Brief Summary of Each Supplement
Progress of Theoretical Physics Supplement No. 182
Stationary Phase and Macrovariable
— From Wave to Particle —
By Reijiro Fukuda
This article tries to elucidate the appearance of the classical Newtonian
behavior of the macroscopic body encountered in the ordinary life, if the
system is treated quantum mechanically. Yet we know that any macroscopic body
consists of many microscopic degrees which are certainly described by
quantum mechanics.
To understand the simultaneous existence of these seemingly contradicting
degrees, the key mechanism is that the stationary phase is present in the
path-integral over the macrovariable, while it is absent for the
microvariable.
Indeed, when a macro-system is treated quantum mechanically,
one can naturally introduce a coordinate called the macrovariable, which is
defined by the average of large number N of microscopic coordinates.
The motion of the macro-system as a whole is noticed
by the change of the macrovariable thus defined.
In the path-integral formula, the macrovariable is shown to be determined
always the stationary phase in the limit N → ∞,
which leads to a deterministic trajectory of the point-like particle
for the macrovariable.
Other microscopic degrees have no stationary point and remain
fluctuating, forming atoms or molecules. These are shown both by
examples and by formal arguments applied to any macroscopic system.
The measurement theory based on the macrovariable as the detector coordinate
is presented, and the ideal result is obtained in the limit
N → ∞.
The correction terms appear for the case of large but finite N,
but they can be calculated systematically in the expansion scheme
of 1/N. These deviations from the ideal result can be detected
experimentally.
As for the reduction mechanism, we propose a model where any object has
a deterministic trajectory in a time scale which is much smaller than
the ordinary time scale we are using in the quantum mechanics.
The wave function is defined by averaging the square-root of the
density over the fluctuating the point-like trajectories. When such
an object is detected by using the macrovariable, the mechanism of
the stationary phase leads to the correct detection probability.
In all the arguments, the requirement that the system is thermodynamically
normal plays an essential role. The conditions for any macroscopic system
to be thermodynamically normal are studied for
N-particle system and also for the field theoretical case.
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