Bifurcation symmetries

The bifurcation of one mode into several branches frequently generates symmetries among the branches. These symmetries are “naturally broken”. After a look at some simple examples, we study a model where the bifurcation symmetry has some features of a flavor symmetry.     

Primeval symmetries

A detailed examination of the Killing equations in Robertson–Walker coordinates shows how the addition of matter and/or radiation to a de Sitter Universe breaks the symmetry generated by four of its Killing fields. The product \(U = a^2 \,{\dot H}\) of the squared scale parameter by the time-derivative of the Hubble function encapsulates the relationship between the two cases: the symmetry is maximal when U is a constant, and reduces to the six-parameter symmetry of a generic Friedmann–Robertson–Walker model when it is not. As the fields physical interpretation is not clear in these coordinates, comparison is made with the Killing fields in static coordinates, whose interpretation is made clearer by their direct relationship to the Poincaré group generators via Wigner–Inönú contractions.     

Clusterization and dynamical symmetries

A variety of dynamical symmetries related to nuclear clusterization is discussed.     

Antihydrogen and fundamental symmetries

The prospects for testing CPT invariance and the weak equivalence principle (WEP) for antimatter with spectroscopic measurements on antihydrogen are discussed. The potential precisions of these tests are compared with those from other measurements. “If there is negative electricity, why not negative gold, as yellow...as our own, with the same boiling point and identical spectrallines...” A. Schuster [1], 1898     

Dressing symmetries

We study the group of dressing transformations in soliton theories. We show that it is generated by the monodromy matrix. This provides a new proof of their Lie-Poisson property. We treat in detail the examples of the Toda field theories and the Heisenberg model. We show that the group of dressing transformations is the classical precursor of the various manifestations of quantum groups in these models, e.g. algebraic Bethe ansatz, non-local currents, or quantum group symmetries. Finally, we define field multiplets supporting a linear representation of the dressing group and we show that their exchange algebras are encoded in the classical double.Communicated by K. Gawedzki     

Monopoles and their symmetries

Properties of spherically symmetric monopoles are discussed. Inversion symmetry is also considered.     

Spatiotemporal and statistical symmetries

The notion of symmetries, either statistical or deterministic, can be useful for the characterization of complex systems and their bifurcations. In this paper, we investigate the connection between the (microscopic) spatiotemporal symmetries of a space-time functionu(x, t), on the one hand, and the (macroscopic) symmetries of statistical quantities such as the spatial (resp. temporal) two-point correlations and the spatial (resp. temporal) average, on the other hand. We show, how, under certain conditions, these symmetries are related to the symmetries of the orbits described byu(x, t) in the characteristic (phase) spaces. We also determine the largest group of spatiotemporal symmetries (in the sense introduced in our earlier work) satisfied by a given space-time functionu(x, t) and indicate how to extract the subgroups of point symmetries, namely those directly implemented on the space and time variables. Conversely, we determine all the functions invariant by a given space-time symmetry group. Finally, we illustrate all the previous points with specific examples.     

Levi-Civita equivariance and Riemann symmetries

We present a novel derivation of all the symmetries of the Riemann curvature tensor. Our approach is entirely geometric, using as it does the natural equivariance of the Levi-Civita map with respect to diffeomorphisms. An important conclusion is thatall symmetries of the curvature tensor have their origin in the principle of general covariance.     

Dynamical symmetries and Nambu mechanics

It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique.     

Connections and symmetries in spacetime

The extent to which a symmetric, metric connection on spacetime determines the metric is given, and some applications to affine collineations are discussed.