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Spin connection vs. Christoffel connection

  1. Feb 7, 2009 #1
    Hi everyone -- I have a question about the relation between the spin connection and the Christoffel connection.

    The spin connection comes from the local (gauge) Lorentz symmetry of how we orient vielbeins at each point in space, it contains a manifold index and two tangent space indices. The Christoffel connection comes from the (also local/gauge) diffeomorphism invariance of general relativity, carrying three manifold indices.

    My question is how are these two connections related, or are they both independent degrees of freedom?

    My background is in particle physics rather than relativity, so I prefer to think about these connections as gauge fields. In this case, I'm confused about the degrees of freedom that people work with when they do perturbative quantum gravity. Why is it that people work with perturbations on the metric rather than perturbations of the Christoffel symbol, which seems to be the "actual" gauge field? (I'm told this has something to do with the Palatini formalism which connects the two?) Further, why don't people treat the spin connection as a physical gauge field?

  2. jcsd
  3. Feb 7, 2009 #2
    My knowledge in this field is limited but there are all sorts of quantum gravity theories that choose different variables as "basic" quantities.

    As for the relation between spin and Christoffel connections, you can think of the Christoffel connection as the connection on an associated bundle induced by a (spin) connection on the principal bundle.
  4. Feb 8, 2009 #3
    See for example Chamseddine hep-th/0511074 (2005) and the discussion in "Beyond the standard model" page 2, Martin Kober ...
  5. Feb 9, 2009 #4
    I have made notes from various sources on this subject if you're interested
    http://www.mathematics.thetangentbundle.net/wiki/Differential_geometry/spin_connection [Broken]
    http://www.physics.thetangentbundle.net/wiki/Gravitational_physics/fermions_in_curved_space [Broken]
    I had to study this stuff to work with superstrings and superspace where the spin connection is graded, but the same sort of ideas hold. Hope you find this useful.
    Last edited by a moderator: May 4, 2017
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