The Lie algebra $$\mathfrak{s}\mathfrak{l}_2 (\mathbb{F})$$ and quasi-deformations

The Lie algebra is “deformed” using twisted derivations satisfying a twisted Leibniz rule. Some particular algebras appearing in this deformation scheme are discussed. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005. Supported by the Liegrits network Supported by the Crafoord foundation     

Parametric Representation of Noncommutative Field Theory

In this paper we investigate the Schwinger parametric representation for the Feynman amplitudes of the recently discovered renormalizable quantum field theory on the Moyal non commutative space. This representation involves new hyperbolic polynomials which are the non-commutative analogs of the usual “Kirchoff” or “Symanzik” polynomials of commutative field theory, but contain richer topological information. Work supported by ANR grant NT05-3-43374 “GenoPhy”.     

Nuclearity and Thermal States in Conformal Field Theory

We introduce a new type of spectral density condition, that we call L ²- nuclearity. One formulation concerns lowest weight unitary representations of and turns out to be equivalent to the existence of characters. A second formulation concerns inclusions of local observable von Neumann algebras in Quantum Field Theory. We show the two formulations to agree in chiral Conformal QFT and, starting from the trace class condition for the conformal Hamiltonian L ₀, we infer and naturally estimate the Buchholz-Wichmann nuclearity condition and the (distal) split property. As a corollary, if L ₀ is log-elliptic, the Buchholz-Junglas set up is realized and so there exists a β-KMS state for the translation dynamics on the net of C^*-algebras for every inverse temperature β > 0. We include further discussions on higher dimensional spacetimes. In particular, we verify that L ²-nuclearity is satisfied for the scalar, massless Klein-Gordon field. Dedicated to László Zsidó on the occasion of his sixtieth birthday Supported by MIUR, GNAMPA-INDAM and EU network “Quantum Spaces–Non Commutative Geometry” HPRN-CT-2002-00280     

On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

New Electroweak Formulation Fundamentally Accounting for the Effect Known as “Maximal Parity-Violation”

The electroweak scheme is wholly recast, in the framework of a relativistic quantum field formalism being a covariant fermion–antifermion extension of the usual one for massive spin- point fermions. The new formalism is able to reread the “maximal P-violation” effect in a way restoring P and C symmetries themselves: it provides a natural “chiral field” approach, which gives evidence of the existence of a pseudoscalar (extra) charge variety anticommuting with the scalar (ordinary) one and just underlying the “maximally P-violating” phenomenology. Its zero-mass limit leads to a strict “chiral” particle theory, which remodels any massless spin- fermion and corresponding antifermion as two mere pseudoscalar-charge eigenstates being the simple mirror images of each other. On such a basis, the (zero-mass) electroweak primary fermions are all redefined to be (only left-handed) “chiral” particles (with right-handed complements just standing for their antiparticles) and to carry at most scalar charges subjected as yet to a maximal uncertainty in sign: it is only by acquiring mass, and by gaining an extra helicity freedom degree, that they now may also manifest themselves as “Dirac” particles, with sharp scalar-charge eigenvalues. The fermion-mass appearance is thus made herein a dynamical condition strictly necessary to obtain actual superselected scalar-charge (and first, electric-charge) eigenstates. A pure “internal” mass-generating mechanism, relying only on would-be-Goldstone bosons (even to yield fermion masses) and no longer including an “external” Higgs contribution, is adopted accordingly. This is shown to be a self-consistent mechanism, which still maintains both renormalizability and unitarity. It involves a P-breaking in the neutral-weak-current sector (due to the Weinberg mixing) while it leaves the charged-current couplings truly P-invariant even in the presence of a (standardly parametrized) CP-violation.     

Mach’s principle and hidden matter

According to the Einstein-Mayer theory of the Riemanniann space-time with Einstein-Cartan teleparallelism, the local Lorentz invariance is broken by the gravitational field defining Machian reference systems. This breaking of symmetry implies the occurrence of “hidden matter” in the Einstein equations of gravity. The hidden matter is described by the non-Lorentz-invariant energy-momentum tensor satisfying the relation . The tensor is formed from the Einstein-Cartan torsion field given by the anholonomy objects, F_Aik=2h_A[i,k], and appears together with Hilbert’s energy-momentum tensor T^* _ik and Poincaré’s pressure λg_ik on the right-hand side of Einstein’s equations so that one has

