Conditional symmetries in parametrized field theories

Abstract

In parametrized field theories, spacelike hypersurfaces and fields which they carry are evolved by a Hamiltonian which is a linear combination of the super-Hamiltonian and supermomentum constraints. We say that a dynamical variable K generates a conditional symmetry of the Hamiltonian when it is linear both in the hypersurface and the field momenta and its Poisson bracket with the Hamiltonian vanishes by virtue of the constraints. Generators are classified by their dependence on the momenta: P-restricted generators depend only on the hypersurface momenta, π-restricted generators depend only on the field momenta, while mixed generators depend on both kinds of momenta. Conditional symmetries in a parametrized Hamiltonian theory are then linked either with ordinary symmetries (isometries, conformal motions, or homothetic motions) of the spacetime background, or with internal symmetries of the fields. In particular, we prove that a generic field with nonderivative gravitational coupling and a quadratic energy density has a P-restricted conditional symmetry if and only if the spacetime background has a Killing vector, while a field with a trace-free energy–momentum tensor has a P-restricted conditional symmetry if and only if the background has a conformal Killing vector. An algorithm allowing us to enumerate all possible mixed conditional symmetries in a given parametrized field theory is explained on an example of the Klein–Gordon field. These results complement our previous proof that canonical geometrodynamics does not possess any conditional symmetry.