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The higher category theory can be employed to generalize the BF action to theso-called 3BF action, by passing from the notion of a gauge group to the notionof a gauge 3-group. The theory of scalar electrodynamics coupled toEinstein-Cartan gravity can be formulated as a constrained 3BF theory for aspecific choice of the gauge 3-group. The complete Hamiltonian analysis of the3BF action for the choice of a Lie 3-group corresponding to scalarelectrodynamics is performed. This analysis is the first step towards acanonical quantization of a 3BF theory, an important stepping stone for thequantization of the complete scalar electrodynamics coupled to Einstein-Cartangravity formulated as a 3BF action with suitable simplicity constraints. It isshown that the resulting dynamic constraints eliminate all propagating degreesof freedom, i.e., the 3BF theory for this choice of a 3-group is a topologicalfield theory, as expected.

Hamiltonian Analysis for the Scalar Electrodynamics as 3BF Theory