Orthogonality of independently propagating states as occurring in the Lee-Oehme-Yang theory

Abstract

We generalize a theorem by Khalfin, originally The complex formed by a neutral meson M 0 ͑i.e., K 0 ,and its antiparticle M 0 is a very important testing ground for discrete symmetries and physics beyond the standard model. Its theoretical treatment relies heavily on the Weisskopf-Wigner approximation ͓1͔ and the LeeOehme-Yang ͑LOY͒ theory ͓2͔ ͑for reviews see, for instance, Refs. ͓3-5͔͒. Central to this approximate theory are the linear combinationswith complex constants p H,L and q H,L , which propagate independently in time. This property is also called ''lack of vacuum regeneration.'' The subscripts H and L refer to heavy and light, respectively. By starting with the additional assumption ofKhalfin demonstrated ͓6,7͔ that the independent propagation In this paper, we generalize Khalfin's theorem by dropping the requirement of CPT invariance.1 It is interesting to relax this requirement in view of the need of improvement of present data for many of the tests of CPT invariance ͓13͔. Because of our generalization, we go beyond the choice of ͉F 1,2 ͘ being a particle-antiparticle pair, though, at present, this choice seems to be the only one with an experimental application in particle physics. 2 We consider a pair of orthonormal electrically neutral states ͉F 1,2 ͘, in analogy to ͉M 0 ͘ and ͉M 0 ͘, and states ͉ f ͘ orthogonal to ͉F 1,2 ͘. Then the time evolution of the states ͉F 1,2 ͘ is, most generally, expressed as ͑3͒Now we want to be more specific about the time evolution ͑3͒. We postulate the existence of two linearly independentwhere p a,b and q a,b are complex constants, such that these states propagate independently in time. Equation ͑4͒ is the analogue of Eq. ͑1͒. This means that the time evolution of ͉P a,b ͘ is described for tу0 by 1 In an unpublished report ͓12͔, quoted as Ref. ͓30͔ in his paper ͓6͔, Khalfin relaxed the assumption of CPT invariance.2 Thus our theorem is not restricted to particle physics. See the remarks at the end of Ref. ͓10͔ for its applicability in the study of communicating metastable states in atomic physics.

Unitarity and real properties of the neutral meson complex

“…Note that this last relation as well as property (18) are valid for any two states |ψ 1 , |ψ 2 ∈ H. The Kalfin's Theorem concerns one of the basic properties of any two state subsystem and, in fact, it is not limited to only such subsystems as the neutral meson complexes. This Theorem states that [11][12][13][14][15][16][17][18],…”

“…In the considered case (24), from ( 16), (17) and ( 24) one can see that A 21 (0) = 0 and A 12 (0) = 0 which by (19) means that without some additional conditions the function f 21 (t) need not be continuous at t = 0. Taking into account that quantum theory requires U (t) to have the form,…”

Unitarity and real properties of the neutral meson complex

“…In the considered case (24), from ( 16), (17) and ( 24) one can see that A 21 (0) = 0 and A 12 (0) = 0 which by (19) means that without some additional conditions the function f 21 (t) need not be continuous at t = 0. Taking into account that quantum theory requires U(t) to have the form,…”

Khalfin's Theorem and Neutral Mesons Subsystem

“…This talk is based on the paper [19]. The aim of the talk is to confront the main predictions of the LOY theory such as ( 16), (17), (20), etc., with predictions following from the rigorous treatment of two state quantum mechanical subsystems and from the properties of the exact effective Hamiltonian for such subsystems. Sec.…”