Covariant Wave Function Reduction and Coherent Decays of Kaon Pairs 1 Bernd

The recently developed relativistically covariant formulation of wave function reduction is illustrated for Lipkin’s proposal to study CP violation in the coherent decay of kaon pairs. Covariant results are obtained in agreement with an amplitude approach proposed in the literature. Work partially supported by the Department of Energy under contracts DE-FG05-87ER40319 and DEFG05-85ER2500. Department of Physics, The Florida State University, Tallahassee, FL 32306, USA. Supercomputer Computations Research Institute, Tallahassee, FL 32306, USA. About thirty years ago Lipkin [1] wrote an interesting paper in which he proposed to exploit quantum mechanical correlations of the EPR [2] type for experimental studies of CP violation in the coherent decay of kaon pairs. His basic idea is quite simple: Consider protonantiproton annihilation into two neutral kaons KK, where the intial state is assumed to be in an s-wave. First, disregard CP violation and take the states of definite CP , called K1 and K2, as basis for the K 0 meson system. Only the K1K2 final state is allowed, whereas the states K1K1 and K2K2 are forbidden. This result follows from parity conservation and Bose statistics. The initial state has odd parity, while Bose statistics forbids odd-parity states for two identical spinless bosons. The K1 decays into two pions, π π or ππ, whereas the K2 does not. Therefore, we have derived that only one of the KK kaons can decay into two pions. With CP violation the shortand long-lived kaons, KS and KL, form a natural (not necessarily orthogonal) basis of the neutral kaon system and are both observed to decay into pions. Lipkin’s observation is that the above derivation requires only parity conservation and Bose statistics, but does not depend on CP conservation. PP annihilation creates the entangled intial state |i〉 = 1 √ 2 [|KL(x)〉 |KS(−x)〉 − |KL(−x)〉 |KS(x)〉] (1) where the x-axis is chosen as the direction of the momenta of the kaons in their CM system; following Lipkin’s notation (x) means that the particle is moving in the positive x direction and (−x) means that it is moving in the negative x direction. Measurement of the decay of one of the kaons into, say, ππ at a spacetime point (t1, x1) [3] with x1 > 0 forces the other into a definite coherent mixture |K r 〉 = α |KS(−x)〉 + β |KL(−x)〉 (2) with coefficients α and β such that at some time t2 the matrix elements of the two terms to the state |π+π−〉 cancel one another. Further, the subscript r stands for “reduced”. States of the