8 Quantum mechanics violations

Research in the theory of quantum gravity [14] led to a proposal for a modification of quantum mechanics time evolution, which might transform a pure initial state into an incoherent mixture [16]. This effect becomes particularly interesting in the kaon system, where quantum oscillations can be accurately measured. Moreover the -factory, where the initial is an antisymmetric coherent state, is a very suitable facility to test this idea. We will not discuss the theoretical models of quantum mechanics violations, which are discussed elsewhere in this handbook [40], but, following the analysis of Ref. [18], we shall analyze some examples of observable effects at DANE .

To describe the time evolution to incoherent states, one has to introduce the formalism of the density matrix. In Ref. [16] it has been proposed to modify the quantum mechanics time evolution equation in the following way:

where is the kaon density matrix, is the usual non-Hermitian kaon Hamiltonian (see eq. (18)) and is the quantum mechanics violating term. For the eigenmatrices of eq. (80) are the usual matrices:

Under reasonable assumptions (probability conservation, not decreasing entropy and strangeness conservation) can be expressed in terms of the three real parameters , and of Ref. [16]. With this parametrization the new eigenmatrices become:

where are the usual CP eigenstates and

The analysis of fixed target experiments has led the authors of Ref. [18] to put stringent bounds on the quantum mechanics violating parameters and :

Quite similar results were already obtained in [17]. In [41] also a bound for () is derived. It is interesting to note that, using the values in eq. (84), the limits on and turn out to be of the order of , which could be the natural suppression factor for these parameters.

These limits have been obtained assuming that there is no CPT violation in the decay amplitudes. In the more general case the effects of , and those of the conventional CPT-violating terms are mixed together.

This situation could be improved at DANE . In effect, quantum mechanics predicts a vanishing amplitude for the transition to the final state , with , independently from possible CPT violations. Therefore, as pointed out in [18], any measurement of equal time events can give a bound for pure quantum mechanics violations. It should be stressed however that the finite experimental resolution will partially wash out these effects (see Fig. 2). Moreover, also the C-even background gives rise to equal time events. Thus only the time distribution, which is different for C-even background and quantum mechanics violations, could help in disentangling them.

In Ref. [18] the consequences of quantum mechanics violation to several DANE \ observables have been analyzed. In particular the explicit formula of the time difference distribution has been derived, discussing the quantum mechanics violating effects in the measurement of . For the time asymmetry of eq. (11) becomes:

In Fig. 7 we have plotted in the usual quantum mechanics case and in the quantum mechanics violating case (following the analysis of [18]), for and , 0 and (close to the predicted DANE sensitivity). The quantum mechanics violating parameters have been chosen to be: and , in order to maximize their effects. As one can see, for very small values of the time difference t the effects of quantum mechanics violations are striking, but probably beyond any realistic experimental resolution. The determination of should not be affected, but an effect could be present in the asymptotic value of the asymmetry where, contrary to the usual quantum mechanics case, also contributes.

  
Figure 7: The time asymmetry for . The full lines correspond to quantum mechanics predictions of eqs. (11) and (12), with (a), 0 (b) and (c). The dashed lines correspond to the upper bounds of quantum mechanics violating parameters and , for the previous values of .

Using the expressions of Ref. [18] and neglecting a possible violation of the rule, we have calculated also the quantum mechanics violating effects for eq. (25). The measured charge asymmetry for semileptonic decays would become:

where are the usual asymmetries, as defined in eq. (23). The effect of the term simulates a C-even background and interestingly, with the limit of eq. (84), this correction turns out to be of the order of the one estimated in the previous section (for ). The term simulates a CPT violation, however, using the bound in eq. (84), this effect turns out to be smaller than the DANE sensitivity.

Concluding, we can say that quantum mechanics violating effects can be neglected in integrated asymmetries at DANE . Nevertheless, if , and were suppressed only linearly by , some effects in time-dependent distributions could be observable. An estimate of DANE sensitivity on these effects would require an accurate simulation of the experimental apparatus, which is beyond the purpose of this work.