Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 7 Oct 1999 16:06:01 +0300 (EET DST)
I glue below that latex file about unitarity.
I hasten to admit that I must abstrac the state
space of configuration space into symbols |m>. The
whole point is to identify general structural principle
behind p-adic S-matrix: therefore detailed realization
is not so essential (and certainly I cannot provide it!).
Best,
MP
*****************************************************
\documentstyle [10pt]{article}
\begin{document}
\newcommand{\vm}{\vspace{0.2cm}}
\newcommand{\vl}{\vspace{0.4cm}}
\newcommand{\per}{\hspace{.2cm}}
\markright{Construction of S-matrix}
\section{Construction of S-matrix}
\markright{Construction of S-matrix}
The derivation of the general form of S-matrix has been a long standing
problem despite the fact that it is known that S-matrix must
follow from Super Virasoro invariance alone and that the condition
$L_0(tot)\Psi=0$ must determine
S-matrix. In the following it will be found that one
indeed ends up with a general expression of stringy S-matrix
using the following ingredients.
a) Poincare and Diff$^4$ invariances.
b) Decomposition of the Virasoro generator $L_0(tot)$ of the entire
universe to a sum of 'free' Super Virasoro generators $L_0(n)$
for various
asymptotic 3-surfaces $X^3_n(a\rightarrow \infty)$
plus interaction terms. 'Free' Super Virasoro generators
are defined by regarding these 3-surfaces as
independent universes characterized by their own absolute minima
$X^4(X^3_n)$ of K\"ahler action.
c) Representation of the solutions of the Virasoro
condition $L_0(tot)\Psi=0$ in a form analogous to the
scattering solution of Schr\"odinger equation.
\vm
Contrary to earlier expectations, it seems that one cannot assign
explicit Schr\"odinger equation with the S-matrix although the
general structure of the solutions of the Virasoro condition
is same as that of scattering solutions of
Schr\"odinger equation in time dependent perturbation
theory and S-matrix is completely analogous to that obtained as
time evolution operator $U(-t,t)$, $t\rightarrow \infty$ in
the perturbation theory for Schr\"odinger equation.
\subsection{Poincare and Diff$^4$ invariance}
Virasoro generators contain mass squared operator.
Poincare invariance of the S-matrix requires
that one must use Diff$^4$ invariant
momentum generators $p_k(a\rightarrow \infty)$
in the definition the Super Virasoro generators and
of S-matrix. At the limit $a\rightarrow \infty$
the generators
of $Diff^4$ invariant Poincare algebra $p_k(a)$
should obey standard commutation
relations.
One can even assume that states
have well defined Poincare quantum numbers and Poincare invariance
becomes exact if one can assume that the states
are eigenstates of four-momentum.
Therefore very close connection
with ordinary quantum field theory results.
\subsection{Decomposition of $L_0$ to free and interacting parts}
At the limit $a \rightarrow \infty$ 3-surfaces $X^3(n)$
associated with particles can be assumed
to behave in good approximation
like their own independent universes. This means that
one can assign to each particle like 3-surface $X^3_n$ its own
$Diff^4$ invariant generators $p_k(n,a\rightarrow \infty)$,
whose action is defined by regarding $X^3_n(a\rightarrow \infty)$
as its own independent universe so that Diff$^4$ invariant translations
act on the absolute minimum spacetime surface $X^4(X^3_n)$
associated with
$X^3_n$ rather than $X^4(X^3)$ associated
with the entire universe $X^3$.
\vm
This means
effective decomposition of the configuration space to
a Cartesian product of single particle configuration spaces
and the gamma matrices associated with various sectors,
in particular those associated with center of mass degrees of freedom,
are assummed to anticommute. It is assumed that each sector
corresponds to either Ramond or NS type representation of Super Virasoro.
The Virasoro generator $L_0(tot)$ for
for entire Universe contains sum of Virasoro generators $L_0(n)$ for
$X^3_n$ plus necessary interaction terms. The
Super Virasoro representation of entire
universe in turn factors into a tensor product of these
single particle Super Virasoro representations.
Quite generally, Super Virasoro
generators for the entire universe can be expressed as sums of
the Super Virasoro generators
associated with various 3-surface $X^3_n$ plus interaction terms.
\subsection{Analogy with
time dependent perturbation theory for Schr\"odinger equation}
Time dependent perturbation theory for ordinary Schr\"odinger equation
is constructed by using energy eigenstates as state basis
and the basic equation is formal scattering solution of
the Schr\"odinger equation
\begin{eqnarray}
\Psi&=&\Psi_0 + \frac{V}{E-H_0+i\epsilon} \Psi \per .
