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General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th
Abstract:
A key point in the spin foam approach to quantum gravity is the
implementation of simplicity constraints in the partition functions of the
models. Here, we discuss the imposition of these constraints in a phase space
setting corresponding to simplicial geometries. On the one hand, this could
serve as a starting point for a derivation of spin foam models by canonical
quantisation. On the other, it elucidates the interpretation of the boundary
Hilbert space that arises in spin foam models.
More precisely, we discuss different versions of the simplicity constraints,
namely gauge-variant and gauge-invariant versions. In the gauge-variant
version, the primary and secondary simplicity constraints take a similar form
to the reality conditions known already in the context of (complex) Ashtekar
variables. Subsequently, we describe the effect of these primary and secondary
simplicity constraints on gauge-invariant variables. This allows us to
illustrate their equivalence to the so-called diagonal, cross and edge
simplicity constraints, which are the gauge-invariant versions of the
simplicity constraints. In particular, we clarify how the so-called gluing
conditions arise from the secondary simplicity constraints. Finally, we discuss
the significance of degenerate configurations, and the ramifications of our
work in a broader setting.
