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\begin{document}

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\def\AUTHORA{Theodore G. Erler}
\def\EMAILA{terler@physics.ucsb.edu}
\def\TITLE{A fresh look at midpoint singularities
in the algebra of string fields.}

\begin{titlepage}

\title{\Large\bf \TITLE}
\author{{\large  \AUTHORA}
\thanks{Email:\EMAILA} \\
\\ Department of Physics
\\ University of California
\\ Santa Barbara, CA}
\maketitle \thispagestyle{empty}

\begin{abstract}
In this paper we indulge in a general discussion of outstanding
conceptual issues in the operator/Moyal approach to open string
field theory, especially with regard to the role of singularities
encountered in this approach. We argue that the singularities are
not artifacts of the formalism, but persist in any basis and have
a concrete physical interpretation in terms of ambiguities in the
star algebra itself, i.e. associativity anomalies. We also study
the free string hamiltonian $L_0$ and explain why it must be
singular when formulated algebraically in terms of the star
product.

\end{abstract}

\end{titlepage}

\section*{I. Introduction}




The business of covariant string field theory began with a simple
insight, due to Witten\cite{Witten}: If a string field is imagined
as a matrix whose indices correspond to the left and right halves
of the string, matrix multiplication generates an associative
algebra suitable for describing string interactions. While this is
an intuitive and fundamental insight, Witten himself was the first
to warn that it should be taken with a grain of salt. A string
field is really a functional of a full and continuous string; to
make it a matrix we would have to split the string into halves,
which brings up all sorts of uncomfortable questions, for example
how to set boundary conditions at the midpoint, how to treat the
midpoint itself, what the left/right splitting does to the kinetic
term, midpoint anomalies, etc. Ghosts complicate the picture even
more, since in the ghost sector star multiplication is not really
matrix multiplication at all, even heuristically.

Nevertheless, the possibility that the algebra of string fields
could be understood in terms of operator algebras is too
compelling to ignore. A vast literature, both old\cite{Old} and
new explores the ramifications of this idea, but for some reason
the approach has always seemed to fall short of the mark. The
general feeling has been that an operator algebraic formulation is
too singular, essentially for reasons mentioned above. Much
preferred has been the more reliable oscillator/conformal field
theory methods\cite{LeClair, Vertices}, and numerical analysis,
particularly level truncation\cite{Level-Truncation}.

Still, over the last two years the matrix string field idea has
been taken further than ever before, due both to a deepening
appreciation of the relationship between noncommutative geometry
and string theory\cite{Aref'eva_review, Douglas} and to the
development of vacuum string field theory (VSFT)\cite{VSFT}, a
model string field theory with a simple enough kinetic term that
it seemed possible to formulate VSFT in a matrix language without
having too much trouble with the
midpoint\cite{Split-strings,Gross-Taylor}. Central to these
developments was the realization, from two different standpoints,
that an operator formulation of the star product could be related
in a well-defined way to the three-string interaction vertex
$|V_3\rangle$ of \cite{Vertices}, which has been taken as the
basis for nearly all numerical and much analytical work in string
field theory. One way to see this was demonstrated by Bars and
Matsuo\cite{Bars-Matsuo}, who started from a regulated split
string\cite{Split-strings,Gross-Taylor,Bars} formalism and
constructed an oscillator vertex which agrees with the usual
$|V_3\rangle$ as the regulator is taken to
infinity\footnote{Bars\cite{Bars} was the first author to
formulate the open string star product explicitly as a Moyal
product, while previous authors had formulated the star product as
matrix multiplication on the space of half-string states. The
Moyal product is, of course, isomorphic to the operator/matrix
product, at least naively. However we will see that, because the
phase space of string field theory is infinite dimensional, the
Moyal perspective has subtle advantages over the split string
approach.}. We call their perspective, which expands the string in
a countable (and when regulated, finite) number of string modes,
the {\it discrete Moyal formalism}. A second approach was
developed by Douglas et al\cite{Moyal}, who found a particular
basis for which $|V_3\rangle$ factorized into a continuous tensor
product of vertices, each describing an operator/Moyal product
with a continuously varying noncommutativity parameter,
$\theta(\k)=2\tanh\frac{\pi\k}{4}$, $0\leq\k<\infty$. We call
their perspective the {\it continuous Moyal formalism}.
%Recently there has been progress in understanding the structure of
%string field theory from both perspectives, but they are not quite
%equivalent, interestingly enough; the discrete Moyal formalism is
%related to the continuous through a singular change of basis.