\end{eqnarray}
\noindent Here $\epsilon$ is infinitesimally small quantity.
$\Psi_0$ ($\Psi$) is eigenstate of $H_0$ ($H$)
with eigen energy $E$. With these assumptions
Schr\"odinger equation is indeed satisfied and one
can construct $\Psi$ perturbatively by developing
right hand side to a geometric series in powers of
the interaction potential
$V$. This expansion defines the perturbative expansion of S-matrix,
when perturbative solution is normalized appropriately.
\vm
Since ordinary Schr\"odinger equation is consistent with the scattering
matrix formalism avoiding elegantly the difficulties with the
definition of the limit $U(-t,t)$, $t\rightarrow \infty$, it
is natural to take this form of Schr\"odinger equation as starting
point when trying to construct explicit form of the 'time' evolution
operator $U$. One can even forget the assumption
about time evolution and require only
that the basic algebraic information guaranteing
unitarity is preserved. This information boils down to the Hermiticity
of free and interacting Hamiltonians and
to the assumption that the spectra
non-bound states for free and interacting Hamiltonians
are identical.
\subsection{Scattering solutions of Super Virasoro conditions}
One ends up with stringy perturbation theory
by decomposing $L_0(tot)$ to a sum of free parts
and interaction term. In this
basis Super Virasoro condition can be expressed as
\begin{eqnarray}
L_0(tot)\vert m\rangle &=& \left[L_0(free) + L_0(int)\right]\vert m\rangle
=0\per .
\end{eqnarray}
\noindent Various terms in this condition are defined in
the following manner:
\begin{eqnarray}
L_0(free)= \sum_n L_0(n)=
\sum_n\left[p^2(n)-L_0(vib,n)\right]\equiv P^2-L_0(vib)\per ,
\nonumber\\\
\begin{array}{ll}
P^2\equiv 0\sum_n p^2(n)\per ; &L_0(vib)\equiv \sum_n L_0(vib,n)\per ;\\
&\\
L_0(n)=p^2(n)-L_0(vib,n)\per .&\\
\end{array}
\end{eqnarray}
\noindent Note that the mass squared operator $p^2(n)$ act
nontrivially only in the tensor factor
of state space associated with $X^3_n$.
\vm
One can write the general scattering solution to this
equation as
\begin{eqnarray}
\vert m\rangle &=&\vert m_0\rangle
-\frac{L_0(int)}{ L_0(free) +i\epsilon} \vert m\rangle \per .
\end{eqnarray}
\noindent
$\epsilon$ is infinitesimal parameter defining precisely the
momentum spacetime integrations in presence of propagator poles.
$L_0(int)$ is defined uniquely by the decomposition of the
$L_0$ associated with the entire universe to a sum of $L_0(n)$:s
associated
with individual 3-surfaces $X^3_n$ regarded
as independent sub-universes plus interaction term.
\vm
$\vert m_0\rangle $ is assumed
to satisfy the Virasoro conditions of the 'free theory'
stating that all particles are on mass shell particles:
\begin{eqnarray}
L_0(n)\vert m_0\rangle&=&\left[p^2(n) -L_0(vib,n)\right]\vert m_0\rangle
=0 \per .
\end{eqnarray}
\noindent These conditions are satisfied if $\Psi_0$ belongs
is expressible as tensor product of solutions of Super Virasoro
conditions for various sectors $X^3_n$.
$\Psi_0$ runs over the entire solution spectrum of 'free'
Super Virasoro conditions.
\vm
The momentum operators $p_k(n)$ are generators of
$p_k(n,a\rightarrow \infty)$
Diff$^4$ invariant translations acting on the 3-surface
$X^3_n(a\rightarrow \infty)$ associated
with particle $n$ regarding it as its own independent universe.
The perturbative
solution of the equation is obtained by iteration and leads to stringy
perturbation theory with $L_0(n)$ appearing in the role of propagators
and $L_0(int)$ defining interaction vertices.
These conditions define Poincare invariant momentum conserving S-matrix
if $L_0(int)$ defines momentum conserving vertices. This should
be the case at the limit $a\rightarrow \infty$.
\vm
An explicit expression for the scattering solution is as geometric series
\begin{eqnarray}
\vert m\rangle &=& \frac{1}{1+X}\vert m_0\rangle\per ,\nonumber\\
\langle m\vert &=& \langle m_0 \vert \frac{1}{1+X^{\dagger}}\per
,\nonumber\\
X&=& \frac{}{L_0+i\epsilon}L_0(int)\per ,\nonumber\\
X^{\dagger}&=& L_0(int)\frac{1}{L_0-i\epsilon}\per .
