Matti Pitkanen (matpitka@pcu.helsinki.fi)
Mon, 27 Sep 1999 07:00:23 +0300 (EET DST)
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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