**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Mon, 27 Sep 1999 13:25:58 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 820] Re: [time 819] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time814] Still about construction ofU"**Previous message:**Matti Pitkanen: "[time 818] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**In reply to:**Hitoshi Kitada: "[time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**Next in thread:**Matti Pitkanen: "[time 820] Re: [time 819] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time814] Still about construction ofU"

You seem to misunderstand my notation. I gave you sufficient help. You should

think with your "p-adic" world by yourself.

Hitoshi

----- Original Message -----

From: Matti Pitkanen <matpitka@pcu.helsinki.fi>

To: Hitoshi Kitada <hitoshi@kitada.com>

Cc: Time List <time@kitada.com>

Sent: Monday, September 27, 1999 1:00 PM

Subject: Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time814]

Still about construction ofU

*>
*

*>
*

*> Just when going to sleep I realized that the half of
*

*> the unitarity conditions still holding true are in *'wrong
*

*> direction'*. The sum of probalities would be one
*

*> for reactions from final states to initial states but not
*

*> for probabilities for reactions from initial to final states!
*

*> Nasty! There is real problem involved!
*

*>
*

*>
*

*> Then I realized that here indeed might be the connection with
*

*> p-adicization and in fact the deep reason for p-adicization. The
*

*> general structure of p-adic valued S-matrix fits completely
*

*> with what one obtains.
*

*>
*

*>
*

*> a) Configuration space decomposes into sectors D_p, p=2,3,5...
*

*> Each quantum jump involves localization in some sector D_p
*

*> so that S-matrix decomposes into 'sum' of matrices
*

*> S(pi-->pj):
*

*>
*

*> S= 'sum' S(pi-->pj). pi is not state label now: I have omitted them.
*

*>
*

*>
*

*> b) S(p_i,pj) is pj-adic valued (or gets values in
*

*> complex extension of p_j-adics). Since S is 'sum' of S-matrices
*

*> belonging to different p-adic number fields one must assume that
*

*> localization to a definite sector D_p occurs in each quantum jump
*

*> since otherwise transition amplitude would be 'sum' of p-adics in
*

*> different p-adic number fields. The localization in D_p
*

*> is what leads to p-adic evolution since sequence of quantum jumps
*

*> corresponds to a sequence of p-adic primes gradually increasing.
*

*> Also unitarity conditions are generalized.
*

*>
*

*>
*

*>
*

*> c) Interpretation:
*

*> Denote by Hilb_0 the space associated with 'free Hamiltonian' (L_0(free))
*

*> and by Hilb the state space associated with 'interacting Hamiltonian'
*

*> (L_0(tot))
*

*>
*

*> i) Diagonal transitions p_i->p_i correspond to
*

*> transitions which lead from Hilb_0 to Hilb_0 (Hilbert space
*

*> associated with free Hamiltonian).
*

*>
*

*> ii) Nondiagonal transitions p_i-->p_j correspond to transitions
*

*> leading from Hilb_0 to Hilb and genuinely outside Hilb_0.
*

*>
*

*>
*

*> d) Consider now diagonal unitarity conditions. S(p_i,pi) describing
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*> diagonal transitions for which p is not changed
*