Despite these interesting developments, skeptics may still suspect
that the operator/Moyal approach is too singular. The discrete
Moyal formalism attempts to deal with midpoint singularities, but
in a seemingly drastic way: one must according to
ref.\cite{Bars-Matsuo,MSFT} carefully define string field theory
as a ``large $N$'' limit of an infinite sequence of theories, each
of which is defined with a finite number $N$ of string modes; in
this way, many subtle midpoint issues can be satisfactorily
resolved. However, even if one could explicitly construct such an
infinite list of regulated theories and show that Witten's axioms
are satisfied (this remains a challenge), one feels that the whole
issue could be avoided by simply not using the operator/Moyal
language to start with. Yet, from the perspective of the
continuous Moyal formalism, the operator description of the
algebra doesn't look that bad; the basis seems less singular than
the discrete basis\cite{Moyal} and there is no obvious need to
regulate. But here too there are subtle singularities whose
relation to the midpoint has not been fully clarified.

In this paper we offer a few simple observations which will
hopefully bring these issues into sharper focus. Our essential
perspective is that all operator/Moyal formulations of the star
product are in a sense ``singular,'' but these singularities have
a physical origin in the definition of the star algebra itself;
they are not a priori the ``fault'' of the operator/Moyal
description. Therefore, the operator/Moyal approach brings a
particularly clear perspective into the role of midpoint
singularities in open string field theory. A fascinating aspect of
these methods is that they also suggest a natural way to formulate
the kinetic operator---the BRST operator and/or the Virasoro
generators---purely algebraically in terms of the star product.
Such formulations have recently featured prominently in studies of
nonperturbative solutions to the string field
equations\cite{Bars-SFE}. Such formulations are necessarily
singular, but for an understandable reason which we can hope to
control. Further investigations into the operator/Moyal approach
will undoubtedly bring a unique perspective on the subtle tension
and interplay between the BRST operator and the star product.

This paper is organized as follows: In the first section, we
review the mixed and discrete Moyal bases, mostly to remind the
reader of important concepts and to fix notation. In the second
section we focus on the continuous Moyal formalism and attempt to
illuminate its midpoint structure from a split string perspective.
In the third we argue that singularities in different
operator/Moyal formalisms can all be traced to physical
ambiguities of the star algebra itself, in particular the
Horowitz-Strominger associativity anomaly\cite{Horowitz}. This
anomaly is one of the more mysterious and elusive aspects of
string field theory, so we try to give a more careful treatment
than in other literature. In the third section we discuss some
interesting additional singularities in the ghost sector. Finally,
we investigate the Hamiltonian in Siegel gauge, $L_0$, formulated
algebraically in terms of the open string star product.