\end{eqnarray}
\subsection{Formulation of inner product using residy calculus}
It is not clear how the dirty looking formulas for the
scattering states containing $\epsilon\rightarrow 0$
can give rise to a finite S-matrix:
the relevant part of
the inner product is proportional to $1/i\epsilon$.
One gets rid of this difficulty by using a proper representation
for the projection operator.
The represention is obtained
by replacing the states $|n(\epsilon)\rangle$ with states
$\vert n(z)\rangle$,
where $\epsilon$ is replaced with complex number $z$.
\begin{eqnarray}
\vert n(z)\rangle&=&\vert n_0\rangle + \frac{1}{L_0(free)+iz}
=\frac{1}{1+X(z)}\vert n_0\rangle\per ,\nonumber\\
X(z)&=& \frac{1}{L_0+iz}L_0(int)\per ,\nonumber\\
X^{\dagger}(z)&=& L_0(int)\frac{1}{L_0-i\bar{z}}\per .
\end{eqnarray}
\noindent Projection
operator can be written in two forms
\begin{eqnarray}
P&=& \frac{1}{2\pi}\oint_C dz\frac{1}{L_0(free) +iz}
\equiv \oint_C dz p(z) \per ,\nonumber\\
P&=& \frac{1}{2\pi}\oint_C d\bar{z} \frac{1}{L_0(free) -i\bar{z}}
\equiv \oint_C d\bar{z} p(\bar{z}) \per ,\nonumber\\
p(z)&=& \frac{1}{2\pi} \frac{1}{L_0(free)+iz}\per , \nonumber\\
p(\bar{z})&=& \frac{1}{2\pi} \frac{1}{L_0(free)-i\bar{z}}\per .
\end{eqnarray}
\noindent $C$ is very small curve surrounding origin containing
no other poles than states annihilated by $L_0$.
By acting on arbitrary state decomposed to eigenstates of $L_0$
one finds
that the integration picks up only the states annihilated by $L_0
(free)$.
\vm
The inner product
for scattering states reads as
\begin{eqnarray}
\langle m\vert P\vert n\rangle&=&
\oint_C dz\oint_C d\bar{z} \langle m(\bar{z})\vert
p(\bar{z})p(z)\vert n(z)\rangle
\per .
\end{eqnarray}
\noindent In this manner the dirty limiting procedure
for defining states and their inner products is replaced with
elegant formalism based on residy calculus and formulation becomes
mathematically more rigorous.
\subsection{Unitarity conditions}
S-matrix is defined between the projections $P\vert n\rangle$
of scattering
states to "free"
states satisfying free Virasoro conditions.
Therefore the Hilbert spaces of "free" and projected
scattering states are at least
formally identical.
This means that off-mass-shell states appear only as intermediate
states in the perturbative expansion of the S-matrix
just as they do in the standard quantum field theory.
\vm
S-matrix is unitary if
outgoing states are orthogonal to each other. This follows from
the definition of S-matrix as
\begin{eqnarray}
S_{n,m}&=& \langle m_0\vert n\rangle\per ,\nonumber\\
\end{eqnarray}
\noindent where $m_0$ is incoming state and $n$ is scattering state
normalized to unity.
Unitary condition reads as
\begin{eqnarray}
\sum_r S_{m,r}(S_{n,r})^*&=& \delta (n,m)\per .
\end{eqnarray}
\noindent where summation is over the "free" states $\vert r_0\rangle$
to which quantum jump occurs.
\noindent Unitarity condition reads explicitely as
\begin{eqnarray}
\sum_r S_{m,r}(S_{n,r})^*&=&\sum_{r} \langle n\vert r_0 \rangle\langle
r_0\vert m\rangle = \langle n\vert m\rangle\per .
\end{eqnarray}
\noindent Here the completeness of the "free" state basis has been used.
Hence unitarity holds true if one has
\begin{eqnarray}
\langle m\vert n\rangle \propto \delta (m,n)\per .
\end{eqnarray}
\noindent provided that the normalization constant
for the outgoing states are finite. In quantum field theories
this is not usually the case and this could be the reason
for why p-adics are necessarily needed.
\vm
In case of Schr\"odinger equation one can prove orthogonality
of the scattering states
by noticing that "free" and scattering state basis are related
by a unitary time development operator, which preserves
the orthonormality
of the incoming states. Now the situation is different. The
combinatorial structure is same as in wave mechanics but
genuine time development operator need not exist and one
must resort to the hermiticity of $L_0(free)$ and $L_0(int)$
plus the general algebraic structure of the scattering states
plus possible additional assumption
in order to prove the unitarity.
