Content-Type: multipart/mixed; boundary="-------------0101290933946"
This is a multi-part message in MIME format.
---------------0101290933946
Content-Type: text/plain; name="01-41.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="01-41.keywords"
String, Locality, Causality
---------------0101290933946
Content-Type: application/x-tex; name="string.final"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="string.final"
\documentclass[12pt]{article}
\usepackage{amsmath,amssymb}
\batchmode
\setlength{\oddsidemargin}{.25in}
\setlength{\textwidth}{6.125in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.5in}
\newtheorem{defn}{Definition}
\newtheorem{lem}{Lemma}
\newtheorem{thm}{Theorem}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newcommand{\pr}{\noindent{\bf Proof}. }
\newcommand{\re}{\noindent{\bf Remark}. }
\newcommand{\res}{\noindent{\bf Remarks}. }
\newcommand{\da}{\dagger}
\newcommand{\pa}{\partial}
\newcommand{\one}{\cO(1)}
\newcommand{\half}{{1 \over 2}}
\newcommand{\al}{\alpha}
\newcommand{\De}{\Delta}
\newcommand{\de}{\delta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\ka}{\kappa}
\newcommand{\La}{\Lambda}
\newcommand{\la}{\lambda}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\Th}{\Theta}
\newcommand{\ep}{\epsilon}
\newcommand{\si}{\sigma}
\newcommand{\Si}{\Sigma}
\newcommand{\cA}{{\cal A}}
\newcommand{\cB}{{\cal B}}
\newcommand{\cC}{{\cal C}}
\newcommand{\cD}{{\cal D}}
\newcommand{\cV}{{\cal V}}
\newcommand{\cO}{{\cal O}}
\newcommand{\cI}{{\cal I}}
\newcommand{\cH}{{\cal H}}
\newcommand{\cS}{{\cal S}}
\newcommand{\cE}{{\cal E}}
\newcommand{\cR}{{\cal R}}
\newcommand{\cT}{{\cal T}}
\newcommand{\cF}{{\cal F}}
\newcommand{\cK}{{\cal K}}
\newcommand{\cG}{{\cal G}}
\newcommand{\cM}{{\cal M}}
\newcommand{\cN}{{\cal N}}
\newcommand{\cW}{{\cal W}}
\newcommand{\cQ}{{\cal Q}}
\newcommand{\cL}{{\cal L}}
\newcommand{\cP}{{\cal P}}
\newcommand{\cU}{{\cal U}}
\newcommand{\cJ}{{\cal J}}
\newcommand{\bbR}{{\mathbb{R}}}
\newcommand{\bbZ}{{\mathbb{Z}}}
\newcommand{\bbC}{{\mathbb{C}}}
\newcommand{\bbN}{{\mathbb{N}}}
\begin{document}
\title{ Locality in Free String Field Theory - II.}
\author{
J. Dimock\thanks{Research supported by NSF Grant PHY0070905}\\
Dept. of Mathematics \\
SUNY at Buffalo \\
Buffalo, NY 14214 }
\maketitle
\begin{abstract}
We study the covariant free bosonic string field theory and
explore its locality (causality) properties. We find
covariant string fields which are strictly local and covariant,
but act on
an unconstrained Hilbert space with an indefinite inner product.
From these we also define observable fields
which act on the physical Hilbert space with an definite inner product.
These are shown to be approximately local.
\end{abstract}
\tableofcontents
\newpage
\section{Introduction}
\subsection{Overview}
We continue to explore the locality properties of
free bosonic string field theory. The guiding question
is whether one can find string fields which satisfy the field equation,
are strictly local (causal), and are Lorentz covariant.
Success in this quest would mean one could define an algebra of local observables of the type
that one usually considers in quantum field theory \cite{Haa92}. We do
not necessarily expect to succeed in this quest, but do expect
that the ways in which we fall short will be of interest.
In an earlier work \cite{Mar93}, \cite {Low94}, \cite{Dim00}, the problem
was considered in
the light cone gauge and string fields were found
which were local with respect to the center of mass coordinate.
However these fields were not Lorentz covariant.
In the present paper we work with a Lorentz covariant formalism
right from the start. It is the so-called ''old covariant
quantization'' in which one quantizes first and then imposes the
constraint.
Before imposing constraints, we are able to construct
string field operators which are Lorentz covariant
and local in the sense that the commutator of two fields
vanishes when the center of mass coordinates are spacelike separated.
However the field operators act on a Hilbert space with an indefinite
inner product. Once the constraints are imposed one obtains
a definite inner product.
On this space we also define covariant field operators called observable
fields.
For these observable fields we establish an approximate locality property.
Our results seem to be consistent with
the treatment of Hata and Oda \cite{HaOd97}
who work in a BRST formalism. A general
account of string field theory can be found in Thorn \cite{Tho89}
Another goal of this work is to solidify the mathematical foundations
of covariant string theory. For earlier work in this direction
see Grundling and Hurst \cite{GrHu93}.
\subsection{Lorentz invariant measures}
We start by developing some facts about Lorentz invariant measures on
the mass shells. (See also \cite{ReSi75}). For any real number $r$ let
\begin{equation}
V_r = \{ p \in \bbR^d -\{0\}: p^2 +r=0 \}
\end{equation}
Here $p^2 = p \cdot p = \sum_{\mu \nu} \eta_{\mu \nu}p^{\mu}p^{\nu} = -(p^0)^2 + |\vec p|^2$
is the Lorentz inner product. The Lorentz group
is all nonsingular linear transformations preserving $p^2$ and it acts
on $V_r$.
First we
define a Lorentz invariant volume element on $V_r$.
Let $\tau = dp^0 \wedge dp^1... \wedge
dp^{d-1}$ be the volume element in $\bbR^d$. With $\al(p)=- p^2$ define $\si$ to be the
unique $d-1$ form on $V_r$ such that
\begin{equation} \label{volume}
d\al \wedge \si = \tau
\end{equation}
Transforming this by a proper Lorentz transformation $\La$
we have $d(\al\circ \La) \wedge \La^* \si = \La^* \tau $.
But $\al \circ \La =\al$ by definition and $\La^* \tau = \tau$ since $ \det \La =1$.
Thus $d\al \wedge \La^* \si = \tau $ and hence $ \La^* \si = \si$.
Now $ \int_{V_r} \ f \ \si$ is
defined for continuous functions $f$ with compact support on
$V_r$. The map $ f \to \int_{V_r} \ f \ \si$ is positive and
hence there is a positive measure $\mu_r$ on $V_r$ such that
\begin{equation} \label{measure}
\int_{V_r} f \ d \mu_r = \int \ f \ \si
\end{equation}
The Lorentz invariance of $\si$ implies the invariance of $\mu_r$.
Now we comment on some specific representations of this
measure, first for $r \geq 0$. In this case the hyperboloid has two sheets which
are
\begin{equation}
V^{\pm}_r = \{ p \in V_r: \pm p^0 >0\}
\end{equation}
\begin{lem} For $r \geq 0$, let $f$ have compact support on $V_r^{\pm}$
and let $\om_r(p) = \sqrt {|\vec p|^2 + r}$. Then
\begin{equation}
\int_{V^{\pm}_r} f \ d \mu_r
= \int f (\pm \om_r(\vec p) , \vec p) \frac{d \vec p}{ 2 \om_r(\vec p)}
\end{equation}
\end{lem}
\pr For $V_r^{\pm}$ we can take global coordinates $\vec p =
(p^1,...,p^{d-1})$. In these coordinates we have
\begin{equation}
\si = \frac{1}{ \pa \al / \pa p^0} dp^1 \wedge ... dp^{d-1} =
\frac{1}{ 2 p^0} dp^1 \wedge ... dp^{d-1}
\end{equation}
This is the form on $V_r^{\pm}$. Pulling it back to $\bbR^{d-1}$ with
the inverse coordinate function $\phi_{\pm}(\vec p ) = ( \pm \om_r(\vec p) , \vec p)$
we have
\begin{equation}
\phi^*_{\pm} (\si) = \frac{\pm 1}{ 2 \om(\vec p)} dp^1 \wedge ...\wedge dp^{d-1}
\end{equation}
Now $\phi_+$ is orientation preserving so
$\int_{V^+_r} \ f \ \si$ is evaluated as $ \int (f \circ \phi_+) \phi^*_+(\si)$.
On the other hand $\phi_-$ is orientation reversing so
$\int_{V^-_r} \ f \ \si$ is evaluated as $- \int (f \circ \phi_-) \phi^*_-(\si)$
In either case we get the stated result.
\bigskip
Another representation uses light-cone coordinates and now we allow
all $r \in \bbR$. Light-cone coordinates $\hat p = \ell (p)$
are defined on $\bbR^d$ by $\hat p =(p^-, \tilde p, p^+)$ where
$p^{\pm} = (2)^{-1/2}(p^0 \pm p^{d-1})$ and $\tilde p = (p^1,...,p^{d-2})$.
For any function $f$ let $\hat f = f \circ \ell^{-1}$ be the expression
in light-cone coordinates.
Also define
\begin{equation}
V_r^{\pm} = \{ p \in V_r : \pm p^+ >0 \}
\end{equation}
either in the original coordinates or in light-cone coordinates
depending on the context.
For $r \geq 0$ these are again the two components of $V_r$, but
for $r<0$ they are just open sets in $V_r$.
\begin{lem} For any r, let $f$ have compact support on $V_r^{\pm}$. Then
\begin{equation} \label{eval2}
\int_{V^{\pm}_r} f \ d \mu_r
= \int \hat f \left(\frac{| \tilde p|^2 +r}{2p^+}, \tilde p ,p^+ \right)
\frac{ d \tilde p dp^+}{ 2 |p^+|}
\end{equation}
\end{lem}
\bigskip
\pr The integral $ \int \ f \ \si$ can be evaluated as
$ \int \ \hat f \ \hat \si$ where $\hat \si = ( \ell^{-1})^* \si$.
