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Hamiltonian for scalar field model of infinite derivative gravity

Theories with an infinite number of derivatives are described by non-local Lagrangians for which the standard Hamiltonian formalism cannot be applied. Hamiltonians of special types of non-local theories can be constructed by means of the (1+1)-dimensional Hamiltonian formalism. In this paper, we consider a simple scalar field model inspired by the infinite derivative gravity and study its reduced phase space by using this formalism. Assuming the expansion of the solutions in the coupling constant, we compute the perturbative Hamiltonian and the symplectic 2-form. We also discuss an example of a theory leading to an infinite-dimensional reduced phase space for a different choice of the form factor.