\vm
Using geometric series expansions and the expression
of the inner product based on residy calculus one can write
unitarity conditions as
\begin{eqnarray}
\begin{array}{l}
\oint_C d\bar{z} \oint_C dz \langle m_0\vert
\frac{1}{1+X^{\dagger}(\bar{z})}p(\bar{z})p(z)
\frac{1}{1+X(z)}\vert n_0 \rangle\ = G(m,n) \per ,\\
\\
G(m,n)= <m\vert P\vert m \rangle \delta (m,n)\per , \\
\\
X(z)= \frac{1}{L_0(free)+iz} L_0(int)\per ,\\
\\
X^{\dagger}(\bar{z})= \frac{1}{L_0(free)-i\bar{z}} L_0(int)\per .\\
\end{array}
\end{eqnarray}
\noindent $G$ is the matrix formed by the wave function
renormalization constants.
\subsection{The conditions guaranteing unitarity}
The solution of the Virasoro condition for $L_0$ has same general
structure
as the scattering solution of Schr\"odinger equation.
The action of ``time development'' operator $U$ means
the replacement of the
superposition of the solutions of ``free'' Super Virasoro conditions with
a superposition of the
corresponding normalized scattering solutions of
the full super Virasoro conditions.
It does not seem however useful to assign explicit Schr\"odinger equation
with Super Virasoro conditions. It is not clear whether this is
even possible.
\vm
The naive expectation that the unitarity of the S-matrix
follows automatically, when scattering solutions are properly normalized
and provided that free and interacting Virasoro generators
$L_0$ can be regarded as Hermitian operators. The fact that
time development operator need not exist might somehow make unitarity
impossible without additional conditions. Potential
difficulties are also caused by the fact that normalization constants
can diverge: this is indeed what they typically do in quantum field field
theories.
\vm
Unitarity does not seem to follow without additional
constraints.
Experimentation with various possibilities guided by
critical comments of Hitoshi Kitada (he pointed out
the possibility of complex formalism and demonstrated
that my first guess did not work) indeed led to a
promising candidate for the additional condition.
The condition for the unitarity
is that $L_0(int)$ annihilates
the projections of the genuine scattering contributions
to the space of "free" states:
\begin{eqnarray}
L_0(int)P\vert n_1\rangle &=&0\per ,\nonumber\\
\vert n\rangle &\equiv &\vert n_0\rangle +\vert n_1\rangle\per .
\end{eqnarray}
\noindent It turns out that these conditions
guarantee unitarity and implies that the
wave function and coupling constant renormalizations are
trivial as indeed expected on basis
of quantum criticality. In real context the condition
forces S-matrix to be trivial but in p-adic case
situation is different. The construction
of the p-adic S-matrix reducew to cohomology theory
realizing Wheeler's idea ``boundary of boundary is zero''
as basic physical law in rather concrete manner.
One can also construct very general family of unitary
S-matrices forming
``category'', which is closed
with respect to direct sum and direct
product.
\subsection{Formal proof of unitarity}
Consider now the formal proof of the unitarity.
Orthogonality condition guaranteing
unitarity can be expressed also as the condition
\begin{eqnarray}
\frac{1}{1+X^{\dagger}} P\frac{1}{1+X}&=&G \per ,\nonumber\\
\nonumber\\
G(m,n) &=&\delta (m,n) \langle m\vert m\rangle\per .
\end{eqnarray}
\noindent This condition can be written in the form
\begin{eqnarray}
\langle m_0\vert n_0\rangle + \langle m_0\vert P \vert n_1\rangle
+\langle m_1\vert P \vert n_0\rangle + \langle m_1\vert P \vert n_1\rangle
= G(m,n)\per .
\end{eqnarray}
The proof of unitarity splits in two basic steps.
\vm
a) Consider first the last term appearing at the
left hand side:
\begin{eqnarray}
\langle m_1\vert P \vert n_1\rangle &=&
\oint_C d\bar{z} \langle m_0\vert
\sum_{k>0}\left[L_0(int)\frac{1}{L_0(free)-i\bar{z}}\right]^k
\frac{1}{L_0(free)-i\bar{z}} P \vert n_1\rangle \per .
