Abstract
The kinematics of loop gravity can be given a manifestly Lorentz-covariant formulation: the conventional -spin-network Hilbert space can be mapped to a space of functions, where Lorentz covariance is manifest. can be described in terms of a certain subset of the projected spin networks studied by Livine, Alexandrov and Dupuis. It is formed by functions completely determined by their restriction on . These are square-integrable in the scalar product, but not in the one. Thus, -spin-network states can be represented by Lorentz-covariant functions, as two-component photons can be described in the Lorentz-covariant Gupta-Bleuler formalism. As shown by Wolfgang Wieland in a related paper, this manifestly Lorentz-covariant formulation can also be directly obtained from canonical quantization. We show that the spinfoam dynamics of loop quantum gravity is locally -invariant in the bulk, and yields states that are precisely in on the boundary. This clarifies how the spinfoam formalism yields an theory on the boundary. These structures define a tidy Lorentz-covariant formalism for loop gravity.
- Received 7 January 2011
DOI:https://doi.org/10.1103/PhysRevD.83.104029
© 2011 American Physical Society