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\begin{document}
\title{ Locality in Free String Field Theory}
\author{
J. Dimock\thanks{Research supported by NSF Grant PHY9722045}\\
Dept. of Mathematics \\
SUNY at Buffalo \\
Buffalo, NY 14214 }
\maketitle
\begin{abstract} Free string field operators are constructed for the open bosonic
string in the light cone gauge in any dimension.
These are naturally localized
by the center of mass coordinate. Relative to this localization
they are shown to have a causal commutator
provided there are no tachyons. For the critical string in d=26 the
result still holds if the tachyon
is suppressed. We also show a causal commutator
relative to the "string light cone".
\end{abstract}
\newpage
\section{Introduction}
We consider the general question of discovering the extent to
which string field theory is local.
The question for interacting strings is completely
unsettled, since the theory itself is unsettled.
One expects such a theory to have a quantized
gravitational field, hence a fluctuating metric and light
cone, and so the very formulation of the question is
likely to be problematic.
On the other hand, for free
strings in a fixed Minkowski metric
one can at least pose the question. In this paper we
find that under certain circumstances there are local string
field operators.
The circumstances are as follows. We consider
open bosonic strings in
the light cone gauge in any dimension spacetime. We assume
that there are no tachyons, or that they have been suppressed.
The field operator is a function $\Phi( X)$ of
parametrized strings $X= X^{\mu}(\sigma)$.
The result is that there is a vanishing commutator
\be [\Phi(X), \Phi (Y) ] = 0 \ee
whenever the center of mass coordinates are space-like
separated, i.e. whenever
\be \left ( \int_0^{ \pi} X(\sigma) d \si - \int_0^{ \pi} Y(\sigma) d\si
\right)^2 >0 \label{cmlc} \ee
The result is perhaps stronger than one might have
expected since (\ref{cmlc}) does not rule out
that a point on $X$ be timelike separated from a point on $Y$.
The result is not really new, but it seems worthwhile
to give a careful mathematical treatment. Indeed we
take this opportunity to give careful treatment of
the whole light cone gauge theory.
An even stronger result has been obtained by Martenic \cite{Mar93} and
Lowe \cite{Low94}. They find that the commutator vanishes
whenever
\be \int_0^{ \pi} \left( X(\sigma) - Y(\sigma) \right)^2 d\si
>0 \label{slc}\ee
a configuration defining the "string light cone".
We also
give a precise version of this result.
One can also state the result in terms of Fourier components
\be X^{\mu}(\si ) = x_0^{\mu} + \sum_{n=1}^{\infty}
\sqrt{2}\ x^{\mu}_n \cos n \si \ee
Then the field is a function $\Phi = \Phi(x_0,x_1,x_2,...) $ of these
components,
the region (\ref{cmlc}) is $ (x_0 - y_0)^2 >0 $ , and the
region (\ref{slc}) is
\be (x_0 - y_0)^2 + 2 \sum_{n=1}^{\infty} (x_{n} - y_{n} )^2 > 0 \ee
In the light cone gauge $(x_n-y_n)^2 \geq 0$ for $ n \geq 1 $
so the second region is larger.
The vanishing commutator for $(x_0-y_0)^2 >0$ embodies a
limitation on how fast string disturbances propagate through
spacetime. The vanishing commutator outside the string
light cone incorporates a limitation on
how fast the various modes can grow.
Throughout the paper we employ a canonical quantization
procedure to pass from classical systems to quantum systems.
This quantization procedure is meant only to motivate
the quantum theory, and should not be construed as a derivation of
the quantum theory. The justification of the quantum
theory should be its consistency with general principles,
its naturality,
and ultimately its success as a model for the real world.
Thus the quantization procedure is not subject to the same
standard of rigor as the development of the quantum theory on
its own terms. Nevertheless we
we will try to be careful.
The organization of the paper is as follows. We first discuss
quantization of particles in light cone coordinates and show
it is equivalent to quantization in standard coordinates.
Then we quantize point fields in light cone coordinates
and show that the result is equivalent to quantization
in standard coordinates. Next we discuss single string
quantization in the light cone gauge. Finally we
define string fields in the light cone gauge and
prove the locality results. Each step builds on the preceding
steps. In an appendix we discuss topics which supplement
the main development. These are: (1.) the characteristic
Cauchy problem for the Klein-Gordon equation, (2.)
canonical field
quantization in light cone coordinates, (3.) the
existence of the light cone gauge for strings.
\section{Point particles and fields }
\subsection{classical particles}
Start with $d$ dimensional Minkowski spacetime $(\bbR^d, \eta )$.
Points are $x = (x^0,x^1,...x^{d-1})$ or $x = (x^0, {\bx})$ and
$x^2 = x \cdot x = \eta_{\mu \nu }x^{\mu} x^{\nu}$ or $x^2
= -(x^0)^2
+ {\bx}^2$
The worldline
$x (\tau)$ of a particle of mass squared $\mu \geq 0$ obeys
$ d^2 x/ d \tau^2 = 0 $ with the constraint
$ (d x/ d \tau )^2 + \mu = 0 $.
This can be written in the form
\bea d x^{\mu}/d \tau &=& p^{\mu} \nn \\
d p^{\mu}/d \tau &=& 0 \label{point} \eea
and constraint becomes
\be p^2 + \mu = 0 \ee
The constraint says that the momentum $p$ lies on the mass shell.
We
restrict to forward directed curves satisfying $d x ^0/d \tau >0$.
These have positive energy $p^0 >0$.
The constraint $-(p^0)^2 + (\bp)^2 + \mu =0 $ is solved by
$p^0 = \om_{\mu} (\bp)
\equiv \sqrt {\bp ^2 + \mu}$.
To put it another way we work in terms of coordinates
${\bf p}= (p^1,...,p^d)$ for the mass shell.
Since $p^0$ is a constant, one can solve $ d x^{0}/d \tau = p^0 $
by $x^0 = p^0 \tau$. The remaining variables are $\bx, \bp$
and if we parametrize in terms of $x^0 $ instead of $\tau $ they
satisfy
\bea d x^{j}/ dx^0 &=& p^{j}/ \om ( \bf p ) \nn \\
d p^{j} / dx^0 &=& 0 \label{oldham} \eea
This is a Hamiltonian system with
Hamiltonian $\om_{\mu}(\bf p)$.
Now consider light cone coordinates on $\bbR^{d}$ given
by $(p^+,p^-, \tilde p)$
where
\bea p^{\pm} &=& \frac {1}{\sqrt 2}(p^0 \pm p^{d-1}) \\
\ \tilde p &=& (p^1, ...p^{d-2})
\eea
In these coordinates the inner product is
$p \cdot a = - p^{+}a^{-} - p^{-}a^{+}+ \tilde p \cdot \tilde a $.
The equations (\ref{point}) become
\bea d x^{\pm } /d \tau = p^{\pm} \ \ \ \ \ && \ \ \ \
d p^{\pm}/d \tau = 0 \nn \\
d x^k /d \tau = p^k \ \ \ \ \ \ && \ \ \ \ d p^k /d \tau = 0 \eea
for $ 1 \leq k \leq d-2$.
The mass shell condition is
\be
-2p^{+}p^{-} + \tilde p^2 +\mu = 0 \ee
Excluding the case $\mu=0, \tilde p = 0 $ we have $p^+ \neq 0$
and can solve the constraint by
$ p^- =(\tilde p^2 + \mu )/2p^+$.
Thus we take $p^+, \tilde p$ as coordinates for the mass shell.
Note that this includes both positive energy ($p^{\pm} >0$)
and negative energy ($p^\pm <0$).
Since $p^{+}$ is constant we may solve
$ d x^{+}/ d \tau = p^{+} $
by $x^+ = p^+ \tau$.
With $x^+,p^-$ determined
the remaining variables are $x^-, p^+, \tilde x, \tilde p $
and if we reparametrize in terms of $x^+$ they satisfy
\bea d x^{-}/ dx^+ = p^-/p^+ \ \ \ \ && \ \ \ \ \
d p^+ / dx^+ = 0\nn \\
d x^{k}/ dx^+ = p^k /p^+ \ \ \ \ \ && \ \ \ \ \
d p^{k} / d x^+ = 0 \label{newham} \eea
The $x^-$ equation can be written $ dx^-/dx^+ = - \pa p^-
/ \pa p^+ $ and the $x^k$ equations can be written
$ dx^k/dx^+ = \pa p^-
/ \pa p^k $.
Thus we have a Hamiltonian system with
canonical variables $(p^+,x^-),(x^k,p^k)$ and
Hamiltonian
\be h_{\mu} = p^- = \frac{\tilde p^2 + \mu}{2p^+},
\label{newhamop} \ee
\subsection{quantum particles}
The standard quantization of the single particle starts
with the Hamiltonian
system (\ref{oldham}). We look for symmetric operators $x^i, p^j$
with $1 \leq i,j \leq d-1$ so that
$[x_i, p_j] =i \de^{ij}$, form the Hamiltonian $\om_{\mu}(\bp)$ and
solve the dynamical equations by
$ x^i(x^0) = \exp(i\om_{\mu}x^0)x^i \exp(-i\om_{\mu}x^0)$,
etc.
Start with the Hilbert space $ L^2 ( \bbR^{d-1}, d \bp) $. Let
$p^j$ be multiplication by the $j^{th} $ coordinate
take $x^j =i \pa / \pa p^j$. Then
$\om_{\mu} (\bp)$ is also a multiplication operator.
Changing to a relativistic normalization we introduce the Hilbert space
\be {\cH}_{\mu} = L^2 ( \bbR^{d-1}, \frac{d \bp}{2\om_{\mu} (\bp )})
\label{oldhilbert}
\ee
The operator of multiplication by $(2\om_{\mu}(\bp))^{1/2}$ is unitary
from the old space to the new space, and we use it to transform
the operators. Then $p^j, \om_{\mu}(\bp)$ are unaffected and
$x^j$ becomes the
Newton - Wigner coordinate operator
\be x^j = i \frac {\pa }{ \pa p^j} - \frac {i p^j}{2(\bp^2 + \mu)} \ee
The choice of the Lorentz invariant measure in
(\ref{oldhilbert})
means we have a simple representation of
the Poincare group on $\cH_{\mu}$, which however we do not need.
