Constrained Hamiltonian analysis of a nonrelativistic Schrödinger field coupled with Chern-Simons gravity

We provide a constrained Hamiltonian analysis of a nonrelativistic Schrödinger field in $2+1$ dimensions, coupled with Chern-Simons gravity. The coupling is achieved by the recently advanced Galilean gauge theory [R. Banerjee, A. Mitra, and P. Mukherjee, Phys. Lett. B 737, 369 (2014); Classical Quantum Gravity 32, 045010 (2015); R. Banerjee and P. Mukherjee, Phys. Rev. D 93, 085020 (2016).] The calculations are repeated with a truncated model to show that deviation from Galilean gauge theory makes the theory untenable. The issue of nonrelativistic spatial diffeomorphism is discussed in this context.