratio will be related to gauge field coupling constants as residues of the related representation functions. Invariants of product representations define unitary Poincaré group representations with masses for free particles in tangent Minkowski spacetime.     

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is possible to reassemble the irreducible summands appearing in the unitary braid group representations from a unitary braided fusion category with possibly different positive multiplicities to get representations that are uniformly equivalent to the ones from a unitary R-matrix. Such an equivalence will be called a localization of the unitary braid group representations. We show that the q = e ^πi/6 specialization of the unitary Jones representation of the braid groups can be localized by a unitary 9 × 9 R-matrix. Actually this Jones representation is the first one in a family of theories (SO(N), 2) for an odd prime N > 1, which are conjectured to be localizable. We formulate several general conjectures and discuss possible connections to physics and computer science.     

Representations of the general covariance pseudogroup

Representations of the general covariance pseudogroup induced by those of its translation-free subgroup are studied.All representations of the finite-component field type are found to belong to a “truncated” class which carries representations of finite-parameter Lie algebras in the fibers of the vector bundle.The conditions for unitarity are discussed and some particular intertwining kernels constructed that implement unitarity in certain subsets of the pseudogroup transformations.     

Representations of MV-Algebras by Hilbert-Space Effects

It is shown that for every Archimedean MV-effect algebra M (equivalently, every Archimedean MV-algebra) there is an injective MV-algebra morphism into the MV-algebra of all multiplication operators between the zero and identity operator on $\ell_{2}(\mathcal{S}_{0})$ , where $\mathcal{S}_{0}$ is an ordering set of extremal states (state morphisms) on M.