\nonumber\\
\end{eqnarray}
\noindent The first thing to observe is that
$\langle m_1\vert$ has operator
$L_0(int)\frac{1}{L_0(free) -i\bar{z}}P$
outmost to the right. Since projection operator
effectively forces
$L_0$ to zero, one can commute $L_0(int)$ past
the operators $1/(L_0(free)-i\bar{z})$
so that it acts directly to $P\vert n_1\rangle$. But
by the proposed condition $L_0(int)P\vert n_1\rangle=0$
vanishes!
\vm
b) Consider next second and third terms at the left hand side
of the unitarity condition. The sum of these terms can
be written as
\begin{eqnarray}
\begin{array}{l}
\langle m_0\vert P \vert n_1\rangle+ \langle m_1\vert P \vert n_0\rangle
\\
\\
= \frac{1}{2\pi}\oint_C dz
\langle m_0 \vert \frac{1}{L_0(free)+iz} \sum_{k>0} X^k
\vert n_0\rangle \\
\\
+ \frac{1}{2\pi}\oint_C d\bar{z} \langle m_0
\vert \sum_{k>0}
(X^{\dagger})^k L_0(int)\vert \frac{1}{L_0(free)-i\bar{z}}
\vert n_0\rangle \per .\\
\end{array}
\end{eqnarray}
\noindent One might naively conclude that the sum of
these terms is zero since the overall sign factors are different
(this looks especially obvious in the dirty $1/i\epsilon$-approach).
This is however the case only if on mass shell states do not
appear as intermediate states in terms $X^k$. Unless this
is the case one encounters difficulties.
\vm
c) One can
project out on mass shell contribution to see what kind
of contributions one obtains: what happens that the conditions
$L_0(int)P\vert m_1\rangle=$ guarantees that these contributions
vanish! Consider the second term in the sum to see how this happens.
The on mass shell contributions from
terms $X^k\vert n_0\rangle$ can be grouped by the following
criterion. Each on mass shell contribution
can be characterized by an integer $r$
telling how many genuinely off mass shell powers of $X$ appear
before it. The on mass shell contributions which
come after r:th X can be written in the form $X^r PX^{k-r}$
The sum over all these terms coming from $\sum_{n>0} X^n$
is obviously given by
$$ X^r P\sum_{k>r} X^{k-r}\vert m_0\rangle = X^r P\vert m_1\rangle
=0
$$
\noindent and vanishes
sinces $X^r$ is of form $...L_0(int)$ and hence
annihilates $P\vert m_1\rangle$.
Thus the condition implying unitarity also implies that
on mass shell states do not contribute to the perturbative
expansion.
\vm
The condition implies not only the unitarity
of the S-matrix but also that wave function renormalization
constants are equal to one so that these cannot serve
as sources of divergences.
\subsection{About the physical interpretation of the conditions
guaranteing unitarity}
The conditions guaranteing unitarity allow a nice physical
interpretation in p-adic context but in real context
they lead to a trivial S-matrix. The necessity
of p-adics is a good news from the point of view of quantum
TGD but still one must keep mind open for a possible
weakening of the conditions.
\vm
As already found, the condition
$$L_0(int)P\vert m_1\rangle=0$$
\noindent guaranteing unitarity implies that wave function
renormalization is trivial. The condition also says
that the effect of the vertex operator on
the "dressed" state $\vert m\rangle $
is same as on the "bare" state $\vert m_0\rangle$:
$$L_0(int)\vert m\rangle =L_0(int)\vert m_0\rangle \per .$$
\noindent A pictorial interpretation
of this is that the contribution
of the virtual particle cloud to any vertex is trivial. This
is very much like vanishing of the radiative corrections to
coupling constants implying that various coupling constants
are not renormalized.
\vm
The invariance of the
p-adic K\"ahler coupling strength under renormalization
group is one of the basic hypothesis of quantum TGD
and there are reasons
to believe that quantum criticality is more or less
equivalent with this property. The condition
$L_0(int)P\vert m_1\rangle=0$ however suggests that this condition is much
more general: all vertices are renormalization group invariants.
In real context this certainly does not make sense since
the coupling constants in the real quantum field
theories for the fundamental
interactions are known to run. In p-adic context situation is however
different. One can interpret RG invariance as the symmetry of
the p-adic S-matrix holding
true in each sector $D_p$ of the configuration space.
The dependence of the
K\"ahler coupling strength on p-adic length scale
$L_p$ means that continuous coupling
constant evolution is effectively replaced with a discrete one.
The dependence of $\alpha_K$
on $p$ dictates the dependence of
the other coupling constants
on p-adic length scale.