Since $ ( \ell^{-1})^* \tau = \tau$ we have $d \hat \al \wedge \hat \si
=\tau$. On $V_r^{\pm}$ we can take coordinates
$(p^+, \tilde p)$ . Since $\tau = dp^- \wedge dp^1 ...\wedge dp^{d-2} \wedge dp^+$
and since
$\hat \al(p)= 2 p^+ p^- - |\tilde p |^2 $.
we find that expressed in these coordinates
\begin{equation}
\hat \si = \frac{1}{ \pa \hat \al / \pa p^-} dp^1 \wedge ... \wedge dp^{d-2} \wedge dp^+ =
\frac{1}{ 2 p^+} dp^1 \wedge ...\wedge dp^{d-2} \wedge dp^+
\end{equation} The inverse coordinate function
is $\phi( p^+, \tilde p) = ( (| \tilde p|^2 +r)/2p^+, \tilde p,p^+)$
defined on the half spaces $\pm p^{+}>0$ in $\bbR^{d-1}$.
The pull back $\phi^*(\hat \si)$ to these half spaces
has the same form. The function $\phi$ is orientation preserving for $V_r^+$
and orientation reversing for $V_r^-$. Thus the integral is
evaluated as
$ \pm \int (\hat f \circ \phi) \phi^*(\hat \si)$ respectively, and in either case we get
the stated result.
\bigskip
For $r \leq 0$ the
sets $V_r^{\pm}$ do not cover all of $V_r$.
However suppose we define $p_j^{\pm} = (2)^{-1/2}(p^0 \pm p^{j})$ with $j = 1,...,d-1$
and define more sets $V_{r,j}^{\pm} = \{ p \in V_r : \pm p^+_j >0 \}$
and $\tilde V_{r,j}^{\pm} = \{ p \in V_r : \pm p^-_j >0 \}$.
On each of these sets we can prove a result similar to (\ref{eval2}).
These sets do cover $V_r$ and by taking a partition of
unity subordinate to this covering we can express any integral
as a sum of integrals of the type (\ref{eval2}).
\begin{lem} \label{decomp} Let $f$ be a function on on $\bbR^d- \{ 0\}$
which is continuous and has compact
support.
Let $f_r$ be the restriction to $V_r$. Then
\begin{equation}
\int f= \int \left( \int_{V_r} f_r d \mu_r \right) dr
\end{equation}
\end{lem}
\pr
Let $U_{r,j}^{\pm} = \{ p \in \bbR^d: \pm p^+_j >0 \}$
and $\tilde U_{r,j}^{\pm} = \{ p \in \bbR^d : \pm p^-_j >0 \}$.
These cover $\bbR -\{0\}$
and by introducing a subordinate partition of unity it suffices
to prove the theorem assuming that $f$ has compact support in one of these
sets, for example $U_r^{+}$ defined by $ p^+ = p_{d-1}^+>0$. Then $f_r$ has
compact support in $V^+_r$ for all $r$.
We make the change of variables
$p^- \leftrightarrow r$ with $r = 2p^+p^- - |\tilde p|^2$
and then we have
\begin{equation} \label{decompose}
\int \hat f(p^-, \tilde p, p^+) dp^- d \tilde p dp^+
=\int \hat f(\frac{| \tilde p|^2+r}{2p^+}, \tilde p, p^+)
\frac{ d \tilde p dp^+ dr}{ 2| p^+|} = \int ( \int_{V_r} f_r d \mu_r) dr
\end{equation}
\newpage
\section{The single string}
\subsection{Pre-constrained}
We now undertake the covariant quantization of the single string. The construction
is mostly standard \cite{GSW87}. However, one novelty is that the center of mass
momentum is treated as a genuine quantum observable with a distribution
of values. Most treatments take a fixed center of mass momentum.
For the open string in $\bbR^d$ the coordinates of the quantum
string should be operators $X^{\mu} (\tau, \sigma)$ define for
$(\tau, \si) \in \bbR \times [0, \pi]$ and satisfying
the wave equation
\begin{equation} \label{waveeqn}
( \frac {\pa^2 X^{\mu}}{\pa \tau^2} - \frac {\pa^2 X^{\mu}}{\pa \sigma^2} ) =0
\end{equation}
with Neumann boundary conditions on $[0,\pi]$.
The operators $ X^{\mu}$ and the string momentum $P^{\mu}= \pa X^{\mu}/ \pa \tau$
are supposed to satisfy the equal $\tau$ commutation relations
\begin{equation} \label{ccr}
[X^{\mu}(\si, \tau ), P^{\nu}(\si', \tau)] = i \pi\de (\si - \si') \eta^{\mu \nu}
\end{equation}
Corresponding to reparametrization invariance we impose the constraints
\begin{equation} \label{constraint}
\left( \frac {\pa X}{\pa \sigma} \pm \frac {\pa X}{\pa \tau} \right)^2
=0
\end{equation}
That is we ask for states which are annihilated by these operators.
Finally we ask that the center of mass $x^{\mu} =
\pi^{-1}\int_0^{\pi} X^{\mu}(\tau, \sigma) d \sigma$
be parametrized in a forward moving direction. This means
we require that the constant center of mass momentum
$p^{\mu}
=dx^{\mu} /d \tau =
\pi^{-1}\int_0^{\pi} P^{\mu}(\tau, \sigma) d \sigma$
should satisfy
\begin{equation}
p ^0 = \frac{dx^0}{d \tau} >0
\end{equation}
Classically one can find solutions of the wave
equation by expanding in eigenfunctions of the Laplacian
with Neuman boundary conditions, that is in a cosine series.
One finds that
\begin{equation} \label{X}
X^{\mu}(\sigma, \tau ) = x^{\mu} + p^{\mu} \tau + i\sum_{n \neq 0}\al^{\mu}_n
e^{-in \tau} \frac{\cos n \sigma }{ n}
\end{equation}
The quantum operator will be given by the same expression.
It formally satisfies the commutation relations (\ref{ccr})
if we ask that $x^{\mu}, p^{\mu} ,\al_n^{\mu}$ be
operators satisfying the commutation relations
\begin{equation}
\begin{split}
[x^{\mu}, p^{\nu}] = & i \eta^{\mu \nu} \\
[\al_m^{\mu}, \al_n^{\nu}] =& m \de_{m+n} \eta^{\mu \nu}\label{CCR1} \\
\end{split}
\end{equation}
Here is a construction of these operators.
First consider the Hilbert space
$\cL^2(\bbR^d)$. On this
space let $x^{\mu}$ be the multiplication operator, let
$p_{\nu} = -i \pa/\pa x^{\nu}$ and let $ p^{\nu} = \sum_{\nu} \eta^{\mu \nu} p_{\nu}$
(the spacetime representation), or else let
$p^{\nu} $ be the multiplication operators and let
$x_{\mu} = i \pa /\pa p^{\mu}$, etc. (the momentum representation).
In either case these satisfy (\ref{CCR1}). A convenient dense domain
for these operators is $\cS(\bbR^d)$,
the Schwartz space of smooth rapidly decreasing functions. This is invariant
under the Fourier transform and connects the two representations.
Next let $\ell^2 = \ell^2( \bbN, \bbC^d)$ be the space of square summable maps
$f: \bbN \to \bbC^d$. This has the usual inner product $(f, g)$ and
also an indefinite inner product $ $.
They are
\begin{equation}
\begin{split}
(f, g ) =& \sum_{n=1}^{\infty} \sum_{\mu } \overline{f_{\mu n} }g_{\mu n}\\
=&
\sum_{n=1}^{\infty}\sum_{\mu \nu} \eta_{\mu \nu} \overline{f_{\mu n} }g_{\nu n}\\
\end{split}
\end{equation}
They are related by $ = (f, J g)$ where $(Jg)_{\mu}
= \eta_{\mu \mu } g_{\mu}$. (Thus $J = \eta$ , but without the geometric
interpretation).
Let $\cF_j(\ell^2)$ be the $j$-fold symmetric tensor product of $\ell^2$
and let $\cF(\ell^2) = \bigoplus_{j=0}^{\infty}\cF_j(\ell^2)$ be the bosonic Fock
space over $\ell^2$. Any unitary operator $U$ on $\ell^2$ induces a unitary
$\otimes ^j U $ on $\cF_j(\ell^2)$ and hence an operator $\Ga (U)$ on $\cF(\ell^2)$.
We define an indefinite inner product on $\cF(\ell^2)$ by
$ =(f, \cJ g)$ where $\cJ = \Ga(J)$.
For any operator $\cO$ on $\cF(\ell^2)$, let $\cO^*$ be the adjoint
with the definite inner product, and let $\cO^{\da}$ be the adjoint with
the indefinite inner product. Hence $(f, \cO g) = (\cO^*f, g)$
and $ = <\cO^{\da }f, g>$ . They are related by
$ \cO^* = \cJ \cO^{\da} \cJ $.
Next define annihilation operators $a(f), b(f)$ on the n-fold symmetric tensor product by
\begin{equation}
\begin{split}
a(f) ( f_1 \otimes ... \otimes f_j) = & \sqrt{j} f_2 \otimes ... \otimes f_j \\
b(f) ( f_1 \otimes ... \otimes f_j) = & \sqrt{j}(f,f_1)\ f_2 \otimes ... \otimes f_j \\
\end{split}
\end{equation}
By restriction these define operators on the symmetric subspace and hence
on $\cF(\ell^2)$.
