Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 8 Jun 1999 10:36:58 +0300 (EET DST)
On Tue, 8 Jun 1999, Hitoshi Kitada wrote:
> Dear Matti,
>
> ----- Original Message -----
> From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
> To: Hitoshi Kitada <hitoshi@kitada.com>
> Cc: <time@kitada.com>
> Sent: Tuesday, June 08, 1999 3:14 PM
> Subject: [time 396] Re: [time 391] Re: [time 388] Re: [time 384] Re: [time
> 380] Re:[time 376] Whatare observers
>
>
> [snip]
>
> > > >
> > > > 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!
> > >
> > > Do you also think the universe evolves along a common time?
> > >
> > Depends on what one means with time. This will be a long list!
> >
> > a) Time development operator U_a is counterpart of
> > exp(iHt) in QM and acts in the space of *'quantum histories'*:
>
> I understand this as the totality of 'quantum histories' in the universe.
Yes, possible quantum histories, possible solutions of field equations.
>
> > a is lighcone proper time but U_a obviously does not have the
> > ordinary interpretation (interaction on at a=0 and off at a--> infty).
> > a is just a formal parameter of operator exponential.
> > One could however say that entire universe=quantum history in a well
> > defined sense evolves with respect to a and hence global time
> > is in question. But this formal time is certainly not what we
> > experience!
> >
> >
> > c) Configuration space possess geometric time (coordinate a):
> > 3-surfaces have cm time coordinate. This is certainly a global time.
>
> Is "cm" the center of mass of the total universe? Universe
understood *classically* as 3-surface with infinite size having
classical energy momentum (Kaehler action). Quantum history
is quantum superposition of these classical universes and
also fermionic degrees of freedom are present.
>
> >
> > c) There is geometric time associated with spacetime surfaces
> > and this is the Einsteinian time. Spacetime sheets (LS:s) have their
> > own geometric times, which can be usually chosen to be Minkowski
> > time in the rest system of spacetime sheet. Not a global time.
>
> Spacetime sheet seems to correspond to the space-time of my notion of LS.
>
Yes
> >
> > d) Subjective time defined by sequence of quantum jumps. Ordered sequence
> > of ordered 'ticks' with no 'metric'. Not a global time.
>
> I am still unclear with your "quantum jumps." You seem to relate it with
> consciousness. What makes quantum jumps happen?
Quantum jumps are moments of consciousness. They just happen. I do
not believe in any further reduction: this kind of reduction would
recreate the problems solves by quantum jump concept.
>
> >
> >
> > > > 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.
> > >
> > > I question what necessitates the unitarity. Unitarity is a notion which
> is
> > > introduced to assist the human reasoning in regards to the understanding
> of
> > > nature.
> > >This is an idealistic notion, which gives us some balance sheets of
> > > probabiltiy as the law of conservation of energy, etc. give. From the
> > > viewpoint that the observer is fundamental that is contrast to your
> view,
> > > the observer has some implicit assumption about what he sees, without
> which
> > > he can not make any judgements/estimates about nature. As Einstein has
> ever
> > > said, we have an image of nature in mind already before we observe
> nature.
> > > What remains is how the image can be applicable and adjustable to
> nature.
> > >
> > >
> > Probability conservation is certainly something which one cannot
> > give up (the nonconservation of probability is identity).
>
> This is the expression of the observer's (your and our) desire. My sense in
> the above statement is that when we organize any theory we have to assume
> such kind of a priori assumptions.
Yes, certainly.
>
> >
> > The requirement that *probability is conserved* for any
> > state together with *linearity of time evolution operator* implies
> > unitarity as one can easily find. <UPsi,UPsi>= <Psi,Psi> for any Psi
> > inplies U^daggerU=1 and hence unitarity.
>
> Here is your basic assumption. The requirement is your requirement and maybe
> all of us. This is not required by nature. What makes it necessary is the
> frame of our mind.
>
OK. You are right. I take probability theory and linearity
of QM at p-adic level as 'God given'.
Perhaps probability theory is replaced with something deeper
some day.
> >
> > When one gives up the idea of LINEAR GLOBAL evolution one
> > gives also the unitarity. In case of LS:s the diffusion
> > of particles to other LS:s would presumably require
> > the use of nonunitary time evolution: OK?.
>
> I might agree in the sense that the loss of information holds in
> observation. But when viewing the universe as a whole as in the case a) and
> the first c) in your statements above, maybe one wants to assume the
> probabilty conserves?
>
Yes, certainly: at least I do so. I assume probability conservation:
it follows from unitarity of U_a automatically. But information
is not conserved: dispersion of configuration space spinor field
in configuratiopn space leads to increases of (potential) information.
**********
Note that in wave mechanics probability density would be
R= Psi^*Psi
whereas information density would be
I= -R*log(R) .
I is maximum when R= 1/V, V--> infinity: that is when one has planewave:
localization of particle gives maximum information gain.
I is minimum (negative infinity) when one has R= 1/deltaV, delta V-->0):
completely localized particle. Localization of already localized particle
gives minimum information gain.
Dispersion is basic characteristic of Schrodinger equation
whereas massless wave equation does not cause dispersion.
This means that time evolution by dispersion creates (potential)
information. Only for energy eigenstates information is not generated
since I is constant of motion in this case.
Note also that fundamental information is information about *position*
in configuration space of 3-surfaces. Do I recall correctly: also
Frieden takes position measurement in special role?
Best,
MP
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