\section*{II. Basic concepts}
A classical open string field $|\Psi\rangle$ is an element of the
Fock space of a particular matter-ghost boundary conformal field
theory(BCFT):
$$|\Psi\rangle\in\mathcal{H}_\mathrm{BCFT}=\mathcal{H}_{matter}
\otimes\mathcal{H}_{ghost}$$ Typically, we are interested in the
BCFT corresponding to an open string living on a space-filling
D-25 brane, in which case we have Neumann boundary conditions in
all directions at the string endpoints. We can then expand the
worldsheet position and momentum operators on the semicircle
$|z|=1,\ \Im(z)>0$ in terms of $\cos n\sigma$ for $\sigma \in
[0,\pi]$:
\begin{eqnarray}&\ &\ \ x(\sigma)=x_0+\sqrt{2}\sum_{n=1}^\infty x_n\cos
n\sigma\ \ \ \ \
x_n=\frac{i}{n\sqrt{2}}(\alpha_n-\alpha_{-n})\nonumber\\
&\ &\pi p(\sigma)=p+\sqrt{2}\sum_{n=1}^\infty p_n\cos n\sigma\ \ \
\ \ \ \ p_n=\frac{1}{\sqrt{2}}(\alpha_n+\alpha_{-n}),\nonumber
\end{eqnarray} where $\alpha_n$ are the standard string mode oscillators
$[\alpha_m,\alpha_n]=m\delta_{m,-n}$ (suppressing Lorentz
indices). The operators $x_n$ and $p_n$ have associated
eigenstates:
\begin{eqnarray}&\ &\langle x_n|=\langle 0|_n
\exp\left[-\frac{n}{2}(x_n^2-2\sqrt{2}i\alpha_n x_n
-\alpha_n^2)\right]\nonumber\\ &\ &\langle p_n|=\langle 0|_n
\exp\left[-\frac{1}{2n}(p_n^2-2\sqrt{2}\alpha_n p_n
+\alpha_n^2)\right]\nonumber\end{eqnarray} where $x_n,p_n$ now
refers to the eigenvalue and $\langle 0|_n$ is the vacuum of the
Hilbert space generated by the $\alpha_n$'s. Focussing for the
moment to the matter sector, we find it extremely convenient to
express a string field as a functional of the {\it even} position
Fourier modes and the {\it odd} momentum Fourier modes:
\begin{equation}\Psi[x_{2n},p_{2n-1}]=\langle p|\prod_{n=1}^\infty \langle
x_{2n}|\langle p_{2n-1}| |\Psi\rangle
\label{eq:field}\end{equation} In this paper we will be interested
in singularities characterized by very large mode numbers, so we
might as well avoid the complication of zero modes and restrict
ourselves to zero momentum\footnote{In the context of the
continuous Moyal formalism, zero modes are discussed in
\cite{Belov}. In the discrete Moyal formalism zero modes do not
present so much of a complication.}. Given two string fields
expressed as in \EQN{field}, we can calculate their star
product\cite{Bars,Moyal} with the equation,
$$\Psi*\Phi[x_{2n},p_{2n-1}]=\mathcal{N}\ \Psi\star\Phi[x_{2n},p_{2n-1}],$$
where $\mathcal{N}$ is a normalization\footnote{This normalization
constant is quite troublesome, since it appears to be infinite. It
was hoped\cite{Moyal} that the constant would cancel between the
matter and ghosts in $D=26$, but the analytic work of
ref.\cite{Belov-Konechny} shows that this does not happen. The
author is not sure how this result should be interpreted, and how
much of a problem it truly represents. It was suggested in
ref.\cite{Moyal} that the normalization might be regulated by
deforming the Witten vertex into a vertex defined with a slightly
different set of conformal maps onto the unit disk, so that the
regulated open string algebra would be {\it non-associative}.
(Specifically, it would be a homotopy associative $A_\infty$
algebra\cite{Tensor}). The necessity of such a deformation would
obviously be quite unfortunate from the standpoint of the
operator/Moyal approach to string field theory.} and $\star$
denotes a canonically normalized Moyal product satisfying,
\begin{equation}[x_{2n},p_{2m-1}]_\star=2i\ T_{2n,2m-1}\label{eq:mixed}
\end{equation} where $T$ is a noncommutativity parameter,
\begin{eqnarray} T_{2n,2m-1}&=&\frac{4}{\pi}\int_0^{\pi/2}d\sigma
\cos 2n\sigma \cos(2n-1)\sigma\nonumber\\ &=&
\frac{2(-1)^{m+n+1}}{\pi}\left(\frac{1}{2m-1+2n}+
\frac{1}{2m-1-2n}\right)\label{eq:T-matrix}\end{eqnarray} This is
the most basic formulation of the open string star algebra in a
Moyal language, often referred to as the ``mixed basis.'' The
continuous and discrete Moyal formulations are more ambitious and
attempt a choice of basis for which the noncommutativity parameter
is diagonal.

