# My COSMION 2004 Slides

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### Transcript of My COSMION 2004 Slides

- 1.General Relativity and the Dirac-Kahler Equation Alexander Yu. Vlasov Federal Radiological Center, IRH 197101, Mira Street 8 St.Petersburg, Russia A. Friedmann Lab for Theoretical Physics 191023, Griboedov Canal 30/32 St.Petersburg, Russia The Dirac Equation D = m, C4 , D = 3 =0 i x , M(4, C), + = 2g . The Dirac-Kahler Equation d d = m, (M) = 4 k=0 k (M), : k (M) 4k (M). D = (d d ), d : k (M) k+1 (M), d : k (M) k1 (M). Original papers: Dirac Equation P. A. M. Dirac, Proc. Roy. Soc. Lond. A117 610 (1928). Dirac-Kahler Equation D. D. Ivanenko and L. D. Landau, Z. Phys. 48 340 (1928), E. Kahler, Randiconti di Mat. Appl. 21 425 (1962). Factorization of the Klein-Gordon Operator m2 = 0 = 3 ,=0 g 2 xx , D2 = , D2 = (dd + d d) = 1,3 = See also http://arXiv.org/abs/math-ph/0403046 E-mail: Alexander.Vlasov@pobox.spbu.ru

2. GR and the Dirac-Kahler Equation A. Yu. Vlasov Matrix, Cliord, Tensor Form of the Dirac Equation D = m = 1 2 3 4 = 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 T 11 21 31 41 12 22 32 42 13 23 33 43 14 24 34 44 Isomorphisms of 16D linear spaces (without algebraic structures, e.g., multiplication): (C4 ) = Cl(4, C) = M(4, C) = C4 C4 ( ) Cl(4, C) Cliord algebra (Dirac-Hestenes equation, spinors as ideals and equivalence classes in the Cliord algebra). (C4 ) Exterior forms (antisymmetric tensors), Grassmann algebra (Dirac-Kahler equation, Dirac type tensor equation). Cl(4, C) = M(4, C) (also as algebras) for any given matrix representation of gamma matrices k. Cl(4, C) = (C4 ) (as linear spaces) i1 ik dxi1 dxik . () M(4, C) = C4 C4 may be formally identied with space of states of two spinor systems ( ). Transformation Properties Exterior forms (M), dierentials d and d are covariant with respect to all linear transformations, group GL(4, R). So the Dirac-Kahler equation is general covariant. Usual Dirac equation is only Lorentz covariant. Lorentz covariance for dierent forms of Dirac equation L SO(3, 1), SL Spin(3, 1), SL, SLS1 L () For the Lorentz group SO(3, 1) GL(4, R) natural transformations of (M) (as tensors) coincide with () with respect to map (). For model with two systems () is: (SL) (S1 L T ), so transformation of is not mixed with auxiliary spinor . The isomor- phisms ( ) of four linear spaces let us consider full group GL(4, R) and for G / SO(3, 1) state becomes entangled with : (L G) (R G). 3. GR and the Dirac-Kahler Equation A. Yu. Vlasov Singularities, Black, White, and Worm- Holes One of prima facie (C. J. Isham) questions in quantum gravity: It was suggested , that the black hole evaporation may be related with a transition from a pure state to a mixed state of a quantum system, but how in principle to consider such a transition? Gauge theories and gravity For a space with a singularity there is a problem with denition of metric on some subspaces. The (pseudo-)Riemannian manifold is the particular example of the anely (or linear) connected space. The linear connection and the metric are two dierent geometrical objects. The linear connection is the particular example in more general theory of connections on a principle bundles with an arbitrary structure group, but this theory has some counterpart in the physics the gauge theory. In the general theory, it is considered a connection on a principal bundle with some Lie group. For a linear connection the structure group is GL(4, R). The important question is the reduction of the structure group to some subgroup. For General Relativity, it is the reduction to the Lorentz group SO(3, 1) it is just the question, if it is possible to choose the atlas, there all transitions between dierent maps may be described by transformations from the considered subgroup. In General Relativity it is related with the question about possibility to use only the transformation from the Lorentz group for transition between dierent coordinate systems and due to a general theorem of dierential geometry, it is possible i globally exist the Minkowski metric and the tetrad eld it is possible geometrical treatment of the equivalence principle in General Relativity. (Before recent Hawkings lecture in Dublin :) 4. GR and the Dirac-Kahler Equation A. Yu. Vlasov r t rg ds2 = g(r)dt2 dr2 g(r) r2 dS2 , g(r) = 1 rg r . Schwarzschild metric t = t f(r)dr g(r) , r = t + dr f(r)g(r) r rg t f(r) = 1 g2(r) sin() = rg 2r ds2 = g(r)(dt 2 dr2 ) + 2f(r)drdt r2 dS2 . 5. GR and the Dirac-Kahler Equation A. Yu. Vlasov ds2 = 1 g(r) dt2 1 g2 (r) dr2 r2 dS2 ds2 = g(r)(dt 2 dr2 ) 2 1 g2(r)drdt r2 dS2 l = 2r rg 1 ds2 = l2 1 l2 + 1 (dt2 l2 dl2 ) + 4l2 l2 + 1 dldt. 1 4(l2 + 1)2 dS2