Matti Pitkanen (matpitka@pcu.helsinki.fi)
Thu, 23 Sep 1999 19:00:21 +0300 (EET DST)
Hi all,
when answering to Hitoshi's questions related to the time evolution
operator U expressible in terms of Virasoro generator L_0, I realized
that I do not have detailed understanding of how time evolution
operator decomposes to a sum interaction terms and single
particle terms.
Below I represent a more rigorous description of what happens and
demonstrate that one indeed obtains stringy perturbation theory. The
formalism is actually a simple generalization of time dependent
perturbation theory and the only new thing is the replacement of V by
L_0(int) and free Hamiltonian H_0 by L_0^0, the 'free Virasoro generator'
plus some formal complications related to fact that one considers entire
universe(;-).
Best,
MP
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\newcommand{\vm}{\vspace{0.2cm}}
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%matti
\subsubsection{Philosophy behind the derivation
of the time evolution operator?}
{\it 1. What one should achieve?}
\vm
The task is to derive explicit form of Schr\"odinger equation
determining the time evolution operator $U$.
It is not difficult to guess what the result must be:
\begin{eqnarray}
i\frac{d}{dt}\Psi&=& H\Psi\per , \nonumber\\
H &\equiv& k\left[-P_T^2 - L_0(vib)-L_0(int)\right]\Psi\per .
\end{eqnarray}
\noindent Here $t$ is a time coordinate having purely group
theoretical meaning. $L_0(int)$ denotes the interaction
part of the Super Virasoro generator and $L_0(vib)$ denotes
the vibrational part of the "free" Virasoro generator
$$L_0^0=P^2-L_0(vib)= \sum_n p^2(n)-L^0(vib,n)\per .$$
\noindent To be honest, the presence of the term $-P_T^2-L_0(vib)$
($P_T^2$ denotes transversal momentum squared operator) in the
expression of $H$ is not quite obvious.
S-matrix becomes exponential of $H$ defining
perturbative expansion for S-matrix. This form of the evolution
equation is indeed precisely the one appearing in time-dependent
perturbation theory defining S-matrix.
The task is to determine
the value of the dimensional constant $k$, which
could be also an operator, and find precise
interpretation for the 'time coordinate' $t$.
\vm
Besides
the evolution equation also the usual Super Virasoro conditions
\begin{eqnarray}
\left[M_n^2 -L^0_0(n)\right]\Psi_0=0\per ,
\end{eqnarray}
\noindent
where $n$ labels the particles present in state,
must be satisfied for the initial states of the time evolution.
In intermediate states these conditions are not met since
virtual particles are off mass shell particles in general.
Note that also more
General Super Virasoro conditions not written here explicitely
must be satisfied by single particle states. It has
turned out to be surprisingly difficult to derive this result
from first principles. The argument below however lead to
a unique time evolution operator.
\vm
{\it 2. Poincare and Diff$^4$ invariance}
\vm
Virasoro generators contain mass squared operator.
Poincare invariance of S-matrix requires
that one must use Diff$^4$ invariant
momentum generators $p_k(a\rightarrow \infty)$
in the definition Super Virasoro generators and
of S-matrix. At the limit $a\rightarrow \infty$
the generators
of $Diff^4$ invariant Poincare algebra should obey standard commutation
relations and time development by $p_0$ should conserve
Poincare quantum numbers when
the time evolution associated with $U$ is
over the entire infinite interval $(-\infty,\infty)$.
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
{\it 3. Schr\"odinger equation from deformation of Diff$^4$ invariant
Poincare algebra?}
\vm
It is obvious that the time evolution should somehow result from
the Virasoro conditions $L_0(tot) \Psi=0$. Time derivative
should somehow result from the mass squared operators $p_kp^k$ associated
with various single particle states in $L_0(tot)$. Linearity
in time derivative is achieved if the time coordinate
associated with time evolution corresponds to
a coordinate transforming like lightcone coordinate
$x_+$ since mass squared operator has the form $p_kp^k= 2p_+p_--p_T^2$
in lightcone coordinates. This in turn suggests that
Schr\"odinger equation should result from the replacement
$$p_+(a\rightarrow \infty)\rightarrow
i\frac{d}{dt}$$
\noindent of the Diff$^4$ invariant momentum generator $p_+$
by a differential operator.
