[time 43] RE: The ordering of spatial states and temporalevents

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 22 Mar 1999 11:26:29 +0200 (EET)

On Mon, 22 Mar 1999, Hitoshi Kitada wrote:

> Dear Matti,
> I am glad to have seen in the communications between you and Stephen that we
> have many common points in your theory and my theory. In this note I just
> comment on some points you find problems:

First answer to the question at the end of the message. TGD comes from
Topological GeometroDynamics. When I published first paper about
TGD in 1982 (or so), I called the theory just GeometroDynamics but
since Wheeler had his own GD we decided to call my theory TGD: probably
David Finkelstein proposed this name. It seems that David has
precognitive talents(:-).

I have some comments about p-adic TGD in relation to the
concept of local system.

a) TGD:eish space time is surface in M^4_+xCP_2 and the hypothesis
is that it decomposes into finite regions obeying effective topology
characterized by p-adic prime p: this can be the case also in time
direction. Thus p-Adic QFT limit must be formulated in the p-adic
counterpart of some finite spacetime region obtained by
mapping real spacetime to its p-adic counterpart in the manner proposed
in previous posting.

b) Real QFT in finite spacetime volume leads to difficulties with
Poincare invariance since Poincare transformations do not leave the
quantization volume invariant. In p-adic QFT Poincare symmetry is
replaced with its p-adic counterpart. What is remarkable that
any p-adic Lie group has hierarchical, fractal decomposition into

 ... subset G_{-n-1) subset G_(-n } ..... subset G_(-1}.

where the elements of G_n are obtained by exponentiations
exp(itJ) of Lie-algebra generators J (having unit p-adic norm)
such that the p-adic norm of t is p^(-n), n>0. There is clearly
a hierarchy of symmetry breakings increasingly smaller
length scales. The reason is that the exponent exp(itJ) does not exists
if argument has p-adic norm larger than 1/p.
This symmetry breaking differs from standard one in that subgroup
has same Lie-algebra as the broken group.

c) For translations this hierarchy corresponds to a hierarchy
of quantization volumes with box of side p^(-n). The marvelous
thing is that Poincare transformations leave this volume invariant
when n is chosen properly! Exact Poincare invariance in finite
volume is achieved! Even time translations leave the
quantization volume with finite time duration invariant.
A possible interpretation is that p-adic QFT describes
the dynamics of particles on spacetime sheet of finite duration:
the energy comes to this spacetime sheeet from larger spacetime sheet
when it begins and flows back when it ends. Perhaps
particle reactions occur on this kind of spacetime sheets
of finite time duration.

d) Note that M^4 metric is of standard form dt^2 -dx^2-... in p-adic
context. Now however there seems to be no sharp difference between
Euclidian and Minkowskian signature of metric.

e) p-Adic Fourier analysis providing representations of Poincare group
seems to exist (this became clear quite recently). The problems
where related to the nonexistence of the exponentials exp(ikx)
when kx has p-adic norm larger than 1/p. One can however define
p-adic planewaves by only requiring that they satisfy the obvious
differential equations stating that they are momentum eigenstates.
The possibility of p-adic pseudoconstants makes it possible
to define planewaves in entire Minkowski space and orthogonality
conditions hold true for the proposed p-adic integral relying
crucially on the canonical identification map between p-adics and reals
inducing ordering of p-adics from that for reals.

f) Also the time development operator P(exp(i int_0^t H_1dt)),
H_1 interaction Hamiltonian, suffers from number theoretical
nonexistence difficulties. This is obvious for diagonal Hamiltonian,
say oscillator Hamiltonian, for which exponent exists only when
interaction time has p-adic norm smaller than some p^n.
 This implies that for a given Hamiltonian and
without any additional restrictions on the states the operator existence
for finite time interval t only. One manner to get rid
of the difficulties is to pose restrictions on the physical
states guaranteing the existence.

