Hamiltonian formalism for space-time noncommutative theories

Space-time non-commutative theories are nonlocal in time. We develop the Hamiltonian formalism for non-local field theories in d space-time dimensions by considering auxiliary $\mathrm{}(d+1\mathrm{})\mathrm{}$-dimensional field theories which are local with respect to the evolution time. The Hamiltonian path integral quantization is considered and the Feynman rules in the Lagrangian formalism are derived. The case of non-commutative ${\phi }^3$ theory is considered as an example.