On the meaning of the Einstein- and the Lorentz-covariant derivation

We define a derivation operation working on hybrid objects which are tensors under theEinstein group of coordinate transformations and under theLorentz group of transformations of reference systems. With these general covariant derivations we formulate the principle of the equivalence of theEinstein- and of theLorentz-covariant representations of the physical laws. This principle means the general covariant constance of the metric tetradsh _i ^A (Weyl's lemma).     

“Superconducting” causal nets

The world is described as a relativistic quantum neural net with a quantum condensation akin to superconductivity. The sole dynamical variable is an operator representing immediate causal connection. The net enjoys a quantum principle of equivalence implying local LorentzSL(2,C) invariance and causality. The past-future asymmetry of its cell is similar to that of the neutrino. A net phase transition is expected at temperatures on the order of theW mass rather than the Planck mass, and near gravitational singularities.     

Asymptotic expansion of the logarithm of the partition function

A method is presented permitting one to find in principle all the non-decreasing terms of the asymptotic expansion of the logarithm of the partition function when the volume of region increases. The constructions are carried out at low activity for lattice systems with generaln-body interactions, and continuous systems with two-body interactions.     

Further studies in aesthetic field theory VII

In this paper we discuss the clock paradox within the framework of the general theory of relativity. It is shown that in generalthe aging asymmetry exists. We also argue that the clock paradox, according to Mach's principle, is essentially a cosmological problem.     

Determinant Bundles, Manifolds with Boundary and Surgery¶II. Spectral Sections and Surgery Rules for Anomalies

Let be a closed fibration of Riemannian manifolds and let , be a family of generalized Dirac operators. Let be an embedded hypersurface fibering over B; . Let be the Dirac family induced on . Each fiber in is the union along of two manifolds with boundary . In this paper, generalizing our previous work[16], we prove general surgery rules for the local and global anomalies of the Bismut–Freed connection on the determinant bundle associated to . Our results depend heavily on the b-calculus [12], on the surgery calculus [11] and on the APS family index theory developed in [13], in particular on the notion of spectral section for the family . Received: 23 October 1996 / Accepted: 28 July 1997     

Unitary Operator of su q (n)-Covariant Oscillator Algebra

The unitary operator of su _q (n)-covariant oscillator algebra is constructed and two types of q-coherent states are obtained explicitly.     

SO(4, 2)-covariant formulation of the Bargmann-Wigner equations

It is shown that the Bargmann-Wigner equations can be written in anSO(4, 2)-covariant form. As well as the Lorentz rotations, theSO(4, 2) group contains a space-inversion and a time-reflection operator (which are different from the usual ones). It also contains the Foldy-Wouthuysen and Cini-Touschek transformations. The spin-s theory for the massive and massless cases, and also a set of Bargmann-Wigner equations corresponding to space-like four-momentum, are all given by the sameSO(4, 2)-covariant equations, and their solutions can be obtained by transforming the solutions corresponding to the special gauge in which the four-momentum vanishes.     

Decomposition of the step operators of theλ √q (n)-covariant oscillator algebra withq as a root of unity

In this paper the decomposition of the creation and annihilation operators ofgl _√q (n)-covariant oscillator algebra is discussed when the deformation parameterq is a (s+1)-th primitive root of unity.     

Gauge independent gravitationally-coupled electromagnetic field

We give a totally 3-covariant formulation of the gravitationally coupled electromagnetic field (with a source) in terms of physical 3-covariant variables, keeping the correspondence with the A.D.M. formulation of the problem. We introduce a 3-covariant time derivation which allows us to discuss the spreading of the transverse and longitudinal components of the electromagnetic field. Then we present the second-order propagation equation and the Hamiltonian 3-covariant equations in a complete electromagnetic gauge independent fashion using the well-known elliptic operator Δ(g _ij ) and we also discuss the dynamical impossibility of disentangling the transverse and longitudinal modes. Thereafter we extend the 3-covariant formulation to the gravitational field and consider the problems of the initial conditions for the full system, as done by York for the vacuum case. Finally, we apply the formalism to the merostatic sourceless problem, showing the connection between the merostatic and the static cases.     