We do take note of the representation of the translation
subgroup which is generated by the momenta and so with
$p^0 = \om_{\mu}(\bp)$ has the form
\be u(a) = \exp(-ip \cdot a ) = \exp (ip^0a^0 -i\bp \cdot \ba) \ee
Now consider quantization in light cone coordinates.
We seek operators $x^-, p^+ $, $x^k, p^k$
for $1 \leq k \leq d-2$ satisfying $[ p^+,x^- ] = i, [x^j,p^k] =
i \de^{jk}$,
then form the Hamiltonian $h_{\mu}$, and generate $x^+$
evolution by the operator $e^{-ih_{\mu}x^+}$.
For the Hilbert space start with $L^2( \bbR^{d-1}, dp^{+} d\tilde p)$,
take $p^+, p^k$ and $h_{\mu}$ to be multiplication operators and then take
$x^- = -i \pa / \pa p^+$ and $x^k = i\pa/ \pa p^k$.
Next restrict to positive energy by restricting $p^+$
to $\bbR^+ = (0, \infty)$. The Hilbert space is then
$L^2( \bbR^+ \times \bbR^{d-2}, dp^{+} d\tilde p)$ and the operators
are as before.
A word on domains is in order here. To form functions of
these operators they should be self-adjoint and not
just symmetric. This is no problem for the operators
on $L^2( \bbR^{d-1})$, they are essentially self-adjoint
on the Schwartz space $\cS ( \bbR^{d-1})$ of smooth rapidly
decreasing functions, and satisfy the commutation relations
there. However when we cut down to
$L^2( \bbR^+ \times \bbR^{d-2})$ there is a problem
with $x^- = -i \pa / \pa p^+$. It is not essentially
self-adjoint on its natural minimal domain
$C^{\infty}_0( \bbR^+ \times \bbR^{d-2})$
and one must specify boundary conditions at $p^+ =0$ to obtain a
self-adjoint extension. However there is no natural
choice, so we stick with the minimal symmetric operator.
Changing to a relativistic normalization we define
Hilbert space
\be \cH_{+ }
= L^2( \bbR^{+ } \times \bbR^{d-2}, \frac{ dp^{+} d\tilde p }{2 p^+}),
\label{newhilbert}
\ee
The operator of multiplication by $ (2p^+)^{1/2}$ is unitary from
the old Hilbert space to the new Hilbert space and we use it to
transform the operators. The operators $ p^+, p^k$ and $h_{\mu}$
are unaffected and the coordinate operators become
\bea x^- &=& - i \frac {\pa }{ \pa p^+} + \frac {i} { 2p^+} \nn \\
x^k &=& i \frac {\pa }{ \pa p^k} \label{xplus}
\eea
again with the minimal domain for $x^{-}$.
Spacetime translations are now written with $p^- = h_{\mu}$
\be u'(a) =
\exp ( i p^{-}a^{+}+ip^+ a^{-} - i \tilde p \cdot \tilde a )
\ee
Next we establish an equivalence between the
standard quantization and the light cone quantization.
Our two choices of coordinates are related by holding $\tilde p$
fixed and exchanging $p^+$ and $p^{d-1}$
The mappings, which are inverse to one another, are
\bea p^{d-1} &=& \frac {1} {\sqrt 2}(p^+ - \frac{\tilde p^2 + \mu}{2p^+} ) \nn \\
p^+ &=& \frac {1}{\sqrt 2}( \om_{\mu}(\tilde p , p^{d-1})+ p^{d-1})
\label{change} \eea
These are diffeomorphisms between $\bbR^+$ and $\bbR$ unless both
$\mu = 0$ and $\tilde p = 0$.
Correspondingly we change coordinates in functions with
an operator $v $ defined by
\be (v \psi) (\tilde p,p^{d-1}) = \psi \left (\frac {1}{\sqrt 2}(
\om_{\mu}(\tilde p,p^{d-1})
+ p^{d-1}), \tilde p \right) \ee
\bpr $v$ is a unitary operator
from $ \cH_{+}$ to $ \cH_{\mu} $
and satisfies
\be v \ u'(a) \ v^{-1} = u( a) \label{equiv} \ee
\epr
\pr
The operator is norm preserving since for $\psi \in \cH_+$
\bea \| v \psi \|^2 &=& \int |
\psi \left (\frac {1}{\sqrt 2}( \om_{\mu}(\tilde p,p^{d-1})
+ p^{d-1}), \tilde p \right) |^2 \frac{d \tilde p dp^{d-1}}{2\om_{\mu} (\tilde p,p^{d-1} )} \nn \\
&=&
\int \int_0^{\infty} |
\psi ( p^+, \tilde p ) |^2
\frac{dp^+ d \tilde p }{2p^+} \nn \\
&=& \| \psi \|^2 \eea
and hence determines a unitary operator . Here we have made the
change of variables $p^{d-1} \to p^+$ and used
$ \om \to (p^+ +(\tilde p ^2 + \mu)/2p^+)/\sqrt{2}$
and
\be \frac { \pa p^{d-1} }{ \pa p^+}
= \frac {1} {\sqrt 2}(1 + \frac{\tilde p^2 + \mu}{2(p^+)^2} )
\ee
The identity (\ref{equiv}) is easily checked.
\bigskip
\re The theorem says that the operator $v$ induces the expected transformations
on the momenta.
We do not claim that $v$ connects the coordinates.
\subsection{quantum fields}
First some notation.
For any Hilbert space $\cH$,
let $\cH_n$ be the n-fold symmetric tensor
product, and let $\cF(\cH) = \oplus_{n=0}^{\infty} \cH_n$ be the
Fock space. Let
$ a^* (g),a(g)$ be the usual creation and annihilation operators
on the Fock space, respectively linear and anti-linear in $g \in \cH$,
and adjoint to each other. As a domain we usually take states
with a finite number of entries denoted $\cF_0(\cH)$. Any
unitary operator $U$ on $\cH$ induces a unitary $\G(U)$ on
$\cF(\cH)$.
Then $\G(U) a^*(g) \G(U)^{-1} = a^* (Ug)$ and
$\G(U) a(g) \G(U)^{-1} = a (Ug)$
For a standard multiparticle theory take the single particle Hilbert
space $\cH_{\mu}$, and form the Fock space $\cF(\cH_{\mu})$.
The single particle time evolution $e^{-i\om_{\mu} x^0}$ becomes
for many particles
$\G(e^{-i\om_{\mu} x^0})$. More generally the single
particle spacetime translation operator $u(a)$ becomes
for many particles
\be U(a) \equiv \G( u(a)) \ee
The single particle wavefunction $e^{-i\om x^0}\psi$ , after Fourier
transformation satisfies the Klein Gordon equation
$(\square - \mu ) \psi =0$ where $\square= \eta^{\mu \nu} \pa_{\mu} \pa_{\nu}$.
Correspondingly for quantum
fields we want solutions $\phi$ of the Klein-Gordon equation
$(\square - \mu ) \phi =0$ which we write in Hamiltonian form as
\bea \pa \phi / \pa x^0 &=& \pi \nn \\
\pa \pi / \pa x^0 &=& -(- \De + \mu) \phi
\eea
We seek operator-valued solutions which satisfy the canonical
commutation relations at equal times , and for which time evolution is
unitarily implemented with positive energy.
Regard the field as a distribution
in space. For real $ g \in C^{\infty }_0(\bbR^{d-1})$ we define
$ \phi( x^0 ,g )= \int \phi ( x^0, \bx ) g(\bx)
d \bx$ on
the multiparticle Hilbert space $\cF(\cH_{\mu})$ by
\bea \phi ( x^0,g)
&=&
a^*( e^{i\om_{\mu} x^0} \tilde g ) + a (e^{i\om_{\mu} x^0} \tilde g ) \nn \\
\pi ( x^0,g)
&=&
ia^* (\om_{\mu} e^{i\om_{\mu} x^0} \tilde g )
-i a ( \om_{\mu} e^{i\om_{\mu} x^0} \tilde g ) \label{qfields}
\eea
where $\tilde g$ is the Fourier transform.
Smearing in space and time, for real $ f \in C^{\infty }_0(\bbR^{d})$ we define
$ \phi(f)= \int \phi (x ) f(x)dx$ by
\be \phi( f) = a(\Pi_{\mu} f) + a^*(\Pi_{\mu} f) \ee
where $\Pi_{\mu}$ is restriction
of the Fourier transform $\tilde f$ to the mass shell:
\bea (\Pi_{\mu} f)( \bp ) &=&
(2 \pi )^{-(d-1)/2} \int e^{i \om_{\mu}( \bp ) x^0 -i \bp \cdot \bx}
f(x^0, \bx) dx^0 d\bx \nn \\
&=& \sqrt{2\pi} \tilde f ( -\om_{\mu} (\bp), \bp) \label{restrict} \eea
The field $\phi(f)$ is well-defined since $\Pi_{\mu}: C^{\infty }_0 \to \cH_{\mu}$.
Also $u(a) \Pi_{\mu} f = \Pi_{\mu} (f( \cdot - a))$
which implies the translation covariance
\be U(a)\phi(f) U(a)^{-1} = \phi (f ( \cdot - a)) \ee
To exhibit the locality of the fields we introduce the
retarded and advanced fundamental solutions $E_{\mu}^{\pm}$ for $\square - \mu$.
For $f \in C^{\infty}_0(\bbR^d)$ we define
$u= E^{\pm} f$ to be the solution of $(\square - \mu)u=f$
which vanishes in the distant past/future. Explicitly
\be (E_{\mu}^{\pm}f) (x) = \frac{-1}{ (2\pi )^{d/2}}
\int_{\G_{\pm} \times \bbR^{d-1}}
\frac {e^{ip \cdot x}}{p^2 +\mu} \tilde f(p) dp \label{prop} \ee
The $p^0$ contour $\G_{\pm}$ is a positive/negative
imaginary translation of the real axis. The expression is
independent of the contour since $\tilde f(p)$ is entire and
rapidly decreasing in real directions. The support of $E_{\mu}^{\pm}f$
is contained in the future/past of the support of $f$ .