We have $a(f) = b(J f)$.
The adjoints satisfy $ a^{\da}(f) = b^*(f)$. We have
$[b(f), b^*(g)] = (f,g)$ and
$[a(f),a^{\da}(g) ] = $.
In $\ell^2$ there is a standard basis $e^{\mu}_m$ defined by
$(e^{\mu}_n)_{\nu m} = \de^{\mu}_{ \nu} \de_{m,n}$.
We define for $n >0$
\begin{equation}
\begin{split}
\al^{\mu}_n =& \sqrt n \ a(e^{\mu}_n) \\
\al^{\mu }_{-n} =& \sqrt n \ a^{\da}(e^{\mu}_n) \\
\end{split}
\end{equation}
These satisfy (\ref{CCR1}) since $ = \eta^{\mu \nu} \de_{n,m}$.
Let $\cD_0$ be the dense subspace of
$\cF(\ell^2)$ generated by applying a finite number of operators $\al_{-n}^{\mu}$
to the no excitation state $\Om_0 =(1,0,0...)$.
Now consider the Hilbert space
\begin{equation} \label{hilbert}
\cL^2(\bbR^d) \otimes \cF(\ell^2)) \approx \cL^2(\bbR^d, \cF(\ell^2))
\end{equation}
Besides the usual inner product this space has an indefinite inner
product inherited from $\cF(\ell^2)$ and defined by
\begin{equation}
<\psi, \chi> = \int <\psi(p), \chi(p)> dp
\end{equation}
The operators $x^{\mu}, p^{\mu}$ in the momentum representation,
and $ \al_n^{\mu}$ all act on this space.
Now we can define the coordinate operator $X^{\mu}(\si, \tau)$ by (\ref{X}).
It is well defined provided we interpret it as a distribution in $\si$ and
to restrict a nice domain like $\cS(\bbR^d) \otimes \cD_0$. It
does satisfy (\ref{waveeqn}),(\ref{ccr}).
On the same domain $(x^{\mu})^{\da }
= x^{\mu}$, $(p^{\mu})^{\da }= p^{\mu}$, $(\al^{\mu}_n)^{\da }= \al^{\mu}_{-n}$.
and hence $(X^{\mu}( \si , \tau))^{\dagger} = X^{\mu}(\si,\tau)$.
We digress to discuss representations of the Lorentz group. First on $\ell^2$
there is a representation defined by $(\La f)_{\mu n} =
\sum_{\nu} \La^{\ \nu}_{\mu \ } f_{\nu n}$
which preserves the indefinite inner product. This induces an
operator $ \Gamma( \La )$ on $\cF(\ell^2)$ which also preserves the
indefinite inner product.
It is not bounded but is at least defined on vectors with a finite number of entries.
Finally for $a \in \bbR^d$ and a proper Lorentz transformation $\La$
we define $U(a, \La)$ on $ \cL^2(\bbR^d, \cF(\ell^2))$
by
\begin{equation}
(U(a, \La)\psi)(p) = e^{-ip \cdot a} \Gamma (\La) \psi( \La^{-1} p )
\end{equation}
This is well-defined if $\psi$ takes values in the domain of $\Ga(\La)$.
The operators $U(a, \La)$ give a representation of
the inhomogeneous Lorentz group which preserves the indefinite inner product
since Lebesgue measure is Lorentz invariant.
We note
that $U(a, \La)^{-1} a(f) U(a, \La) = a (\La^{-1} f)$.
Since
$\La^{-1} e^{\mu}_n =
\sum_{\nu} (\La^{-1})_{ \nu \ }^{\ \mu} e^{\nu}_n= \sum_{\nu} \La^{ \mu \ }_{\ \nu} e^{\nu}_n$
this implies
\begin{equation} \label{trans}
U(a, \La)^{-1} \al_n^{\mu}\ U(a, \La) =
\sum_{\nu} \La^{ \mu \ }_{\ \nu} \al^{\nu}_n
\end{equation}
We also have
$ x^{\mu} \rightarrow
\sum_{\nu} \La^{ \mu \ }_{\ \nu} x^{\nu} + a^{\mu} $,
and $p^{\mu} \rightarrow \sum_{\nu} \La^{ \mu \ }_{\ \nu} p^{\nu} $ and thus
\begin{equation}
U(a, \La)^{-1} X^{\mu}(\si, \tau)\ U(a, \La) =
\sum_{\nu} \La^{ \mu \ }_{\ \nu} X^{\nu}(\si, \tau) + a^{\mu}
\end{equation}
Now we turn to the constraint operators (\ref{constraint}).
Passing to the Fourier components one finds the operators
\begin{equation} \label{L}
\begin{split}
L_0 &= \half \ p^2 + \sum_{n=1}^{\infty}\al_{-n} \cdot \al_n
\\
L_m &= \al_m \cdot p \ + \ \half \sum_{n \neq m,0}\al_{m-n} \cdot \al_n \ \ \ \ \ \ m \neq 0
\\
\end{split}
\end{equation}
These are well defined on $\cS(\bbR^d) \otimes \cD_0$ and satisfy
$L_m^{\da} = L_{-m}$.
Instead of asking for states $\psi$ satisfying
$L_n \psi = 0$ for all $n$ we make the standard modification and ask for states
satisfying
\begin{equation}
\begin{split}
(L_0 - 1) \psi =& 0 \\
L_m \psi =& 0 \ \ \ \ \ \ \ \ \ \ \ m>0 \label{first}
\end{split}
\end{equation}
As usual when quantizing a parametrized theory, the dynamics
are contained in the constraints. Suppose we
define the operator $M^2$ (not really a square) on a dense domain
in $\cF(\ell^2)$ by
\begin{equation}
M^2 = 2(N-1)
\end{equation}
where $N$ is the excitation operator which can be written
in any of the following forms
\begin{equation}
\begin{split}
N = & \sum_{n=1}^{\infty}\al_{-n} \cdot \al_n \\
= & \sum_{n,\mu \nu} n\ \eta_{\mu \nu}\ a^{\da} (e^{\mu}_n) a (e^{\nu}_n) \\
= & \sum_{n, \mu} n\ b^* (e^{\mu}_n) b (e^{\mu}_n) \\
\end{split}
\end{equation}
The constraint $(L_0-1)\psi = 0$ can be written $(p^2 + M^2)\psi = 0$.
In the spacetime representation we have
\begin{equation}
\label{KG} (- \square + M^2 )\psi =0 \end{equation}
It the Klein - Gordon equation for an $\cF(\ell^2)$ valued function, and gives the
evolution in time. The operator $M^2$ is identified as
a mass operator. The next result shows that $M^2$ is self-adjoint
and has spectrum ${-2,0,2,4,6...}$ with finite multiplicity.
\begin{lem}
$N$ is self-adjoint and has spectrum $0,1,2,...$ with finite multiplicity.
\end{lem}
\pr Let $\{ N_{\mu,n} \} $ be a finite sequence of
positive integers indexed by $\mu= 0,1,..., d-1$ and $n=1,2,...$,
with at most finitely many $N_{\mu , n} \neq 0$.
For each such sequence we define a vector
\begin{equation}
\psi( \{ N_{\mu,n} \} ) = \prod_{\mu,n}
\frac{(b^*(e_n^{\mu}))^{ N_{\mu,n}} }{\sqrt{ N_{\mu,n} !}} \Om_0
\end{equation}
This is a orthonormal basis for $\cF(\ell^2)$ and they are eigenfunctions
of $N$ since
\begin{equation}
N \ \psi( \{ N_{\mu,n} \} ) = (\sum_{\mu,n} n N_{\mu,n} )\ \psi( \{ N_{\mu,n} \} )
\end{equation}
This gives the self adjointness and the spectrum. The finite multiplicity
follows since for any positive integer $n^*$ there are only a finite number of sequences with
$\sum_{\mu,n} n N_{\mu,n} = n^*$.
\subsection{Reconfigured}
The constraint
(\ref{KG}) cannot be satisfied in $ \cL^2(\bbR^d, \cF(\ell^2))$. To impose
the constraint we will have to modify the Hilbert space.
To begin we write this space as a direct integral
over the various mass shells. For the moment our purpose
is just to motivate Definition \ref{defn}
below, so we pass over various technicalities such as the exact
definition of the direct integral in this case. (See however
Nielsen \cite{Nie80} ).
Let $\| \cdot\|$ denote the norm
in the Fock space $\cF(\ell^2)$ defined by the definite inner product.
For $\psi \in \cL^2(\bbR^d, \cF(\ell^2))$
the norm squared can be written as
\begin{equation}
\int_{\bbR^d} \| \psi (p)\|^2 dp =
\int_{-\infty}^{\infty} \left( \int_{V_r} \| \psi ( p)\|^2 d \mu_r(p) \right) dr
\end{equation}
This follows from (\ref{decompose}) extended to $\cL^2$ functions.
Then we can make the identification
\begin{equation} \label{1}
\cL^2(\bbR^d, \cF(\ell^2), dp ) =
\int ^{\oplus} \cL^2(V_r, \cF(\ell^2), d \mu_r)\ dr
\end{equation}
where
$\psi \in \cL^2(\bbR^d, \cF(\ell^2), dp ) $
is identified with the map $r \to \psi_r$ ( the restriction of $\psi$ to $V_r$).
