Commutators and Propagators

Philipp Bader, Sergio Blanes, Nikita Kopylov

We consider the numerical integration of the Schrödinger equation with a time-dependent Hamiltonian given as the sum of the kinetic energy and a time-dependent potential. Commutator-free (CF) propagators are exponential propagators that have shown to be highly efficient for general time-dependent Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians of the said structure, showing a considerably improved performance. We obtain new fourth- and sixth-order CF propagators as well as a novel sixth-order propagator that incorporates a double commutator that only depends on coordinates, so this term can be considered as cost-free. The algorithms require the computation of the action of exponentials on a vector similar to the well-known exponential midpoint propagator, and this is carried out using the Lanczos method. We illustrate the performance of the new methods on several numerical examples.

Paul Richard Halmos

Alfredo Luis, Jan Peřina

We examine the preservation of Poisson brackets and commutation relations in contradirectional propagation. It is shown that the relation between the input and the output amplitudes is a canonical transformation. This result is translated to the quantum case where preservation of the commutation relations is demonstrated in the case of linear propagation equations for the amplitudes.

Fred Alan Wolf, Daniel P. Sheehan

Causality in quantum field theory means the vanishing of commutators for spacelike separated fields (VCSSF). I will show that VCSSF is not tenable. For VCSSF to be tenable, and therefore, to have both retarded and advanced propagators vanish in the elsewhere, a superposition of negative energy antiparticle and positive energy particle propagators, traveling forward in time, and a superposition of negative energy particle and positive energy antiparticle propagators, traveling backward in time, are required. Hence VCSSF predicts non‐vanishing probabilities for both negative energy particles in the forward‐through‐time direction and positive energy antiparticles in the backwards‐through‐time direction. Therefore, since VCSSF is unrealizable in a stable universe, tachyonic propagation must occur in denial of causality.