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

*Thu, 23 Sep 1999 19:00:21 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 799] Still about construction of U"**Previous message:**Stephen P. King: "[time 797] Re: [time 796] Pratt's Paper"

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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\documentstyle [10pt]{article}

\begin{document}

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

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

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

%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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