Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 6 Jul 1999 17:51:06 +0300 (EET DST)
Why the Universe looks classical?
TGD predicts that quantum states are *quantum superpositions of classical
universes*: 3-surfaces Y^3 on lightcone boundary M^4_+xCP_2,
or equivalently, of the unique spacetime surfaces X^4(Y^3), defined as
absolute minima of Kaehler action.
[By the nondeterminism of Kaehler action the minima are actually
degenerate but one can modify the definition of 3-surfaces Y^3 by
allowing 'association sequences' consisting of the union of Y^3
on lightcone boundary and spacelike 3-surfaces X^3_i
not belonging to the lightcone boundary and having mutual time like
separations.]
This raises the question which is trouble for all theories in which one
makes spacetime dynamical objec.
*Why the universe looks so classical?*
Why our subjective experience suggests so strongly that spacetime is the
arena of physics and dynamical objects are particles in spacetime rather
than 3-surfaces themselves? A possible answer to this question came
almost accidentally, when I was pondering the precise definition of
subsystem concept crucial for the formulation of strong NMP..
1. Configuration space spinor fields
Configuration space spinor fields can be regarded as functions from
configuration space to the Fock space of fermions from which all known
elementary particles and a lot of exotics are constructed.
Very roughly: configuration space
spinor field assigns to each 3-surface Y^3 a state, which corresponds
to the many particle state of an ordinary quantum field theory.
One can hence write the state formally as
|TGD State> = SUM (Y^3) |Ordinary State(Y^3)> .
Here a formal continuous summation over Y^3 is understood. When only
single Y^3 is present state behaves like ordinary state of quantum field
theories since spacetime surface is now sharply defined.
2. Subsystem concept
Subsystem concept must be consistent with the definition used in
QFT. This requires that entangled subsystem corresponds to
*parallel superposition of subsystems* S(Y^3) entangled with
its complement. Thus one can say that subsystem a la TGD is function
assigning to Y^3 a subsystem a la QFT.
This means that one can write the initial state
UPsi_i of quantum jump
in entangled form as
UPsi_i = SUM(Y^3)C_nN(Y^3) |n(Y^3)>|N(Y^3)>
C_nN(Y^3) are entanglement coefficients depending on surface
Y^3.
[One can also imagine other, nonlocal manners to construct
superposition of entangled state, but the hypothesis that
*physics is local at configuration space level* suggests that this
expression has special significance.]
3. What quantum measurement means for configuration
space spinor fields?
Moments of consciousness mean quantum jumps to the eigenstates of
density matrix for some subsystem which defines entanglement structure
in the manner described above.
Hence the state UPsi_i *must become unentangled state* in quantum jump.
Since entanglement coefficients depend on Y^3 in general and since sum
is actually continuous integral there is only single possibility:
*Localization to single Y^3!*
This would mean that in each quantum jump localization to single
classical spacetime surface would occur and time development by
quantum jumps would effectively define hopping around
the configuration space! Thus the classicality of the world
of subjective experience would follow from the basic structure
of the configuration space spinor fields. Classical time development
would be like Brownian motion in infinite-dimensional
configuration space.
This looks nice but cannot be true. Complete localization
of the configuration space spinor fields is not consistent with
the basic symmetries of TGD (Super Virasoro, Super Kac Moody,
Super Canonical symmetries). It breaks also Poincare symmetry.
4. Localization in zero modes is enough for classicality
One must somehow milden the hypothesis about complete localization
and this is indeed possible. The point is that
configuration space of 3-surfaces has fiber space structure.
a) Fiber corresponds to ordinary quantum fluctuating degrees of freedom
and metric in these degrees of freedom is vanishing (contravariant metric
defines propagator for small perturbations of 3-surface).
b) Base space corresponds to *zero modes* in which configuration space
line element vanishes. Zero modes characterize size and shape
of 3-surface Y^3 and also the classical Kahler field associated
with it. Zero modes are purely TGD:eish concept resulting from
the nonpointlike nature of particles. Zero modes have the role
of fundamental order parameters in the spirit
of the Haken's theory of self organization. Configuration
space isometries act in the fiber and hence leave zero modes invariant.
A very attractive hypothesis is that quantum jumps involve only
*localization in zero modes*.
