Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 7 Jun 1999 12:05:39 +0300 (EET DST)
On Mon, 7 Jun 1999, Hitoshi Kitada wrote:
> Dear Matti,
>
>
>
> >
> >
> > On Sun, 6 Jun 1999, Hitoshi Kitada wrote:
> >
> >
> > > Dear Stephen,
> >
> > snip
> >
> >
> > > > > The inner product for configuration space spinor fields reduces
> > > to inner
> > > > > product ofm configuration space spinors integrated over entire
> > > > > configuration space of 3-surfaces. Inner product of spinors is
> > > just Fock
> > > > > space inner product for fermions (oscillator operators create
> > > the state).
> > > > >
> > > > > In your case you have single phi and inner product must be inner
> > > product
> > > > > for some subsystem (LS?). Hence situation is different from that
> > > > > in TGD.
> > > >
> >
> > [Stephen]
> >
> > > > I am getting confused. :( We need to ask Hitoshi about these
> > > details...
> > >
> > [Hitoshi]
> >
> > > If you argue in LS theory, the inner products are of an infinite
> > > number, proper to each Local System. I.e. LS theory considers an
> > > infinite number of Hilbert spaces describing the inner state of each
> > > observer's system. The outside of an observer's system is not
> > > described by Hilbert spaces. Only a part of the outside that is an
> > > object of an observation is described by a Hilbert space structure.
> > >
> > > In LS theory, the phenomena arise by the participation of the
> > > observer. In this sense, my standpoint is the same as the Wheeler's
> > > "participatory universe."
> > >
> > > The total state \phi of the universe is not considered in a Hilbert
> > > space. It represents just the state of the total universe, which
> > > does not evolve. No inner product is considered regarding \phi.
> >
> >
> > Yes. I understand. This resembles the approach of Joel Henkel
> > to the quantum description of biossystems. He also considers collection
> > of Hilbert spaces in his nonunitary QM. I had long discussions with
> > Joel for some time ago. We found that the decomposition of
> > Hilbert space to 'pieces' corresponds in TGD to the decomposition
> > of quantum TGD to padic quantum TGD:s. Breaking of real unitarity
> > is possible test for Joel's approach as also for TGD
> > (but for different reasons) and probably any
> > theory giving up the idea about the quantum state of entire
> > universe.
>
> My theory considers an LS, say L, as being equipped with the Hamiltonian H_L
> of L itslef. With respect to this Hamiltonian H_L, the local system L evolves
> according to exp(-iH_L) and thus this evolution preserves the unitarity
> insofar as that local system is the object of observation. Namely an observed
> system is always considered as a closed system.
>
> Breaking of the unitarity when considered in a larger LS, L', would occur but
> it occurs only when the observer could detect the larger system L'. Unless the
> observer knows L', he has to assume that the unitarity of the evolution
> exp(-itH_L) of the system L under consideration holds because his concerns are
> not extended beyond the observed system L. In other words, any observer
> observes an object with assuming the "ideal" unitarity of the observed
> system's evolution. This is a restatement of the usual assumption in actual
> observations/experiments, which is necessary for any theoritical
> considerations to be possible.
In Henkel's theory there are also 'local systems' obeying unitary time
development. But there is also nonunitary time development.
This is related to symmetry breaking. For instance, different vacuum
expectations of Higgs field or some order parameter would correspond
to nonequivalent Hilbert spaces and non-unitary time development can
lead from realization to another. The interaction with surroundings
somehow induces the nonunitary time development. The weak point of
Henkel's approach is how to realize this time development concretely.
In TGD approach different sectors D_p of configuration space (p prime)
correspond to quantum theories in different p-adic number fields and
at first it seems that unitarity is broken down dramatically!
One can however define generalized S-matrix which obeys generalized
unitarity. S-matrix for transitions leading
from any D_p1 to D_p is C_p valued.
The restriction of S-matrix to transitions D_p to D_p IS unitary!
Despite the fact that S-matrix elements D_p to D_p1 , p_1 neq p can
be nonvanishing!! Sounds highly paradoxical!
The point is that total p-adic probability for transition D_p to D_p1, p_1
neq p can *vanish*! This is something genuinely p-adic and highly
paradoxal and nonsensical from the real point of view.
Therefore p-adic unitarity allows something which
is not unitary in real context.
There is however question of interpretation. One can consider
either p-adic sums of p-adic probabilities or sums of real
counterparts of p-adic probabilities as predictions of theory.
How the physical situation determines which definition of
probabilities one must use? The answer to this question
leads to the concept of monitoring and resolution of monitoring.
Concrete example about conceptual beauty of p-adics illustrating the
concept of monitoring.
a) Elementary particle can have S-matrix for which
the total *p-adic* decay rate is vanishing. Elementary particle
is p-adically stable: individual p-adic probabilities
for decays to various many particle states can be nonvanishing
and only their *p-adic* sum vanishes.
b) The interpretation is
that the p-adic sum of p-adic decay probabilities measures the decay rate
of elementary particle when the *resolution of monitoring of
final states is not able to distinguish between final states*.
b) When monitoring is able to distinguish between subspaces
of final states situation changes. For instance,
one could be able to measure charges of final state
particles or measure momenta with some resolution.
Optimal resolution
is able to distinguish between all final states.
The measured decay rate, which is *sum over the real counterparts of
p-adic decay probabilities* to those subspaces of the Hilbert space of
final states defined by resolution and is *nonvanishing* in general.
Breaking of unitarity in real level means that real probabilities
are not derivable from unitary S-matrix.
c) Thus p-adics make possible to avoid the introduction of decay widths
and complex energies, which are mathematically ugly
concepts. Elementary particles could be
stable in absence of monitoring not able to
resolve between various final states.
>
> [snip]
>
> > > > Thus I am proposing many \phi! :)
> > >
> > > To each obsevation, there corresponds a proper universe. In this
> > > sense, there are many \phi, where \phi is used in different meaning
> > > from the \phi in the above.
> > >
> >
> > Yes. This resembles Henkel's approach.
>
> It seems resembling, but I consider a theoretical framework applicable to the
> actual situation of observations. Explanation of actual situations seems
> requiring us/me to assume that there exist many universes which vary in
> accordance with each observation.
>
Best,
MP
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