Finding room for antilinear terms in the Hamiltonian

TL;DR

The paper demonstrates how quantum systems can be engineered to include antilinear terms in their Hamiltonians by constructing equivalent systems where these terms appear as linear operators, preserving physical predictions.

Contribution

It introduces a method to embed antilinear Hamiltonian components into quantum systems through system equivalence, expanding the theoretical framework of quantum Hamiltonian design.

Findings

Equivalent systems can incorporate antilinear terms as linear operators.

The construction preserves physical equivalence and measurement outcomes.

It reveals unconventional features like vacuum degeneracy and distinguishes Hamiltonian from energy observable.

Abstract

Although the Hamiltonian in quantum physics has to be a linear operator, it is possible to make quantum systems behave as if their Hamiltonians contained antilinear (i.e., semilinear or conjugate-linear) terms. For any given quantum system, another system can be constructed that is physically equivalent to the original one. It can be designed, despite the Wightman reconstruction theorem, so that antilinear operators in the original system become linear operators in the new system. Under certain conditions, these operators can then be added to the new Hamiltonian. The new quantum system has some unconventional features, a hidden degeneracy of the vacuum and a subtle distinction between the Hamiltonian and the observable of energy, but the physical equivalence guarantees that its states evolve like those in the original system and that corresponding measurements produce the same results. The same construction can be used to make time-reversal linear.