Uncertainty Principles in Krein Space

TL;DR

This paper extends the uncertainty principle to Krein spaces, involving a fundamental symmetry operator and nonlinear functions of operators, generalizing classical Heisenberg relations.

Contribution

It introduces new uncertainty relations in Krein spaces, including nonlinear and operator-dependent forms, broadening the scope beyond Hilbert space formulations.

Findings

Uncertainty relations involve Krein space symmetry operator J.

Existence of classes of non-self-adjoint operators with non-zero commutator implying uncertainty.

Classical Heisenberg uncertainty principle is recovered as a special case.

Abstract

Uncertainty relations between two general non-commuting self-adjoint operators are derived in a Krein space. All of these relations involve a Krein space induced fundamental symmetry operator, $J$, while some of these generalized relations involve an anti-commutator, a commutator, and various other nonlinear functions of the two operators in question. As a consequence there exist classes of non-self-adjoint operators on Hilbert spaces such that the non-vanishing of their commutator implies an uncertainty relation. All relations include the classical Heisenberg uncertainty principle as formulated in Hilbert Space by Von Neumann and others. In addition, we derive an operator dependent (nonlinear) commutator uncertainty relation in Krein space.