The indefinite inner product on $\cF(\ell^2)$ induces the same
on $ \cL^2(V_r, \cF(\ell^2), d \mu_r)$ and we have, again by (\ref{decompose}),
\begin{equation} \label{2}
<\psi , \chi> =
\int_{-\infty}^{\infty} <\psi_r, \chi_r> \ dr
\end{equation}
Thus the decomposition can be regarded as a decomposition of indefinite inner
product spaces. The operators $M^2, L_m$ act on $ \cL^2(V_r, \cF(\ell^2), d \mu_r)$
and we have the decompositions
\begin{equation} \label{3}
\begin{split}
L_0-1 =& \int ^{\oplus}\frac12 (-r + M^2)\ dr\\
L_m = &\int ^{\oplus} L_m \ dr\\
\end{split}
\end{equation}
This means for example that $(L_m \psi)_r = L_m \psi_r$.
Since the Lorentz group acts on $V_r$ the
operators $U(a,\La )$ act on $ \cL^2(V_r, \cF(\ell^2), d \mu_r)$, and
they preserve the indefinite inner product since the measure is
Lorentz invariant.
Denoting this representation by $U_r(a , \La)$ we
have
\begin{equation} \label{4}
U(a, \La) = \int^{\oplus} U_r(a, \La)\ dr
\end{equation}
To impose the constraint we first make a minimal
reconfiguration of the
Hilbert space so it can accept the constraints.
Since $M^2$ has spectrum in $-2,0,2,4,...$ the constraint $(L_0-1)\psi =0$
only has a chance for $r$ in this set.
Accordingly we pick out these values from the direct integral and form a direct
sum.
At this stage we also impose the
forward moving condition $p_0 >0$ as much as possible.
For $r \geq 0$ we pick out the positive energy part $V_r^+$
of the mass shell. This is not possible for $r<0$
without losing the representation of the Lorentz group, and so we leave it alone.
We these ideas in mind we make the following definitions
after (\ref{1}), (\ref{2}), (\ref{3}), (\ref{4}).
\begin{defn} \ \label{defn}
\begin{enumerate}
\item
The Hilbert space for the single string is
\begin{equation}
\begin{split} \label{hrdef}
\cH =& \bigoplus_{r= -2,0,2,... } \ \ \cH_r \\
\cH_r =&
\cL^2( V^{(+)}_r, \cF(\ell^2), d\mu_r) \\
\end{split}
\end{equation}
where $ V^{(+)}_r = V_r^+$ for $r \geq 0$ and $ V^{(+)}_r = V_r$ for $r < 0$ .
\item For
$\psi, \chi$ in $\cH$ with components $\psi_r, \chi_r$
an indefinite inner product is defined by
\begin{equation}
<\psi , \chi> =
\sum_{r } <\psi_r, \chi_r>
\end{equation}
\item A representation of the inhomogeneous Lorentz group
is defined by
\begin{equation}
U(a, \La) = \bigoplus_r U_r(a, \La)
\end{equation}
\item The constraint operators are defined by
\begin{equation}
\begin{split}
L_0-1 = & \bigoplus_r \frac12 (-r + M^2) \\
L_m = & \bigoplus_r L_m \\
\end{split}
\end{equation}
\end{enumerate}
\end{defn}
Let us be more precise about the domains of
$N, M^2, L_m$. We will define them as closed operators on $\cH_r$ and then
the above equations define them as closed operators
on $\cH$. As we have noted $N$ (or $M^2$) is
self-adjoint on $\cF(\ell^2)$ and we define $N$ on $\cH_r$ by
$(N\psi)(p) = N \psi(p)$ with domain
\begin{equation}
D(N) = \{ \psi \in \cH_r : \psi(p) \in D(N) \ a.e.\ p \ , \
\int \| N\psi(p) \|^2 d\mu_r(p) \ < \infty \}
\end{equation}
Similarly for each $p$ let $L_m(p)$ be the closure of the operator (\ref{L})
defined on $\cD_0 \subset \cF(\ell^2)$.
(It is closable since the adjoint
$L_m^*(p) = \cJ L_{-m} \cJ $ is densely defined).
Then define $L_m$ on $\cH_r$
by $ (L_m \psi)(p) = L_m(p) \psi(p)$ with domain
\begin{equation}
D(L_m) = \{ \psi \in \cH_r : \psi(p) \in D(L_m(p)) \ a.e.\ p \ , \
\int \| L_m(p) \psi(p) \|^2 d\mu_r(p) \ < \infty \}
\end{equation}
With these domains we have:
\begin{lem}
$N$ and $M^2$ are self adjoint, and $L_m$ is closed.
\end{lem}
\pr Start with the second. Let $\psi_j \in D(L_m)$ satisfy $\psi_j \rightarrow \psi$ and $ L_m \psi_j
\rightarrow \chi$. Then there exist a subsequence such that for almost every $p$ we
have $\psi_{j_k}(p) \rightarrow \psi(p)$ and $L_m(p)\psi_{j_k}(p) \rightarrow \chi(p)$.
(This is a corollary of the proof that vector-valued $\cL^p$ spaces are complete
\cite{HiPh57}).
Since $L_m(p)$ is closed it follows that $\psi(p) \in D(L_m(p))$ and
$L_m(p) \psi(p) = \chi(p)$ for a.e. $p$.
Hence $\psi \in D(L_m)$ and $L_m \psi = \chi$. Thus $L_m$ is closed
The same argument shows that $N$ is closed. It is also symmetric and
since it has a dense set of analytic vectors, for example $\cC^{\infty}_0 \otimes \cD_0$,
it is self-adjoint.
\subsection{Constrained}
Now let
$\cH'$ be the subspace of $\cH$ satisfying the constraints (\ref{first}).
We have
\begin{equation} \label{prime}
\begin{split}
\cH' =& \bigoplus_{r= -2,0,2,... } \cH_r' \\
\cH_r' = & \{ \psi \in \cH_r: (-r + M^2)\psi =0,\ L_m \psi = 0 \ \rm{for} \ m>0 \} \\
\end{split}
\end{equation}
Note that a function $\psi \in \cH_{-2}$ is in $\cH'_{-2}$ iff both $N\psi=0$
and $L_m\psi=0$ which is true iff $\psi$ takes values in $\cF_0(\ell^2) \approx \bbC$.
Thus $\cH'_{-2} = L^2( V_{-2}, \cF_0(\ell^2), d\mu_{-2})$. These are
the \textit{tachyons}.
Next we consider the \textit{isotropic} or \textit{spurious} elements
in $\cH'$ which are defined
by
$\cH'' = \cH' \cap (\cH')^{\perp}$. Here the orthogonal subspace is defined by
the indefinite inner product.
Vectors in $\cH''$ satisfy $< \psi, \psi > = 0$.
The subspace has the form
\begin{equation}
\begin{split}
\cH'' = & \bigoplus_r \cH''_r \\
\cH''_r = & \cH_r' \cap (\cH_r')^{\perp}\\
\end{split}
\end{equation}
Now let $ \cH^{phys} = \cH' / \cH'' $. We identify
\begin{equation}
\begin{split}
\cH^{phys} = & \bigoplus_r \cH^{phys}_r \\
\cH^{phys}_r =& \cH'_r / \cH''_r \\
\end{split}
\end{equation}
The indefinite inner product on $\cH'$
lifts to $ \cH^{phys} $ and is the direct sum of the
the inner products on $\cH_r^{phys}$ lifted from $\cH_r'$.
Now we establish the famous no-ghost theorem. Our method is to
reduce the result to a statement pointwise in $p$ and them
quote the literature.
\begin{lem} For $d=26$,
$<.,.>$ is positive definite on $\cH^{phys}_r$ and $\cH^{phys}$.
\end{lem}
\pr It suffices to prove the result on $\cH^{phys}_r$.
The space $ \cH_r = \cL^2( V_r^{(+)}, \cF(\ell^2), d\mu_r)$
can be regarded as a constant fiber direct integral
\begin{equation}
\cH_r =
\int^{\oplus}_{V^{(+)}_r} \cF(\ell^2) d \mu_r(p)
\end{equation}
Indeed the former can be taken as the definition of the latter, so this just
amounts to a change in notation.
We have the decompositions $(-r+M^2) = \int^{\oplus}(-r+ M^2) d \mu_r(p)$
and $L_m = \int^{\oplus} L_m(p) d \mu_r(p)$. The constrained space can
be characterized as
\begin{equation}
\begin{split}
\cH'_r = &
\int^{\oplus}_{V^{(+)}_r} \cH'(p) d \mu_r(p) \\
\cH'(p) = & \{ \psi \in \cF(\ell^2): (p^2 + M^2)\psi =0, L_m(p) \psi = 0 \ \rm{for} \ m>0 \} \\
\end{split}
\end{equation}
This means $\psi \in \cH_r'$ iff $\psi(p) \in \cH'(p)$ for a.e. $p \in V^{(+)}_r$.
Since eigenvalues of $M^2$ have finite multiplicity,
$\cH'(p)$ is finite dimensional. Note also $\cH'(p) \subset \cD_0$.
Next we have
$(\cH'_r )^{\perp} =
\int^{\oplus}(\cH'(p))^{\perp} d \mu_r(p) $ and it follows that
\begin{equation}
\begin{split}
\cH''_r = &
\int^{\oplus}_{V^{(+)}_r} \cH''(p) d \mu_r(p) \\
\cH''(p) =& \cH'(p) \cap \cH'(p)^{\perp} \\
\end{split}
\end{equation}
Now for $\psi, \chi \in \cH_r'$ we have $\psi- \chi \in \cH_r''$
if and only if $\psi(p), \chi(p) \in \cH'(p)$ satisfy
$\psi(p)- \chi(p) \in \cH''(p)$ for a.e. $p$. Thus equivalence classes
can be defined pointwise which we write as
\begin{equation}
\begin{split}
\cH^{phys}_r = &
\int^{\oplus}_{V^{(+)}_r} \cH^{phys}(p) d \mu_r(p)\\
\cH^{phys} (p) =& \cH'(p) /\cH''(p) \\
\end{split}
\end{equation}
Now it suffices to prove that the inner product is positive
definite on $\cH_{phys} (p)$. A proof of this can
be found in Frenkel, Garland, and Zuckerman
\cite{FGZ86}. They also compute the dimension of this space
and show it depends only on $p^2$. For the original proofs see Brower \cite{Bro72} and
Goddard and Thorn \cite{GoTh72}.
\begin{lem} $U(a, \La)$ determines a unitary representation of
the inhomogeneous Lorentz group on $\cH^{phys}$.
