8 research outputs found
Nearly Instantaneous Alternatives in Quantum Mechanics
Usual quantum mechanics predicts probabilities for the outcomes of
measurements carried out at definite moments of time. However, realistic
measurements do not take place in an instant, but are extended over a period of
time. The assumption of instantaneous alternatives in usual quantum mechanics
is an approximation whose validity can be investigated in the generalized
quantum mechanics of closed systems in which probabilities are predicted for
spacetime alternatives that extend over time. In this paper we investigate how
alternatives extended over time reduce to the usual instantaneous alternatives
in a simple model in non-relativistic quantum mechanics. Specifically, we show
how the decoherence of a particular set of spacetime alternatives becomes
automatic as the time over which they extend approaches zero and estimate how
large this time can be before the interference between the alternatives becomes
non-negligible. These results suggest that the time scale over which coarse
grainings of such quantities as the center of mass position of a massive body
may be extended in time before producing significant interference is much
longer than characteristic dynamical time scales.Comment: 12 pages, harvmac, no figure
Decoherent histories analysis of the relativistic particle
The Klein-Gordon equation is a useful test arena for quantum cosmological
models described by the Wheeler-DeWitt equation. We use the decoherent
histories approach to quantum theory to obtain the probability that a free
relativistic particle crosses a section of spacelike surface. The decoherence
functional is constructed using path integral methods with initial states
attached using the (positive definite) ``induced'' inner product between
solutions to the constraint equation. The notion of crossing a spacelike
surface requires some attention, given that the paths in the path integral may
cross such a surface many times, but we show that first and last crossings are
in essence the only useful possibilities. Different possible results for the
probabilities are obtained, depending on how the relativistic particle is
quantized (using the Klein-Gordon equation, or its square root, with the
associated Newton-Wigner states). In the Klein-Gordon quantization, the
decoherence is only approximate, due to the fact that the paths in the path
integral may go backwards and forwards in time. We compare with the results
obtained using operators which commute with the constraint (the ``evolving
constants'' method).Comment: 51 pages, plain Te
Trajectories for the Wave Function of the Universe from a Simple Detector Model
Inspired by Mott's (1929) analysis of particle tracks in a cloud chamber, we
consider a simple model for quantum cosmology which includes, in the total
Hamiltonian, model detectors registering whether or not the system, at any
stage in its entire history, passes through a series of regions in
configuration space. We thus derive a variety of well-defined formulas for the
probabilities for trajectories associated with the solutions to the
Wheeler-DeWitt equation. The probability distribution is peaked about classical
trajectories in configuration space. The ``measured'' wave functions still
satisfy the Wheeler-DeWitt equation, except for small corrections due to the
disturbance of the measuring device. With modified boundary conditions, the
measurement amplitudes essentially agree with an earlier result of Hartle
derived on rather different grounds. In the special case where the system is a
collection of harmonic oscillators, the interpretation of the results is aided
by the introduction of ``timeless'' coherent states -- eigenstates of the
Hamiltonian which are concentrated about entire classical trajectories.Comment: 37 pages, plain Tex. Second draft. Substantial revision
Decoherent Histories Approach to the Arrival Time Problem
We use the decoherent histories approach to quantum theory to compute the
probability of a non-relativistic particle crossing during an interval of
time. For a system consisting of a single non-relativistic particle, histories
coarse-grained according to whether or not they pass through spacetime regions
are generally not decoherent, except for very special initial states, and thus
probabilities cannot be assigned. Decoherence may, however, be achieved by
coupling the particle to an environment consisting of a set of harmonic
oscillators in a thermal bath. Probabilities for spacetime coarse grainings are
thus calculated by considering restricted density operator propagators of the
quantum Brownian motion model. We also show how to achieve decoherence by
replicating the system times and then projecting onto the number density of
particles that cross during a given time interval, and this gives an
alternative expression for the crossing probability. The latter approach shows
that the relative frequency for histories is approximately decoherent for
sufficiently large , a result related to the Finkelstein-Graham-Hartle
theorem.Comment: 42 pages, plain Te