Matti Pitkänen (matpitka@pcu.helsinki.fi)
Thu, 11 Nov 1999 17:05:19 +0200
Dear Hitoshi et al,
Below my reply on your comments about "v=x/t problem".
>
> Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
>
> Subject: [time 980] Re: [time 978] a fundamental question on QM time
>
> snip
>
> > > 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.
>
> You are right. I think usual QM is the fountain of general problems of QM.
>
> >
> > 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.
>
> Certainly in relativistic case.
>
> >
> > 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.
>
> I think this S-matrix is the one we have discussed before?
>
Yes.
> > 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.
>
> Is this a composite one consisting of many jumps in a)?
>
Yes. Each quantum jump decomposes to Psi_i-->UPsi-->Psi_f..
> >
> >
> > c) Geometric time development defined by absolute minimization
> > of Kaehler action. One can assign to spacetime sheets classical
> > momenta and they are conserved.
>
> Then you have two kinds of time as you have claimed?
>
Yes. Geometric time and experienced subjective time.
> >
> >
> > 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.
>
> If these momenta are classical, they are free from the uncertainty
principle?
Basically. Note however that 3-surface rather than point like particle
is the basic dynamical object.
> >
> > 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.
>
> So the final state is also classical?
In the sense described above. Quantum aspects are related to quantum
fluctuations
of 3-geometry which are microsopic (in the sense that they occur perhaps
only below some
length scale, most naturally p-adic lenght scale). Of course, also fermionic
degrees of freedom
are present and usually these are regarded as nonclassical (components of
configuration space
spinors correspond to Fock states of fermions).
It seems that the property of being exact momentum eigenstate
is broken by localization in zero modes in the sense that planewaves are
restricted
inside cube of size of p-adic length scale multiplied by some power of p.
a) p-Adic momenta have nice hierarchy:
P= p^k*n, n = SUM_k n_kp^k integer which can in principle be infinite as
real integer.
b) Pinary cutoff for n so that n is finite integer means that there is
finite momentum resolution.
The planewaves with finite pinary cutoff can be realized in cube of
finite size
L=p^m L_p, m integer charactezing pinary cutoff.
c) The smaller the pinary cutoff the larger the size of the box. This is in
accordance
with Uncertainty Principle. The better the momentum resolution the poorer
the
spatial resolution. Pinary cutoff which is crucial for the mapping
taking real geometric structures to their p-adic counterparts seems to
also define the resolution in momentum space and make imposssible the
concept of exact momentum eigenstate.
> > 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?
>
> Maybe, but this seems to depend on what assumption you adopt.
>
This is an open problem.
> > 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.
>
> Then in your context how is the uncertainty principle understood?
>
Uncertainty principle is realized at configuration space level: for the¨
position of 3-surface in imbedding space basically rather than for
position of pointlike particle in 3-space.
Momenta (or ´Diff^4 invariant momentum generators' ) act on
space of configuration space spinor fields and
momentum eigenstates cannot be completely localized
in configuration space. One could also define position
coordinates for 3-surface as group theoretical parameters:
the position coordinate is x if point is obtained from
fixed reference point by applying group operation exp(xp)
to it.
> > 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)
>
> Is only the "final state" observable in your context?
This is question about how contents of conscious experience are determined.
a) Density matrix of subsystem is fundamental *quantum observable* in the
sense of
quantum measurement theory.
b) The zero modes (size and shape of 3-surface,
classical Kahler field ) in which sharp localization occurs in quantum jump
could
be regarded as *classical observables*. Interesting possibility is that
conscious experience gives direct information about these. A more
general view is that moment of consciousness gives conscious information
about the values of zero modes of both Psi_i and Psi_f. Perhaps kind of
comparison is involved.
Best Wishes,
Matti
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