tag:blogger.com,1999:blog-10614348.post2130541445519579093..comments2024-01-22T11:26:37.599-08:00Comments on TGD diary: Correlated Polygons in Standard Cosmology and in TGDMatti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-10614348.post-20920246793978266822016-04-25T23:53:16.401-07:002016-04-25T23:53:16.401-07:00Interesting finding. It seems that the idea is to ...Interesting finding. It seems that the idea is to represent the effect of time reparametrization of time coordinate as a modification of G and Lambda. Since the metric is scaled by a conformal factor in conformal transformation this brings in Schwarzian derivative in the transformation of cosmological term. The re-defined cosmological term emerges from the change of time coordinate and contaisn third time derivative although transformation formula for tensor involves only first derivatives of coordinate variables.<br /><br />This is tricky game: one replaces G and Lambda with time dependent constants: I do not believe that this helps much. The appearence of Schwartzian derivative suggests that there might be however some deeper involved<br /><br />The cosmological constant term would be analogous to energy momentum tensor in conformal field theories. One interpretation in TGD could be that string world sheets and magnetic flux tubes accompany each other and magnetic energy gives rise to analog of dark energy characterized by cosmological constant. Second interpreration could be that the appearance of small four-volume term in twistor lift of Kaehler action corresponds to this term. Volume term generalizes bosonic string action (area).<br /><br />In TGD framework this relates to the extension of 2-D conformal symmetries to their 4-D analogs. One has supersymplectic symmetries with structure of conformal algebra and extened conformal symmetries at light-cone boundary and at light-like orbits of partonic 2-surfaces and also ordinary conformal symmetries at string world sheets. <br /><br />Somehow all this should extend to a 4-D analog of conformal symmetry. Most naturally to quaternionic structure and quaternonic analyticity, which can be formulated in terms of Cauchy-Riemann-Fueter conditions (http://tgdtheory.fi/public_html/tgdquantum/tgdquantum.html#twistorstory ). Quaternionic analyticity would be a property of preferred extremals.<br /><br /> I wish I could make this intuition more explicit. The generalization of conformal field theory should be algebraic continuation from complex plane to quaternions.Matpitka6@gmail.comhttp://tgdtheory.com/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-77011816900244083362016-04-25T16:27:36.695-07:002016-04-25T16:27:36.695-07:00Dark Energy and the Schwarzian Derivative
http://...Dark Energy and the Schwarzian Derivative<br /><br />http://arxiv.org/abs/1403.5431<br /><br />6 Conclusion<br />In this paper I have shown how the Schwarzian derivative enters the formula for the change of the density of dark energy under temporal re-parameterisations and how the Schwarzian tensor enters when considering conformal rescalings of the metric. I have illustrated this by considering a ΛCDM cosmology. It is striking that a similar behavour crops up in the change of the stress tensor of a two-dimensional CFT under conformal transformations. This seems to hint at a deeper connection between dark energy and CFT’s in 3+1 spacetime dimensions. In this connection it would be interesting to see whether or how the Schwarzian derivative enters the transformation formulae for the stress tensor.Anonymousnoreply@blogger.com