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Neutral Kaon decay without symmetry violation: A nonstandard theoretical speculation

It is shown that if antiparticles are realized in quantum field theory by negative frequency states, which nevertheless have positive energy density, the resulting theory provides a qualitative explanation for the experiments on the neutral K mesons, without assumming any symmetry violation.


Introduction
The existence of the positron was predicted on the basis of the negative energy solutions of the Dirac equation, but then a physical interpretation of negative energy states was not found in the framework of single-particle theory. So, in the usual construction of quantum field theory antiparticle states have positive frequencies, and we have an axiomatic positivity condition on the free antiparticle spectrum [1,2,3].
But there is also an alternative. In 1941 Stueckelberg [4] gave a classical picture of pair creation and annihilation where the electron-positron pair is described by a single world line which is reflected in time by electromagnetic interaction. The electron part of the world line has dt/dτ > 0, where τ is the proper time, the positron part has dt/dτ < 0. For particles the classical energy variable p 0 is positive, for antiparticles negative. Stueckelberg also devised a quantum theory which is in correspondence to this classical picture by introducing the proper time as an evolution parameter into quantum theory, with the mass as the conjugate invariant quantity, masses become the eigenvalues of the operator i∂ τ . Particles are decribed by positive frequency solutions, antiparticles have negative frequencies. It is well known [5] that this manifest covariant quantum theory solves the problems of localization and zitterbewegung which arise when we consider only the time-development in a 3 × 1 decomposition. But Stueckelberg's classical and quantum picture, further extended by Feynman [6] to spin 1/2, could not solve the negative energy problem. But there exists a solution to this problem, in the way that, negative frequency states are interpreted in a consistent way with positive energy density, with corresponding pictures in classical theory, quantum mechanics and quantum field theory. This theory is used to analyse neutral K-meson decay with the result that no symmetry violation needs be assumed in order to expain the experimental observations.

Positivity of the energy density
We start with classical mechanics [4]. The constrainṫ on the trajectories of classical point particles allows for solutions with negativeẋ 0 = cdt/dτ , i.e. t = −τ for a particle at rest. These solutions were usually discarded, postulating t = τ for a particle at rest, until Stueckelberg showed that these solutions may be interpreted as antiparticles. With respect to the physical interpretation it is important to note that the solutions of the canonical equations with dt/dτ < 0, which also have negative canonical energy p 0 < 0, possess a positive energy density [7] which is positive due to the two negative signs of p 0 andẋ 0 . (We use the sign convention + − −− for the metric.) With respect to quantum theory, Klein-Gordon theory has an energy density which is positive irrespective of the sign of the eigenvalues of the operator p 0 = ih c ∂ ∂t . A solution to the positivity problem for spin 1/2 was first indicated by Arshansky and Horwitz [9]. The argument is from representation theory. Since the energy density operators [8] always have a positive spectrum the question of a positive energy density is a matter of choosing a positive definite scalar product which preserves the sign of the eigenvalues when expectation values are taken. The idea by Arshansky and Horwitz [9] was to start with two-component spinors which transform under Wigner's unitary induced representation or its conjugate representation, for which we have a manifest positive norm [10]. Then the embedding of the two-component spinors into a space of four-component spinors is done in a norm-preserving way, leading to the positive norm on the embedded subspace [11]. We note that the functional form (4) of the norm, with a sign included for antiparticles, is the same as in standard quantum field theory, where antiparticle states are realized with positive frequencies [1], p. 188. This construction was carried out rigorously [13]. The use of negative frequencies is not in contradiction to the axiomatic approach, since it is well known that positivity is independent of all other axioms [2,3].
In original Dirac and Klein-Gordon theories the charge conjugation is an antilinear transformation. In the parametrized theory this charge conjugation transformation is extended to parameter-dependent wave-equations, whereby it retains its antilinearity, and, due to a positive definite scalar product, is also antiunitary. For clarity we denote this antiunitary charge conjugation byC.C transforms τ into −τ and p µ into −p µ with x invariant. The parity transformation P remains unitary and the time inversion transformation T antiunitary, since both can first be defined on the particle sector alone and then carried over to the antiparticle sector with help of the charge conjugation transformation. The time-inversion carries t into −t and τ into −τ , so that particles remain particles. The combinedCP T -transformation, which leaves τ invariant, is unitary. Since antiunitary transformations do not have eigenvalues and eigenstates we cannot defineC orCP eigenstates in our theory, but we can defineCT andCP T eigenstates. The action of the symmetry transformations on spin 0 and spin 1/2 wave functions is identical with that of quantum mechanics. The infinitesimal generator i∂ τ of translations of the invariant parameter is invariant under all these transformations, whereas the infinitesimal generator i∂ t of time translations is invariant only under P and T , changing sign underC andCP T . As a consequence any Hamiltonian must be indefinite in aCP T -symmetric theory.
We consider some relevant subsystem Ψ i of a physical system, with a finite number i = 1, .., n of states, for which an effective evolution equation is assumed. The development of Ψ i in time is governed by an effective whereas the evolution in proper time τ is described by an effective mass matrix M ij with If the invariance of the subsystem underCP T is assumed, the mass matrix is invariant underCP T , as i∂ τ is, whereas the HamiltonianH ij is transformed into −H ij underCP T , as is i∂ t → −i∂ t . The mass matrix may be choosen to be positive definite, the Hamiltonian necessarily is indefinite.

The neutral K-meson system
For all definitions which are not standard we use a tilde. The K 0 meson is a spin 0 particle consisting of a d and as quark, which have spin 1/2. SinceCP is not unitary, we useCP T to define the antiparticle to the K 0 meson, which we denote by K 0 , bỹ so that we haveCP T eigenstates We describe the K 0 and K It follows that |K 1 and |K 2 are eigenstates of M, just as in standard theory. Now in standard theory the assumption that both Hamiltonian and mass matrix are invariant under the CP -transformation leads to the conclusion that |K 1 and |K 2 are also eigenstates of the Hamiltonian, and cannot mix in time. The experimentally observed decay of the longest-living state into both CP -eigenstates hence implies that the physical states K L and K S are distinct from | K 1 and | K 2 and thus CP -symmetry must be violated [14]. This is different in our theory.CP T -invariance in our theory implies that our HamiltonianH changes sign underCP T , so we havẽ TransformingH into the basis Ψ ′ = |K 1 , |K 2 yields The structure ofH ′ is a general consequence of the Hamiltonian beingCP Todd. If A and B are any twoCP T -eigenstates with eigenvalues a, b ∈ {+1, −1} then so that only states with differentCP T -eigenvalues yield nonzero matrix elements. From (11) we see that the states |K 1 and |K 2 always mix in time ifH is nonzero. TheCP T -eigenstates are not preserved in time, the eigenstates of the Hamiltonian are notCP T -eigenstates. Hence, if we start with an arbitrary initial state and wait until only the slowest-decaying eigenstate has survived, we will always observe that this state decays into both channels ofCP T -eigenstates. So we are not forced to assumeCP T -orCPviolation in order to explain qualitatively, from first principles, the experimentally observed phenomenon of the long-living K L decaying into different CP T -eigenstates. The dynamical mixing of the |K 1 and |K 2 also offers an explanation for the fact that CP -violation practically is not seen in the decay of the physical K S [14]: The short decay time provides no time for the mixing process to happen. A numerical study of a simple model system confirms the mechanism as described [11].