Gauge theory in deformed $$ \mathcal{N} $$ = (1, 1) superspace

We review the non-anticommutative Q-deformations of = (1, 1) supersymmetric theories in four-dimensional Euclidean harmonic superspace. These deformations preserve chirality and harmonic Grassmann analyticity. The associated field theories arise as a low-energy limit of string theory in specific backgrounds and generalize the Moyal-deformed supersymmetric field theories. A characteristic feature of the Q-deformed theories is the half-breaking of supersymmetry in the chiral sector of the Euclidean superspace. Our main focus is on the chiral singlet Q-deformation, which is distinguished by preserving the SO(4) ∼ Spin(4) “Lorentz” symmetry and the SU(2) R-symmetry. We present the superfield and component structures of the deformed = (1, 0) supersymmetric gauge theory as well as of hypermultiplets coupled to a gauge superfield: invariant actions, deformed transformation rules, and so on. We discuss quantum aspects of these models and prove their renormalizability in the Abelian case. For the charged hypermultiplet in an Abelian gauge superfield background we construct the deformed holomorphic effective action. The text was submitted by the authors in English.     

Ortho- and Para-helium in Relativistic Schrödinger Theory

The characteristic features of ortho- and para-helium are investigated within the framework of Relativistic Schr?dinger Theory (RST). The emphasis lies on the conceptual level, where the geometric and physical properties of both RST field configurations are inspected in detail. From the geometric point of view, the striking feature consists in the splitting of the -valued bundle connection into an abelian electromagnetic part (organizing the electromagnetic interactions between the two electrons) and an exchange part, which is responsible for their exchange interactions. The electromagnetic interactions are mediated by the usual four-potentials A _μ and thus are essentially the same for both types of field configurations, where naturally the electrostatic forces (described by the time component A ₀ of A _μ) dominate their magnetostatic counterparts (described by the space part A of A _μ). Quite analogously to this, the exchange forces are as well described in terms of a certain vector potential (B _μ), again along the gauge principles of minimal coupling, so that also the exchange forces split up into an “electric” type ( ) and a “magnetic” type ( ). The physical difference of ortho- and para-helium is now that the first (ortho-) type is governed mainly by the “electric” kind of exchange forces and therefore is subject to a stronger influence of the exchange phenomenon; whereas the second (para-) type has vanishing “electric” exchange potential (B ₀ ≡ 0) and therefore realizes exclusively the “magnetic” kind of interactions ( ), which, however, in general are smaller than their “electric” counterparts. The corresponding ortho/para splitting of the helium energy levels is inspected merely in the lowest order of approximation, where it coincides with the Hartree–Fock (HF) approximation. Thus RST may be conceived as a relativistic generalization of the HF approach where the fluid-dynamic character of RST implies many similarities with the density functional theory.     

Algebraic Theory of Linear Viscoelastic Nematodynamics

This paper consists of two parts. The first one develops algebraic theory of linear anisotropic nematic “N-operators” build up on the additive group of traceless second rank 3D tensors. These operators have been implicitly used in continual theories of nematic liquid crystals and weakly elastic nematic elastomers. It is shown that there exists a non-commutative, multiplicative group N₆ of N-operators build up on a manifold in 6D space of parameters. Positive N-operators, which in physical applications hold thermodynamic stability constraints, do not generally form a subgroup of group N₆. A three-parametric, commutative transversal-isotropic subgroup of positive symmetric nematic operators is also briefly discussed. The special case of singular, non-negative symmetric N-operators reveals the algebraic structure of nematic soft deformation modes. The second part of the paper develops a theory of linear viscoelastic nematodynamics applicable to liquid crystalline polymer. The viscous and elastic nematic components in theory are described by using the Leslie–Ericksen–Parodi (LEP) approach for viscous nematics and de Gennes free energy for weakly elastic nematic elastomers. The case of applied external magnetic field exemplifies the occurrence of non-symmetric stresses. In spite of multi-(10) parametric character of the theory, the use of nematic operators presents it in a transparent form. When the magnetic field is absent, the theory is simplified for symmetric case with six parameters, and takes an extremely simple, two-parametric form for viscoelastic nematodynamics with possible soft deformation modes. It is shown that the linear nematodynamics is always reducible to the LEP-like equations where the coefficients are changed for linear memory functionals whose parameters are calculated from original viscosities and moduli.      