If one can assume that entanglement coefficients C_nN quite generally
*depend on zero modes only*, then localization in zero modes is enough
to achieve final state of required type. Localization in zero modes
sharpens the hypothesis about localization into definite sector D_p
of configuration space obeying effective p-adic topology since sectors
D_p are induced by the division of zero modes to corresponding sectors.
If localization in zero modes indeed occurs, then one can understand
classicality. The macroscopic parameters characterizing 3-surface
are sharply defined after quantum jump. For instance, the
sizes, shapes and mutual distances of spacetime sheets characterizing
material objects are fixed compeletely. Also Kahler field is fixed. Only
small quantum fluctuations are possible. The world of conscious
experience
looks essentially classical. Localization also implies that QFT based
subsystem concept applies to the the final states of the quantum jumps but
not for the initial states U Psi_i.
As far as the application of strong NMP is considered, all that is
needed is to compare negentropy gains for states located at Y^3:
this means enormous simplification of the theory. The probability that
localization occurs in Y^3 is given by the modulus squared of
configurations space spinor field and sequence of quantum
jumps is expected to lead to those regions of configurations space
whether this probability density is largest.
5. Symmetries--> independence of entanglement coefficients on
fiber degrees of freedom
The independence of the entanglement coefficients C_nN
on fiber degrees of freedom very probably follows from the symmetries
of the theory: I do not have any real proof for this however.
If the states |n> and |N> are gauge invariant, then isometry invariance
of entanglement coefficients means essentially that coefficients
are invariant under the infinite dimensional group of Super Canonical
transformations and this very probably implies that they can depend
on zero modes only since fiber is coset space defined by the group of
canonical transformations of the lightcone boundary M^4_+ xCP_2.
6. Connection with Haken's theory of self organization and Higgs
mechanism
What is important that quantum jumps define hopping motion
in zero modes defining the universal order parameter space
and thus the dynamics for the world of conscious experience can be
modelled as hopping around in the space of zero modes. This means
that Haken's classical theory of self organization generalizes almost
as such. The hopping around zero modes corresponds to dissipation leading
gradually to those values of zero modes for which configuration
space spinor field is very large and corresponds
to a maximum of vacuum functional, which is exponent for the
absolute minimum of Kaehler action. This justifies the construction
of QFT limit of TGD in spacetime surfaces which correspond to
maxima of vacuum functional.
Order parameters are parameters completely analogous to Higgs fields
and localization means that the values of these fields are sharp
in any final state of the quantum jump instead of being
quantum superpositions over several values. The drift to
the maximum of Kaehler function provides mechanism
analogous to that leading to the minimum of Higgs potential.
Symmetry breaking mechanism associated with self organization
generalize directly to TGD context.
Thus localization in zero modes is consistent with the standard
quantum description of order parameters. There is also connection
with Joel Henkel's theory in which one introduces
state spaces parametrized by zero modes and assumes that states
are localized in zero modes. TGD however provides fundamental
mechanism describing the motion in order parameter space without
introducing unitarity breaking but replacing real unitarity with
p-adic unitarity. Note however that real unitarity is broken also
in TGD.
7. Are parameters characterizing various degenerate absolute
minima of Kaehler action zero modes?
There are some discrete parameters characterizing the various
degenerate absolute minima associated with a surface Y^3 on lightcone
boundary. It seems that also these parameters must be identified
as zero modes. In these degrees of freedom the complete localization
of configuration space spinor field is not however necessary.
The point is that these degrees of freedom are *discrete* and
quantum jump can quite well occur to superpositions of degenerate
3-surfaces X^4(Y^3) such that entanglement coefficients in this
subset are constant. This would make possible partial localization
essential for the earlier argument for the arrow of psychological time
in which volitional quantum jumps at time t selects one branch
from the superposition of branches of the multifurcation.
Localization in zero modes might help to understand
the arrow of psychological time. Psychological time could be perhaps
identified as an order parameter, which more or less gives center of
mass time for cognitive spacetime sheet with finite time duration.
Physical time arrow could be related to a gradual drift of cognitive
spacetime sheet on lightcone: there are of course also other
possibilities and one must keep mind open.
With Best,
Matti Pitkanen
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