Abstract Discussions of quantum mechanics often loosely claim that time evolution logically must be unitary, in order for the probabilistic interpretation of the amplitudes of the state vector to make sense at all times. We discuss from first principles whether this claim is true: if we assume only that the time-evolution operator is linear, then does the stronger requirement that it be unitary follow from the other axioms of quantum mechanics? The answer is subtle. We discuss two mathematically distinct but physically equivalent formulations of the axioms of quantum mechanics, and consider generalizing each to postulate only that time evolution is linear. Within one formulation, the unitarity of time evolution follows logically from the other axioms – but within the other formulation, it does not. Allowing the time-evolution operator to be (a priori) arbitrarily linear does not change the physical observables in one formulation of quantum mechanics, but changes the other formulation to a distinct (internally consistent) physical theory that allows new phenomenology like (e.g.) faster-than-light communication. Therefore, the unitarity of time evolution is arguably better thought of as a logically independent and experimentally falsifiable axiom of quantum mechanics, not as a tautological consequence of the other axioms.Abstract Discussions of quantum mechanics often loosely claim that time evolution logically must be unitary, in order for the probabilistic interpretation of the amplitudes of the state vector to make sense at all times. We discuss from first principles whether this claim is true: if we assume only that the time-evolution operator is linear, then does the stronger requirement that it be unitary follow from the other axioms of quantum mechanics? The answer is subtle. We discuss two mathematically distinct but physically equivalent formulations of the axioms of quantum mechanics, and consider generalizing each to postulate only that time evolution is linear. Within one formulation, the unitarity of time evolution follows logically from the other axioms – but within the other formulation, it does not. Allowing the time-evolution operator to be (a priori) arbitrarily linear does not change the physical observables in one formulation of quantum mechanics, but changes the other formulation to a distinct (internally consistent) physical theory that allows new phenomenology like (e.g.) faster-than-light communication. Therefore, the unitarity of time evolution is arguably better thought of as a logically independent and experimentally falsifiable axiom of quantum mechanics, not as a tautological consequence of the other axioms.

arXiv:2309.04095Foundations of PhysicsFeb 1, 20250 citations

Functional Integral Approach to $C^*$ -algebraic Quantum MechanicsFunctional Integral Approach to $C^*$-algebraic Quantum Mechanics

J. LaChapelle

The algebraic approach to quantum mechanics has been key to the development of the theory since its inception, and the approach has evolved into a mathematically rigorous $C^\ast$ -algebraic formulation of the axioms. Conversely the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a recently developed notion of functional integration that allows to construct functional integral representations of $C^\ast$ -algebras. The merger is a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role.The algebraic approach to quantum mechanics has been key to the development of the theory since its inception, and the approach has evolved into a mathematically rigorous $C^\ast$-algebraic formulation of the axioms. Conversely the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a recently developed notion of functional integration that allows to construct functional integral representations of $C^\ast$-algebras. The merger is a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role.

May 27, 20152 citations

Simple New Axioms for Quantum Mechanics

Nicolaas P. Landsman

arXiv:quant-ph/9604008International Journal of Theoretical PhysicsJan 1, 19988 citations

Functional Integral Approach to

C^*

-algebraic Quantum Mechanics

The algebraic approach to quantum mechanics has been key to the development of the theory since its inception, and the approach has evolved into a mathematically rigorous

C^\ast

-algebraic formulation of the axioms. Conversely the functional approach in the form of Feynman path integrals is far from mathematically rigorous: Nevertheless, path integrals provide an equally valid and useful formulation of the axioms of quantum mechanics. The two approaches can be merged by employing a recently developed notion of functional integration that allows to construct functional integral representations of

C^\ast

-algebras. The merger is a hybrid formulation of the axioms of quantum mechanics in which topological groups play a leading role.