**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Thu, 7 Oct 1999 16:06:01 +0300 (EET DST)*

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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}

**Next message:**Hitoshi Kitada: "[time 900] Re: [time 899] Virasoro conditions,renormalization groupinvariance,unitarity,cohomology"**Previous message:**Matti Pitkanen: "[time 898] Re: [time 896] V|m_1>=0 instead of VP|m_1>=0"**In reply to:**Hitoshi Kitada: "[time 897] Re: [time 896] V|m_1>=0 instead of VP|m_1>=0"**Next in thread:**Hitoshi Kitada: "[time 900] Re: [time 899] Virasoro conditions,renormalization groupinvariance,unitarity,cohomology"

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