\end{lem}
\pr $U_r(a, \La)$ is defined on all of $\cH_r'$.
It preserves $\cH'_r$ since
$[L_m, U_r(a, \La)] = 0$ by(\ref{trans}).
If $\chi \in (\cH_r')^{\perp}$ then
for $\psi \in \cH_r'$
we have
$< \psi, U_r(a, \La)\chi > = < U_r(a,\La)^{-1} \psi,\chi >=0$
and so $ U_r(a, \La)\chi \in (\cH_r')^{\perp}$.
Thus $ U_r(a, \La)$ preserves $\cH''_r$ and so it lifts
to $\cH^{phys}_r$. Since it is still inner product preserving
it is unitary.
\subsection{Physical States}
We want to exhibit some non-trivial smooth elements of
$\cH'_r \subset \cL^2(V^{(+)}_r,\cF(\ell^2), d \mu_r)$.
These then determine well-behaved vectors in $\cH_r^{phys}$ and $\cH^{phys}$. We start
with the fact the for each $p$ the spaces $\cH'(p)$ are non-trivial (finite dimensional )
vector spaces.
\begin{lem} \label{lemma}
$\Ga(\La)$ is a bijection from $\cH'( p)$
to $\cH'(\La p)$ .
\end{lem}
\pr
First note that $\Ga(\La)$ is defined on $\cH'(p)$
since $\cH'(p) \subset \cD_0$.
Then we use (\ref{trans})
to conclude that $\Ga( \La)^{-1} M^2\ \Ga( \La) =M^2 $ and that
$\Ga( \La)^{-1} L_m(p)\ \Ga( \La) =
L_m(\La^{-1} p) $.
This gives the result.
\begin{lem}
For any $q \in V^{(+)}_r $ there is a neighborhood
$U$ of $q$ in $V^{(+)}_r $ and a smooth family $\La(p,q)$ of Lorentz transformations
defined for $p \in U$ such that $\La(p,q) \ q = p$.
\end{lem}
\pr The proof uses the following well-known fact (see for example \cite{War83}). Let
$G$ be a Lie group with closed subgroup $H$. Let $\pi: G \rightarrow G/H $ be
the projection onto the coset space. Then there are local smooth
sections. That is for any point in $G/H$ there is a neighborhood $U$
and a smooth map $s: U \rightarrow G$ such that $\pi \circ s = id $.
In our case
let $\cL$ be the proper Lorentz transformations ($\cL$ = a component of $SO(d-1,1)$).
Fix $q$ and let $H$ be the subgroup
which leaves $q$ invariant. (e.g. if $r>0$ then $H = SO(d-1)$).
Since $\cL$ acts transitively on $V^{(+)}_r$ we have that the map
$\La \to \La q$ from $\cL $ to $V^{(+)}_r$ lifts to a diffeomorphism
$\La H \to \La q$ from the coset
space $\cL/H$ to $V^{(+)}_r$. This identifies the
two spaces.
Now from the general result there is a neighborhood $U$ of $H$ in $\cL/H$
and a map $s: U \to \cL$ such that
$s(\La H ) H = \La H$ for all $\La H \subset U$. Equivalently
we can regard $U$ as a neighborhood of $q$ in $V^{(+)}_r$ and
have a map $s: U \to \cL$ satisfying $s(p)q =p$ for all $p \in U$. Defining $\La(p,q)
= s(p)$ we have the result.
\bigskip
\begin{lem} Let $q, U$, and $\La(p,q)$ be as above. \label{states}
\begin{enumerate}
\item $\Ga(\La(p,q))$ is a bijection from $\cH'(q)$
to $\cH'(p)$.
\item There exist $\psi \in \cC^{\infty}(U, \cF(\ell^2))$
such that $\psi(p) \in \cH'(p)$ for all $p \in U$.
\item There exist $\psi \in \cC^{\infty}_0(V^{(+)}_r, \cF(\ell^2))$
such that $\psi(p) \in \cH'(p)$ for all $p \in V_r$, i.e. $\psi \in \cH_r'$.
\end{enumerate}
\end{lem}
\pr The first follows from lemma \ref{lemma}. For the second let
$\psi_0 \in \cC^{\infty}_0( U , \cH'(q))$ and take $\psi(p) = \Ga( \La(p,q)) \psi_0(p)$.
Multiplying by
$\chi \in \cC^{\infty}_0(U)$ gives a functions satisfying the third condition.
By adding functions for different neighborhoods $U$ we get a rich class
of functions. \bigskip
\newpage
\res
\begin{enumerate}
\item There is another way to look at this result. Let $\xi$ the set of all
pairs
$\{(p,\psi) \}$ where $p \in V_r$ and $\psi \in \cH'(p)$,
and let $\xi_U$ be the subset of pairs with $p \in U$.
We have smooth mappings $(p , \psi) \to (p,\Ga( \La(p,q)^{-1} ) \psi)$
from $\xi_U$ to $U \times \cH'(q)$ and hence from $\xi_U$ to
$U \times \bbC^s$ where $s$ is the dimension of $\cH'(q)$.
Covering $V_r^{(+)}$ by such charts gives
$\xi$ the structure of a smooth vector bundle.
Such bundles have smooth sections which our result.
\item If $r>0$ then the neighborhood $U$ can be
taken to be all of $V^+_r$. In this case we can take $q = (\sqrt{r},0,0,..)$
and let $\La(p,q)$ be the standard boost to $p$. ( See Weinberg \cite{Wei95},
equation (2.5.24) for the explicit formula).
\end{enumerate}
\section{String field theory}
We develop the string field theory by
taking the dynamical equation satisfied by the
single string wave equation,
specializing to real solutions, treating these solutions as
a classical Hamiltonian system, quantizing this
system, and then finally imposing the constraints.
This is ''second quantization'' , and the exact status
of this process is
always a little ambiguous \ldots which quantization was the genuine quantization?
Whatever attitude one takes one ends up at the same place.
In any case the quantization process is just meant to be suggestive
of a true quantum theory.
Our formulation of the problem uses techniques which have been useful
in the study of quantum field theory in curved spacetime
\cite{Ish78}, \cite{Dim80}, \cite{Dim92}.
\subsection{String field equation}
We start by defining real elements of $\cF(\ell^2)$. These are vectors
satisfying $C \psi = \psi$ where $C$ is some conjugation on $\cF(\ell^2)$.
A conjugation is an anti-linear isometry satisfying $C^2 = 1$. We
also want our conjugation to satisfy $[C,M^2]=0$ and $[C,\cJ]=0$.
Then also $\overline{<\psi, \chi> }= $.
For example one could take $ C_0 = \Ga(c_0)$ where $c_0$ is
the usual conjugation $c_0 \psi = \bar \psi $ on $\ell^2$.
In the following we just suppose that some $C$ satisfying the
above conditions has been chosen.
Now we study the Klein-Gordon equation:
\begin{equation}
(- \square + M^2) U=0
\end{equation}
for functions $U: \bbR^d \to \cF(\ell^2) $.
Given real $F_0,G_0 \in \cC^{\infty}_0(\bbR^{d-1}, \cF(\ell^2))$
there is a unique smooth real solution $U$ such that $U=F_0$
and $\pa U/ \pa x^0 = G_0$ on some surface $x^0 = t_0$, called
a Cauchy surface.
The solution has compact support on any other Cauchy surface $x^0 =t$.
Such solutions will be called
\textit{regular}.
Associated with this equation there is a real bilinear form.
For any functions
$U(x) = U(x^0, \vec x)$ and $V(x) = V(x^0, \vec x)$ it is defined by
\begin{equation}
\sigma_t (U,V) = \int_{x_0 = t }( __-
< \frac{\pa U}{ \pa x^0}(x) ,V(x)>)\ d \vec x
\end{equation}
Green's identity states that for $t>s$
\begin{equation}
\begin{split}
&\sigma_t (U,V)-\sigma_s (U,V) = \\
&\ \ \ \ \ \ \int_{s < x^0 < t}
(____
- <((-\square +M^2) U)(x),V(x)> )
dx \\
\end{split}
\end{equation}
Thus if $U,V$ are regular solutions then $\sigma_t(U,V)$ is independent of
$t$ and is just denoted $\sigma(U,V)$.
This form is skew symmetric and non-degenerate on the space of
regular solutions, i.e. it is symplectic. This symplectic form will be
the basis on quantization.
But first we develop some facts about fundamental solutions
for our KG equation. The following results are standard
for the scalar KG equation.
Advanced and retarded fundamental solutions $E^{\pm}$ are
operators on functions
$F \in \cC^{\infty}_0(\bbR^d , \cF(\ell^2))$ defined by
\begin{equation}
(E^{\pm} F)(x) = \frac{1}{(2 \pi)^{d/2}} \int_{\Ga_{\pm} \times \bbR^{d-1}}
\frac{ e^{ip\cdot x}}{p^2 + M^2} \
\tilde F(p) dp
\end{equation}
The $p^0$ contour $\Gamma_{\pm} $ is the real line shifted slightly above/below
the real axis. The exact choice does not matter because $\tilde F$ is entire
and rapidly decreasing in real directions.