Now we compute the commutator. With $E_{\mu}= E_{\mu}^{+} - E_{\mu}^{-}$
we have
\be [ \phi(f_1), \phi (f_2) ] =
2i Im (\Pi_{\mu}f_1, \Pi_{\mu}f_2)
= i \ee
where $ = \int f_1(x) f_2(x) dx$. The last step follows by
working backwards and identifying the $p^0$ integral as an integrals
around $p^0 = \pm \om_{\mu}(\bp) $.
Because of the support properties of $E^{\pm}$ the commutator
vanishes if $f,g$ have spacelike separated supports.
\bigskip
Now we turn to fields in light cone coordinates.
Let
$\square' =-2 \pa_{+}\pa_{-} + \De$ be the D'Alembertian in light cone coordinates.
The single
particle wave function $e^{-ih_{\mu} x^+} \psi$ after Fourier transformation
satisfies $(\square' - \mu) \psi = 0$. Correspondingly we look for
a quantum field operator $\phi'(x^+,x^-, \tilde x)$
satisfying $(\square ' - \mu ) \phi' =0$
and also some commutation relations.
The canonical commutation relations
in terms of
\be \phi' (x^+,g) = \int \phi'(x^+,x^-, \tilde x)g(x^-,\tilde x)
dx^- d \tilde x \ee
are
\be [ \phi'(0,\frac {\pa g_1}{\pa x^-}), \phi'(0, \frac {\pa g_2}{\pa x^-})]=
\frac { i}{2} \int g_1(x^-, \tilde x ) \frac {\pa g_2}{\pa x^-}(x^-, \tilde x )
dx^{-} d\tilde x \label{lccom} \ee
We explain in Appendix \ref{A2} why this is the appropriate choice.
To fulfill these conditions we form the multiparticle light cone
Hilbert space $\cF(\cH_+) $, and define on this space for
real $g \in C^{\infty}_0(\bbR^{d-1})$:
\be \phi' (x^+,g)
=
a^*( e^{ih_{\mu}x^+} g^\# ) + a(e^{ih_{\mu}x^+} g^\# )
\label{lcfield}
\ee
Here $g^\#$ is defined on $\bbR^+ \times \bbR^{d-2}$ by
\be g^\#(p^+, \tilde p ) = \tilde g(-p^+, \tilde p ) \ee
and $\tilde g(p^+, \tilde p )$ is the Fourier transform of $g(x^-, \tilde x)$.
\bpr The field $\phi' (x^+,g)$ is well defined
for real $g \in C^{\infty}_0(\bbR^{d-1})$ provided
$\tilde g(0, \tilde p)=0$.
It satisfies the field equation $(\square ' - \mu ) \phi' =0$
in the sense of distributions.
The fields $ \phi'(0,\pa g /\pa x^-)$ are well-defined and
have the commutator (\ref{lccom}).
\epr
\pr The function $\tilde g$ is in Schwartz space and if $\tilde g(0, \tilde p) =0$
it satisfies $|\tilde g (p^+, \tilde p)| = \cO(|p^+|)$ near the origin.
These facts are sufficient to
guarantee $ g^\# \in \cH_{+}$. Thus $\phi'(x^+,g)$ is well defined
for $\tilde g(0, \tilde p) =0$ and
$\phi' (x^+, \pa g/ \pa x^-)$ is well defined for any $g$.
The field equation is the statement that
\be 2\frac {\pa }{ \pa x^+}\phi' (x^+, \frac {\pa g}{ \pa x^{-}})
+ \phi' (x^+, (\De-\mu) g) =0 \ee
This is easily checked.
For the commutator we have
\bea [ \phi'(0,\frac {\pa g_1}{\pa x^-}), \phi'(0, \frac {\pa g_2}{\pa x^-})]
&=& (p^+ g_1^\#,p^+ g_2^\#)_{\cH_+} - (p^+ g_2^\#,p^+ g_1^\#)_{\cH_+} \nn \\
&=& - \frac {1}{2} \int \overline{ \tilde g_1(p^+,\tilde p ) }
p^+ \tilde g_2 (p^+,\tilde p) dp^+ d \tilde p \nn \\
&=& \frac { i}{2} \int g_1 \frac {\pa g_2}{\pa x^-}
dx^- d\tilde x \eea
In the second step we have used
$g^\#(p^+,\tilde p) = \tilde g (- p^+, \tilde p) = \overline{ \tilde g(p^+, -\tilde p)}$
to combine two integrals over $p^+ > 0$ into an integral over the whole line.
This completes the proof.
\bigskip
Now we
smear in all the coordinates and consider
\[\phi' (f) =
\int \phi'(x^+, x^-, \tilde x)f(x^+,x^-, \tilde x)dx^+ dx^- d \tilde x \]
The expression (\ref{lcfield}) becomes the operator on $\cF(\cH_+) $
\be \phi' (f) = a^*( \cP_{\mu} f) + a(\cP_{\mu} f) \ee
where $\cP_{\mu}$ is the restriction operator in the new
coordinates:
\be (\cP_{\mu} f ) ( p^+,\tilde p) = \sqrt{ 2 \pi }
\tilde f (- \ \frac{\tilde p^2 + \mu}{2p^+},- p^+, \tilde p) \label{pf}
\ee
\bthm For any real $f \in C^{\infty} _ 0(\bbR^{d})$:
\begin{enumerate}
\item $\phi' (f)$ is well defined for $d \geq 3, \mu \geq 0$
(and $d = 2$ if $\mu >0$).
\item $\phi' (f)$ satisfies the field equation
$\phi' ((\square ' - \mu )f)=0$
\item With spacetime translations represented by $ U'(a) = \G (u'(a))$
we have the covariance
\be U'(a) \phi'(f) U'(a)^{-1} = \phi' (f ( \cdot - a)) \ee
\item The commutator $ [\phi'(f_1), \phi'(f_2) ]$
vanishes if $f_1,f_2 $ have space-like separated supports.
\end{enumerate}
\ethm
\pr
To show the field is well-defined we need to know that
$\cP_{\mu}f$ is in $\cH_{+}$.
Since $\tilde f$ is in Schwartz space we have
that for any
$N$ there is a Schwartz norm $ \| f \|_{\cS} $
and a constant $C$ such that
\be |\tilde f(p^-,p^+, \tilde p) | \leq
C \| f \|_{\cS} ( 1 + |p^-| + |p^+| + | \tilde p |)^{-N}
\label{schwartz}
\ee We can assume that the norm has the form
the form
\be \| f \|^2_{\cS} = \sum_{\al, \beta} \int
|(x^{\al} \pa ^{\beta} f) (x)|^2 dx\ee
with a finite sum over multi-indexes $\al, \beta$.
For $|p^+| \geq 1$ we drop the $|p^-|$ and get
\be | (\cP_{\mu } f) (p^+, \tilde p ) | \leq
C \| f \|_{\cS} ( 1 + |p^+| + | \tilde p |)^{-N} \ee
For $|p^+| \leq 1$ first bound $\cP_{\mu}f$
by $(1+|p^-|)^{-1/8}(1 + |\tilde p |)^{-N+1}$ and then
use $|p^-| \geq |\tilde p|^2/2| p^+ |$ to get
\be | (\cP_{\mu } f) (p^+, \tilde p ) | \leq
C \| f \|_{\cS} |p^{+}|^{1/8} |\tilde p|^{-1/4}
(1 + |\tilde p |)^{-N+1} \ee
Combining these we have for a constant $C$ independent of $\mu \geq 0$
\be
\int \int_0^{\infty}|( \cP_{\mu } f ) (p^+, \tilde p) |^2
\frac{ dp^{+} d\tilde p }{2 p^+}
\leq C \| f \|^2_{\cS} < \infty \label{pmubd} \ee
which shows $\cP_{\mu}f$ is in $\cH_{+}$. (If $d=2$, then $\tilde p$
does not exist, and for a lower bound on $p^-$
we use $|p^-| \geq \mu/2|p^+|$. ).
The field equation follows from $ \cP_{\mu} ((\square ' - \mu )f) = 0$.
The covariance follows from $u'(a)\cP_{\mu}f = \cP_{\mu}( f ( \cdot - a)) $.
The commutator is evaluated by returning to standard coordinates.
Let
\be \hat f (x) = f( \frac{x^0 +x^{d-1}}{ \sqrt 2},\frac {x^0 -x^{d-1}}{ \sqrt 2},
\tilde x ) \label{fhat} \ee
be the test function in standard
coordinates.
One can check the identity
\be v\ \cP_{\mu}f = \Pi _{\mu} \hat f \label{id} \ee
Since $v$ is unitary the
commutator is evaluated as
\bea [\phi'(f_1), \phi'(f_2) ] &=&2i Im (\cP_{\mu}f_1, \cP_{\mu} f_2 )_{\cH_+} \nn \\
&=&2 i Im (\Pi_{\mu} \hat f_1, \Pi_{\mu} \hat f_2 )_{\cH_{\mu}} \nn \\
&=& i< \hat f_1, E_{\mu} \hat f_2 > \label{key} \eea
If $-2(x^+-y^+)(x^--y^-) + (\tilde x - \tilde y )^2 >0$ for
all $x \in supp f_1,
y\in suppf_2$, then $ -(x^0-y^0)^2 + (\bx -\by)^2 >0$ for all
$ x \in supp \hat f_1,
y \in supp \hat f_2$ and the commutator vanishes by the standard
result.
This completes the proof.
\bigskip
The next result shows the
complete equivalence of the light cone field and the standard
field.
\bthm Let $V: \cF ( \cH_{+} ) \to \cF ( \cH_{\mu})$
be the unitary operator $V = \G( v) $. Then
\bea V \phi'(f) V^{-1} & = & \phi( \hat f) \nn \\
V U'(a) V^{-1} & = & U( a) \eea
\ethm
\pr The first identity follows from (\ref{id})
and the second identity follows from (\ref{equiv}).
\section {Strings}
Our discussion of strings is more or less standard, see
\cite{GGRT73}, \cite{GSW87}, \cite{Hat92}.
We start with a discussion of the classical
free relativistic string. This is described by a Hamiltonian system
and some
constraints. We use the constraints to eliminate redundant variables and
obtain a reduced Hamiltonian system with independent variables.
