1202.4638 (Julio César Arce)

Unification of the conditional probability and semiclassical
interpretations for the problem of time in quantum theory    [PDF]

Julio César Arce

We show that the time-dependent Schr\"odinger equation (TDSE) is the
phenomenological dynamical law of evolution unraveled in the classical limit
from a timeless formulation in terms of probability amplitudes conditioned by
the values of suitably-chosen internal clock variables, thereby unifying the
conditional probability interpretation (CPI) and the semiclassical approach for
the problem of time in quantum theory. Our formalism stems from an exact
factorization of the Hamiltonian eigenfunction of the clock + system composite,
where the clock and system factors play the role of marginal and conditional
probability amplitudes, respectively. Application of the Variation Principle
leads to a pair of exact coupled pseudo-eigenvalue equations for these
amplitudes, whose solution requires an iterative self-consistent procedure. The
equation for the conditional amplitude constitutes an effective "equation of
motion" for the quantum state of the system with respect to the clock
variables. These coupled equations also provide a convenient framework for
treating the back-reaction of the system on the clock at various levels of
approximation. At the lowest level, when the WKB approximation for the marginal
amplitude is appropriate, in the classical limit of the clock variables the
TDSE for the system emerges as a matter of course from the conditional
equation. In this connection, we provide a discussion of the characteristics
required by physical systems to serve as good clocks. This development is seen
to be advantageous over the original CPI and semiclassical approach since it
maintains the essence of the conventional formalism of quantum mechanics,
admits a transparent interpretation, avoids the use of the Born-Oppenheimer
approximation, and resolves various objections raised on them.

View original: http://arxiv.org/abs/1202.4638