Also for $x \in\bbR^d$ let $J^{\pm}(x) = \{ y: (x-y)^2 <0, \pm(x^0-y^0) >0 \} $
be the
past or future of $x$. For $A \subset \bbR^d$
define $J^{\pm} (A) = \cup_{x \in A} J^{\pm}(x)$.
\begin{lem} For $F \in
\cC^{\infty}_0(\bbR^d , \cF(\ell^2))$ we have
$(-\square + M^2) E^{\pm} F=F$ and $supp(E^{\pm} F) \subset J^{\pm} (suppF)$.
\end{lem}
\pr
Let $P_r$ be the operator on $\cF(\ell^2)$ which is the
projection onto the eigenspace $M^2 = r$.
On the range of $P_r$ we have $E^{\pm}F = E^{\pm}_r F$ where
\begin{equation}
(E_r^{\pm} F)(x) =\frac{1}{ (2 \pi)^{d/2}} \int_{\Ga_{\pm} \times \bbR^{d-1}}
\frac{ e^{ip\cdot x}}{p^2 + r}
\ \tilde F(p) dp
\end{equation}
These are the advanced/retarded fundamental solutions
for the Klein-Gordon equation with mass $r$ and they satisfy $(-\square + r)E_r^{\pm}F=F$
and $ supp (E_r^{\pm} F) \subset J^{\pm}( supp F)$.
Now $ E^{\pm}$ are fundamental solutions since $P_r (- \square + M^2 ) E^{\pm}F
= (- \square + r ) E^{\pm}_r P_rF =P_rF$. For the support property we have
$supp(E^{\pm} F) \subset \cup_r supp (P_r E^{\pm} F)$ .
But $P_r E^{\pm} F = E_r^{\pm}P_r F $ and
$supp( E_r^{\pm}P_r F) \subset J^{\pm} (supp (P_rF)) \subset J^{\pm}(supp F)$
and hence the result.
\bigskip
The propagator function is defined by $E = E^+ - E^-$.
Then $U = EF$ is a regular solution. In fact we have:
\begin{lem} \
\begin{enumerate}
\item $U$ is a regular solution of $(- \square + M^2)U =0$ iff it can be written $U = EF$
with $F \in
\cC^{\infty}_0(\bbR^d , \cF(\ell^2))$
\item $F \in \cC^{\infty}_0(\bbR^d , \cF(\ell^2))$ satisfies $EF=0$ iff
$F= (- \square + M^2)H$ for some
$H \in \cC^{\infty}_0(\bbR^d , \cF(\ell^2))$
\end{enumerate}
\end{lem}
\pr Let $U$ be a regular solution and let $ \theta = \theta (x^0)$
be smooth and satisfy $\theta = 1 $ for $x^0$ sufficiently positive
and $\theta = 0$ for $x^0$ sufficiently negative. Define
$F=(- \square +M^2) ( \theta U) = -(-\square + M^2) (( 1- \theta ) U)$.
Then $F$ has compact
support. Next note that $ \theta U = E^+ F$ since the difference satisfies the $KG$
equation and vanishes in the distant past.
Similarly $- (( 1- \theta ) U) = E^-F$.
Taking the difference of the last two
equations gives $U=EF$ . This proves the first claim.
For the second suppose $F = (- \square + M^2 )H$.
Then $EF= (- \square + M^2 )EH =0$. On the other hand if $EF=0$, then $H = E^{\pm}F $
has compact support and $ (- \square + M^2 )H = F$.
\bigskip
The next identity establishes a connection
between any solution and its
values on any Cauchy surface (all in the sense of distributions).
We define
\begin{equation}
____ = \int ____ dx
\end{equation}
\begin{lem} For $F \in \cC^{\infty}_0(\bbR^d , \cF(\ell^2))$
and any regular solution $U$ :
\begin{equation} \label{connection}
\sigma(U,EF) = ____
\end{equation}
or equivalently For $F,G \in \cC^{\infty}_0(\bbR^d , \cF(\ell^2))$
\begin{equation} \label{connection2}
\sigma(EF,EG) = = -
\end{equation}
\end{lem}
\pr By Green's identity we have for $t<0$
\begin{equation}
\si_0 (U,E^+ F) - \si_t (U,E^+ F) = \int_{t dx
\end{equation}
Letting $t \to -\infty$ we get an expression for $\si_0 (U,E^+ F) $.
Similarly we get an expression for $\si_0 (U,E^- F) $ . They
are
\begin{equation}
\sigma_0(U,E^{\pm}f) = \pm \int_{ \mp x^0 >0} ____ dx
\end{equation}
Take the difference to obtain the result.
\bigskip
We next want to make a connection with the single string Hilbert space
$\cH=\oplus_r \cH_r$.
Given $F \in \cC^{\infty}_0(\bbR^d , \cF(\ell^2))$
we define $\Pi F \in \cH$
by specifying that $(\Pi F)_r \in \cH_r$ is
obtained by taking the Fourier transform, projecting onto the
subspace $M^2=r$ with $P_r$, and then restricting to $V^{(+)}_r$.
More precisely for $p \in V^{(+)}_r$ we define
\begin{equation}
(\Pi F)_r(p) = \sqrt{ 2 \pi} \ P_r \tilde F ( p )
\end{equation}
We will need to exclude tachyons, so we restrict to
functions $F$ which take values in
\begin{equation}
\cF_+ (\ell^2) \equiv (\cF_0(\ell^2))^{\perp} \equiv \bigoplus_{j \geq 1} \cF_j(\ell^2)
\end{equation}
We have $M^2 \geq 0$ on this subspace. Hence for such $F$ , $(\Pi F)_{-2} =0$ and
hence $\Pi F \in \cH_+$ the no-tachyon subspace of $\cH$:
\begin{equation}
\cH_+ = \bigoplus_{r \geq 0} \cH_r
\end{equation}
\begin{lem} For (real) $F,G \in \cC^{\infty}_0(\bbR^d , \cF_+(\ell^2))$
\begin{equation} \label{basiceq}
\sigma(EF,EG) = = 2 {\rm Im} <\Pi F, \Pi G>
\end{equation}
\end{lem}
\bigskip
\pr Only the second identity needs proof.
We compute with $E_r = E_r^+ - E_r^-$
\begin{equation}
\begin{split}
=& \sum_{r \geq 0}< E_rP_rF, P_r G >\\
=& \sum_{r\geq 0}2\ {\rm Im} <(\Pi F)_r, (\Pi G)_r>
= 2 \ {\rm Im} <\Pi F, \Pi G> \\
\end{split}
\end{equation}
The second step follows since for $r \geq 0 $ and any $F,G$
\begin{equation}
\begin{split}
= & -\int_{\Gamma_+ - \Gamma_-}
< \tilde F( \bar p) , \tilde G(p)>\frac {1} { p^2 +r} dp \\
= &-2 \pi i\{\int < \tilde F(\om_r(\vec p), \vec p) ,
\tilde G(\om_r(\vec p),\vec p)>
\frac { d \vec p }{2\om_r(\vec p)} -c.c. \} \\
= & \ 2\ {\rm Im} < \sqrt{2\pi} \tilde F | V_r^+, \sqrt{2\pi} \tilde G | V_r^+> \\
\end{split}
\end{equation}
In the second step we have evaluated the contour integral
by taking residues at $p^0 = \pm \om (\vec p)$. We have also
used
$ \overline{< \tilde F(p) , \tilde G(p)>} =<\cC \tilde F(p) ,\cC\tilde G(p)>
= < \tilde F(-p) , \tilde G(-p)>$ for $p$ real , a consequence of the reality of
$F,G$.
\subsection{String field operator}
Now we quantize solutions of the string field equation.
We take as our phase space the space of all regular solutions $\Phi$
of $(-\square + M^2) \Phi =0$ with symplectic form
$\sigma( \Phi, \Phi')$ defined
previously. For each solution $U$ there is a function $\Phi \to
\sigma(\Phi, U)$ on the phase space.
We quantize these functions by replacing them by operators on a complex Hilbert
space, also denoted $\sigma(\Phi, U)$,
which are required to satisfy
\begin{equation} \label{comm1}
[\sigma (\Phi, U), \sigma (\Phi, V)] = i \sigma (U, V)
\end{equation}
This looks more familiar if we identify solutions with
their data on some Cauchy surface. Then the operators
are $\si(\Phi_0,\Pi_0;F_0,G_0) = \Phi_0(G_0) - \Pi_0(F_0)$
and the commutator is written
$ [\sigma (\Phi_0, \Pi_0; F_0,G_0), \sigma (\Phi_0, \Pi_0; F'_0,G'_0)]
= i \sigma (F_0,G_0;F'_0,G'_0)$. As a special case we have the standard
$ [ \Phi_0(G_0), \Pi_0(F_0)] = i$.
The full spacetime field operator is obtained from the operators
$\sigma(\Phi, U)$ just as in the classical case.
Following (\ref{connection}) we define the field operator
as a distribution by
\begin{equation} \label{connection3}
\Phi (F) = \sigma ( \Phi,EF)
\end{equation}
Then $\Phi(F)$ satisfies the field equation and has
a local commutator as the next result shows.
\begin{lem} let $\sigma (\Phi, U)$ be a family of operators indexed
by regular solutions $U$ of the KG equation and satisfying
(\ref{comm1}). Then the operators $\Phi (F) = \sigma ( \Phi,EF) $ defined
for $F \in \cC^{\infty}_0( \bbR^d, \cF(\ell^2))$
satisfies
\begin{equation}
\begin{split}
\Phi( (-\square + M^2 )F) = & 0 \\
[ \Phi (F), \Phi(G)] =& -i < F,EG > \\
\label{basic}
\end{split}
\end{equation}
Furthermore every operator valued distribution $\Phi(F)$
satisfying (\ref{basic}) arises in this way.
