Hitoshi Kitada (hitoshi@kitada.com)
Tue, 9 Nov 1999 01:36:11 +0900
Dear All,
Let me try to propose a question which seems to be a fundamental
question on quantum mechanical time and seems not to have been paid an
appropriate attention in conventional physical theories.
In classical Newtonian mechanics, one can define (mean) velocity v by
v=x/t of a particle that starts from the origin at time t=0 and arrives
at position x at time t, if we assume that the coordinates of space and
time are given in an a priori sense. This definition of velocity and
hence that of momentum do not produce any problems, which assures that
in classical regime there is no problem in the notion of space-time.
Also in classical relativistic view, this seems to be valid insofar as
we discuss the motion of a particle in the coordinates of the
observer's.
Let us consider quantum mechanical case where the space-time coordinates
are given a priori. Then the velocity of a particle should be defined as
something like v=x/t. This is a definition, so this must hold in exact
sense if the definition works at all. However in QM case, the
uncertainty principle prohibits the position and momentum from taking
exact values simultaneously. That principle is based on the notion of
position and momentum operators that satisfy the canonical commutation
relation. In the point that the space and momentum are given by
operators, the definition of velocity v above does not apply to QM.
Further the uncertainty principle tells that there is a minimum value
for the product of the variances of the position and momentum from their
expected values, and thus tells that there is an absolute independence
between the notions of position and momentum. I.e. the principle does
not say anything about the relation like x=tv, but instead just tells
that they have to be away from their expectation values.
This observation seems to suggest that, if given a pair of a priori
space and time coordinates, QM becomes contradictory, and that the
independent quantities, space and momentum operators, have to be taken
as the fundamental quantities of quantum mechanics. As time t can be
defined as a ratio x/v in this view, time is a redundant notion that
should not be given a role independent of space and momentum.
This argument seems to give another support to the view that an a priori
notion of time is not a basic notion but the notion of independent space
and momentum operators are basic ones. In this view there can be found a
relation like x=tv as an approximate relation that holds to the extent
that the relation does not contradict the uncertainty principle.
(http://www.kitada.com/time_I.tex)
The quantum jumps that are assumed as an axiom on observation in usual
QM theory may arise from the classical nature of time that determines
the position and momentum in precise sense simultaneously. This nature
of time may urge one to think jumps must occur and consequently one has
to observe definite eigenstates. In actuality what one is able to
observe is scattering process, but not the eigenstates as the final
states of the process. Namely jumps and eigenstates are ghosts arising
based on the passed classical notion of time. Or in more exact words,
the usual QM theory is an overdetermined system that involves too many
independent variables: space, momentum, and time, and in that framework
time is not free from the classical image that velocity is defined by
v=x/t.
Best wishes,
Hitoshi
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