*> satisfies p_i-adic valued version of ordinary unitarity conditions.
*

*> This means that in p-adic context D_pi behaves as its own sub-universe.
*

*>
*

*> In real context this would mean that S(pi,pj) would vanish: essentially
*

*> the same result what you deduced from scattering solution.
*

*> In p-adic context the concept of p-adic probability comes in rescue.
*

*>
*

*>
*

*> e) Consider next non-diagonal unitarity conditions.
*

*> S(p_i,pj), pj neq pi, satisfies unitarity conditions but since p_i is
*

*> different from p_j these transitions are nondiagonal and *all inner
*

*> products of rows of S(p_i,pj) vanish*. In particular,
*

*>
*

*> *total p-adic probabilities for transitions from D_pi to D_pj vanish!!*
*

*>
*

*>
*

*> This would *not* make sense in real context but is completely OK in p-adic
*

*> context since the concept of negative p-adic number does not make sense.
*

*>
*

*>
*

*>
*

*> f) Thus total p-adic probability for transitions leading from D_p (Hilb_0)
*

*> to D_pj, pj neq pi (to the complement of Hilb_0 in Hilb) vanishes.
*

*>
*

*> The real probabilities do not of course vanish and are calculated
*

*> by normalizing the real counterparts of p-adic probabilities.
*

*> Real probabilities defined in this manner are not deducible from
*

*> real S-matrix.
*

*>
*

*> g) Thus it would seem that you have pointed out the deep reason for
*

*> why p-adicization is needed! Decomposition of the configuration space
*

*> to sectors D_p would provide concrete realization for the
*

*> Hilb_0-Hilb relation. What one should show that the
*

*> p-adicization of the kernela of U defined by scattering
*

*> solution indeed leads to a p-adic valued S-matrix satisfying the
*

*> unitarity conditions.
*

*>
*

*> Note that p-adiczation might also be involved with the problems
*

*> caused by the infinite value of renormalization constant Z of
*

*> Psi.
*

*>
*

*> What do you think?
*

*>
*

*>
*

*>
*

*> I add a latex file about section about generalized unitarity conditions
*

*> associated with the 'Super S-matrix' expressible as 'sum'
*

*> of p-adic valued S-matrices. This topic can be found in 'p-Adiciation
*

*> of quantum TGD' in 'TGD inspired theory of consciousness..' at my homepage
*

*> and also in 'p-Adic TGD'.
*

*>
*

*>
*

*> Best,
*

*> MP
*

*> ********************************************************************
*

*>
*

*>
*

*>
*

*> \documentstyle [10pt]{article}
*

*> \begin{document}
*

*> \newcommand{\vm}{\vspace{0.2cm}}
*

*> \newcommand{\vl}{\vspace{0.4cm}}
*

*> \newcommand{\per}{\hspace{.2cm}}
*

*>
*

*>
*

*>
*

*> \subsection{Generalized unitarity conditions}
*

*>
*

*> Unitarity conditions
*

*> generalizing the conservation of probability to quantum context.
*

*> What makes the problem
*

*> nontrivial is that generalized unitarity relations should
*

*> they apply in the entire configuration space having decomposition
*

*> into regions $D_p$, $p=2,3,...$, so that the values
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*> of the inner products
*

*> in various sectors belong to different number fields $R_p$
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*> so that sums of p-adic numbers belonging to different $R_p$:s
*

*> are involved!
*

*>
*

*>
*

*>
*

*>
*

*> \vm
*

*>
*

*> The trivial solution of the problem would be based on
*

*> the assumption that time development operator defined as
*

*> the exponential of the Virasoro generator $L_0$ does not
*

*> cause dispersion from given sector $D_p$ to other sectors.
*

*> The decomposition of the configuration
*

*> space into non-communicating
*

*> sectors $D_p$ does not look physically plausible since
*

*> one would lose the beatiful
*

*> consequences of quantum jumps between quantum histories
*

*> picture (the problem of fixing the initial values at big bang
*

*> is circumvented).
*

*> Furthermore, dispersion
*

*> between different sectors is expected
*

*> to occur since $L_0$ is the infinite-dimensional
*

*> counterpart of the Laplacian
*

*> associated with the Schr\"odinger equation. An important point
*

*> is that the action of $L_0$ is that of a differential
*

*> operator and p-adic numbers do not enter at this stage.
*

*>
*

*>
*

*> \vm
*

*>
*

*> The solution of the unitarity problem is based on the
*

*> trivial looking observation
*

*> that $1$ and $0$ can be regarded as common elements of
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*> $R$ and all p-adic
*

*> number fields $R_p$. Assume the existence
*

*> of a preferred basis of states with the property that each state
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*> is localized into some sector $D_p$ of $CH$: this means
*

*> that the total
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*> state space is degeneralized direct sum of form
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*>
*

