# [time 823] Re: [time 820] P-adic Physics

Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 27 Sep 1999 15:15:18 +0300 (EET DST)

On Mon, 27 Sep 1999, Stephen P. King wrote:

> 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. :-)

Good to hear bout you again. I do not want to waste my time with
info wars since truth is always the first victim. I have made a firm
decision that the fights with Sarfatti were the last ones for me.

I think that best manner to achieve this is to respect each
other's style of thinking.

Best,
MP

>
> 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
> >
> > 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>
> > > 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?
> > > >
> > > >
> > > >
> > > > 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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