\vm
Expressing S-matrix as
$$S=1+T\per , $$
\noindent the conditions guaranteing unitarity can be written
in the form
$$ T+T^{\dagger}+ T^{\dagger}T=0\per .$$
\noindent As already found,
the conditions guaranteing unitarity imply that
much stronger conditions
$$T+T^{\dagger}=0$$
\noindent and
$$T^{\dagger}T=0$$
\noindent hold true. These conditions obviously state
that $iT$ is hermitian and nilpotent matrix.
The rows and columns of $iT$
are orthogonal vectors with vanishing length squared
such that diagonal components of $T$ are real.
These conditions do not certainly make sense in real context
since real or complex valued hermitian
nilpotent matrices are impossible mathematically.
p-Adic probability
concept however allows in principle to circumvent the
difficulty.
\vm
Somewhat loosely speaking, the conditions satisfied
by $T$ imply that the absorptive
parts of the forward scattering amplitudes
given by $T+T^{\dagger}$
vanish identically. Therefore scattering amplitudes
would be analytic functions lacking the cuts
characterizing the scattering amplitudes in real context.
By unitarity the absorptive
parts are proportional to $TT^{\dagger}$ which
therefore also
vanishes: this means
vanishing total reaction rates. Thus the conditions
$L_0(int)P\vert m_1\rangle=0$ imply trivial S-matrix in
real context.
\vm
The content of the
conditions $L_0(int)P\vert m_1\rangle=0$
is that the total p-adic probability for
the scattering from a state
$\vert m_0\rangle $ to the states $\vert n_0\rangle \neq
\vert m_0\rangle $ vanishes. This means
that the p-adic probability for the diagonal
scattering $\vert m_0\rangle\rightarrow \vert m_0\rangle$
is exactly one. As far as total scattering rates are
considered, p-adic many-particle states behave therefore like
many-particle states of a free field theory.
\vm
This mechanism implies an elegant description of elementary
particles (see the chapter
"p-Adic mass calculations: New Physics"
of \cite{padTGD}). In real context the concept of elementary
particle has some unsatisfactory features: the reason
is basically that the concept of free particle is in
conflict with the nontriviality of the interactions.
For instance, in case of unstable particles
one is in practice forced to introduce decay
widths $\Gamma$ making particle energies complex:
$E\rightarrow E+i\Gamma$. This kind of mathematical
trickery takes into account the finite lifetime of the
particle in a rather ugly manner.
p-Adic decay widths however vanish and particles behave
like stable particles as far as total p-adic decay rates
are considered. Real decay widths are of course
nonvanishing and are in TGD framework parameters related to
the time evolution by quantum jumps rather than
unitary time evolution by $U$ and real decay widths have
absolutely nothing to do with the energy of the particle.
\vm
It has been also found that total p-adic probabilities
for the transitions between sectors $D_{p_1}$ and $D_{p_2}$
$p_1\neq p_2$ of the configuration space must vanish
by internal consistency requirements.
The proposed scenario generalizes this hypothesis
from the level of the configuration space sectors to the
level of quantum states.
One consequence of the generalized
hypothesis is that the total p-adic probability for
a transition changing the values of the zero mode coordinates
vanishes although S-matrix elements for the transitions
changing the values of the zero modes and even the value
of $p$, are non-vanishing.
\vm
Real scattering probabilities can be deduced from
the p-adic
probabilities by canonical identification map followed
by normalization to one and total
reaction rates are determined by real probabilities.
Unitarity does not make sense in the real context since in
general it is not possible to assign S-matrix
to real reaction rates. There are however
reasons to expect that
at the limit of large p-adic prime real unitarity is
good approximation although total p-adic reaction
rates must still vanish. p-Adic unitarity provides
also an elegant solution to the infrared divergences
leading to infinite total reaction
rates and forward scattering
amplitudes.
\subsection{$T$-matrix defines p-adic cohomology}
$iT$ is hermitian nilpotent matrix. Therefore
one can regard $iT$ as an exterior
derivative operator defining
cohomology. The construction of the cohomology defined by $iT$
reduces to the task of finding those vectors of the state
space which are mapped to zero by $iT$ but which do not belong
to the zero norm subspace defined by $iT$. There is a
nice parallel with supersymmetric theories:
Hermitian super charges are nilpotent operators. Also
the BRST charges appearing in the quantization of Yang Mills
theories and defining physical states as
BRST cohomology are nilpotent and Hermitian.
BRST charges appear also in the construction of
physical states satisfying Super Virasoro conditions.
