Stephen P. King (stephenk1@home.com)
Mon, 27 Sep 1999 08:10:42 -0400
Dear Matti and Hitoshi,
Perhaps we need to step back and take stock of that ideas have led us
to this point. Lance and I have been talking on the phone about
causality and clocking, toward, I hope, a way of understanding how it is
that the "space-times" that are 'observed' by Local Systems are related
to each other. Please let us leave our emotional attachments at the
gate. The Truth is beyond any finite model that we could construct! Let
us focus on better understanding of how we see our worlds and learn from
each other. :-)
Kindest regards,
Stephen
Matti Pitkanen wrote:
>
> On Mon, 27 Sep 1999, Hitoshi Kitada wrote:
>
> > You seem to misunderstand my notation. I gave you sufficient help. You should
> > think with your "p-adic" world by yourself.
>
> Sorry. It seems that it is you who misunderstands something. This is not
> about notations. It is about physics and mathematical content. I
> understand you notation quite well. I understand also your point,
> it is correct and I am
> grateful that you made it. In real context
> scattering solution leads to trivial S-matrix if one assumes
> Hilb_0=Hilb. On the other hand, Hilb_0 subspace of Hilb leads to problems
> with unitarity. This can be seen directly.
>
> You do not however see my position.
>
> a) I have general formulation of S-matrix as sum of p-adic valued
> S-matrices: I have talked with Stephen about one year in this group. I
> want to undestand what forces p-adicity.
>
> b) I have now very promising general formula for S-matrix in terms
> of scattering solution. This solution has same general structure
> as that given by stringy perturbation theory and is in accordance
> with physical intuitions. But I have problem with it.
>
>
> The first thing to do is to try to see the connection
> between a) and b). And there is! This is just what I have been
> seeking: the deep reason for p-adicization.
>
> It is sad that you are not willing to take p-adic concepts seriously.
> In any case I am grateful for you help and mention you in
> appropriate part of my book unless you have something against it.
>
> Best,
>
> Matti Pitkanen
>
> >
> > 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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