If hamiltonian contains strongly interacting part this kind of situation
is expected to happen. Color interaction is good example of this kind of
situation. This suggests number theoretical mechanism of color
confinement: in sufficiently larger time scales physical states are color
singlets since otherwise the color part of interaction Hamiltonian would
be so large that time development operator does not exist
as a p-adic number.

g) Still a comment related to Lorentz invariance.
 In TGD each quantum jump is preceided by an interaction
of 'time development' operator U_a , a goes to infinity,
which is exponent of Virasoro generator L_0: similar
operator appears also in string models but the time parameter
is interal time of the string world sheet.
U_a acts in the space of quantum histories so that the usual
intuitive picture as an operator carrying the state from
t=-infty to t=infty does not make sense. The problems with
Lorentz invariance are circumvented now because a is
lightcone proper time and hence Lorentz invariance. Poincare invariance
is broken and I must admit that I do not understand how strong the
breaking can be.


Matti Pitkanen

P.S. I finally visited you homepage and well as Stephen's homepage and
found them very inspiring. I will study them more closely.

> >
> >
> >On Sat, 20 Mar 1999, Stephen P. King wrote:
> >
> >> Dear Matti,
> >>
> >> Matti Pitkanen wrote:
> >> >
> >> snip
> >> [SPK]
> >> > The paper co-authored by Hitoshi and Lance Fletcher
> >> > (http://www.kitada.com/time_III.html) explains all of the basic thinking
> >> > involved in LS. It is rather revolutionary and goes against the grain of
> >> > conventional physical thinking, but, that all said, it does provided a
> >> > starting point with which to address many other difficulties in modeling
> >> > consistently our world.
> >> > An example, the primitive ideas are examined:
> >> >
> >> > "1.We begin by distinguishing the notion of a local system consisting of
> >> > a finite number of particles. Here we mean by "local" that the
> >> > positions of all particles in a local system are understood as defined
> >> > with respect to the same reference frame."
> >> >
> >> > Here we do not assume any particular properties of the
> "particles"
> >> > other than what is explicitly stated and use the standard definition of
> >> > a "particle"; some entity existing at the locus of an set of
> >> > coordinates, but we do not assume any properties yet...
> snip
> >> > [MP]
> >> > There is clear analogy with many sheeted spacetime. In TGD elementary
> >> > particles correspond to so called CP2 type extremals of size of order
> 10^4
> >> > Planck lengths. They have metric with Euclidian signature but lightlike
> >> > curve as M^4_+ projection. These tiny 3-surfaces are glued by
> topological
> >> > sum to 3-surface which is roughly like a piece of Minkowski space with
> >> > size of order Compton length and possessing outer boundary. This process
> >> > leads to massivation of elementary particle described by p-adic
> >> > thermodynamics. It seems that one could think CP2 type extremal as a
> >> > local system and piece of M^4_+ as spacetime. Am I correct?
> >>
> >> Yes! I would like to understand this "topological sum" better. Could
> >> you explain it to us?
> >
> >Topological sum for two n-dimensional surfaces is formed as follows.
> >Cut balls D^n from both surfaces. The boundaries of these
> >holes are spheres S^(n-1). Connect the boundaries of the
> >holes by the cylinder D^1xS^(n-1) along its ends. In two dimensional case
> >you remove disks from the two surfaces and connect them by a cylinder by
> >gluing its ends to the boundaries of the holes.
> >
> >I looked at the references you mentioned. There seems to be also a
> >difference. Local clocks are introduced at each spacetime point
> >by replacing X^4 with X^4xR^6 (R^6 phase space of particle).
> >I do not attach local system to *every point* of background spacetime.
> >Local system would represent topological nonhomegenuity of
> >spacetime surface in TGD approach. But these spacetime sheet 'glued'
> >to background space represent in good approximation their own universes
> >and in good approximation one can construct their physics discarding
> >the interactions with external world. In this manner one obtains
> >QCD, low energy hadronic physics, nuclear physics, atomic physics,...
