We investigate the invariant canonical transformation of a spatially covariant scalar-tensor theory of gravity, called the XG theory. Under the invariant canonical transformation, the forms of the action or the Hamiltonian and the primary constraints are preserved, but the action or the Hamiltonian is not invariant. We derive the Hamiltonian in a nonperturbative manner and perform the Hamiltonian analysis for full theory. We confirm that the theory has at most 3 degrees of freedom as long as the theory has the spatial diffeormorphism symmetry. Then, we derive the invariant canonical transformation by using the infinitesimal transformation. The invariant metric transformation of the XG theory contains a vector product as well as the disformal transformation. The vector product and the disformal factor can depend on the higher order derivatives of the scalar field and the metric. Furthermore, we discover additional canonical transformation which leaves the theory invariant. Using the invariant transformation, we study the relation between the Horndeski theory and a beyond Horndeski theory, called the GLPV theory, and find that we cannot obtain general GLPV theory from the Horndeski theory through the invariant canonical transformation we found.

• Received 22 April 2016

DOI:https://doi.org/10.1103/PhysRevD.94.104054