\vm
{\it 4. The existence of preferred coordinates make
direction of $p_+$ unique}
\vm
The selection of
preferred lightlike directions defined
by $p_+$ and $p_-$ in the algebra of Diff$^4$ invariant
momentum generators breaks manifest Poincare invariance.
There exists however preferred coordinates:
for given 3-surface $Y^3$ the Minkowski coordinates for which time axis
corresponds to the direction of the classical four-momentum and
one spatial axis correspond to the direction of
the classical spin,
are preferred coordinates $(x_+,x_-,x_T)$
and fixed up apart from $SO(2)$
rotation. If one allows Lorentz transformations acting nontrivially
in $(x_+,x_-)$-plane,
the set of nonequivalent
coordinate choices of this kind is parametrized by $SO(3,1)/SO(1,1)\times
SO(2)
=M^4_+$ (future lightcone!)
and each 3-surface corresponds to a unique point of this space.
Both the construction of configuration space geometry and
the mapping
of real quantum TGD to its p-adic counterpart rely crucially
on the selection of these preferred coordinates.
General consistency requires that
each quantum jump involves localization to
a set of 3-surfaces having same
preferred $M^4$ coordinates and hence same
directions of classical four-momentum
and spin. This choice implies preferred basis in the
space of Diff$^4$ invariant momentum
geneators $p_k(a\rightarrow \infty)$.
\subsubsection{Explicit derivation of the Schr\"odinger
equation}
Consider now how one could derive the Schr\"odinger
equation.
a) The unitary time evolution operator $U$ should obey Sch\"odinger
type evolution equations derivable from Virasoro condition
$$L_0 (tot)\Psi=0$$
\noindent for the configuration space spinor field describing entire
Universe.
\vm
b) $L_0(tot)$ must have decomposition into single particle Super
Virasoro generators $L_0(n)$
\begin{eqnarray}
L_0(tot)&=& \frac{K}{2}\left[ DD^{\dagger} +D^{\dagger}D\right]
- \sum_n L_0(n,vib) +L_0(int)\per ,\nonumber\\
D&=&\sum_nD_n\per ,\nonumber\\
D_n&=& p^k(n) \gamma_k(\epsilon,n)\per .
\end{eqnarray}
\noindent $L_0(int)$ describes
the interactions between spacetime sheets and defines the
informational time evolution.
$L_0(n,vib)$ is
the vibrational part of the free Super Virasoro generator associated
associated spacetime sheet labelled by $n$.
$D$ corresponds to the sum of the single particle Dirac operators $D_n$
and $\gamma_k(\epsilon, n)$ can correspond
to either Ramond or NS representation depending on
whether one has $\epsilon=0$ or $\epsilon=1/2$.
One can require that the gamma matrices
$\gamma_k(n)$ associated with different particles (spacetime
sheets) anticommute with each other. Anticommutators of $\gamma_k(n)$ are
either proportional identity matrix or a projector to the sector
labelled by $n$. The first option is suggested
by the approximate decomposition of the configuration space to
Cartesian product of single particle configuration spaces. One
can however question the idea that this decomposition occurs in cm
degrees of freedom. The proportionality of the nonvanishing
anticommutators
to projectors $P_n$ is favoured by the requirement that
time evolution operator gives rise to string perturbation theory
and only this option will be considered in the sequel.
\vm
d) The action of cm part of $L_0(tot)$ decomposes to a sum over
squares of single particle Dirac operators:
\begin{eqnarray}
L_0(tot)&=& \frac{K}{2}\sum_n\left[ D_nD_n^{\dagger}
+D_n^{\dagger}D_n\right]
- \sum_n L_0(n,vib) +L_0(int)\per , \nonumber\\
\frac{1}{2}\left[ D_nD_n^{\dagger} +D_n^{\dagger}D_n\right]
&=&p_k(n)p^k(n)P_n\per .
\end{eqnarray}
\noindent $P_n$ denotes the projector to a particular Virasoro sector,
which presumably corresponds to single spacetime sheet.
