Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 9 Nov 1999 08:16:37 +0200 (EET)
On Tue, 9 Nov 1999, Hitoshi Kitada wrote:
> 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 is very real problem.
a) Your definition of velocity is classical and applies in practice
to point like particles. In case of field momentum it does not work.
b) One could try to avoid the problem is by defining
velocity as a parameter characterizing symmetry transformation, in this
case Lorentz boost. Given velocity would only characterize
the transformation relating too states. In this case velocity
appears only in the transformation formula defining how energy and
momentum are changed in Lorentz boost. Velocity of particle can be defined
in terms of the components of four momentum.
c) This definition is not operational definition in style "v=x/t"
but it can be used to assign velocity parameter to quantum
particle. Note that velocity is purely geometric quantity
in Minkowski geometry since velocity corresponds to hyperbolic
angle.
>
> 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.
>
I regard this as important question. Following monolog does not
provide "final" solution to the problem!
In TGD framework there are several time developments.
a) "Time development" U in single quantum jump (I am speaking
about TGD now) defining S-matrix is considered. It seems that there
is not need to assign Schrodinger evolution to this time development:
just S-matrix characterizes it. S-matrix conserves four-momenta.
Negentropy Maximization Principle need not be consistent with momentum
conservation and could force final states to consists of wave packets
around average momentum.
b) Time development by quantum jumps. Poincare invariance guarantees
conservation of four-momentum in quantum jump, that is U connects
states with same Poincare quantum numbers. This is what particle
physicist needs. NMP might imply that momentum is not precisely
conserved in the sequence of quantum jumps so that the evolution
of the Universe is not restricted by momentum conservation.
c) Geometric time development defined by absolute minimization
of Kaehler action. One can assign to spacetime sheets classical
momenta and they are conserved.
The basic problem is how quantum mechanical Poincare quantum numbers
relate to the momenta and energies measured in laboratory using classical
physics notions.
a) In TGD framework one can assign to quantum particles
four-momenta. Quantum state corresponds to a superposition of spacetime
surfaces. Quantum particle corresponds to spacetime sheets moving on this
surface classically so that one can assing to particles
also classical momenta and velocities.
b) Since final state of quantum jump is superposition of
*macroscopically* equivalent spacetime surfaces (localization in "zero
modes"), it seems that the particle like spaceetime sheets must have
sharp directions and values of velocities. This in case that particle
orbits are of macroscopic size. What "macroscopic" means is presumably
defined by p-adic length scale hypothesis: macroscopy begins at length
scale L(p).
c) If the four-velocities for classical spacetime sheets
are same as for corresponding quantum particles, one achieves
quite nice correspondence at purely kinetical level. Feynmann
diagrams have precice geometric realization.
d) If one requires that classical masses
are same as quantum masses, the correspondence is even more tight.
An interesting question is should one also require that classical
conserved quantities are identical for various spacetime sheets for all
spacetime surfaces in superposition.
e) Also the masses of particles are determined classically in
particle physics experiments, say by putting charged particle in
magnetic field. Could one require that the classical mass of particle is
same as quantum mass? One should be very cautious here since the mass of
particle results from small p-adic thermal mixing of massless
particle with 10^(-4) Planck mass excitations. Therefore
it would seem that particle mass as we usually define it
is quantum statistical parameter.
In any case, the fact that classical physics is genuine part of
quantum theory in TGD framework, seems to provide solution
to the problem. The basic technical problem seems to be about how
precise correlation between classical and quantum numbers results from
consistency arguments (localization in zero modes being the most important
one and implying the "classicality" of the final states
of quantum jump)
> 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
>
>
-Matti Pitkanen
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