Modular Conjugation and the Implementation of Supersymmetry

Any -graded C ^*-dynamical system with a self-adjoint graded-Kubo-Martin-Schwinger (KMS) functional on it can be represented (canonically) as a -graded algebra of bounded operators on a -graded Hilbert space, so that the grading of the latter is compatible with the functional. The modular conjugation operator plays a crucial role in this reconstruction. The results are generalized to the case of an unbounded graded-KMS functional having as dense domain the union of a net of C ^*-subalgebras. It is shown that the modulus of such an unbounded graded-KMS functional is KMS.      

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C ^*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor ^*-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C ^*-algebra. As an application, we associate an ergodic G-action on a unital C ^*-algebra to an inclusion of Rep(G) into an abstract tensor C ^*-category . If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G. If a tensor C ^*-category has a Hecke symmetry making an object ρ of dimension d and μ-determinant 1, then there is an ergodic action of S _μ U(d) on a unital C ^*-algebra having the as its spectral subspaces. The special case of is discussed.     

An Index for 4 Dimensional Super Conformal Theories

We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S ³ × time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in AdS ₅ × S ⁵ and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS ₅, but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the theory. We note that entropy for some small supersymmetric AdS ₅ black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the Yang Mills theory.     

Algebrodynamics over complex space and phase extension of the Minkowski geometry

First principles should predetermine physical geometry and dynamics both together. In the “algebrodynamics” they follow solely from the properties of biquaternion algebra and the analysis over . We briefly present the algebrodynamics over Minkowski background based on a nonlinear generalization to of the Cauchi-Riemann analyticity conditions. Further, we consider the effective real geometry uniquely resulting from the structure of multiplication and found it to be of the Minkowski type, with an additional phase invariant. Then we pass to study the primordial dynamics that takes place in the complex space and brings into consideration a number of remarkable structures: an ensemble of identical correlated matter pre-elements (“duplicons”), caustic-like signals (interaction carriers), a concept of random complex time resulting in irreversibility of physical time at macrolevel, etc. In partucular, the concept of “dimerous electron” naturally arises in the framework of complex algebrodynamics and, together with the above-mentioned phase invariant, allows for a novel approach to explanation of quantum interference phenomena alternative to recently accepted wave—particle dualism paradigm. The text was submitted by the author in English.     

Quadratic Deformations of Lie–Poisson Structures

In this letter, first we give a decomposition for any Lie–Poisson structure associated to the modular vector. In particular, splits into two compatible Lie–Poisson structures if . As an application, we classified quadratic deformations of Lie– Poisson structures on up to linear diffeomorphisms. Research partially supported by NSF of China and the Research Project of “Nonlinear Science”.     

Entropy functionals of Kolmogorov-Sinai type and their limit theorems

Two functionals and are introduced forC ^*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov-Sinai-type theorems hold for these functionals and . Our functionals and are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya.     

Convergence of the Wick Star Product

We construct a Fréchet space as a subspace of where the Wick star product converges and is continuous. The resulting Fréchet algebra _ħ is studied in detail including a ^*-representation of _ħ in the Bargmann-Fock space and a discussion of star exponentials and coherent states.     

Combinatorial Point for Fused Loop Models

Integrable loop models associated with higher representations (spin ℓ/2) of are investigated at the point . The ground state eigenvalue and eigenvectors are described. Introducing inhomogeneities into the models allows to derive a sum rule for the ground state entries. Supported by ANR program “GIMP” ANR-05-BLAN-0029-01, European networks “ENIGMA” MRT-CT-2004-5652, “ENRAGE” MRTN-CT-2004-005616, and ESF program “MISGAM”.     

A remark on extensions of quasi-free derivations on the CAR-algebra

Let δ be a quasi-free derivation of the CAR algebra, and let be a closed *-derivation which is an extension of δ. We use Price's techniques from [6] to show that if the polynomials in the linear field operators a(f)→a ^* (f) in D( ) is a core for , then is quasi-free.     

<Emphasis Type="Italic">q</Emphasis>-Analogue of Modified KP Hierarchy and its Quasi-Classical Limit

A string basis is constructed for each subalgebra of invariants of the function algebra on the quantum special linear group. By analyzing the string basis for a particular subalgebra of invariants, we obtain a “canonical basis” for every finite dimensional irreducible -module. It is also shown that the algebra of functions on any quantum homogeneous space is generated by quantum minors. Supported by the Australian Research Council and Chinese National Natural Science Foundation project number: 10471070