Smooth Loops, Generalized Coherent States and Geometric Phases

A description of generalized coherent states and geometric phases in the light of the general theory of smooth loops is given.Comment: LATeX file, 11 page

The principle of relative locality

We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry, such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature.Comment: 12 pages, 3 figures; in version 2 one reference added and some minor modifications in sects. II and III mad

Quasigroups, Asymptotic Symmetries and Conservation Laws in General Relativity

A new quasigroup approach to conservation laws in general relativity is applied to study asymptotically flat at future null infinity spacetime. The infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to the Poincar\'e quasigroup and the Noether charge associated with any element of the Poincar\'e quasialgebra is defined. The integral conserved quantities of energy-momentum and angular momentum are linear on generators of Poincar\'e quasigroup, free of the supertranslation ambiguity, posess the flux and identically equal to zero in Minkowski spacetime.Comment: RevTeX4, 5 page

Smooth Loops and Fiber Bundles: Theory of Principal Q-bundles

A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher ''nonassociative'' symmetries.Comment: 20 page

On the notion of gyrogroup

together with CnoFile{1619832}{} Einstein's relativistic velocity addition oplus in the c-ball bold R^3_c of the Euclidean 3-space bold R^3 is nonassociative. Hence, the Einstein groupoid (bold R^3_c,oplus) is not a group. Abstracting the key features of Einstein's groupoid (bold R^3_c,oplus) [see W. Krammer and H. K. Urbantke, Results Math. {bf 33} (1998), no.~3-4, 310--327; [msn] MR1619832 (99i:83003) [/msn]; see the preceding review], the notion of gyrogroup [A. A. Ungar, Found. Phys. {bf 27} (1997), no.~6, 881--951; [msn] MR1477047 (98k:83002) [/msn]] was introduced by the reviewer [Amer. J. Phys. {bf 59} (1991), no.~9, 824--834; [msn] MR1126776 (92g:83003) [/msn]] to describe a grouplike object that shares analogies with groups. The analogies stem from the relativistic effect called Thomas gyration (or, precession), and are emphasized by the prefix "gyro" that is extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, gyrotranslations, etc. Furthermore, the reviewer introduced the gyrosemidirect product group as the product of a gyrogroup and a group that results in a group, in full analogy with the semidirect product group between two groups. The reviewer gave a brief history of the idea of a product of a nongroup groupoid and a group which results in a group, tracing it to Tits, Karzel and Kikkawa [A. A. Ungar, Aequationes Math. {bf 47} (1994), no.~2-3, 240--254; [msn] MR1268034 (95b:30076) [/msn]]. par However, the authors of the paper under review show that the idea should be traced to Sabinin's work, which has been rediscovered by Kikkawa. The authors justifiably claim that the reviewer did not give appropriate credit to Sabinin's pioneering work in this area. They describe the connections between gyrogroup theory and loop theory demonstrating that a gyrogroup is a Bol loop with Bruck identity, and that the reviewer's gyrosemidirect product is a construction that Sabinin has considered in a more general case since 1972 [P. O. Mikheev and L. V. Sabinin, in {it Quasigroups and loops: theory and applications}, 357--430, Heldermann, Berlin, 1990; [msn] MR1125818 [/msn]]. Finally, the authors remark that the discovery of gyrogroups in mathematical physics shows the vitality of quasigroup and loop theory