This is then quantized.
\subsection{classical strings} \label{classical strings}
A single string is specified
by the world sheet $X: [0, \pi] \times \bbR \to \bbR^d$
which is forward directed in the sense that $\pa X^0/ \pa \tau
>0$. In the conformal gauge the string satisfies the wave
equation
\be
( \frac {\pa^2}{\pa \tau^2 }- \frac {\pa^2}{\pa \sigma^2 })X^{\mu} ( \tau,
\si) =0 \ee
and the constraint
\be ( \frac { \pa X}{\pa \tau } \pm \frac { \pa X}{\pa \sigma })^2 = 0 \ee
The constraint need only be satisfied at $\tau = 0$; it
is preserved by time evolution.
For the open string we also require Neumann boundary conditions
\be \frac { \pa X}{ \pa \sigma} (\tau, 0) =
\frac { \pa X}{ \pa \sigma} (\tau, \pi) =0 \ee
We write the equation as a first order system
\bea \pa X^{\mu}/ \pa \tau & = & \Pi^{\mu} \nn \\
\pa \Pi^{\mu}/ \pa \tau & = &
\frac {\pa^2 X^{\mu}}{ \pa \sigma ^2 } \label{first} \eea
with the constraint
\be (\Pi \pm \frac { \pa X}{\pa \sigma })^2 = 0
\label{second}\ee
We are especially interested in the center of mass
variables
\bea x^{\mu}(\tau) &=&
\frac {1}{\pi} \int_0^{\pi} X^{\mu} (\tau ,\si) d\si \nn \\
p^{\mu}(\tau) &=&
\frac {1}{\pi} \int_0^{\pi} \Pi^{\mu} (\tau ,\si) d\si \eea
which satisfy $d x^{\mu} / d \tau =p^{\mu}$ and
$d p^{\mu} / d \tau = 0$.
Our system of equations (\ref{first}), (\ref{second})
is invariant under conformal
diffeomorphisms on two dimensional Minkowski space.
For a generic class of solutions, which entail $p^+ >0$,
we can use this fact
to reparametrize so that:
\be X^+(\tau,\si) = p^+ \tau \label{lcg} \ee
a condition which defines the light cone gauge. We discuss
this step in detail in Appendix \ref{A.3}. Hereafter
we assume that (\ref{lcg}) holds, and hence also that
$\Pi^+(\tau, \si) = p^+$
Now the constraint says
\be 2 p^+ ( \Pi^- \pm \frac { \pa X^-}{\pa \sigma }
) = \sum_{k=1}^{d-2} ( \Pi^k \pm \frac { \pa X^k}{\pa \sigma })^2\ee
or equivalently
\bea
2 p^+ \Pi^{-} &=& \sum_{k=1}^{d-2} ( \Pi^k)^2 +
(\frac { \pa X^k}{\pa \sigma })^2 \nn \\
2 p^+ \frac { \pa X^-}{\pa \sigma }
&=& 2 \sum_{k=1}^{d-2} \Pi^k \frac { \pa X^k}{\pa \sigma }
\eea
Next we expand $X^{\mu},\Pi^{\mu}$ in eigenfunctions of
$(- \pa^2 / \pa \si^2)$
with Neumann boundary conditions, i.e. in a cosine series. We
have
\bea X^{\mu}(\tau,\si ) &=& x^{\mu}(\tau) + \sum_{n=1}^{\infty}
\sqrt{2} x^{\mu}_n (\tau) \cos n \si \nn \\
\Pi^{\mu}(\tau, \si ) &=& p^{\mu}(\tau) + \sum_{n=1}^{\infty}
\sqrt{2} p^{\mu}_n(\tau) \cos n \si \label{string} \eea
where
\bea
x_n^{\mu}(\tau) &=& \frac{\sqrt{2}}{\pi} \int_0^{ \pi} X^{\mu} (\tau, \sigma )
\cos n \si d \sigma \nn \\
p_n^{\mu}(\tau) &=& \frac{\sqrt{2}}{\pi} \int_0^{ \pi} \Pi^{\mu} (\tau, \sigma )
\cos n \si d \sigma \eea
The gauge condition says that $x^+ = p^+ \tau$ and
that $ x^+_n = 0$ and hence $ p^+_{n} = d x^+_n/ d \tau = 0$.
The constraints express $p^-, p_n^-, x_n^-$ as functions
of $p^+,\tilde p, \tilde p_n, \tilde x_n$ where $\tilde p = (p^1,...,p^{d-2})$,
etc. In particular we have
\be p^- =
\frac {1} { 2 p^+ }( \tilde p^2 +
\sum_{n=1}^{\infty} (\tilde p_n^2 + n^2 \tilde x_n^2 ))
\label{pm} \ee
The remaining variables are
$x^-, p^+, \tilde x, \tilde p, \tilde x_n, \tilde p_n $
and they satisfy
\bea d x^{-}/ d \tau = p^{-} &\ \ \ \ & d p^+ /d \tau = 0 \nn \\
d x^{k}/ d \tau = p^{k} &\ \ \ \ & d p^k /d \tau = 0 \nn \\
d x^{k}_n/d \tau = p_n^k & \ \ \ \ &
d p_n^{k} /d \tau = - n^2 x^k_n
\label{taueqn} \eea
This is not a Hamiltonian system, but it is if we drop
$(p^+,x^-)$ and take a Hamiltonian
\be \hat H =
p^+p^- = \frac{1}{2}( \tilde p^2 +
\sum_{n=1}^{\infty} (\tilde p_n^2 + n^2 \tilde x_n^2 )) \ee
Now
changing variables from $\tau $ to $x^+ = p^+ \tau $
we get
\bea d x^{-}/ d x^+ = p^{-}/p^+ &\ \ \ \ & d p^+ /d x^+ = 0 \nn \\
d x^{k}/ d x^+ = p^{k}/p^+ &\ \ \ \ & d p^k /d x^+ = 0 \nn \\
d x^{k}_{\ell}/d x^+ = p_{n}^k/p^+ & \ \ \ \ &
d p_{n}^{k} /d x^+ = (-n^2/p^+) x^k_{n}
\label{xeqn} \eea
This is a Hamiltonian system with canonical variables
$(p^+,x^-), (x^k,p^k), (x^k_n,p^k_n)$ and Hamiltonian
$H = p^{-}$ given by (\ref{pm}).
\subsection{quantum strings}
Now we quantize the above system.
We seek operators
$x^-, p^+, x^k, p^k, x^k_{n} , p^k_{n} $ for $1 \leq k \leq d-2,\ 1 \leq n$
which at equal times satisfy the canonical commutation relations:
\footnote { Since we are considering a reduced Hamiltonian system
one might think that quantization should be based on Dirac brackets
rather than Poisson brackets as above. These turn out to be the
same \cite{HRT76}. }
\bea
[ p^+, x^- ] &=& i \nn \\
\ [x^j , p^k] &=& i \de_{jk} \nn \\
\ [x^j_{n} , p^k_{m}] &=& i \de_{jk} \de _{nm}
\eea
Furthermore they should satisfy the dynamical equations (\ref{xeqn}) and
$x^+$-evolution should be implemented with
a Hamiltonian which has spectrum bounded below.
We start by considering only $ x^k, p^k, x^k_{n} , p^k_{n} $
and solving the equations (\ref{taueqn}).
Just as an $N$-component scalar field on $\bbR^{d}$ has a simple
definition on the Fock space over $L^2 (\bbR^{d}, \bbC^N)$
or its Fourier transform,
so the transverse modes of a string in $\bbR^d$
parametrized by $[0,\pi]$ have a
simple definition on
the Fock space over $L^2([0,\pi],\bbC^{d-2})$
or its cosine transform $\ell^2(\bbN, \bbC^{d-2})$.
This is the same as $\ell^2(\bbN \times (1,...,d-2) )$
and we call it simply $\ell^2$ so that
\be \ell^2 = \{ \{\psi^k_{n}\}: \sum_{k=1}^{d-2}
\sum_{n=1}^{\infty} |\psi ^k_{n}|^2 <
\infty \} \ee
We have excluded the zero modes, which are treated separately.
Thus the Hilbert space is not just the Fock space $\cF(\ell^2)$
but rather
\be L^2(\bbR^{d-2} ) \otimes \cF ( \ell^2 ) \ee
In this space let $p^k= p^k \otimes I$ be the multiplication operator
and let $x^k = i \pa / \pa p^k \otimes I$, just as for the single particle.
Also let $e^k_n$ be the orthonormal
basis in $\ell^2$ consisting of the characteristic functions
of the points $(n,k) $.
Let $a^k_n = I \otimes a(e^k_{n})$
and let $(a^k_n)^* = I \otimes (a(e^k_{n}))^*$
and define $x^k_n = ( 2n)^{-1/2} ((a^k_n)^* + a^k_n)$
and $p^k_n = i( n/2)^{1/2} ((a^k_n)^* - a^k_n)$.
These satisfy the commutation relations, and the solution to (\ref{taueqn})
with these operators at $\tau = 0$ is
\bea
x^{k} (\tau ) &=& x^k + p^k \tau \nn \\
p^{k} (\tau ) &=& p^k \nn \\
x^k_{n} (\tau ) &=&
(2n)^{-1/2} ( e^{in\tau}(a^k_{n})^*+
e^{ -in\tau}a^k_{n}) \nn \\
p^k_{n} (\tau ) &=&
i (n/2)^{1/2} ( e^{in\tau}(a^k_{n})^*-
e^{ -in\tau}a^k_{n})
\label{dfn} \eea
We can recover the full string operator $X^k(\tau, \si)$
as in (\ref{string}). In the literature these
formulas are usually written in terms of $\al^k_n = -i \sqrt{n} a^k_n$
and $\al^k_{-n} = i \sqrt{n} (a^k_n)^*$.
Now $\tau$-evolution is written $x_n^k(\tau) = e^{i\hat H \tau}
x_n^k e^{-i\hat H \tau}$, etc. where the evolution operator is
$ e^{-i\hat H \tau}= e^{-iat} ( e^{-i(\tilde p)^2t/2} \otimes \G( e^{-int} )) $
for any constant $a$. The generator is
\be \hat H = \frac {\tilde p ^2}{2} +N -a \ee
Here $N = I \otimes N$ is
defined by $N = d \G(n) \equiv i d/dt \ \G( e^{-int} )|_{t=0}$.