Super and BRST charges
are presumably not representable as matrices but it is perhaps
p-adicity what makes the representation as infinite-dimensional
matrix possible. Super symmetric situation suggests
that state space has decomposition into states labelled
by "T-parity" instead of R-parity: states with R-parity
zero are states which do not belong to the image of $iT$
and the states with belong to the image
of $iT$ have T-parity one.
\vm
In case that T-cohomology
is trivial, the states in these two spaces are in one-one
correspondence. In a more general case,
state space decomposes to the direct sum
$V_0+ V_1+ iTV_1$, where $V_0$ corresponds
to cohomologically nontrivial subspace mapped to zero by
$iT$ and $V_1$ corresponds to the states which are not
mapped to zero by $iT$. Apart from
a multiplicative constant, $iT$ can defined as a
"projection operator" to the space of exact states:
$$iT= \sum_k \vert Te_k\rangle \langle e_k\vert \per ,$$
\noindent where $e_k$ are p-adic zero norm states.
The rows of $T$ span a linear
subspace for which every vector has vanishing norm
and $T$ maps state space to this zero-norm subspace.
Thus the construction of the matrices $T$ reduces
to that of finding zero-norm subspaces of the
entire state space.
\vm
Physically the cohomologically nontrivial
states belonging to $V_0$ and
mapped to zero by $iT$
(closed but not exact states) are
noninteracting states remaining invariant under the "time
evolution" operator $U$. These
states are obviously natural candidates for the fixed
points of the time evolution by quantum jumps.
\subsection{About the construction of $T$-matrices}
$iT$ matrices are hermitian nilponent matrices and it
not at all clear whether this kind of matrices exist
at all. Certainly they do not exist in real context.
It is quite easy to construct p-adic
vectors having vanishing length squared.
Possible problems are related to the
orthogonality requirement for the rows
of $T$.
\vm
It is easy to check
that nilpotent hermitian p-adic valued
$2\times 2$ matrices exist. Assume that
$p ~ mod ~ 4 =3$ so that $i=\sqrt{-1}$ is not ordinary
p-adic number. The most
general form of this matrix is
\begin{eqnarray}
iT&=& \left(\begin{array}{lr}
a & b\\
\bar{b} &-a\\
\end{array} \right)\per , \nonumber\\
b&=& b_1+ib_2\per , \nonumber\\
a&=& \sqrt{-\vert b\vert^2}= \sqrt{-b_1^2-b_2^2}\per .
\end{eqnarray}
\noindent By hermiticity $a$ must be "p-adically real"
number. This is indeed possible
in p-adic context but both $b_1$ and $b_2$ must be
obviously nonvanishing.
\vm
One can construct infinite
number of $2N\times 2N$-dimensional matrices $iT$
as direct sums $iT_1\oplus iT_2\oplus ...$ and tensor products
$iT_1\otimes iT_2\otimes...$ of two-dimensional $iT$-matrices
and $iT$-matrices constructed from them.
$T=0$ is also acceptable $T$-matrix in $1\times 1$-dimensional
case and one can include this matrix to direct sum to
obtain $2N+1\times 2N+1$-dimensional T-matrices.
Clearly, the "category" constructed in this manner is closed with
respect to $\oplus$ and $\otimes$ operations.
This category is also closed under tensor multiplication
by Hermitian matrices since the tensor product of arbitrary Hermitian
matrix with Hermitian and nilpotent matrix is also Hermitian
and nilponent: Hermitian nilpotency is infectuous disease!
More concretely, by taking arbitrary Hermitian matrix
and multiplying its elements with Hermitian and nilponent matrix
one obtains new Hermitian and nilponent matrix.
Also the sum of commuting
Hermitian and nilpotent matrices has same properties.
It should be noticed that all possible T-matrices
form also a ``category'' in the proposed sense.
\vm
An interesting question is whether all $(2N+1)\times (2N+1)$
dimensional matrices are direct sums of $2N\times 2N$-dimensional
matrix and $1\times 1$ dimensional zero-matrix. The
study of 3-dimensional case suggests that this is indeed the
case.
In 3-dimensional case it seem that no $T$-matrices exist.
One can write the solution ansatz as
\begin{eqnarray}
iT&=&
\left(\begin{array}{lll}
a_1 & b_1 & b_2\\
\bar{b}_1 & a_2 & b_3\\
\bar{b}_2 &\bar{b}_3&a_3 \\
\end{array} \per , \right)\nonumber\\
\nonumber\\
b_i&=& b_{i1}+ib_{i2} \per , \per i=1,2,3\per ,\nonumber\\
a_1&=& \epsilon_1\sqrt{-\vert b_1\vert^2-\vert b_2\vert^2 }\per
,\nonumber\\
a_2&=& \epsilon_2\sqrt{-\vert b_1\vert^2-\vert b_3\vert^2 }\per
,\nonumber\\
a_3&=& \epsilon_3\sqrt{-\vert b_2\vert^2-\vert b_3\vert^2 }\per .
