A Few Stops on the Road to T-violation
Bryan W. Roberts
University of Southern California
www.usc.edu/bryanroberts
The laws governing quantum fields distinguish between the past and the future. This property is called the “failure of time reversal invariance” or “T-violation.” I would like to discuss one peculiar feature of the discovery of T-violation in the 20th century. Namely, there was no complete theory of weak interactions, let alone a complete standard model, when the discovery of T-violation was announced in 1964. Yet in spite of our relative ignorance about the underlying physics, various tests for T-violation were designed and achieved. This is strikingly different from Einstein’s three tests for general relativity, which were described and tested with a more or less complete theory of gravitation already in hand. This paper aims to clarify how such discoveries in a state of ignorance can be achieved, by illustrating three templates for determining T-violation without knowing what the underlying system is like.
Our discussion will be restricted tests for T-violation in ordinary unitary evolution. I will not discuss the well-known time asymmetry arising in the Born rule, or in the non-unitary evolution of quantum fields near evaporating black holes. We will begin by discussing the original path to T-violation undertaken by Christenson, Cronin, Fitch and Turlay, in their 1964 study of weakly interacting neutral kaon decay. Their approach rests, I claim, on the following fact.
- T-Violation by Curie’s Principle. If an initial state evolves unitarily to some final state, and if one of those two states is preserved by a linear transformation while the other is not, then the system fails to be invariant under that transformation. In particular, if a system can be shown to violate the linear CP transformation in this way, then it follows from the CPT theorem that it is also T-violating.
After illustrating this template with an example, I will turn to a second path to T-violation. This makes use of a related principle, similar in character to Curie’s principle, but which is “probabilistic” in form. For lack of a better name, I will call it the “Reversal Principle.”
- T-Violation by the Reversal Principle. If an initial and final state are both preserved by the time reversal operator, and if the probabilities of transitioning from one to the other after some time t are not the same, then the system is T-violating.
I will illustrate in simple terms how this strategy was used in direct tests for T-violation in neutral kaon oscillations. Finally, I will discuss a third path to T-violation, which typically involves a search for exotic matter, such as fundamental particles with an electric dipole moment. Mathematically, this approach turns out to be closely related to Wigner’s proof of Kramers Degeneracy. It rests on the following principle.
- T-Violation by non-degeneracy. Under appropriate circumstances, a system described by a Hamiltonian that is non-degenerate will be T-violating.