It is called the number operator, although the terminology is
different from field theory where $d \G(1)$
would be called the number operator.
It can also be written
\be N = \sum_{k,n} n (a^k_n )^*
(a^k_n )
= \frac {1} { 2 }
\sum_{k,n}: (p^k_{n})^2 + n^2 (x^k_{n})^2:
\ee
Thus the quantum $\hat H$ is obtained from the classical
$\hat H$ by Wick ordering and (possibly) adjusting by a
constant.
Now add the last pair of variables $p^+,x^-$. Taking our cue from the
particle quantization we tensor in
$ L^2 ( \bbR^+, dp^+/ 2 p^+ )
$ and define
\bea \cH & =& L^2 ( \bbR^+, \frac { dp^+} { 2 p^+} ) \otimes
L^2(\bbR^{d-2},d \tilde p ) \otimes \cF ( \ell^2 ) \nn \\
&=& L^2 ( \bbR^+\times \bbR^{d-2}, \frac { dp^+ d \tilde p} { 2 p^+} )
\otimes \cF(\ell^2) \nn \\
&=& L^2 ( \bbR^+\times \bbR^{d-2},
\cF(\ell^2), \frac { dp^+ d \tilde p} { 2 p^+} ) \label{Hilbert}
\eea
Here we have listed some naturally isomorphic Hilbert spaces.
The last is the square integrable $\cF(\ell^2)$-valued functions on
$ \bbR^+\times \bbR^{d-2}$.
Now let $p^+$ be the multiplication operator and $x^- =- i \pa / \pa p^+
+ i / 2p^+ $ just as in (\ref{xplus}).
Classically $p^-$ is $\hat H / p^+$, and we take this as the
quantum definition as well so
\be p^- = \frac{ \hat H }{ p^+} = \frac { {\tilde p} ^2 +2(N-a )} { 2 p^+}
\label{pminus} \ee
All these operators can be thought of as acting on
$C^{\infty}_0( \bbR^+\times \bbR^{d-2}, \cF_0(\ell^2))$, a
dense domain in $\cH$.
For $\tau $ evolution we take
$x^- ( \tau) = x^- + p^- \tau$ and $p^+ (\tau) = p^+$
and then (\ref{taueqn}) is solved.
Now we replace $\tau $ by $x^+/p^+$ to get a solution of
(\ref{xeqn}).
We have
\bea
x^- ( x^+) &=& x^- + (p^-/p^+)x^+ \nn \\
p^+ (x^+) &=& p^+ \nn \\
x^{k} (x^+ ) &=& x^k +(p^k/p^+) x^+ \nn \\
p^{k} (x^+ ) &=& p^k \nn \\
x^k_{n} (x^+ ) &=&
(2n)^{-1/2} ( e^{inx^+/p^+}(a^k_{n})^*+
e^{ -inx^+/p^+}a^k_{n}) \nn \\
p^k_{n} (x^+ ) &=&
i (n/2)^{1/2} ( e^{inx^+/p^+}(a^k_{n})^*-
e^{ -inx^+/p^+}a^k_{n})
\eea
The $x^+$-evolution is implemented by $e^{-i\hat H x^+/p^+}$
which is written $e^{-iHx^+}$ where
\be H = \frac{ \hat H }{ p^+} = p^- \ee
Spacetime translations only change the center of mass,
or to put it another way, the total momentum is the center
of mass momentum. So spacetime translations are
represented by the unitary operators
\be U(a) =
\exp ( i p^{-}a^{+} +ip^+ a^{-} - i \tilde p \cdot \tilde a )
\ee
with $p^-$ as above. The mass operator is
\be M^2 = 2p^+p^- -\tilde p^2 = 2( N - a ) \ee
and the spectrum is $\{ -2a, -2a+2,-2a+4,...\}$ with increasing
multiplicity.
Then there is the question of whether there is a unitary representation
of the Poincare group with the the translation subgroup as above.
Here there is a well-known anomaly in the commutator of
the generators of the Lorentz transformation. The anomaly
vanishes if and only if $d=26$ and $a=1$. One expects that
in this case the unitary representation of the Poincare group
exists, although there is apparently no rigorous proof. With $a=1$ the
spectrum starts at $-2$ which is the tachyon.
\subsection{string wave function}
The string wave function $\psi $ is in $ \cH =
L^2 ( \bbR^+ \times \bbR^{d-2}, \cF(\ell^2), dp^+ d \bp / 2 p^+ )$.
But the center of mass coordinates $x^k,x^{-}$
have a momentum space representation, and the
the mode coordinates $x^k_n$ have a harmonic oscillator representation.
We also want to change to a coordinate representation in which
these are multiplication operators.
First we take the inverse Fourier transform so that
$x^-,x^k$ are multiplication operators.
At the same time we pass to the Schrodinger picture and incorporate
the $x^+$ evolution. Thus we define
\be \psi (x^+,x^-, \tilde x) = (2\pi)^{-(d-1)/2} \int
\int_0^{\infty} e^{-ip^-x^+ -i p^+x^- +i\tilde p \tilde x}\psi (p^+, \tilde p )
\frac { dp^+ d \tilde p} {( 2 p^+)^{1/2}} \label{configwf} \ee
where $p^-$ is defined by (\ref{pminus}).
If we assume that
$ \psi \in C^{\infty}_0 ( \bbR^+ \times \bbR^{d-2},\cF_0( \ell^2))$,
then $ \psi (x^+,x^-, \tilde x)$ is a smooth
$\cF(\ell^2)$ valued function on $\bbR^d$. As such it satisfies
the equation
\be ( \square ' - 2(N-a) ) \psi =0 \label{anewkg} \ee
which generalizes the Klein Gordon equation.
On the other hand we can represent all the $x^k_n$ as
multiplication operators as follows. (See for example \cite{ReSi75}, p228).
Consider the Gaussian random process indexed by $(n,k)$ with
covariance $(2n)^{-1}$. This consists of
a probability measure space $(Q, \mu)$
and Gaussian random variables $x^k_n$ such that $\int x_n^k x_{n'}^{k'}d \mu
=(2n)^{-1}\de_{nn'}\de_{kk'}$. The space Q can be thought of as a space of
sequences $\{x_n^k\}$ and the functions $x_n^k$ as the coordinates.
There is a unitary operator $S$ from $L^2(Q, d \mu) $ to $\cF(\ell^2)$
such that $S \cdot 1 = \Om_0 =$ the Fock vacuum, and
$ S^{-1} x_n^k S = x_n^k$,
where on the left $x_n^k = (2n)^{-1/2} ( (a_n^k)^* + a_n^k) $
and on the right it is multiplication by $x_n^k$. Using $S$,
states and operators in $\cF(\ell^2) $ can be represented
in $L^2(Q, d\mu)$. The operators $p_n^k, N$ become differential
operators, about which more later.
If we make both changes at once the wave functions
are a functions from $\bbR^d $ to $L^2(Q, d \mu)$.
Hence they are functions on $ \bbR^d \times Q $ and
can be written
\be \psi ( x, \tilde x_1, \tilde x_2, ....) \ee
where $x=(x^+,x^-, \tilde x)$.
\section{String fields}
\subsection{center of mass localization}
First consider the multi-string quantum mechanics.
The single string Hilbert space is $ \cH =
L^2 ( \bbR^+ \times \bbR^{d-2}, \cF(\ell^2),
dp^+ d \tilde p / 2 p^+ ) $ and
$x^+$-evolutions is given by $e^{-iHx^+}$ where
$ H = (\tilde p ^2 +2(N -a))/2p^+ $. For many strings we
take the Fock space $\cF(\cH) $ and the evolution operator is
$\G(e^{-iHx^+})$. More generally $\cU (a) = \Gamma( U(a))$
gives a representation of the translation group on the Fock space.
We want to define a field operator on this Hilbert space,
and we proceed by analogy with point field theory.
The field operator should be a operator version of real-valued
wave
functions $ \psi(x)$ as in (\ref{configwf}). Real means
real under the isomorphism $\cF(\ell^2) \cong L^2(Q, d\mu) $.
The wave function can be regarded
as a $ \cF(\ell^2)$-valued distribution in the
sense that
\be = \int ( F(x), \psi (x) )_{\cF} \ dx \ee
is a continuous function on real-valued
$F \in C^{\infty}_0( \bbR^{d}, \cF_0(\ell^2))$.
Correspondingly we
ask for symmetric operators $\Phi(F)$ on $\cF(\cH) $ defined,
linear, and continuous in real $F \in C^{\infty}_0
( \bbR^{d}, \cF_0(\ell^2))$.
The field operator should satisfy the
same equation as the wave function, that is
$ ( \square ' -2( N-a) ) \Phi = 0 $
This should be understood in the sense of distributions, namely
\be \Phi ( ( \square ' - 2(N-a )) F) = 0 \label{newkg}
\ee
To fulfill these requirements follow (\ref{lcfield}) and
define for real $G \in
C^{\infty}_0 ( \bbR^{d-1}, \cF_0(\ell^2))$ an operator $ \Phi (x^+,G) $
on $\cF(\cH)$ by
\be \Phi (x^+,G) = a^*(e^{ i Hx^{+}} G^\#) +
a(e^{ i Hx^{+}} G^\#)
\ee where $G^\#(p^+, \tilde p) = \tilde G(-p^+, \tilde p) $.
Just as for point fields, this can be motivated by a
canonical quantization procedure for solutions of
$ ( \square ' -2( N-a) ) \Phi = 0 $.
Also as for point fields the operator is well defined under
certain restrictions on $G$, but we do not tarry on this point.