Nonstandard GLh(n) quantum groups and contraction of covariant q-bosonic algebras

GL_h(n) × GL_h(m)-covariant h-bosonic algebras are built by contracting the GL_q(n) × GL_q(m)-covariant q-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of the corresponding R _h-matrices. Whenever n = 2, and m = 1 or 2, it is proved by using U_h(sl(2)) Clebsch-Gordan coefficients that they can also be expressed in terms of coupled commutators in a way entirely similar to the classical case. Some U_h(sl(2)) rank-(1/2) irreducible tensor operators, recently constructed by Aizawa in terms of standard bosonic operators, are shown to provide a realization of the h-bosonic algebra corresponding to n = 2 and m = 1.     

Periodic fields in five-dimensional relativity

The concept of periodic fields in the framework of five-dimensional relativity is introduced in a 5-covariant manner. An analysis is given of periodic scalar, vector and bivector fields. The resulting 4-dimensional theories are examined, and it is found that some of them correspond to familiar physical theories in space-time, whereas others require further investigation.     

gl q,q* (n)-Covariant multimode oscillators and q-symmetric states

In this paper theql _q (n) oscillator algebra is extended to the complex deformation parameter case [gl _q,q* (n) algebra], and q-symmetric states forgl _q,q* (n)-covariant multimode oscillator system are investigated.     

Maps Between Deformed and Ordinary Gauge Fields

In this paper, we introduce a map between the q-deformed gauge fields defined on the GL_q(N) -covariant quantum hyperplane and the ordinary gauge fields. Perturbative analysis of the q-deformed QED at the classical level is presented and gauge fixing à la BRST is discussed. Another star product defined on the hybrid (q,h) -plane is explicitly constructed.     

Uniqueness of the Bethe ansatz of theXXZ chain and statistical interaction

We prove that there is only one form of the Bethe ansatz with the generalXXZ chain. We give an explanation of theXXZ chain in terms of statistical interactions.     

Cluster properties of lattice and continuous systems

Various strong decay properties are proved for lattice systems with generaln-body interactions, and for continuous systems with two-body andn-body interactions. The range of the potentials is finite or infinite.Equipe de Recherche du C.N.R.S. 174.     

The general anharmonicx 2q+2 damped oscillator and the coherent state representation

We obtain here a perturbative solution of the generalx ^2q+2 anharmonic damped oscillator in the coherent state representation. The solution does not contain any secular term and shows, explicitly, the damping and the anharmonic effects.     

Covering and gluing of algebras and differential algebras

Extending work of Budzyński and Kondracki, we investigate coverings and gluings of algebras and differential algebras. We describe in detail the gluing of two quantum discs along their classical subspace, giving a C^*-algebra isomorphic to a certain Podleś sphere, as well as the gluing of U_q^1/2(sl₂)-covariant differential calculi on the discs.     

General triplectic quantization

The general structure of the Sp(2)-covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism, the so-called triplectic quantization, as presented in our previous paper with A.M. Semikhatov is further generalized and clarified. We present new unified expressions for the generating operators which are more invariant and which yield a natural realization of the operator V^a and provide for a geometrical explanation for its presence. ThisV^a operator provides then for an invariant definition of a degenerate Poisson bracket on the triplectic manifold being non-degenerate on a naturally defined submanifold. We also define inverses to non-degenerate antitriplectic metrics and give a natural generalization of the conventional calculus of exterior differential forms which, e.g., explains the properties of these inverses. Finally we define and give a consistent treatment of second class hyperconstraints.     

GLq(N)-covariant braided differential bialgebras

We study the possibility of defining the (braided) comultiplication for the GL _q (N)-covariant differential complexes on some quantum spaces. We discover suchdifferential bialgebras (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BM _q (N) with both multiplicative and additive coproducts. The latter case is related (forN = 2) to theq-Minkowski space andq-Poincaré algebra.     

Calculation of generalp-adic Feynman amplitude

The generaln-point masslessp-adic Feynman amplitude with arbitrary parameters of analytic regularization for each line is calculated. This result is presented in the form of a sum over hierarchies of a given graph. The structure of ultraviolet and infrared divergences ofp-adic Feynman amplitudes is characterized and the startriangle uniqueness identity in thep-adic case is derived.Supported by Alexander von Humboldt-Stiftung