The matrix $T$ is obviously important, so let's say a few more
words about it. We can think of $T$ as a linear map between two
half order Sobelev spaces, one of ``odd moded'' sequences,
$\mathcal{H}_{odd}$, and the other of ``even moded'' sequences
$\mathcal{H}_{even}$:
$$T:\mathcal{H}_{odd}\to\mathcal{H}_{even},$$
with,
\begin{eqnarray}&\
&\langle a,b\rangle=\sum_{n=1}^\infty (2n-1)a_{2n-1}b_{2n-1} \ \ \
\ a,b\in\mathcal{H}_{odd}\nonumber\\ &\ &\langle
s,t\rangle=\sum_{n=1}^\infty 2n s_{2n} t_{2n}\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \
s,t\in\mathcal{H}_{even}\label{eq:Sobelev}\end{eqnarray} As an
operator on $\mathcal{H}_{odd}$, $T$ has an
inverse\cite{Bars,Associative}, a matrix often called $R$ in the
literature:
\begin{eqnarray}R_{2m-1,2n}&=&\frac{4}{\pi}\int_0^{\pi/2}d\sigma
\cos(2m-1)\sigma[\cos 2n\sigma-(-1)^{n}]\nonumber\\
&=&\frac{4n(-1)^{n+m}}{\pi
(2m-1)}\left(\frac{1}{2m-1+2n}-\frac{1}{2m-1-2n}\right)\nonumber\\
&=&\frac{(2n)^2}{(2n-1)^2}T_{2n,2m-1}
\end{eqnarray} and
\begin{equation}R:\mathcal{H}_{even}\to\mathcal{H}_{odd},\ \ \ \ \
RT=1_{odd},\ \ \ TR=1_{even}\label{eq:T-inv}\end{equation} $T$ has
a natural domain $D(T)$ when extended to be a linear operator on
the Banach space of bounded sequences $\ell^\infty$. $D(T)$ is a
much larger space than $\mathcal{H}_{odd}$, and surprisingly $T$
has a zero mode in $D(T)$ \cite{Associative}:
\begin{equation}v_{2n-1}=\frac{2\sqrt{2}}{\pi}\frac{(-1)^{n+1}}{2n-1},
\ \ \ \ Tv=0\
\end{equation} This zero mode is not normalizable\footnote{The
normalization is chosen, as in \cite{Associative} so that
$\sum_{n=1}^\infty v_{2n-1}^2=1$.} in $\mathcal{H}_{odd}$,
consistent with the fact that $T$ is invertible there. Outside of
$\mathcal{H}_{odd}$ the equation $RT=1$ is simply no longer true.
Only when we are considering the action of bounded operators on a
Hilbert space can we multiply operators unambiguously before
evaluating their action on a particular state. In our case, on any
Hilbert space larger than $\mathcal{H}_{even}$, $R$ will not be a
bounded operator, so \EQN{T-inv} is not meaningful. The situation
is similar to the equation $\frac{1}{x}x=1$: when acting on a
suitably well-behaved space of functions, the equation makes
perfect sense; but if we allow it to act on a delta function, for
example, the equation is simply wrong: $\frac{1}{x}(x\delta(x))=0
\neq \delta(x)$. However, given the fact that $RT=1$ in
$\mathcal{H}_{odd}$, we can by extension naturally define
$RT\equiv 1$ on all of $D(T)$; but then, in some sense, we loose
associativity in operator/vector multiplication,
$$R(Tv)\equiv RTv=0\ \ \ \ (RT)v=v$$
These ``associativity anomalies''\cite{Associative} are deeply
connected with midpoint singularities, and play an important role
in our analysis.

Alternatively, we could define an operator $\bar{T}$, with the
same entries as $T$, but which conversely maps from even moded
sequences to odd moded sequences. In this case, however, it is
more natural to define the even/odd mode Hilbert spaces
$\mathcal{H}'_{even}$ and $\mathcal{H}'_{odd}$ to be $-\half$
order Sobelev spaces. On $\mathcal{H}'_{even}$, $\bar{T}$ has an
inverse, $\bar{R}$; but outside of $\mathcal{H}'_{odd}$, $\bar{R}$
has a zero mode,
\begin{equation}v'_{2n-1}=\frac{2\sqrt{2}}{\pi}(2n-1)
(-1)^{n+1}\label{eq:R-zero}\end{equation} as is readily seen from
\EQN{T-inv}. Again we run into problems with a zero mode. Note
that the completion of $\mathcal{H}'_{even}$ and
$\mathcal{H}'_{odd}$ with respect to the $\ell^\infty$ norm is the
dual space of $\mathcal{H}_{even}$ and $\mathcal{H}_{odd}$,
respectively; we write
$\overline{\mathcal{H}'_{even}}=\mathcal{H}^*_{even}$ and likewise
for $\mathcal{H}'_{odd}$.