At the limit of a vanishing interaction term Schr\"odinger
equation must reduce to single
particle super Virasoro conditions.
\begin{eqnarray}
KM_n^2(n)- \sum_n L_0(n,vib)&=&0\per . \nonumber\\
\end{eqnarray}
\noindent In intermediate states these conditions are
not satisfied since
virtual particles are not in general on mass shell particles.
\vm
e) Virasoro conditions must somehow give rise to Schr\"odinger
evolution equation. To achieve this one must define precisely
the action of the
operators $p_k(n)$. S-matrix must be Poincare and General
Coordinate invariant and this suggests that one should define
the operators $p_k(n)$ as $Diff^4$ invariant Poincare
translation generators $p_k(n,a)$ at the limit $a\rightarrow \infty$ ($a$
denotes lightcone proper time). At
this limit Poincare invariance is not broken and $p_k(n)$
are expected to commute mutually. This guess is not
quite correct. In order to obtain Schr\"odinger equation
some of the generators $p_k$ must be proportional to time
derivative and this time derivative must appear
linearly in the operator $p_k(n)p^k(n)$. This requirement
fixes the scenario completely
\vm
f) The time parameter $t$ appearing in the Schr\"oedinger equation
for $U$ must correspond to a purely group theoretical parameter.
For given 3-surface one can identify almost unique
preferred lightcone coordinates
$(p_+,p_-,p_T=(p_x,p_y))$ and for given final state of quantum jump
every 3-surface corresponds to same choice of preferred coordinates.
Using these coordinates
in the algebra of translation generators, one can
express mass squared operator as
\begin{eqnarray}
p_kp^k&=& 2p_+p_- -p_T^2 \per .
\end{eqnarray}
\noindent Suppose not that $p_-$ and $p_T$ act as $Diff^4$ invariant
translation generators but that $p_+$ acts as sum of the $Diff^4$
invariant
translation generator and a derivative operator acting on
time parameter characterizing the Schr\"odinger evolution.
\begin{eqnarray}
p_-&=&p_-(a\rightarrow \infty) \per ,\nonumber\\
p_T&=&p_T(a\rightarrow \infty) \per ,\nonumber\\
p_+&=& i\frac{d}{dt} \per .
\end{eqnarray}
\noindent These conditions hold separately for all values
of $n$, although the label $n$ is not explicitely written
in the formulas above. The crucial step is the deformation
of the generator $p_+$ by adding the time derivative
and one should find deeper justification for this operation.
\vm
g) Poincare invariance allows to choose physical states
as eigenstates of the $Diff^4$ invariant momentum generators.
One can write $\Psi$ in the form
\begin{eqnarray}
\Psi&=& exp(iP_+t)\hat{\Psi}\per ,\nonumber\\
P_+&=&\sum_n p_+(n) P_n \per .
\end{eqnarray}
\noindent In this representation
the action of each mass squared operator decomposes to
\begin{eqnarray}
p_k(n)p^k(n)P_n\hat{\Psi}&=& \left[2p_+(n) p_-(n)-p_T^2(n)\right]P_n
\hat{\Psi}
\per ,
\end{eqnarray}
\noindent where $p_n^2$ does not satisfy mass shell condition
now.
\begin{eqnarray}
L_0(tot)\Psi&=& \left[L_0^0 + L_0(int)\right]\Psi =0 \per , \nonumber\\
L_0^0&=& \sum_n \left[2p_+(n)p_-(n) -p_T^2(n) -L_0^0(n)\right]P_n\per .
\end{eqnarray}
\noindent This gives the desired Schr\"odinger equation
\begin{eqnarray}
i\frac{d\Psi}{dt} &=& \frac{1}{2KP_-}\left[
p_T^2+L_0(vib)+L_0(int)\right]
\Psi \per ,\nonumber\\
\frac{1}{P_-}&=&\sum_n \frac{P_n}{p_-(n)} \per .
\end{eqnarray}
\noindent Thus the solution of the Sch\"odinger
equation leads to a unique time evolution operator.
\vm
h) One ends up with stringy perturbation theory by expressing
the Schr\"odinger equation in eigenstate basis for $p_+(n)$. In this
basis the time evolution equation can be transformed to the
form
\begin{eqnarray}
\left[\sum_n \left[p^2(n)-L_0(vib,n)\right] - L_0(int)\right]\Psi =0 \per
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