A new perturbative expansion of the time evolution operator associated with a quantum system

Abstract:
In two fundamental papers published in 1949, Dyson developed a perturbative expansion of the time evolution operator that has been adopted extensively in any field of physics. Dyson expansion has a transparent physical interpretation in terms of time ordered elementary processes which makes its application particularly appealing, especially in quantum field theory. On the other hand, for many applications, Dyson expansion has severe drawbacks, as a low convergence rate and the lack of unitarity of its truncations. Later, Magnus introduced an expansion of the evolution operator such that each of its truncations retains the property of being unitary. We will present a new perturbative expansion of the evolution operator associated with a (in general, time-dependent) quantum Hamiltonian that enjoys remarkable properties. In particular, each of its truncations has property of being unitary and it generalizes Magnus expansion, so opening the possibility of achieving computational advantages and of satisfying specific requirements. In the case of a time-independent Hamiltonian, the link with standard perturbation theory for linear operators will be discussed.