Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 27 Sep 1999 17:04:23 +0300 (EET DST)
On Mon, 27 Sep 1999, Hitoshi Kitada wrote:
> Dear Matti and Stephen,
>
> This is not a war as Matti seems to like to refer to it. Why westerners tend
> to or like to see things as wars or fights or quarrels or troubles or anything
> bad ...? I just want to make you ask your questions about my formulation in
> exact words. You did not ask any questions about what I wrote, whose
> understanding needs many questions for you to understand the things behind it,
> but you just state what you think without exact relation with my words and
> formulation.
I was referring to "conversations with Sarfatti" when talking about info
wars. My style of thinking is pattern recognition: I do not
see things in terms of formulas as you do. This certainly causes
difficulty in communication.
>
> Exact words are necessary to express your thought or theory. At the present
> stage of exposition of your theory, you just state your impressions about the
> universe.
Much more. But of course, the precise formulation of S-matrix is
dream for a long time. I am just trying to catch the basic pattern.
I can be happy if I can convince myself
that it is possible to get unitary S-matrix, that
p-adics are necessary to understand the S-matrix, and stringy
perturbation theory results from these ideas.
> It cannot get any understanding if you would stay in self
> satisfaction with your theory. I find some common points in yours with my
> theory, especially in the idea of 3-surfaces attached at each point which
> looks like resembling my local system. But I find at the same time some
> difference: The state function \Psi of your universe seems to be in a
> resonance state, if we track exactly your theory. My formulation is a step
> toward clarifying these differences and common points as well. To see them, it
> is necessary to express your theory precisely using mathematical words first.
> But your mathematics looks confusing topologies of Hilbert spaces, which seems
> the cause of your misunderstanding:
>
Might be: detailed construction of Hilbert spaces involved is also
in temporal distance of decades. I am really happy if I can
construct convincing general picture about what is involved.
In any case, generalization of S-matrix to 'sum' or collection
of p-adic valued S-matrices is 'must' to me, not just
confusing topologies: this should be clear from the long discussions
with Stephen.
> > On the other hand, Hilb_0 subspace of Hilb leads to problems
> > with unitarity.
>
> You seem understand what you mean in this statement inside yourself. But to
> make others understand what you have in mind, you need to express it in exact
> words so that others can understand. The smaller Hilbert space \HH_+ (=Hilb_0
> in your notation) is a dual space of the larger Hilbert space \HH_- and
> (\HH_+,\HH,\HH_-) constitute a Gelfand triple, which Stephen often referred
> to. There are topological problems in my formula in [time 804], whose
> understanding requires considering this triple.
My excuse for not exprerssing myself clearly is
that I have just nice idea and I am trying to understand what is precisely
involved when one goes from single particle level to many particle
level. This is trial and error process.
Is it possible to explain the basic idea of Gelfand triple
in words? What is the role of \HH?
> Did you notice this point when
> you wrote your *decisive* statement above? The point which you see wrong in my
> formulation is not clear to me also.
I have *never* claimed that anything is wrong with your formulation:
quite the contrary: I realized that there is something wrong with
my naive picture! And this is what I try to understand!
My statement was just the conclusion *assuming* that
Hilb_0 is subspace of Hilb and putting in identitity operators
for Hilb_0 and Hilb. Identity operator of Hilb_0 is *projector*
in Hilb and this implies that only the *rows* of S-matrix
defined *as I naively define it* satisfies those unitary
conditions. Columns do not however satisfy orthonormality
conditions.
What is your opinion about this argument. Is it correct or does it
somehow fail and how?
> You stick to your own words, and do not
> try to make them express in other words. In the present discussion about your
> formulation, you are speaking with me, so you should know my words. My
> impression is that you neglect my words by sticking to your own expression,
> which lacks exactness necessary to make points of arguments clear but makes
> them more obscure than before. You need to take time to understand what I
> wrote. You respond in a few minutes after my posts. I believe I do not write
> such superficial things.
This is certainly the case! For me the this is discussion rather than
publishing an article in Handbuch der Physik(;-).
Best,
MP
>
> Best,
> Hitoshi
>
> ----- Original Message -----
> From: Stephen P. King <stephenk1@home.com>
> To: Matti Pitkanen <matpitka@pcu.helsinki.fi>
> Cc: Hitoshi Kitada <hitoshi@kitada.com>; Time List <time@kitada.com>
> Sent: Monday, September 27, 1999 9:10 PM
> Subject: [time 822] Re: [time 820] P-adic Physics
>
>
> > 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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