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

*Fri, 24 Sep 1999 12:21:48 +0300 (EET DST)*

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Dear All,

Thanks to Hitoshi's earlier questions, I got in fluctuating state

in my viewes about precise definition of the 'time evolution'

operator U.

Just when I thought that I had built a satisfactory picture (yesterday's

posting) for how Schrodinger equation emerges I realized that it is not

needed at all and that it only produces troubles with manifest Poincare

invariance!

One can start directly from manifestly Poincare invariant Super Virasoro

conditions and construct scattering solutions for them using precisely

the same horribly receipe as is used to construct scattering solutions of

Schr\"odinger equation. Only the general algebraic structure

guaranteing unitarity is borrowed

from scattering solutions of Schrodinger equation and there is now need

to assing explicit Schrodinger equation with the definition

of S-matrix. Formalism is manifestly covariant.

The action of U replaces solution of 'free' Super Virasoro conditions

with the corresponding scattering solution of real Virasoro conditions.

The definition of 'free' Super Virasoro is extremely elegant and

relies on decomposition of 3-surface to separate 3-surfaces.

Interaction terms result when one one expresses real super Virasoro

generators for X^3 as sums of free Super Virasoro generators for

surfaces X^3_n representing particles plus necessary interaction terms.

I hope that this is indeed the final formulation but might be wrong.

Criticism is wellcome.

Best,

MP

\documentstyle [10pt]{article}

\begin{document}

\newcommand{\vm}{\vspace{0.2cm}}

\newcommand{\vl}{\vspace{0.4cm}}

\newcommand{\per}{\hspace{.2cm}}

%matti

\subsubsection{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 be the condition giving rise to the

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 inputs.

a) Poincare and Diff$^4$ invariances.

b) Decomposition of the Virasoro generator $L_0$ of the entire

universe to sum of 'free' Super Virasoro generators 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 charactized by their own absolute minima

of K\"ahler action.

c) Representation of the solutions of the Virasoro

condition for $L_0$ 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 S-matrix although the

general structure of the solutions of the Virasoro conditions

is same as that associated with 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.

\vm

{\it 1. Poincare and Diff$^4$ invariance}

\vm

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 would be obtained.

\vm

At the limit $a \rightarrow \infty$ 3-surfaces

describing 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 $p_k(a\rightarrow \infty)$,

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 associated with

$X^3_n$ rather than with entire 3-surface $X^3$. This means

effective decomposition of the configuration space to

a Cartesian power 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 $L_0$ for

for entire Universe contains sum of mass squared operators for

$X^3_n$ plus 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.

\vm

{\it 2. Analogy with

time dependent perturbation theory for Schr\"odinger equation}

\vm

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-V+i\epsilon} \Psi \per .

\end{eqnarray}

\noindent Here $\epsilon$ is infinitesimally small quantity.

If $E$ simultaneously is an eigen energy for the solution

$\Psi_0$ of the free Hammilton $H_0$ and for the solution

$\Psi$ of the entire Hamiltonian, Schr\"odinger equation is indeed

satisfied and one can construct the entire solution perturbatively by

developing right hand side to a geometric series in powers of 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 find Schr\"odinger equation for 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.

\vm

{\it 3. Scattering solutions of Super Virasoro conditions}

\vm

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)\Psi= \left[L_0(free) + L_0(int)\right]\Psi=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]= P^2-L_0(vib)\per ,

\nonumber\\\

\begin{array}{ll}

P^2=\sum_n p^2(n)\per ; &L_0(vib)= \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}

\Psi &=&\Psi_0 + \frac{L_0(int)}{ L_0(free) - L_0(int)+i\epsilon}

\Psi \per .

\end{eqnarray}

\noindent

$\epsilon$ is infinitesimal parameter defining precisely the

momentum spacetime integrations in presence of propagator poles.

$\Psi_0$ is assumed to satisfy the Virasoro conditions of the 'free

theory' stating that all particles are on mass shell particles:

\begin{eqnarray}

L_0^0(n)\Psi_0&=&\left[p_n^2 -L_0(vib,n)\right]\Psi_0=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 basis of 'free' Virasoro

conditions.

\vm

The momentum operators $p_k(n)$ are generators of

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.

These conditions define Poincare invariant momentum conserving S-matrix

if $L_0(int)$ defines momentum conserving vertices.

$L_0(int)$ is defined uniquely by the decomposition of the

$L_0$ associated with the entire universe to a sum of $L_0$:s associated

with individual 3-surfaces $X^3_n$ regarded

as independent sub-universes plus interaction term.

\vm

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. Potential

difficulties are caused by the fact that normalization constants

can diverge: this is indeed what they do in quantum field field

theories typically. It might be that p-adic valued

S-matrix is the only manner to avoid this difficulty.

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.

\end{document}

Best,

MP

**Next message:**WDEshleman@aol.com: "[time 800] Orthogonal Words"**Previous message:**Matti Pitkanen: "[time 798] Informational time evolution U"**Next in thread:**Hitoshi Kitada: "[time 801] Re: [time 799] Still about construction of U"

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