Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 24 Sep 1999 12:21:48 +0300 (EET DST)
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
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\newcommand{\vm}{\vspace{0.2cm}}
\newcommand{\vl}{\vspace{0.4cm}}
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%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
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