Teitelboim's PhD thesis (and a few subsequent articles) discusses how the surface deformation algebra and "ultralocality" allows us to derive the matter part of the Hamiltonian and Momenta constraints. Teitelboim worked through the Scalar, Vector, and (covariant rank-2) Tensor cases.

Surprisingly enough, it works also for Spin-1/2 fields! We find
\[\delta\psi\sim-\int\{\psi,\mathcal{H}_{i}^{\text{Dirac}}(y)\}\,\delta N^{i}(y)\,\mathrm{d}y\]
where
\[\mathcal{H}_{i}^{\text{Dirac}}\sim \pi\partial_{i}\psi +\frac{1}{8}\pi\partial^{j}(\gamma_{i}\gamma_{j}-\gamma_{j}\gamma_{i})\psi\]
possibly up to an additional term like the second with derivatives acting to the right. This generates the Lie derivative of the Dirac spin-1/2 field
\[\delta\psi = \mathcal{L}_{\delta N}\psi = (\partial_{i}\psi)\delta N^{i} - \frac{1}{8}(\partial_{i}(\delta N_{j}) - \partial_{j}(\delta N_{i}))\gamma^{i}\gamma^{j}\psi.\]
(At least, if I've done my calculations correctly...)

But that's amazing! #GeneralRelativity #AlgebraOfSurfaceDeformations #DiracField