\end{lem}
\pr The field equation
follows from $E (- \square + M^2) F =0$
and the commutator follows from the identity (\ref{connection2}).
For the converse given $\Phi(F)$
we define $ \sigma ( \Phi,U) = \Phi(F)$
for any $F$ such that $U = EF$.
To see that this is well defined we have to show that if $EF_1 = EF_2$
then $\Phi(F_1) = \Phi(F_2)$, or if $EF=0$ then $\Phi(F) = 0$.
But we have seen that $EF=0$ implies $F = (-\square + M^2) H$ and hence the result follows.
The operators $ \sigma ( \Phi,U)$ have the commutator (\ref{comm1})
again by the
identity (\ref{connection2}).
\bigskip
\res
\begin{enumerate}
\item
Since $\sigma (U, V) $ is a symplectic form, representations of
(\ref{comm1}) do exist on general principles.
Thus string field theories exist. Furthermore the spacetime field $\Phi(F)$
defined by (\ref{connection3}) is strictly local because if $supp ( F)$ and $supp (G)$
are spacelike separated, then $supp (EF)$ and $supp (G)$ do not overlap
and hence $[\Phi(F), \Phi(G)] = -i =0$.
All this holds without suppressing the negative mass part of the equation!
\item
However this is not the end of the story. We actually
want the particular representation in which time translation is unitarily implemented
with positive energy. (One can think of this as the forward moving condition
again). Choosing a particular representation requires a complex structure
or a "one-particle structure" on phase space.
These are equivalent to expressing the symplectic form $\sigma(EF,EG)$ as the
imaginary part of an inner product on some complex Hilbert
space. But if we suppress the tachyon
then this has already been accomplished in (\ref{basiceq}) where
it is written as $ 2\ {\rm Im} <\Pi F, \Pi G> $ . Furthermore it
is this choice which is associated with positive energy as we shall see.
\end{enumerate}
These considerations lead to the following definition. Tachyons are
completely suppressed.
We bypass $\si(\Phi, U)$ and go directly to operators
$\Phi(F)$ satisfying (\ref{basic}).
Also we enlarge the class of test functions from compact support
to the Schwartz space
of smooth rapidly decreasing functions.
The Hilbert space is the Fock space over the no-tachyon single string Hilbert space
$\cH_+$:
\begin{equation}
\cK = \cF(\cH_+)
\end{equation}
This has the indefinite inner product $< \Psi, \Xi> = (\Psi, \Ga(\cJ) \Xi)$.
States in $\cK$ with finitely many entries are denoted $\cK_f$.
\begin{defn} (The String Field).
For $ F \in \cS(\bbR^d , \cF_+(\ell^2))$ we have $\Pi F \in \cH_+$ and
we define on $\cK_f$
\begin{equation}
\Phi (F) = a^{\dagger} (\Pi F) + a (\Pi F )
\end {equation}
\end{defn}
\begin{thm} \
\begin{enumerate}
\item The string field satisfies $\Phi(F)^{\da} = \Phi(F)$, the field
equation $\Phi((-\square + M^2) F) =0$, and has the commutator
$[ \Phi (F), \Phi(G)] = -i < F,EG >$.
\item There is positive energy representation $\cU(a, \La)$
of the inhomogeneous Lorentz group on $\cK$ such that
\begin{equation}
\cU(a, \La)\Phi(F) \cU(a, \La)^{-1}
= \Phi (F_{a, \La})
\end{equation}
where $F_{a, \La}(x) = \Ga(\La) F(\La^{-1}(x - a))$.
\end{enumerate}
\end{thm}
\pr The field equation is satisfied since $\Pi (-\square + M^2 )F=0$.
The commutator is evaluated as
\begin{equation}
[\Phi(F), \Phi(G) ]= 2i\ {\rm Im} <\Pi F, \Pi G> = -i
\end{equation}
since the identity (\ref{basiceq}) holds for $\cS$ as well as $\cC^{\infty}_0$.
The representation is defined by $\cU(a, \La) = \Ga ( U(a,\La))$.
We compute
\begin{equation}
\begin{split}
\cU(a, \La)\Phi(F) \cU(a, \La)^{-1}
= & a^{\dagger} (U(a, \La)\Pi F) + a (U(a, \La)\Pi F )\\
= & a^{\dagger} (\Pi F_{a, \La}) + a (\Pi F_{a, \La})\\
= & \Phi (F_{a, \La}) \\
\end{split}
\end{equation}
As noted we have the following corollary:
\begin{cor} ( Locality). If $F,G$ have spacelike separated supports
$[ \Phi (F), \Phi(G)] = 0$. \label{vanishing}
\end{cor}
\bigskip
Now we impose the constraint, and just as for the single string
this will give us a positive definite inner product.
Let $\hat L_m$ be the Fourier transform of $L_m$, that is
$\hat L_m$ is given by (\ref{L}) but with $p_{\mu} = -i \pa / \pa x^{\mu}$.
We would like to select states which are annihilated by
$ L_m \Phi $ for $m>0$. However, just as for
the Gupta- Beuler quantization of the electromagnetic
field \cite{StWi74}
we must compromise and only impose the condition on the
negative frequency part of the field defined by
$\Phi_{-} (F) = a(\Pi F)$. This is defined and
anti-linear on complex test functions. We look for
states annihilated by $(\hat L_m\Phi_-)(F) \equiv \Phi_-( \hat L_{-m}F)$.
This is fulfilled by taking the subspace
\begin{equation}
\cK' = \cF( \cH'_+) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cH'_+ = \cH' \cap \cH_+
\end{equation}
All $\Psi \in \cK'$ satisfy the required $a(\Pi \hat L_{-m} F)\Psi =0$
since if $\psi \in \cH'_+$ then
\begin{equation}
< \Pi \hat L_{-m} F, \psi> = < L_{-m} \Pi F, \psi> = < \Pi F, L_m\psi> =0
\end{equation}
Thus $\cK'$ is our constrained space, something we might have guessed
directly.
Next let $\cK'' = \cK ' \cap (\cK')^{\perp}$ be the isotropic vectors in $\cK'$.
and define
\begin{equation}
\cK^{phys} = \cK'/ \cK''
\end{equation}
This space inherits an indefinite inner product from $\cK'$.
\begin{lem} For $d=26$
the inner product on $\cK^{phys}$ is positive definite and we
have
the identification of Hilbert spaces:
\begin{equation}
\cK^{phys} =\cF(\cH^{phys}_+) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cH^{phys}_+= \cH'_+/\cH''_+
\end{equation}
\end{lem}
\pr $\cH''_+ = \cH'' \cap \cH_+$ is a closed subspace of $\cH'_+$ and so we can
write $\cH'_+ = \cH''_+ \oplus \cM $ where $\cM$ is the orthogonal
complement with respect to the definite inner product.
The projection onto $\cM$ has kernel $\cH''_+$ and gives
an identification of $\cM$ with $\cH^{phys}_+$ which preserves
the indefinite inner product. Thus the inner product is
positive definite on $\cM$, and of course zero if
either entry is in $\cH''_+$.
Now we have the identification of Hilbert spaces \cite{GlJa71}
\begin{equation}
\cK' = \cF(\cH''_+ \oplus \cM ) = \cF(\cH''_+) \otimes \cF(\cM)
\end{equation}
Under this identification $\Ga(\cJ) = \Ga(\cJ) \otimes \Ga(\cJ) $
and so the induced indefinite inner product satisfies
$< \Psi_1 \otimes \Psi_2, \Psi'_1 \otimes \Psi'_2> =
< \Psi_1, \Psi'_1>< \Psi_2, \Psi'_2> $.
Splitting $ \cF(\cH''_+)= \cF_0(\cH''_+)\oplus \cF_+(\cH''_+)$
and using $\cF_0(\cH''_+) \approx \bbC$ we have
\begin{equation}
\cK' = \cF(\cM) \ \oplus ( \cF_+(\cH''_+) \otimes \cF(\cM))
\end{equation}
with the natural indefinite inner product. Every component
of $\cF_+(\cH''_+) \otimes \cF(\cM)$ has at least one factor
in $\cH''_+$ and so we can identify
\begin{equation}
\cK'' = \cF_+(\cH''_+) \otimes \cF(\cM)
\end{equation}
Thus
\begin{equation}
\cK^{phys} = \cF(\cM) = \cF(\cH^{phys}_+)
\end{equation}
These identifications preserve the indefinite inner product.
Since the inner product is positive definite on $\cH^{phys}_+$
it is positive definite on $\cK^{phys} $.
\bigskip
For certain test functions the
string field operator $\Phi(F)$ on $\cK$ determines an
operator on $\cK^{phys}$. We define
\begin{defn}
$F \in \cS(\bbR^d, \cF_+(\ell^2))$
is a \emph{constrained} test function if $\Pi F \in \cH'_+$
\end{defn}
To get real constrained test functions it is useful to pick
a particular conjugation on $\cF(\ell^2)$.
It is $C_1 = \Ga(c_1)$
where $c_1$ on $\ell^2$ is defined by
$(c_1 f)_{0n} = \bar f_{0n}$ and $(c_1 f)_{kn} = -\bar f_{kn}$
for $k=1,...,d-1$. For the next result real means $C_1 \psi = \psi $.
\begin{lem}
Non-trivial (real) constrained test functions exist
\end{lem}
\pr Take $r \geq 0$ and choose $\psi_0 \in \cC^{\infty}_0 (V_r^+,\cF(\ell^2))$
so that $\psi_0(p) \in \cH'(p)$ for all $p \in V_r^+$. Then $\psi$ defines an element of
$\cH_r'$ and hence an element of $\cH'_+$.
