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Canonical invariance of spatially covariant scalar-tensor theory

We investigate invariant canonical transformations of a spatially covariant scalar-tensor theory of gravity, called the XG theory, by which the action or the Hamiltonian and the primary constraints keep their forms invariant. We derive the Hamiltonian in a non perturbative manner and complete the Hamiltonian analysis for all regions of the theory. We confirm that the theory has at most 3 degrees of freedom in all regions of the theory as long as the theory has the symmetry under the spatial diffeormorphism. 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 derivative terms of the scalar field and the metric. In addition, we discover the invariant canonical transformation which transforms the momentum of the metric. Using the invariant transformation, we study the relation between the Horndeski theory and the GLPV theory, and find that we can not obtain the arbitrary GLPV theory from the Horndeski theory through the invariant canonical transformation we have found.