Now smearing in $x^{+}$ as well, we define
for real $F \in C^{\infty}_0 ( \bbR^{d}, \cF_0(\ell^2))$
\be \Phi (F) = a^*(\cP F) + a( \cP F) \ee
where
\be (\cP F) ( p^+,\tilde p) = \int
\exp ( i Hx^{+} )
\tilde F( x^+,-p^+, \tilde p) dx^+ \ee
and where $\tilde F$ denotes the Fourier transform in the last
$d-1$ variables. Just as for point fields we have
\bthm For real $F \in C^{\infty} _ 0(\bbR^{d}, \cF_0(\ell^2))$:
\begin{enumerate}
\item $\Phi (F)$ is well defined for $d \geq 3, a \leq 0$
(and $d = 2$ if $ a < 0$).
\item $\Phi (F)$ satisfies the field equation
$\Phi ((\square ' - 2(N-a) )F)=0$
\item
Under spacetime translations: $ \cU(a) \Phi(F) \cU(a)^{-1} =
\Phi (F ( \cdot - a)) $
\end{enumerate}
\ethm
\pr
For the proof it is convenient to choose an explicit orthonormal basis
for $\cF(\ell^2)$. The basis is the occupation number basis
and is indexed by sequences
of non-negative integers
$ \al =\{ \al_{n,k} \}$ with $1 \leq n, 1 \leq k
\leq d-2 $ and
finitely many non-zero
entries.
The basis elements are
\be \chi_{\al} = \prod_{n.k}
\frac{ [(a_n^k )^*]^{\al_{nk}}}{\sqrt{\al_{nk}!}} \Om_0 \ee
We have
\be N \chi_{\al} = (\sum_{nk} n \ \al_{nk})
\chi_{\al} \ee
For $\psi \in \cF(\ell^2)$ let $\psi_{\al}
= (\chi_{\al},\psi)$ be the components in these
basis elements. Similarly $ \cF(\ell^2)$-valued functions $F(x)$
have components
$F_{\al} (x) = (\chi_{\al}, F(x))$.
Now for $\cP F(p^+, \tilde p )$ we can compute the components as
\bea (\cP F)_{\al} (p^+, \tilde p) &=&
\sqrt{2 \pi} \tilde F_{\al}(- \frac{\tilde p^2 +
\mu (\al) }{2p^+},-p^+, \tilde p) \nn \\
&=& ( \cP_{\mu (\al)} F_{\al} )(p^+, \tilde p)
\eea
where
\be \mu (\al) = 2 ( (\sum_{nk} n \ \al_{nk}) - a) \ee
To show that $\Phi(F)$ is well defined we must show that
$\cP F$ is in $\cH$. To see this we expand $\cP F$ in the above
basis and use (\ref{pmubd}) noting that $a \leq 0$ implies $\mu(\al) \geq 0$
(and for $d=2$ that $a < 0$ implies $\mu(\al) >0$).
We obtain for the $\cH$ norm:
\bea & \ & \int \int_0^{\infty} \| (\cP F )(p^+,\tilde p) \|^2 \frac{ dp^{+} d\tilde p }{2 p^+} \nn \\
& = & \sum_{ \al}
\int \int_0^{\infty}|( \cP_{\mu (\al) } F_{\al}) (p^+, \tilde p) |^2 \frac{ dp^{+} d\tilde p }{2 p^+} \nn \\
& \leq & C \sum_{ \alpha} \| F_{\al} \|^2_{\cS} \nn \\
& = & C \| F \|^2_{\cS} < \infty \eea
Here in the last step we refer to the
$\cF(\ell^2)$-valued version of the Schwartz norm (\ref{schwartz})
and use $x^{\beta} (\pa ^{\gamma} F_{\al})(x)
=(x^{\beta} \pa ^{\gamma} F)_{\al}(x)$.
The field equation is satisfied since
$ \cP ( \square ' - 2(N-a )) F =0$ and
the covariance follows from $U(a) \cP F = \cP (F(\cdot-a ))$.
Each of these statements can be reduced to the point field
case by expanding in the basis.
This completes the proof.
\bigskip
Now we define advanced and retarded fundamental
solutions for the standard coordinate operator $ \square -2( N-a)$ .
As for the point field these are defined for
$F \in C^{\infty} _ 0(\bbR^{d}, \cF_0(\ell^2))$ by
\be ( E^{\pm}F) (x) = \frac {-1}
{(2 \pi)^{d/2}} \int_{\G_{\pm} \times \bbR^{d-1}}
\frac {e^{ipx }}{ p^2 + 2( N -a) } \tilde F (p) dp \ee
We have that
\be (E^{\pm} F)_{\al} = E^{\pm}_{ \mu (\al)} F_{\al} \ee
where $E^{\pm}_{\mu}$ is defined in (\ref{prop}). This shows that the support
of $E^{\pm} F$ is contained in the future/past of the
support of $F$. Define as before $E = E^+ - E^- $.
If $F_1, F_2 \in C^{\infty}_0 (\bbR^d, \cF_0( \ell^2))$ are test functions
in light cone coordinates , let $ \hat F_1, \hat F_2$
be the expression in standard coordinates as in (\ref{fhat}).
The main result is:
\bthm (Locality) For $F_1, F_2$ as above
\be [ \Phi (F_1), \Phi (F_2) ] = <\hat F_1, E \hat F_2> \ee
The commutator vanishes if $F_1,F_2$ have space-like separated
supports.
\ethm
\pr If $F$ is real, then $F_{\al} = (\chi_{\al}, F)$ is real
since in the $L^2(Q, d \mu)$ representation $F$ is real
by definition and the $\chi_{\al}$ correspond to Hermite
polynomials which are real.
Then by our earlier result (\ref{key})
\bea [ \Phi (F_1), \Phi (F_2) ] &=& 2 i Im( \cP F_1,\cP F_2)_{\cH} \nn \\
&=& \sum _{\al} 2i Im
( \cP_{\mu (\al )} F_{1\al}, \cP_{\mu (\al)}
F_{2\al})_{\cH^+} \nn \\
&=& \sum _{\al} <\hat F_{1\al} ,
E_{\mu (\al )} \hat F_{2\al }> \nn \\
&=& <\hat F_1,E \hat F_2>. \eea
If $supp \hat F_1$ and $supp \hat F_2$ are space-like separated
then $supp \hat F_{1\al}$ and $supp \hat F_{2\al}$ are space-like
separated (since $supp \hat F_{\al} \subset supp \hat F$) ,
hence
$ <\hat F_{1\al} ,
E_{\mu (\al )} \hat F_{2\al }>=0 $ and the commutator vanishes.
\bigskip
\res
\benum
\item The result still holds for the critical theory
$d=26, a =1 $ provided the tachyon is suppressed, i.e.
provided $F_{1,0} = F_{2,0} = 0$ . In this case $\al=0$ does not
contribute and $\mu(\al) \geq 0$ still holds.
If we combine the locality result with a treatment of
Lorentz covariance for $d=26, a=1$, it may be possible to show that the
field operators $\Phi(F)$ generate local $C^*$ algebras
satisfying all the Haag-Kastler axioms.
\item We expect that the locality result would hold for any free string
theory in the light cone gauge, i.e. closed strings , superstrings, etc.
For critical superstrings in d$=10$ the tachyon is absent from the start
and one would not have to remove it \cite{Low94}
There is some speculation about interacting strings in \cite{LSU94}.
\item
One can also formulate
the question in a covariant gauge. However the covariant
string field operator then contains certain projection operators
which are difficult to analyze, and so there is not
result as yet. See also \cite{GrHu93}
\eenum
\subsection{string light cone }
Now we go for the stronger result of \cite{Mar93}. In the
Hilbert space $\ell^2$ we consider the finite dimensional
subspaces $\ell^2_N$ which are spanned by the
vectors $e^k_n$ with $ 1 \leq k \leq d-2$ and $1 \leq n \leq N$.
Correspondingly the Fock space
$\cF(\ell^2 )$ has the subspace $ \cF(\ell^2_N )$.
We are going take our test functions (and hence fields)
to be $ \cF(\ell^2_N )$-valued for some arbitrary $N$. Since the
union of the spaces $ \cF(\ell^2_N )$ is dense in $ \cF(\ell^2 )$
we do not expect to miss anything important with this restriction
There is a unitary operator $S$ from $L^2 ( \bbR^{N(d-2)}, d\mu_N)$
to $ \cF(\ell^2_N )$ where $d \mu_N$
is the Gaussian measure
\be d \mu_N(\tilde x^1, ..., \tilde x^N) =
\prod_{n=1}^N (n/\pi)^{(d-2)/2} e^{- n\ \tilde x_n^2 }\ d \tilde x_n \ee
The operator is determined by its
action on polynomials which is
\be S \left( P( \tilde x_1, ..., \tilde x_N) \right)
= P( \tilde x_1, ..., \tilde x_N) \Om_0 \ee
where on the right $\tilde x_n$ is the Fock space operator.
This is the finite variable version of the previously defined $S$.
Under the inverse mapping operators on $\cF(\ell_N^2)$
transform as
\bea S^{-1} x_n^k\ S &=& x_n^k \nn \\
S^{-1} p_n^k \ S &=& - i \frac {\pa}{\pa x_n^k}
+i n x_n^k \nn \\
S^{-1} N \ S &=& L \eea
where we define the differential operator
\be L =\frac{1}{2} ( \sum _{k= 1}^{d-2} \sum_{n = 1}^N -
(\frac {\pa} { \pa x^k_{n}} )^ 2
+ 2n x_n^k \frac {\pa} { \pa x^k_{n}})
\ee
The result for the string light cone is:
\bthm Let $F_1,F_2 \in C^{\infty}_0 (\bbR^d, \cF (\ell_N^2))$
and suppose that the transforms $ S^{-1} \hat F_1$, $ S^{-1} \hat F_2$
are in $ C_0^{\infty}( \bbR^d \times \bbR^{N(d-2)}) $
and have spacelike separated supports with respect to the
Minkowski metric on $\bbR^d \times \bbR^{N(d-2)}$ . Then
\be [ \Phi (F_1), \Phi (F_2) ] = 0
\ee
\ethm
\bigskip
\pr The commutator is evaluated as
\[ \int ( (S^{-1}\hat F_1 )(x),( S^{-1}E\hat F_2)(x)) dx \]
\be = \int (S^{-1} \hat F_1 )(x, \tilde x_1, ..., \tilde x_N)
( S^{-1}E \hat F_2)(x,\tilde x_1, ..., \tilde x_N) dx \ d\mu_N(
\tilde x_1 ...\tilde x_N) \label{stringcom}
\ee
Now both operators $N,L$ are essentially self-adjoint on the polynomial
domains. One can show that the
full domain $D(L)$ includes
$ C_0^{\infty}( \bbR^{N(d-2)})$, and so $D(N)$
includes $S[C_0^{\infty}( \bbR^{N(d-2)})]$.