\end{eqnarray}
\noindent The constraint that $a_i$ are ``p-adically real
numbers'' is nontrivial.
There 6 unkowns $b_i$. The square roots defining
$a_i$ are unique only modulo sign factors $\epsilon_i$.
Formally
there are 6 orthogonality conditions which can
be written as
\begin{eqnarray}
a_1+a_2&=& -\frac{b_2\bar{b}_3}{b_1}\per ,\nonumber\\
a_1+a_3&=& -\frac{b_1b_3}{b_2}\per ,\nonumber\\
a_2+a_3&=& -\frac{b_1b_2}{b_3}\per .\nonumber\\
\label{eqs}
\end{eqnarray}
\noindent One one writes formally
$b_i$ in the form $b_i=x_i^{1/2} exp(i\phi_i)$,
where $x_i^{1/2}$ is taken to be real: exponential
factor is however not p-adic exponent function.
One finds that
the equations stating the vanishing of the inner products
give same condition for the phases $exp(i\phi_i)$
defined as
$$ exp(i\phi_i)\equiv \frac{b_i}{x_i^{1/2}}$$.
\noindent The equation reads as
\begin{eqnarray}
exp(i\phi_1)&=&exp(i\phi_2)exp(-i\phi_3)\per .
\end{eqnarray}
\noindent The equation has now number theoretic contents and it
is not at all obvious that solutions exist. Thus
the number of equations reduces to $4$.
\vm
Taking the squares of both sides of the equations
\ref{eqs} one
obtains equations for the moduli of the $x_i\equiv \vert b_i\vert^2$.
The squares of the equations \ref{eqs} give
\begin{eqnarray}
2a_1a_2x_1&=&x_2x_3+x_1(2x_1+x_2+x_3) \per ,\nonumber\\
2a_1a_3x_2&=&x_1x_3+x_2(2x_2+x_3+x_1) \per ,\nonumber\\
2a_2a_3x_3&=&x_1x_2+x_3(2x_3+x_1+x_2) \per .
\end{eqnarray}
\noindent These equations are clearly cyclically
symmetric.
\vm
By taking squares again
one obtains 3 equations, which are degree 4 homogenous
polynomials in variables $x_i$.
The three cyclically symmetric equations read
\begin{eqnarray}
P_i(x_1,x_2,x_3) &=& 4x_i^2(x_i +x_{i+1})(x_i+x_{i+2})\nonumber\\
&-&\left[ x_{i+1}x_{i+2} +x_i(2x_i+x_{i+1}+x_{i+2})\right]^2=0\per .
\nonumber\\
\end{eqnarray}
\noindent where one has $i+3\equiv i$.
$P_i$ is homogenous polynomial in its arguments
having the general form
$$P_i(x_1,x_2,x_3)= 4x_i^4+...+ x^2_{i+1}x^2_{i+2} \per .$$
\noindent
From a given solution (if it exists) one obtains
a new solution by multiplying it with a p-adic number allowing
p-adically real square root. This means that one can
scale $x_i$ simultaneously by a square of ''p-adically
real'' number.
One can fix the solution by fixing say $x_3$
to some arbitrarily chosen value. This means that one has
3 equations and 2 unkowns. This suggests that the three
polynomial equations do not allow any solutions.
This would mean that ``irreducible'' $3\times 3$-matrices $iT$ do
not exist. An interesting conjecture is that 2-dimensional
$T$-matrices are the only irreducible $T$-matrices
and hence together
with $1\times 1$-dimensional zero matrix
generate the category of all $T$-matrices. This would be
in line with the fact that fermionic oscillator
operators are used to construct Fock states.
\vm
To sum up, the conditions
$L_0(int)\vert m_1\rangle=0$ make sense only p-adically
and force the theory to be as close to free theory as
it can possibly be. An especially attractive feature is
the reduction of the construction of
the p-adic S-matrix to a generalized cohomology theory.
What is especially nice and perhaps of practical importance
is that allowed S-matrices form ``category'' with respect
to direct sum and direct product operations.
TGD based construction of S-matrix
could realize Wheeler's great dream that physics
could be reduced to the
almost trivial statement ''boundary has no boundary''!
Of course, one can regard
the success of the real unitarity as an objection against
p-adic approach and one must therefore keep mind
open for the weakening of the conditions.
\end{document}
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