New method of applying conformal group to quantum fields

Most of previous work on applying the conformal group to quantum fields has emphasized its invariant aspects, whereas in this paper we find that the conformal group can give us running quantum fields, with some constants, vertex and Green functions running, compatible with the scaling properties of renormalization group method （RGM）. We start with the renormalization group equation （RGE）, in which the differential operator happens to be a generator of the conformal group, named dilatation operator. In addition we link the operator/spatial representation and unitary/spinor representation of the conformal group by inquiring a conformal-invariant interaction vertex mimicking the similar process of Lorentz transformation applied to Dirac equation. By this kind of application, we find out that quite a few interaction vertices are separately invaxiant under certain transformations （generators） of the conformal group. The significance of these transformations and vertices is explained. Using a particular generator of the conformal group, we suggest a new equation analogous to RGE which may lead a system to evolve from asymptotic regime to nonperturbative regime, in contrast to the effect of the conventional RGE from nonperturbative regime to asymptotic regime.

Nonlinear Conformal Electromagnetism

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manifold of dimension n, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n=4 has very specific properties for the computation of the Spencer cohomology, we prove that there is thus no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.

Zeeman-Like Topologies in Special and General Theory of Relativity

This is a short review article in which we discuss and summarize the works of various researchers over past four decades on Zeeman topology and Zeeman-like topologies, which occur in special and general theory of relativity. We also discuss various properties and inter-relationship of these topologies.

A Contribution to the DLF-Theory: On Singularities of the SU(2,2)-Action in U(1,1)

Segal’s chronometric theory is based on a space-time D, which might be viewed as a Lie group with a causal structure defined by an invariant Lorentzian form on the Lie algebra u(2). Similarly, the space-time F is realized as the Lie group with a causal structure defined by an invariant Lorentzian form on u(1,1). Two Lie groups G, GF are introduced as representations of SU(2,2): they are related via conjugation by a certain matrix Win Gl(4). The linear-fractional action of G on D is well-known to be global, conformal, and it plays a crucial role in the analysis on space-time bundles carried out by Paneitz and Segal in the 1980’s. This analysis was based on the parallelizing group U(2). In the paper, singularities’ general (“geometric”) description of the linear-fractional conformal GF-action on F is given and specific examples are presented. The results call for the analysis of space-time bundles based on U(1,1) as the parallelizing group. Certain key stages of such an analysis are suggested.

A Novel Conformation Investigation on Newly Synthesized Compound of Diethyl Puerarin-7-yl Phosphate

A novel compound, diethyl puerarin-7-yl phosphate, was synthesized through a simplified Atheron-Todd reaction for the first time. The structure of this compound was elucidated by IR, ESI-MS and NMR. Two conformations of the compound were testified by 2D NMR （HSQC and HMBC） and dynamic NMR. Furthermore, we carried out the conformational analysis using chemical calculation by the Gaussian 03. Finally, we obtained two preferred conformations and energy values.

Mbius Homogeneous Hypersurfaces with Three Distinct Principal Curvatures in S^(n+1)

Let x ： M^n→S^n＋1 be an immersed hypersurface in the （n ＋1）-dimensional sphere S^n＋1. If, for any points p,q ∈ M^n, there exists a Mobius transformation Ф ： S^n＋l →S^n＋1 such that Ф o x（M^n） = x（M^n） and Ф o x（p） = x（q）, then the hypersurface is called a Mobius homogeneous hypersurface. In this paper, the Mobius homogeneous hypersurfaces with three distinct principal curvatures are classified completely up to a Mobius transformation.