Matti Pitkanen (email@example.com)
Sat, 21 Aug 1999 11:13:03 +0300 (EET DST)
Hi Stephen et all,
I read the paper of Smolin et al and found it quite interesting.
1. Disappearence of time
a) The problem of the disappearing time in quantized General Relativity is
familiar to me and I have used it as one argument in favour of 3-space
as surface identication.
b) In fact, also in TGD all calculations of S-matrix elements reduce
effectively to functional integrals over 3-surfaces belonging to the
lightcone boundary. This is due to General Coordinate Invariance and the
determinism of Kaehler action: the determinism is broken by degeneracy of
absolute minima (cognitive spacetime sheets) but this implies only
additional summation in functional integral.
c) Cognitive spacetime sheets and evolution by quantum jumps however
save psychological time. Actually the necesssary anatomizing of cognitive
spacetime sheets forces to leave boundary of lightcone even if one could
calculate everything else without leaving it.
d) Although calculation of S-matrix elements reduces to functinal
integral on 'lightcone boundary', entire spacetime surfaces are
absolutely essential for construction of theory and for Poincare invariant
S-matrix: this is one of the latest results.
i) Diff^4 does not commute with ordinary Poincare:
absolute minima associated with translates of 3-surfaces are not
translates of absolute minima.
ii) The so called Diff^4 invariant representations of Poincare
transformations translate/rotate 3-surface X^3 in lightcone proper time
a=constant hyperboloid infinitesimally to Y^3 and replace spacetime
surface X^4(X^3) by X^4(Y^3).
iii) For finite values of lightcone proper time a X^4(Y^3) is
deformation of X^4(X^3) instead of pure Poincare transform having same
iv) Diff^4 invariant Poincare algebra closes and reduces to ordinary
Poincare algebra only at the limit a--> infty (lightcone boundary is
infinitely far in past). This algebra acting on 3-surfaces in infinite
future must be used to define momentum eigenstates and makes Poincare
invariance possible. The price paid for is that theory cannot be
constructed on lightcone boundary alone.
2. Computational nonconstructability as loophole of no-time arguments
Smolin saw the (computational) nonconstructibility of spate spaces of GRT
as a loophole killing the argumentation leading to the disappearence of
time in GRT. Smolin suggests that the theory should be discretized to
overcome the difficulty. I think that models are necessary discrete but
that theories are unavoidably computationally nonconstructible but
must allow computational models as approximations.
Smolin also suggests that standard quantum mechanics fails in
quantization of GRT. It is easy to agree with this.
I however see deeper reason for the disappearence of time: the
assumption that spacetime surface is abstract pseudo-Riemannian manifold
leads automatically to the disappearence of time in quantized theory.
The hypothesis that spacetime is 4-surface saves the situation.
Geometric time corresponds more or less to cm time coordinate of
3. Basic ideas and concepts
The following concepts and ideas appear in paper.
1. Non-constructibility of state spaces.
2. Spin networks
3. Discrete time evolution: successor of spin net work state:
emergence of discretized time.
4. Evolution at the level of state space
There are surprisingly many resemblances with my own approach
and somewhat self-centeredly I will compare Smolin's approach to
my own in the following.
a) Non-constructibility of state spaces
I would not be surprised if also the real state space of quantum TGD
would have this property. Of course, the reduction of state
construction to that of representations of infinite-dimensional
symmetry groups might help here. On the other hand, all possible
topologies 3-topologies imbeddable to M^4_+x CP_2 are involved.
b) Spin networks
My point of view is that theories and models are different
thing. It is possible to discover general principles of theories
although calculation requires discretization and approximations.
Pinary cutoff is basic feature of reals to p-adics correspondences
and it could be that pinary cutoff might be what makes things
calculable and makes possible models of arbitrarily high accuracy.
Pinary cutoff means replacement of 3-surfaces, spacetime surfaces,
configuration space,... with lattice analogous to spin network.
c) Discretization of time evolution
The discrete evolution of spin networks suggested by Smolin et al
resembles time evolution by quantum jumps although I understood that
this was meant to be more like discrete time evolution of Schrodinger
equation (although it was mentioned that next state need not be unique).
Time would be measured as number of steps occurred but there is very long
way to the geometric time and spacetime of General Relativity from this.
d) Evolution of state space and self-organization
Smolin et al suggested also evolution of state space
and self-organization at cosmological scales. This indeed what TGD
predicts. The effective p-adic topology characterizing p-adicized state
space increases in the long run. The maximal accuracy of the
representation of world prodived by the experiences determined by pinary
cutoff becomes better and better and discretization becomes more and more
precise. Self-organization by quantum jumps is basic feature of also TGD
approach: entire geometric time development is replaced with a new one so
that self-organization is not property of single geometric time evolution
In TGD however the real configuration space and configuration space
spinor fields are on the background: every quantum jump
involves the steps Psi_i-->UPsi_i--Psi_f and UPsi_i representes
real configuration space spinor field dispersed in entire
configuration space, superposition of all possible parallel spacetime
surfaces. Final state Psi_f is located in single sector D_p of
configuration space and spacetimes in superposition are macroscopically
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