August 31, 2004; additional features of the N-E-S Seminar: meets at the SPb University (Saint Petersburg, Russia)

August 20, 2003: Chronometric Bulletin #3.

Big news of the last few months: two more worlds have "mathematically shown" themselves. All three worlds have been talked about since ancient times (see http://agniyoga.org): Dense, D (where we currently live in), L ("light", the Subtle), F (the Fiery). These letters are used below to denote their simplest mathematical models.

D is just the Segal's chronometric world. D is a universal cover of the matrix group U(2) together with a bi-invariant (Lorentzian) inner product on it. It has a constant positive scalar curvature.

Actually, the mathematical knowledge of L, F was already available in early 80s (published in several articles, mostly in Russian; articles [1,4] from the "About the Instructor" file from my webpage have been translated into English). I was then less aware of their physical significance.

There are three "very special" dim=4 symmetric spaces which have none-zero curvature [1, Corollary on p.256]. This year I have realized that the three groups with respective bi-invariant Lorentzian metrics seem to be parts of one whole.

The Fiery world F is a universal cover of the matrix group U(1,1) (with the respective bi-invariant inner product). It is of constant negative scalar curvature.

The world L (also based on a certain Lie group, a solvable one) has a name of "plain wave Einstein equations solution", it is a very special space-time among worlds of General Relativity. The curvature tensor is not zero but the scalar curvature is zero (it has been discussed in detail in my paper [4]). No surprise that it plays a role of a mediator between D and F (we go to L when we "die").

Roughly speaking, the three worlds share the same causal structure. (F, L can be locally realized as dim = 4 orbits in U(2)).

The causal structure of the simply connected version of F is acceptable (no closed causal curves). It follows from a similar property of the simply connected version of SU(1,1) (that last one is a non-trivial result obtained in "Lie Groups, Convex Cones, and Semigroups" by J. Hilgert, K.H. Hofmann, and J. D. Lawson; Clarendon Press. Oxford, 1989).

Physics-related conjecture is that the three worlds have to be considered together, and, as a result, our mathematical description of D can not be quite adequate without L, F (especially, without F). That is why I compare the finding to that of QM "hidden variables".

Purely technically, the mathematical theory of particles (and of their interactions) has to be developed anew(presumably, along the D-guidelines, see Segal's publications; the "Chronometry" file on my webpage is a survey). Indeed, two more covariant fundamental operators now come into play (they originate from F-,L-Laplace-Beltrami operators). By now, only D-,M_0- operators have been used (M_0 here is for the Minkowski space-time).