Stephen P. King (stephenk1@home.com)
Mon, 07 Jun 1999 14:16:44 -0400
Dear Matti,
I am reminded of an on-going conversation that my off-line friend Paul
Hanna and I have been having concerning how to generalize the definition
of velocity (and motions of "particles" in general) into the language of
probabilities. My ideas here are very rough so I beg for your patience
and critique. In [time 392] you said:
[MP]
"In quantum field theory situation is different since it is not possible
to interpret time evolution as evolution in any kind of configuration
space (the required assignment of the space of quantum states to single
point of 3-space does not make sense). Problems are also caused by the
fact that probability density is not scalar quantity anymore but time
component of a 4-vector."
[SPK]
Could we not see this as a problem and consider its implications? If
the probability density is the time component of a 4-vector, what would
the spatial components be?
and:
[MP]
"b) One can also worry about General Coordinate Invariance.
In case of a nonrelativistic Schroedinger equation the information
is Galilei invariant. In case of QFT Lorentz invariance
is lost since probability density behaves like a component
of a four-vector."
[SPK]
Can we construct a Lorentz group for the 4-vector of a single particle
(as an observer), such that it would have an *illusion* of a preferred
inertial frame? Thus a *space-time* is identified with the range of
possible states (wrong word!) of the group and such for any particle. We
modify this construction to account for the space-times of different
types of particles, e.g. photons, neutrinos, electrons, etc.
I effectively reversing the usual way that physics is done, instead of
assuming an a priori space-time and then figuring out the group
theoretic behaviors of objects "in it", I am saying that we consider the
group theoretic properties of a particle as defining, a postiori, the
particular space-time that it would have. Does this make any sense?
and further:
[MP]
"One is accustomed to speak about communication as information flow.
Therefore one could wonder whether it is possible to define the concept
of information current somehow in quantum TGD framework.
U_a, a-->infty is indeed defined as a time evolution operator associated
with Virasoro generator L_0 playing the role of Hamiltonian.
Hence it should be possible to formally associate with
the time evolution U_a a conserved probability current having time
component I^a plus spatial components in the degrees of
freedom characterized by the coordinates of the reduced configuration
space. This assignment would be completely analogous to that performed
for
the ordinary Schroedinger equation and the Lorentz invariance of
the lightcone proper time coordinate a would make this assignment
possible."
[SPK]
Could we think of this U_a as identified with individual LSs, such that
it would represent the "perceived" space-time and act as the external
counterpart to the internal unitary propagator group that defines the
LSs internal clock. This notion would make explicit the subject-object
duality existing between each LS and its set of observables.
[MP]
"In p-adic context n= Log_pR is pseudo constant
for finite values of the integer n and this would mean
that information current would be conserved locally in p-adic sense.
This would *not* imply the conservation of information even
in the case that n is pseudo constant everywhere.
[SPK]
This "everywhere" is not absolute, but bounded relative to a given p?
Thus we do not have a unitarity violation problem when changing
framings[1], since each framing (defined as a p-adic space-time patch)
would have its own convex set of information that is "conserved".
[1] This "framing" notion relates to how the composition of an LS is
altered when we shift between perspectives, like the situation discussed
by Hitoshi in Section 9 of http://www.kitada.com/bin/time_I.pdf and what
I understand of your discussion of biological p-adic physics [time 377]
etc.
[MP]
If this indeed works, one could assign
with a given time evolution U_a transfer of information in
the reduced configuration space of 3-surfaces. The zero modes
characterizing the nondeterminism of the K\"ahler action
contain information about the moments for multifurcations in the
time development of the spacetime surface and this
gives hopes of approximate reduction of this
information flow to an effective information flow
occurring at the level of 'quantum average effective spacetime'."
[SPK]
I am thinking of the motions of "particles" (consistent with Hitoshi's
center-of-mass definition of the exteriors of LSs) as being modelable,
in a fundamental sense, by a "probability density current" quantity.
It is "probabilistic" since its possible directions are not a priori
definite but are defined by the observational act, which is an
interaction subject to finite constraints between pairs of LSs. It is a
"density" because it is a quantity that is not a priori single valued
nor a priori localized to a single point, and a "current" because there
is a continuous "flow" to the individual components due to the existence
of a potential, which I think exists due to the non-absoluteness of the
optimization. We can see that if the potential vanishes, so does the
possibility of a change in the velocity and the information is identical
everywhere in the U^T (this is another way of saying that U^T is a bound
state and is at absolute equilibrium with all proper subsets of itself).
Perhaps these notion is nonsensical! :) I am just trying to work out
how it is that we observe an illusion of continuous motions and can
communicate about these to each other when we can prove that they are
not real, e.g. they are an illusion! :)
see:
http://lobelia.physics.wisc.edu/duncan/phys531/wavemechanics/node2.html
Onwards to the Unknown,
Stephen
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