We have seen that such functions exist in lemma \ref{states}.
Next define
\begin{equation}
\psi( \om_r(\vec p), \vec p) = \psi_0( \om_r(\vec p), \vec p)
+C_1 \psi_0( \om_r(\vec p), -\vec p)
\end{equation}
This satisfies $ C_1\psi( \om_r(\vec p), \vec p)=\psi( \om_r(\vec p), -\vec p)$
and is still an element of $\cH'_r$. This is so since
$C_1 \al^0_n C_1 = \al^0_n$ and $C_1 \al^k_n C_1 = -\al^k_n$
and hence $C_1 L_m( \om_r(\vec p), \vec p) C_1 = L_m( \om_r(\vec p),- \vec p)$.
We will find $F$ so that $\Pi F= \psi$.
First write $\psi(\om_r(\vec p), \vec p) = h(\vec p)$ for a function
$h \in \cC^{\infty}_0( \bbR^{d-1}, \cF(\ell^2))$. (Or $\bbR^{d-1} - \{ 0\}$ if $r=0$).
Let $\chi \in \cC^{\infty}_0 (\bbR)$ be real and satisfy $\chi(0) =1$.
We define $F$ by specifying that the Fourier transform be
\begin{equation}
\tilde F (p^0, \vec p) = (2 \pi)^{-(1/2) } \chi(-(p^0)^2 + |\vec p|^2 +r )h( \vec p)
\end{equation}
Then $\tilde F$ is smooth and has compact support and
hence $F \in \cS(\bbR^d, \cF(\ell^2))$. Since $h( \vec p) \in Ran
P_r$
we have $(\Pi F)_r = \psi$ and $(\Pi F)_{r'} = 0$ for $r' \neq r$ as required.
Since elements of $Ran P_r$ have no zero component in Fock space, this is true
of $h( \vec p) $ and hence $F(x)$. Thus $F$ takes values
in $\cF_{+}$. Finally we have $C_1 \tilde F (p) = \tilde F (-p)$
and hence $C_1 F(x)= F(x)$ so $F$ is real.
\bigskip
Recall that $\cK_{f}$ is the subspace of $\cK$ with a
finite number of entries. Similarly define $\cK'_f$
and $\cK''_f$ and $\cK^{phys}_f =
\cK'_f/\cK''_f$. One can identify $\cK^{phys}_f$
with a dense subspace of $\cK^{phys}$.
\begin{thm} (Observable fields for $d=26$)
\begin{enumerate}
\item
Let $F \in \cS(\bbR^d, \cF_+(\ell^2))$ be a constrained test function. Then
$\Phi(F)$ on $\cK_f$ lifts to an operator $\Phi(F) $ on $\cK^{phys}_f$ called
an observable field
\item These satisfy
$[ \Phi (F), \Phi(G)] = -i < F,EG >$.
\item
The representation $\cU(a, \La)$ on $\cK$ lifts to a unitary representation
$\cU(a, \La)$ on $\cK^{phys}$ and
\[\cU(a, \La)\Phi(F) \cU(a, \La)^{-1}
= \Phi (F_{a, \La}) \]
\end{enumerate}
\end{thm}
\pr Since
$\Pi F \in \cH'_+$
we have that $\Phi(F)$ preserves $\cK'_f$.
It also preserves $\cK''_f$ since if $\Psi \in \cK''_f$
and $\Xi \in \cK_f'$
then
$< \Xi, \Phi(F) \Psi>= < \Phi(F)\Xi, \Psi>=0$.
Since $\cK'_f$ is dense in $\cK'$
we have $< \Xi, \Phi(F) \Psi>=0$ for all $\Xi \in \cK'$
and hence $ \Phi(F) \Psi \in \cK''_f$.
Hence $\Phi(F)$ acts on $\cK^{phys}_f$.
The commutator follows from the commutator on $\cK_f$
For the covariance first note that $F$ is constrained if and only if $F_{a, \La}$
is constrained.
This follows from the identity
$\Pi F_{a, \La} = U(a, \La) \Pi F$ and the fact that $U(a, \La)$
preserves $\cH'_+$.
The operator $\cU(a, \La)$ preserves $\cK'_f$ since
$U(a, \La)$ preserves $\cH'_+$. We argue as before that
it also preserves $\cK''_f$ and so it lifts.
The unitarity follows since it is inner product preserving,
and the identity lifts from the identity on $\cK_f$.
\bigskip
\re According to this theorem the observable fields
have a local commutator.
But can the fields themselves be localized? That is, are
there constrained
test functions $F \in \cC^{\infty}_0(\bbR^d, \cF_+(\ell^2))$?
Or is there some other way to get strictly localized operators?
These are open questions.
Without strictly localized fields we cannot get a
vanishing result like Corollary \ref{vanishing}. The best we
can do is the following approximate result. If $F,G$ are
constrained then it estimates the commutator for observable
fields on $\cK^{phys}$. Otherwise it refers to fields on $\cK$.
\begin{cor} let $F,G \in \cS(\bbR^d, \cF_+(\ell^2))$,
and let $a$ be in the spacelike region $|a^0| < (1-\ep)|\vec a|$.
Then as $|a| \to \infty$, we have for any $n>0$
\begin{equation}
[\Phi(F_a), \Phi(G)] = -i =\cO( |a|^{-n} )
\end{equation}
\end{cor}
\pr Since $EG$
is bounded we have
\begin{equation}
\begin{split}
|| \leq &\one \int_{supp(EG)} \|F(x-a)\| dx\\
\leq & \one \int_{supp(EG)} (1 + |x-a|)^{-n-d-1} dx\\
\leq & \one d(a, supp (EG))^{-n} \\
\leq & \cO ( |a|^{-n}) \\
\end{split}
\end{equation}
In the last step we use the fact that $supp (EG)$ is contained
in a set of the form $\{ x \in \bbR^d: |\vec x| \leq x^0 +C \}$.
We omit the details.
\newpage
\begin{thebibliography}{99}
\bibitem {Bog74} J. Bognar, Indefinite Inner Product Spaces,
Springer-Verlag (1974).
\bibitem{Bro72} R.C. Brower, Spectrum-generating algebra and
no-ghost theorem for for the dual model, Phys. Rev. D6, (1972),
1655.
\bibitem {Dim80} J. Dimock, Algebras of local observables on
a manifold, Commun. Math. Phys. 77, (1980), 219-228.
\bibitem {Dim92} J. Dimock, Quantized electromagnetic field on
a manifold, Rev. Math. Phys. 4, (1992), 223-233.
\bibitem {Dim00} J. Dimock, Locality in Free String Field
Theory, J. Math. Phys. 41, (2000), 40-61.
\bibitem {FGZ86} I.B. Frenkel, H. Garland, G. J. Zuckerman,
Semi-infinite cohomology and string theory, Proc. Nat. Acad. Sci. 83,(1986)
8442-8446.
\bibitem {GlJa71} J. Glimm, A. Jaffe, Quantum field theory
models, in Statistical Mechanics and Quantum Field Theory,
C. DeWitt, R. Stora, eds. , Gordon and Breach, New York, (1971).
\bibitem {GoTh72} P. Goddard, C. Thorn, Compatibility of the dual
pomeron with unitarity and the absence of ghosts in the dual
resonance model, Phys. Lett. 40B, (1972), 235.
\bibitem{GSW87} M. Green, J. Schwartz, E. Witten, Superstring Theory,
Cambridge University Press (1987).
\bibitem {GrHu93} H. Grundling, C.A. Hurst, The operator
quantization of the open bosonic string field algebra, Commun. Math. Phys.
(1993), 473-525.
\bibitem{Haa92} R. Haag, Local Quantum Physics, Springer-Verlag (1992).
\bibitem {HaOd97} H. Hata, H. Oda,
Causality in covariant string field theory, Phys. Lett. B 394,
(1997), 307-314.
\bibitem {HiPh57} E. Hille and R.S. Phillips, Functional analysis
and semi-groups, American Mathematical Society, Providence, (1957).
\bibitem{Ish78} C. Isham, Quantum field theory in curved spacetimes:
a general mathematical framework, in Differential Geometrical Methods
in Mathematical Physics II, K. Bleuler, H. Petry, A. Reetz, eds,
Springer-Verlag (1978).
\bibitem {Low94} D. Lowe, Causal properties of free string field
theory, Phys. Lett. B326 (1994), 223-230.
\bibitem {Mar93} E. Martinec, The light cone in string theory,
Class. Quant. Grav. 10 (1993) L187-L192.
\bibitem{Nie80} O. Nielsen, Direct integral theory, Marcel Dekker,
New York, (1980).
\bibitem {ReSi75} M.Reed, B.Simon, Methods of Modern Mathematical Physics II,
Academic Press, New York, (1975).
\bibitem {StWi74} F. Strocchi, A. Wightman, Proof of the charge superselection
rule in local relativistic quantum field theory, J. of Math. Phys. 15, (1974),
2198-2224.
\bibitem{Tho89} C. Thorn, String field theory, Phys. Rep. 175, (1989), 1-101.
\bibitem{War83} F. Warner, Foundations of Differentiable Manifolds
and Lie Groups, Springer-Verlag (1983).
\bibitem{Wei95} S. Weinberg, The Quantum Theory of Fields I, Cambridge
University Press (1995).
\end{thebibliography}
\end{document}
---------------0101290933946--
__