> >
> >Also our starting point could be seen as same. The failure of Newtonian
> >spacetime applied in standard QM in General Relativistic context.
> >My solution is to give up totally the idea about physical state as
> >time=constant snapshot and describe quantum state as quantum history
> >and try to understand the emergence of psychological time ('clocks')
> >as a problem of consciousness theory: why the contents of conscious
> >experience is located around some value(s possibly) and why this value
> >of time tends to increase at least locally. Here the nondetermism of
> >Kahler action, which generated this discussion, is in fundamental role.
> >Quantum jumps for which nondetermism is located in finite time interval
> >give conscious information about that time interval and hence conscious
> >experiences with time localized experiences become possible. Without
> >nondeterminism experience would contain information diffused over entire
> >intial and final quantum histories.
> >
> >
> >
> >
> >What troubled me in local system approach were the following points.
> >Dirac equation for atoms gives predictions which are verified
> >experimentally and replacing Dirac equation with Schrodinger equation does
> >not seem promising.
> My point in choosing Schroedinger equation is in that Dirac eqaution cannot
> treat many body problem. As far as I know, it is an equation for one partcile
> (an electron) in an exterior field. This is the reason why I had to choose
> Schroedinger equation as a starting point of a description of nature inside a
> local system. The relation among plural number of local systems is derived
> from this equation, and I obtained some halfly-relativistsic equatIon.
> But if we can take a standpoint that Stephen stated in [time 22] as follows,
> we might be able to start with Dirac or any other relativistic QM equation.
> form [time 22]:
> >> Also electromagnetic aspect is a quanutm mechanical one, where classical
> >> treatment breaks (e.g. the stability of matter (atoms, molecules,...) does
> not
> >> hold in classical electromagnetic theory). Further Weyl seems to be able to
> > >treat only one body problem (i.e. it can treat only the external forces,
> and
> > >cannot conisder the interactions between plural number of particles, which
> is
> > >essentially the same as Einstein's GR.)
> >
> >While the emmition and absorbtion of photons is a QM phenomena, the
> >light cone structures are not "internal" to LS as classical
> >trajectories. They are, as Hitoshi said before, perspectivist, e.g. a
> >means to observe. Weyl's treatement of only a single body, IMHO, is not
> >a problem in the context that such is an LS seen from the outside; it
> >would be a classical "particle."
> >When thinking about this and we shift from internal LS QM behaviour to
> >external LS classical behaviour we must keep track of what is happening.
> >It can be very tricky! :)
> Here Stephen considers that what is observed is just "one" LS. If we take this
> view, we may have no problem in the description of the behavior of an LS when
> observed as a single classical particle. (Though still a problem seems to
> remain in the description of physics inside an LS.)
> >Secondly, Why R^6, why not only R^3 if local system
> >obeys nonrelativistic QM?
> The description related with R^6 is used in the introduction of some of my
> papers to explain my theory. I used it to indicate that what is fundamental
> in QM is the six componets: three configurations x_1, x_2, x_3 and three
> components of momentum p_1, p_2, p_3 canonically conjugate to x_1, x_2, x_3
> for the case of a single particle. This is some rhetorical expression. Since
> p_1, p_2, p_3 are determined by x_1, x_2, x_3 by canonical conjugacy, what is
> necessary is just R^3 as you pointed out. I chose my expression just to
> emphasize the point above, and might have been slightly away form the
> rigorousness in the expression.
> >
> >
> >>
> >> [MP]
> >> > Hierarchy continues: for instance, quark like 3- surfaces are glued to
> >> > hadronic 3-surfaces, and so on. At human length scales my body is a
> >> > 3-surface with outer boundary identifiable as my skin glued to a larger
> >> > 3-surface.
> snip ...
> By the way, may I ask the non-abbreviated terms for TGD? What is
>the original words?
> Best wishes,
> Hitoshi

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