Hitoshi Kitada (hitoshi@kitada.com)
Mon, 27 Sep 1999 13:25:58 +0900
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
> 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
> of the inner products
> in various sectors belong to different number fields $R_p$
> 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
> $R$ and all p-adic
> number fields $R_p$. Assume the existence
> of a preferred basis of states with the property that each state
> is localized into some sector $D_p$ of $CH$: this means
> that the total
> state space is degeneralized direct sum of form
>
> $$H= \oplus_p H_{p}\per .$$
>
> \noindent This direct sum indeed makes sense since configuration space
> 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
> 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
> >
> >
> >
>
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