Now suppose $H \in C^{\infty}_0 (\bbR^d, \cF (\ell_N^2))$
is such that $ H(x) \in S[C_0^{\infty}( \bbR^{N(d-2)})]$
for all $x$,
as for $\hat F_1, \hat F_2$ in the hypotheses the theorem. Then
$ H(x) \in D(N)$
and one can also deduce that $(E^{\pm} H)( x) \in D(N)$.
It follows that $(S^{-1}E^{\pm} H)( x) \in D(L)$. Similarly
it is in $D(L^r)$ for any $r$ and from this we can deduce
that it is a smooth function.
Now we can compute
\bea && ( \square - 2(L-a) ) S^{-1} E^{\pm } H \nn \\
&& = S^{-1}( \square - 2(N-a) ) E^{\pm } H \nn \\
&& = S^{-1} H \eea
Furthermore $S^{-1} E^{\pm}H$ vanishes in the distant past.
It follows that
\be S^{-1} E^{\pm} \hat H = \cE^{\pm} ( S^{-1} H )
\ee
where $\cE^{\pm}$ are the fundamental retarded and advanced
solutions for the operator
\be \square -2( L -a) =
- (\frac {\pa} { \pa x^0} )^ 2 + \sum_k
(\frac {\pa} { \pa x^k} )^ 2
+ \sum_{n,k}
(\frac {\pa} { \pa x^k_{n}} )^ 2
-2nx_n^k \frac {\pa} { \pa x^k_{n}} +2a \ee
This is the wave operator in $\bbR^d \times \bbR^{N(d-2)}$
perturbed by some lower order terms. But it is well known
that lower order terms do not affect the domain of dependence or
the domain of influence.
Thus the support of $ \cE^{\pm}( S^{-1} H ) $
is contained in the future/past of the support $S^{-1} H$
as defined by the metric
\be - (dx^0) ^2 + \sum_{k=1}^{d } (dx^k)^2 + \sum _{k= 1}^{d-2} \sum_{n = 1}^N
(dx^k_{n})^2 \ee
Returning now to (\ref{stringcom})
we have that the support of
$ S^{-1} E^{\pm} \hat F_2 = \cE^{\pm}
(S^{-1} \hat F_2) $
is contained in the future/past of the support of $ S^{-1} \hat F_2$.
Hence it
vanishes on the support of $S^{-1} \hat F_1$ by our assumption,
and so the commutator vanishes.
\appendix
\section{Appendix}
\subsection{a characteristic Cauchy problem} \label{A1}
One does not generally expect a characteristic Cauchy problem
to be well-posed. However if one restricts the class of solutions,
one can obtain existence and uniqueness theorems. Here we
show how this obtains for the Klein-Gordon equation $(\square - \mu)u=0$
in $\bbR^d$ based on
developments in the text.
We restrict the discussion to the case $\mu >0$.
We consider the class of regular solutions of the Klein Gordon
equation, defined to be smooth functions $u(x^0,\bx)$ with
data $u(0, \bx) = f(\bx)$ and $(\pa u / \pa x^0)(0, \bx)= g(\bx)$
in the Schwartz space
$\cS(\bbR^{d-1})$. Any such solution can be written as a sum of
positive and negative frequency parts:
\bea u(x^0, \bx) & =& (2\pi)^{-(d-1)/2} \int
e^{-i\om (\bp)x^0 + i\bp \cdot \bx} h_{+}(\bp)
\frac { d\bp}{2\om(\bp)} \nn \\
&+& (2\pi)^{-(d-1)/2} \int e^{+i\om (\bp)x^0 + i\bp \cdot \bx} h_{-}(\bp)
\frac { d\bp}{2\om(\bp)} \label{u} \eea
Here $h_{\pm}$ are given by
\be h_{\pm} (\bp) = \om(\bp)\tilde f(\bp) \pm i \tilde g(\bp) \ee
and are also in $\cS(\bbR^{d-1})$
We also consider such solutions in light cone coordinates defined
by
\be \check u(x^{+}, x^{-}, \tilde x)
= u \left(\frac{1}{\sqrt{2}}(x^{+} + x^{-}) , \tilde x ,
\frac{1}{\sqrt{2}}(x^{+} - x^{-})\right) \ee
We are interested in the Cauchy problem for the light-like
(characteristic) surface $ x^0 = -x^{d-1} $ or $x^{+} =0$.
Does the restriction $U (x^-, \tilde x ) \equiv
\check u (0,x^{-} , \tilde x)$ uniquely determine a solution?
First we have the following:
\blem Let $u$ be a regular solution of $(\square - \mu)u=0$.
Then the restriction $U(x^-, \tilde x)$ to $x^+ = 0$ is in
$\cS(\bbR^{d-1})$. Furthermore for any multi-index $\al$
and any $r$ there is a constant $C$ such
that the
Fourier transform satisfies \be | \pa^{\al}\tilde U(p^{+}, \tilde p)|
\leq C|p^+|^r \label{bound}\ee
\elem
\bigskip
\pr We have for $ U(x^{-} , \tilde x)$ the expression
\bea
&& (2\pi)^{-(d-1)/2} \int
\exp(-i( \frac{\om + p^{d-1}}{\sqrt{2}})x^{-} )
e^{ i\tilde p \cdot \tilde x} h_{+}(\tilde p , p^{d-1})
\frac { d\tilde p d p^{d-1}}{2\om(\tilde p , p^{d-1})} \nn \\
&+&(2\pi)^{-(d-1)/2} \int
\exp(i( \frac{\om + p^{d-1}}{\sqrt{2}})x^{-} )
e^{ i\tilde p \cdot \tilde x} h_{-}(\tilde p ,- p^{d-1})
\frac { d\tilde p d p^{d-1}}{2\om(\tilde p , p^{d-1})}
\eea
Now make the change of variables $p^{d-1} \to p^{+}$ as in (\ref{change})
and we find
\bea
&& (2\pi)^{-(d-1)/2} \int \int_0^{\infty} e^{-ip^{+}x^{-} + i\tilde p \cdot \tilde x}
h_{+}( \tilde p, \frac {1} {\sqrt 2}(p^+ - \frac{\tilde p^2 + \mu}{2p^+} ) )
\frac { dp^{+} d \tilde p}{2p^{+}} \nn \\
&+& (2\pi)^{-(d-1)/2} \int \int_0^{\infty} e^{ip^{+}x^{-} + i\tilde p \cdot \tilde x}
h_{-}( \tilde p, -\frac {1} {\sqrt 2}
(p^+ - \frac{\tilde p^2 + \mu}{2p^+} ))
\frac { dp^{+}d \tilde p}{2p^{+}}
\eea
>From this we identify
\be \tilde U(p^{+} , \tilde p) = \left \{ \barr{rl}
- h_{+}( \tilde p, -\frac {1} {\sqrt 2}(p^+ - \frac{\tilde p^2 + \mu}{2p^+} ))
/2p^{+} & \ \ \ p^{+} >0 \\
0 & \ \ \ p^{+}=0 \\
h_{-}( \tilde p, -\frac {1} {\sqrt 2}(p^+ - \frac{\tilde p^2 + \mu}{2p^+} ) )
/2p^{+} & \ \ \ p^{+} <0 \\
\earr \right. \label{ninetythree} \ee
We show that $\tilde U$ is rapidly decreasing at infinity, and is rapidly
vanishing as $p^+ \to 0$.
Since $h_{\pm}$ are Schwartz
functions we have
\be | \tilde U(p^{+} , \tilde p)| \leq C ( 1 + |\tilde p | +
|p^+ - \frac{\tilde p^2 + \mu}{2p^+}| )^{-N} |p^+|^{-1} \label{anotherbound}\ee
for any $N$.
If $\tilde p^2 + \mu \leq (p^+)^2$
we have $|p^+ - \frac{\tilde p^2 + \mu}{2p^+}| \geq |p^+|/2$
and the rapid decrease follows from (\ref{anotherbound}).
If $ \mu / 4 \leq (p^+)^2 \leq \tilde p^2 + \mu $
we drop this factor and get
$ | \tilde U(p^{+} , \tilde p)| \leq C ( 1 +|\tilde p |)^{-N} $
which gives the rapid decrease in this region.
Finally if $(p^+)^2 \leq \mu/4$ then
$|p^+ - \frac{\tilde p^2 + \mu}{2p^+}| \geq \mu/ 4 |p^+|$
and we obtain
\be | \tilde U(p^{+} , \tilde p)|
\leq C ( 1 + |\tilde p |)^{-{N/2}} |p^+|^{N/2-1} \ee
This gives the rapid decrease and also establishes the bound (\ref{bound})
for $\al = 0$.
The same bounds hold for all partial derivatives of $\tilde U$ for $p^+ \neq 0$.
In particular they all converge to zero as $p^+ \to 0$. From this
one can deduce that $\tilde U$ is smooth.
Hence $\tilde U$
is in $\cS(\bbR^{d-1})$ and so is $U$.
\bigskip
This result tells us how to pose the existence and uniqueness result.
\bthm Let $g(x^{-}, \tilde x) \in \cS(\bbR^{d-1})$ have a
Fourier transform $\tilde g(p^{+} , \tilde p)$ which satisfies (\ref{bound}).
Then there is a unique regular solution of
$(\square - \mu) u =0$ such that the restriction to $x^+ =0$ satisfies
$U(x^-, \tilde x) = g(x^-, \tilde x) $ \ethm
\bigskip
\pr Let $u$ be such a solution with $g=0$. Then
$h^{\pm}=0$ by (\ref{ninetythree}) and so $u=0$.
This proves the uniqueness.
