Some approaches to quantum gravity suggest (Hawkings) that quantum field
theory should be modified in such a way that pure quantum-mechanical states
evolve into mixed states, which would necessarily entail a violation of
CPT. This possibility may be analyzed using the formulation of
open quantum-mechanical systems coupled to an unobserved
environment(J.Ellis et al.). In this framework, the observed system is
described by a density matrix 
that obeys a modified quantum Liouville
equation
where H is the usual quantum-mechanical Hamiltonian. The extra term
would induce a loss of quantum coherence in the
observed system, and hence a violation of CPT. Since it is
conjectured to arise from quantum-gravitational effects, the magnitude
of 
may be at most 
, where
is the gravitational mass scale obtained from
Newton's constant: 
. An equation of the form
(72) is supported by one interpretation of string theory,
but could have more general applicability.
In the case of the neutral kaon system, if the conservation of energy
and strangeness are assumed, the open-system equation
introduces three CPT-violating parameters 
, 
and 
,
which are distinct from 
- 
mass and lifetime differences
, 
. The open-system evolution equation (72) is solved
perturbatively in the small parameters , , and
. A second-order formula for 
is derived to:
where 
, 
, 
are scaled variables 
, 
,
, with 
the decay
width difference between 
and 
.
Other definitions used are:
A first-order formula for 
is
given by:
Since the open quantum mechanic formalism changes the time evolution
of states, the best limits on the CPT violation parameter
, and are obtained by comparing measurements at early decay times
(CPLEAR) and large decay times ( -experiments):
where 
is the
-violation parameter measured in decays to 
,
and 
is the semileptonic asymmetry measured in decays.