*> $$H= \oplus_p H_{p}\per .$$
*

*>
*

*> \noindent This direct sum indeed makes sense since configuration space
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*> spinor
*

*> fields themselves are complex valued.
*

*>
*

*> \vm
*

*>
*

*> Since S-matrix is defined by the inner products
*

*> of configuration space spinor fields, it must decompose
*

*> into a formal direct sum of S-matrices $S(p_i,p_j)$ mapping the states
*

*> restricted to $D_{p_i}$ to $D_{p_j}$:
*

*>
*

*> \begin{eqnarray}
*

*> S&=& \oplus_{i,j} S(p_i,p_j) \per .
*

*> \end{eqnarray}
*

*>
*

*> \noindent $S(p_i,p_j)$ must be
*

*> $R_{p_j}$-valued since it is defined by an integral restricted to
*

*> $D_j$. Unitarity relations can be written as
*

*>
*

*>
*

*> \begin{eqnarray}
*

*> \sum_{k} S(p_i,p_k) S^{\dagger}(p_j,p_k)= Id_{p_i}\delta_{p_i,p_j} \per .
*

*> \end{eqnarray}
*

*>
*

*>
*

*> \noindent Here $Id_{p_i}$ denotes identity operator in sector
*

*> $D_{p_i}$.
*

*> The definition of $S^{\dagger}=S^{*T}$ involves transpose
*

*> and complex conjugation.
*

*> For a fixed value of $k$ both terms of the product are $p_k$-adic
*

*> numbers so that the sum is well defined and must vanish always
*

*> for $i\neq j$. This makes certainly sense.
*

*>
*

*> \vm
*

*>
*

*> For $i=j$ the condition states probability conservation.
*

*> For $k=i$ one obtains standard probability conservation:
*

*>
*

*> \begin{eqnarray}
*

*> S(p_i,p_i) S^{\dagger}(p_i,p_i)= Id_{i} \per ,
*

*> \end{eqnarray}
*

*>
*

*> \noindent so that there are no problems.
*

*> For $k\neq i$
*

*> one has
*

*>
*

*> \begin{eqnarray}
*

*> S(p_i,p_k) S^{\dagger}(p_i,p_k)= 0\per , \per k\neq i \per .
*

*> \end{eqnarray}
*

*>
*

*> \noindent In real context this would require $S(p_i,p_k)=0$ for
*

*> $k\neq i$. In p-adic context this is not necessary so thanks to
*

*> the rather miraculous properties of p-adic probability
*

*> discussed already earlier.
*

*> The total probability for the dispersion from sector $D_i$ to sector
*

*> $D_k$, $k\neq i$ can indeed vanish although
*

*> the real counterparts of the individual dispersion probabilities
*

*> can be nonvanishing!
*

*> Thus the concept of monitoring dependent real probabilities
*

*> emerges at the level of the basic quantum TGD: transition from
*

*> $D_i$ to $D_k$, $k\neq i$ can occur only if the sector $D_k$ is
*

*> is monitored in nontrivial manner which means
*

*> that the states in $D_k$ are divided into two classes
*

*> at least and experiment somehow monitors to which of these classes
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*> the final state belongs.
*

