Ph. Spindel

Extended supersymmetric σ-Models on group manifolds (I). The complex structures

Abstract We investigate the possibilities of N-extended rigid supersymmetries in two-dimensional actions, in particular parallelized group manifolds. We give a full classification of all such manifolds with N = 2 or N = 4. We explicitly construct the supersymmetries which are related to complex or quaternionic structures. The N = 4 manifolds are related to Wolf spaces. Only flat spaces allow N > 4. Depending on the dimension N = 1, 2, 4, 8 mod 8.

Extended supersymmetric σ-models on group manifolds (II). Current algebras

Abstract The supersymmetries of σ-models on group manifolds are generated by currents, of which we calculate the operator product expansions. For flat spaces with N > 4 there appear more operators than the usual dimension 2, 3 2 and 1 currents. For N = 2 on any even dimensional group manifold we find the Ademollo et al. algebra. For N = 4 we find a class of new algebras. In all cases except SU(2) ⊗ U(1) and SU(3) the algebra splits in a direct product of several of these new N = 4 algebras, each having a complicated energy-momentum tensor. By including background charges one can restrict the algebras to dimension 2, 3 2 and 1 currents. On the non-compact Wolf spaces one can obtain zero central charge.

Complex Structures on Parallelized Group Manifolds and Supersymmetric

Abstract We determine the number of supersymmetries of σ -models having a Wess-Zumino term in their lagrangian. We show that, working on parallelized group manifolds, the obstruction due to the holonomy group of the target space is avoided, but nevertheless N ⩽4 except on tori. We give an explicit description of the supersymmetries beyond N =1, which are related to complex structures. Finally, the group manifolds are classified according to N .

\sigma

The creation of the universe is regarded as a self-consistent process in which matter is engendered by the space-time varying cosmological gravitational field and vice versa. Abundant production can occur only if the mass of the particles so created is of the order of the Planck mass (= κ−12. We conjecture that this is the origin of the fundamental length scale in field theory, as it is encountered, for example, in present efforts towards grandunification. The region of particle production is steady state in character. It ceases when the produced particles decay. The geometry of this steady state is characteristic of a de Sitter space. It permits one to estimate the number of ordinary particles presently observed, N. We find log N = O (mτdecay) = O(g−2) = O(102), with the usual estimate of g = O(10− at the Planck length scale. This is not inconsistent with the experimental estimate N ⋍ O(1090). After production, cosmological history gives way to the more conventional scheme of free expansion. The present paper is a self-contained account of our view of cosmological history and the production of matter in a varying gravitational field. Special care has been taken to describe the vacuum correctly in the present context and to perform the necessary subtractions of zero-point effects.

Models

Loop algebras of parallel transformations are defined on seven spheres. These algebras can be extended with eight fermionic charges and the Virasoro algebra to a soft superalgebra using the geometry of cosets of supergroups, or to a nonassociative infinite‐dimensional N=8 d=2 superconformal algebra.

Cosmogenesis and the origin of the fundamental length scale

Abstract We study, using Rindler coordinates, the quantization of a charged scalar field interacting with a constant (Poincare invariant), external, electric field in (1+1) dimensionnal flatspace: our main motivation is pedagogy. We illustrate in this framework the equivalence between various approaches to field quantization commonly used in the framework of curved backgrounds. First we establish the expression of the Schwinger vacuum decay rate, using the operator formalism. Then we rederive it in the framework of the Feynman path integral method. Our analysis reinforces the conjecture which identifies the zero winding sector of the Minkowski propagator with the Rindler propagator. Moreover, we compute the expression of the Unruhs modes that allow us to make a connection between the Minkowskian and Rindlerian quantization schemes by purely algebraic relations. We use these modes to study the physics of a charged two level detector moving in an electric field whose transitions are due to the exchange of charged quanta. In the limit where the Schwinger pair production mechanism of the exchanged quanta becomes negligible we recover the Boltzman equilibrium ratio for the population of the levels of the detector. Finally we explicitly show how the detector can be taken as the large mass and charge limit of an interacting fields system.

Loop algebras and superalgebras based on S 7

Abstract It is well known that the presence of a constant electric field in Minkowski space induces pair production from vacuum. The population ratios of pairs of different mass so produced is Boltzmann in character, the temperature being a /2 π ; where a is the acceleration due to the field. These ratios are maintained subsequent to production when the interaction of the charged particles with the vacuum of neutral radiation is taken into account, i.e. particles are born in a thermal equilibrium which is maintained dynamically. We attempt to explain this consistency in terms of the tunneling of the wave function behind the classical turning point.

Quantum Charged Fields in (1+1) Rindler Space

We analyze the emergence of a minimal length for a large class of generalized commutation relations, preserving commutation of the position operators and translation invariance as well as rotation invariance (in dimensions higher than one). We show that the construction of the maximally localized states based on squeezed states generally fails. Rather, one must resort to a constrained variational principle.

Thermal properties of pairs produced by an electric field: A tunneling approach

We compute the Pauli–Jordan, Hadamard, and Feynman propagators for the massive metrical perturbations on de Sitter space. They are expressed both in terms of mode sums and in invariant forms.

About maximally localized states in quantum mechanics

We provide an exhaustive classification of self-dual four-dimensional gravitational instantons foliated with three-dimensional homogeneous spaces, i.e. homogeneous self-dual metrics on four-dimensional Euclidean spaces admitting a Bianchi simply transitive isometry group. The classification pattern is based on the algebra homomorphisms relating the Bianchi group and the duality group SO(3). New and general solutions are found for Bianchi III.