Lie-Poisson Deformation of the Poincar\'e Algebra

We find a one parameter family of quadratic Poisson structures on ${\bf R}^4\times SL(2,C)$ which satisfies the property {\it a)} that it is preserved under the Lie-Poisson action of the Lorentz group, as well as {\it b)} that it reduces to the standard Poincar\'e algebra for a particular limiting value of the parameter. (The Lie-Poisson transformations reduce to canonical ones in that limit, which we therefore refer to as the `canonical limit'.) Like with the Poincar\'e algebra, our deformed Poincar\'e algebra has two Casimir functions which we associate with `mass' and `spin'. We parametrize the symplectic leaves of ${\bf R}^4\times SL(2,C)$ with space-time coordinates, momenta and spin, thereby obtaining realizations of the deformed algebra for the cases of a spinless and a spinning particle. The formalism can be applied for finding a one parameter family of canonically inequivalent descriptions of the photon.Comment: Latex file, 26 page

On the physics behind the form factor ratio μpGEp(Q2)/GMp(Q2)\mu_p G_E^p (Q^2) / G_M^p (Q^2)

We point out that there exist two natural definitions of the nucleon magnetization densities : the density $\rho_M^K (r)$ introduced in Kelly's phenomenological analysis and theoretically more standard one $\rho_M (r)$. We can derive an explicit analytical relation between them, although Kelly's density is more useful to disentangle the physical origin of the different $Q^2$ dependence of the Sachs electric and magnetic form factors of the nucleon. We evaluate both of $\rho_M (r)$ and $\rho_M^K (r)$ as well as the charge density $\rho_{ch}(r)$ of the proton within the framework of the chiral quark soliton model, to find a noticeable qualitative difference between $\rho_{ch}(r)$ and $\rho_M^K (r)$, which is just consistent with Kelly's result obtained from the empirical information on the Sachs electric and magnetic form factors of the proton.Comment: 12 pages, 5 figures. version to appear in J. Phys. G.: Nucl. Part. Phy

Almost Special Holonomy in Type IIA&M Theory

We consider spaces M_7 and M_8 of G_2 holonomy and Spin(7) holonomy in seven and eight dimensions, with a U(1) isometry. For metrics where the length of the associated circle is everywhere finite and non-zero, one can perform a Kaluza-Klein reduction of supersymmetric M-theory solutions (Minkowksi)_4\times M_7 or (Minkowksi)_3\times M_8, to give supersymmetric solutions (Minkowksi)_4\times Y_6 or (Minkowksi)_3\times Y_7 in type IIA string theory with a non-singular dilaton. We study the associated six-dimensional and seven-dimensional spaces Y_6 and Y_7 perturbatively in the regime where the string coupling is weak but still non-zero, for which the metrics remain Ricci-flat but that they no longer have special holonomy, at the linearised level. In fact they have ``almost special holonomy,'' which for the case of Y_6 means almost Kahler, together with a further condition. For Y_7 we are led to introduce the notion of an ``almost G_2 manifold,'' for which the associative 3-form is closed but not co-closed. We obtain explicit classes of non-singular metrics of almost special holonomy, associated with the near Gromov-Hausdorff limits of families of complete non-singular G_2 and Spin(7) metrics.Comment: Latex, 26 page