*>
*

*>
*

*>
*

*>
*

*> \vm
*

*>
*

*> If the final state of the quantum jump would contain components in several
*

*> sectors $D_p$ of $CH$, the corresponding transition probability would be
*

*> sum of p-adic probabilities belonging
*

*> to different p-adic number fields. Since this does not make
*

*> sense, a selection of $p$ must take place in each quantum jump.
*

*> Time development
*

*> by quantum jumps defines thus a sequence $...\rightarrow p_1\rightarrow
*

*> p_2 \rightarrow ...$. From this it is rather easy to deduce that
*

*> generalized
*

*> unitarity conditions picture actually leads to evolution!
*

*>
*

*> \end{document}
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*>
*

*> On Mon, 27 Sep 1999, Hitoshi Kitada wrote:
*

*>
*

*> > Dear Matti,
*

*> >
*

*> > Matti Pitkanen <matpitka@pcu.helsinki.fi> wrote:
*

*> >
*

*> > Subject: [time 816] Re: [time 815] A summary on [time 814] Still about
*

*> > construction ofU
*

*> >
*

*> >
*

*> > >
*

*> > >
*

*> > >
*

*> > > Thank you for good posting. Your are right in that Hilbert space
*

*> > > is extended. One however obtains S-matrix for which other half
*

*> > > of unitary condition with summation over intermediate states of
*

*> > > extended Hilbert space is satisfied and this makes
*

*> > > S-matrix physical. Other half of unitarity conditions
*

*> > > involving sum over the intermediate states in smaller Hilbert space is
*

*> > > lost.
*

*> > >
*

*> > > See below.
*

*> >
*

*> > skip
*

*> >
*

*> > > > This is not your expectation. Why this happened? There are two
*

possible

*> > reasons:
*

*> > > >
*

*> > > > 1) The first is that we have assumed that both of \Psi and \Psi_0 are
*

in

*> > the
*

*> > > > Hilbert space \HH. If we assume \Psi_0 is in \HH, then \Psi must be
*

*> > outside \HH.
*

*> > >
*

*> > > This is certainly the case since Psi contains superposition of
*

*> > > off mass shell states. p^2-L_0(vib)=0 is not satisfied for Psi.
*

*> > > If this were not the case, the entire equation would be nonsensical
*

*> > > since right hand side would be of form (L_0(int)/+ie)Psi.
*

*> > > Thus we have Hilbert spaces which we could call Hilb_0 and Hilb.
*

*> > >
*

*> > >
*

*> > > One the other hand. Psi is image of on mass shell state under
*

Psi_0-->Psi

*> > > and S-matrix is defined as matrix elements
*

*> > >
*

*> > > SmM== <Psi_0(m),Psi (M)>.
*

*> > >
*

*> > > One restricts outgoing momenta to on mass shell momenta in inner
*

product.

*> > > This means projection of Psi (m) to the space Hilb_0 spanned by Psi_0:s
*

*> > > when one calculates inner products defining S-matrix.
*

*> > >
*

*> > > One obtains unitarity relations
*

*> > >
*

*> > > sum_N SmN (SnN)^* = delta (m,n)
*

*> > >
*

*> > > from completeness in Hilb: sum_N |N> <N|=1
*

*> > >
*

*> > > but NOT
*

*> > >
*

*> > > sum_m smM (SmN)*.
*

*> > >
*

*> > > since Hilb_0 completeness relation sum_m |m><m|=1 are not true in Hilb
*

*> > > but become sum_m |m><m>= P, P projector to Hilb_0.
*

*> > >
*

*> > > But this seems to be enough! One obtains S-matrix with orthogonal
*

*> > > rows: this gives probability conservation plus additional conditions.
*

*> >
*

*> > The probability conservation (i.e. unitarity of scattering operator) is
*

not so

*> > easy to prove. I just gave an outline. If one would want to get a rigorous
*

*> > proof, it might require several years.
*

*> >
*

*> > > Colums are however not orthogonal.
*

*> >
*

*> > I am not familiar with Dirac notation, but I believe I did not make
*

mistakes

*> > in my formulae, insofar as about its formality.
*

*> >
*

*> >
*

*> > >
*

*> > >
*

*> >
*

*> > Best wishes,
*

*> > Hitoshi
*

*> >
*

*> >
*

*> >
*

*>
*

**Next message:**Matti Pitkanen: "[time 820] Re: [time 819] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time814] Still about construction ofU"**Previous message:**Matti Pitkanen: "[time 818] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**In reply to:**Hitoshi Kitada: "[time 817] Re: [time 816] Re: [time 815] A summary on [time 814] Still about construction ofU"**Next in thread:**Matti Pitkanen: "[time 820] Re: [time 819] Re: [time 817] Re: [time 816] Re: [time 815] A summary on [time814] Still about construction ofU"

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