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: ...
    • = formal device: anticommuting c-numbers are hard to visualise and impossible to measure
    • extremely useful for description of fermions
  • Berezin work: ...
  • ... 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!