For existence
we define $h_{\pm}$ by
\be h_{\pm} (\tilde p, p^{d-1}) = \mp 2p^+ \tilde g (p^+, \tilde p ) \ee
evaluated at
\be p^+ = \frac {1} {\sqrt 2}
( \pm \om(\tilde p , p^{d-1}) - p^{d-1})
\ee
This is a smooth function and we have
\be | h_{\pm} (\tilde p, p^{d-1})| \leq C
(1+ |\om(\tilde p , p^{d-1}) \mp p^{d-1}| + |\tilde p| )^{-N} \ee
To see it is rapidly decreasing
consider several cases. For $ \mp p^{d-1} \geq 0$ the above bound suffices,
and if $ -1 \leq \mp p^{d-1} \leq 0 $ just the
$(1 + |\tilde p| )^{-N}$ suffices.
Finally if $ \mp p^{d-1} < -1$ we use $|\tilde g(p^+, \tilde p)|
\leq C|p^+|^r$ to obtain
\bea | h_{\pm} (\tilde p, p^{d-1})| &\leq&
C |\om(\tilde p , p^{d-1}) \mp p^{d-1}|^{r+1} \nn \\
& \leq & C | \frac{\tilde p^2 + \mu}{2p^{d-1}}|^{r+1} \eea
A convex combination of this bound and a
$ (1+ |\tilde p| )^{-N} $ bound yields the rapid decrease in this region.
The same bounds hold for all the partial derivatives of $h_{\pm}$.
Hence $h_{\pm} \in \cS(\bbR^{d-1})$.
Now $u$ defined by (\ref{u}) is a regular solution, and
we we have chosen $h_{\pm}$ so that $\tilde U$
computed in (\ref{ninetythree}) is just $\tilde g$.
Hence $U = g$.
\subsection{canonical quantization for fields} \label{A2}
We quantize the space of solutions of the Klein-Gordon
equation $(\square - \mu )u =0$.
There is a natural symplectic form on this space given by
\be \si ( u_1,u_2) = \int_{\Si} ( u_1 \frac {\pa u_2 } {\pa n} -
u_2 \frac {\pa u_1 }
{\pa n} ) d \sigma
\ee
where $\Si$ is a space-like hypersurface with normal $ n_{\mu}$ and
and forward normal derivative $ \pa / \pa n =
n^{\mu} \pa / \pa x^{\mu}$ .
The form is independent of the choice of the surface.
Quantization can be regarded as
asking for a field operator $\phi$ such that
\be [ \si(\phi, u_1), \si(\phi, u_2)] = i \si(u_1,u_2) \label{ccr} \ee
for any solutions $u_1,u_2$.
This formulation generalizes nicely to manifolds, see for
example \cite {DiKa87}
In the standard formulation one takes the surface $\Si$ to be
$x^0 = 0$ in which case $\pa / \pa n = \pa / \pa x^0$.
Solutions $u(x)$ are identified with their data
$u(0,\bx)= f(\bx)$ and $ (\pa u / \pa x^0)(0,\bx) = g(\bx)$ and
the form becomes
\be \si ( f_1,g_1;f_2,g_2) = \int ( f_1(\bx) g_2 (\bx) - g_1(\bx) f_2(\bx)) d \bx
\ee
For quantization one takes $\si (\phi, \pi ; f,g) =\phi (g) - \pi (f)$
with the standard commutator $ [\phi(f) , \pi(g) ] = i \int f(\bx)g(\bx) d \bx$, and obtains the announced
\be [ \si ( \phi, \pi; f_1,g_1),\si ( \phi, \pi; f_2,g_2)]= i\si ( f_1,g_1;f_2,g_2)\ee
For the light cone formulation the surface $\Sigma$ is $x^+ =0$.
This is lightlike, but can be regarded as a limit of spacelike
hypersurfaces. Solutions are written in
light cone coordinates $\check u(x^+, x^-, \tilde x )$ and
are identified with their
restriction $\check u(0 ,x^-, \tilde x ) = f(x^-, \tilde x)$ to this surface.
As we have seen in Appendix \ref{A1}, this is justified under certain circumstances.
The forward derivative is $\pa / \pa n = \pa / \pa x^{-}$,
and the symplectic form becomes after an integration by parts
\be \si ( f_1,f_2) = 2\int f_1(x^-, \tilde x) \frac{ \pa f_2}{\ \pa x^-}(x^-, \tilde x)
dx^- d \tilde x \ee
There is now no natural split into coordinates and momenta.
For quantization we seek for a field operator $\phi(x^-, \tilde x )$ such that
the smeared version $\si (\phi, f) = 2 \phi (\pa f / \pa x^{-} )$ satisfies (\ref{ccr}), that
is so that
\be [ \phi(\frac {\pa f_1}{\pa x^-}), \phi( \frac {\pa f_2}{\pa x^-})]=
\frac { i}{2} \si (f_1, f_2) \ee
\subsection{existence of the light cone gauge} \label{A.3}
Let $X^{\mu}( \tau,\si)$ be an open string. Thus
it satisfies the wave equation $\pa^2 X^{\mu}/ \pa \tau^2 -\pa^2 X^{\mu}/
\pa \si^2
=0$ and the constraint $(\pa X/ \pa \tau \pm \pa X/ \pa \si)^2 =0$ and
the forward moving condition $\pa X^0/ \pa \tau >0$ .
We assume $X^{\mu}(\tau,\si)$ is defined on the whole plane and
impose Neumann boundary conditions by requiring that as a function of
$\si$ it is
even and periodic with period $2\pi$.
We investigate the circumstances under which one can find a conformal change
of coordinates
so that in the new coordinates $X^{+}(\tau,\si) = p^+ \tau$.
We work in coordinates
$ u = \tau + \sigma ,
v = \tau - \sigma $
in terms of which the equation is $\pa^2 X^{\mu} / \pa u \pa v =0 $
and the constraint is $(\pa X / \pa u)^2 = (\pa X / \pa v)^2 = 0$.
The general solution
of the equation
has the form $X^{\mu}( u,v ) =f^{\mu}(u) + g^{\mu}(v)$.
or in the original coordinates $X^{\mu}( \tau,\si ) =f^{\mu}(\tau + \si) +
g^{\mu}(\tau - \si)$. The Neumann boundary
conditions are equivalent to the condition that
$f^{\mu} = g^{\mu}$ up to a constant (and we may as well
take them equal), and that $f'$ is periodic with
period $2\pi$. The latter condition implies that
\be f^{\mu}(u) = a^{\mu} u + h^{\mu}(u) \ee
where $h^{\mu}$ is periodic with period $2\pi$ and
\be a^{\mu} = (2\pi)^{-1} \int_{-\pi}^{\pi}(f^{\mu})'(s) ds
= (2\pi)^{-1}( f^{\mu}(\pi) - f^{\mu}(-\pi) ) \ee
A short
calculation shows that $2a^{\mu} = p^{\mu}$.
The constraint equation becomes $(f')^2 =0$ and the forward
moving condition becomes $(f^0)' >0$.
Thus we have
\be (f^0)' = \left( \sum_{k=1}^{d-1} ((f^k)')^2 \right)^{1/2} \ee
It follows that
\be (f^0)' \geq |(f^{d-1})'| \ee
and hence that
\be (f^{+})' \geq 0\ee
We make the further mild restriction that
\be (f^{+})' > 0 \label{condition} \ee This is
a condition on initial data. It implies that $a^+ >0$.
Now we can state:
\bthm Let $X$ be an open string so that
\be X^{\mu} (\tau,\si) = f^{\mu} ( \tau + \sigma) + f^{\mu} ( \tau - \sigma)
\ee
with $f'$ smooth,
$f'$ periodic, $(f')^2 = 0$, and $(f^0)'>0$. If also $(f^+)'>0$
then there is
a conformal diffeomorphism $(\tau,\si ) \to (\tilde \tau, \tilde \si )$
such that in the new coordinates
\be \tilde X^{\mu} (\tilde \tau, \tilde \si ) =
\tilde f^{\mu} ( \tilde\tau +\tilde \sigma) +
\tilde f^{\mu} ( \tilde\tau - \tilde\sigma)
\ee
where $\tilde f$ satisfies the same conditions and in addition
\be \tilde X ^{+}( \tilde \tau, \tilde \si )
= p^+ \tilde \tau \ee
\ethm
\pr The condition (\ref{condition}) and the periodicity imply
that $ (f^+)'> \ep >0$, and it follows that
$ \tilde u = \al(u) \equiv (a^+)^{-1} f^{+}(u)$ is a diffeomorphism on $\bbR$.
With this coordinate change we define
\be \tilde f^{\mu} (\tilde u) = f^{\mu}(\al^{-1}( \tilde u)) = f^{\mu} ( (f^{+})^{-1} (a^+\tilde u) ) \ee
Then $\tilde f^{+} (\tilde u) = a^+\tilde u$
Now $\tilde u = \al (u), \tilde v = \al (v)$
is a conformal diffeomorphism on $\bbR^2$ with the
Minkowski metric $du dv$. Then
$ X^{\mu}( u, v) =
f^{\mu} ( u)
+ f^{\mu} ( v)$ becomes
in the new coordinates
\be \tilde X^{\mu}( \tilde u,\tilde v) =
\tilde f^{\mu} ( \tilde u)
+ \tilde f^{\mu} ( \tilde v) \ee
Hence
\be \tilde X^{+}( \tilde u,\tilde v) = a^+( \tilde u + \tilde v) \ee
and introducing $ \tilde \tau = (\tilde u + \tilde v )/ 2 $ and
$\tilde \si = (\tilde u - \tilde v )/2 $ we get
that
\be \tilde X ^{+}(\tilde \tau, \tilde \sigma ) = 2a^{+} \tilde \tau
= p^+ \tilde \tau \ee
as desired.
It remains to show that $\tilde f$ has same properties as $f$.
To see that $\tilde f'$
is periodic note that
$ \al(u + 2 \pi ) = \al (u) + 2\pi $
hence
$ \al ^{-1} (\tilde u + 2 \pi ) = \al ^{-1} (\tilde u)
+ 2 \pi $
and the periodicity follows from
\be (\tilde f^{\mu})'(\tilde u ) =
\frac { (f^{\mu})' ( \al^{-1}( \tilde u )) }
{ \al' ( \al^{-1}( \tilde u )) } \ee
The other conditions follow as well.
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\end{document}