<< Cosmo.WebHome %TOC% ---- %INCLUDE{"GrassmannCosmoFun2009Head"}% ---+ Mon 14 Sep 2009 * NICOLAI 14.09.2009 * Grassmann algebra key to supersymmetry * anticommuting c-numbers * Grassmann algebra splits into even (bosonic) and odd (fermionic) parts * QFT: %$ \{\phi(x),\phi\dagger(y)\} = \hbar \delta(x-y) $% ... * = formal device: anticommuting c-numbers are hard to visualise and impossible to measure * extremely useful for description of fermions * Berezin work: %$ \frac{\partial}{ \partial\theta_i} \theta_j = \delta^i_j, $% ... * ... graded Lie algebras * supersymmetry - bosons - forces, fermions - matter * susy QM - q(t), \phi(t), lagrangian L(q,\dot{q},\phi,\dot{\phi}} * requires superpotential W(q) * Poisson brackets \{.,.\} or Dirac brackets * \{.,.\} = (i\hbar)^{-1} [commutator (fermions) or anticommutator (otherwise)] * Hamiltonian * Ward identities for correlators * Witten index *susy QFT - susy needs to be merged with symmetries like Poincare and internal symmetries * superalgebra = graded Lie algebra * N-extended susy for i,j=1,...,N merges spactime and internal symmetries when N \ge 2 * representations (supermultiplets) * N = 8 supergravity - most symmetric - was favoured in 1970's * finiteness or not? * Einstein gravity is perturbatively non-renormalisable \Gamma^{(2)}_{div}... * for 2-loop; 3-loop or greater infinities * implies many people go to superstrings instead * counterterms needed to renormalise? * Counterterm gymnastics - N=8 supergravity * 2-loop corrections require e.g. 10^5 feynman diagram calculations * 3-loop corrections require e.g. 10^{20} feynman diagram calculations * using very different techniques, the latter is doable * on-shell field theory (p^2 != m^2) * S-matrix theory * square Yang Mills amplitudes using KLT (Kawai Lewellen Tye (1986?)) rules * Z.Bern http://www.livingreviews.org/lrr-2002-5 * 4-loop finiteness * get about 50 Mondrian-like diagrams * Why do we get a finite theory here? Big mystery * sociologically: * superstringers relied on: * N=8 supergravity divergent for \ge 3 loops * N=8 supergravity not viable for phenomenology * but (gauged) N=8 supergravity has a stationary point with residual SU(3) \times U(1) symmetry * Anthony LASENBY * http://en.wikipedia.org/wiki/Geometric_algebra * a geometric algebra approach can lead to a spherical model, with Omega_total about 1.01 to 1.02! -- Main.BoudRoukema - 15 Sep 2009
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