Let us now briefly describe the discrete Moyal basis, which gives
the oldest and in some sense the best understood Moyal formulation
of the open string star product\cite{Bars,Bars-Matsuo}. There are
actually two convenient choices of basis, the discrete even and
the discrete odd: \begin{eqnarray}\mathrm{even:} &\
&x_{2n}=x_{2n}^D\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \
p_{2n-1}=\half \sum_{m=1}^{\infty}p_{2m}^D T_{2m,2n-1}\nonumber\\
\mathrm{odd:}&\ & x_{2n-1}^D=\half \sum_{m=1}^\infty
R_{2n-1,2m}x_{2m}\ \ \ \ \ \ \
p_{2n-1}^D=p_{2m-1}\label{eq:discrete}
\end{eqnarray} Substituting these relations into \EQN{mixed}, bearing in mind
\EQN{T-inv}, we find the simple result\footnote{Almost all of the
work in the discrete Moyal formalism has focused on the even
basis. The reason for this is mostly historical, since it was the
basis first considered in ref.\cite{Bars}. Ironically, this basis
is also perhaps the most problematic when unregulated.}
$$[x_{2n}^D,p_{2m}^D]_\star=i\delta_{2n,2m}\ \ \ \ \ \ \
[x_{2n-1}^D,p_{2m-1}^D]_\star=i\delta_{2n-1,2m-1}$$ An innocent
but very important assumption in this construction is that the
transformations \EQN{discrete} are invertible. As our previous
discussion makes clear, they are indeed invertible provided that
the odd indices operate in $\mathcal{H}_{odd}$ and the even
indices operate in $\mathcal{H}_{even}$. This is only true
provided that states in the open string star algebra satisfy
certain conditions. Whether or not these conditions are physically
reasonable is not clear, but it was shown for instance in
ref.\cite{Bars-Matsuo} that the perturbative vacuum is a
well-defined functional after the change of basis \EQN{discrete}
but the sliver\cite{K&P} is not.

Whatever the precise definition of the algebra of string fields
ends up being, it is clear that at least sometimes it is useful to
think about fields for which the transformation \EQN{discrete} is
not invertible. This is where problems arise. Consider, for
example, the half-string momentum functional rewritten in the even
basis: \begin{equation}P_L=\int_0^{\pi/2}d\sigma
p(\sigma)=\frac{\sqrt{2}}{\pi}\sum_{n=1}^\infty
\frac{(-1)^{n+1}}{2n-1}p_{2n-1}\ \ \Longrightarrow\ \
P_L=\sum_{n=1}^{\infty}0 p^D_{2n}=0\label{eq:P_L}\end{equation}
This unfortunate result is a direct consequence of the fact that
$T$ has a zero mode. In ref.\cite{Associative} this singularity
was explained as arising from the fact that the discrete even
basis imposes Neumann boundary conditions at the midpoint, making
it impossible to describe a string configuration with a midpoint
discontinuity. Therefore, the generator of this discontinuity,
$P_L$, vanishes. The discrete odd basis has an analogous problem:
\begin{equation} D=\frac{2\sqrt{2}}{\pi}\sum_{n=1}^\infty
\frac{(-1)^{n+1}}{2n-1}x_{2n-1}^D\ \ \Longrightarrow\ \
D=\sum_{n=1}^\infty 0x_{2n}=0 \label{eq:D}\end{equation} This
singularity occurs because the discrete odd basis can describe an
unphysical string configuration where the midpoint has
disconnected from the rest of the string\cite{Associative}. In the
next section we will discover analogous singularities to these in
the continuous Moyal formalism.

The strategy for dealing with midpoint singularities in the
discrete Moyal formalism has been to carefully regulate/deform the
theory into a theory containing only a finite number of string
modes. Then the analogue of $T$ is a finite dimensional invertible
matrix, and our previous remarks concerning subtleties of infinite
dimensional Hilbert spaces and zero modes does not apply. At the
end of our calculations, we take the number of string modes to
infinity to recover physical results. Much more detailed
discussion of this approach can be found elsewhere\cite{Bars,
Associative, Bars-Matsuo, MSFT}, but this is sufficient background
for our purposes.

We should mention another basis which has sometimes been discussed
in the literature\cite{PP,MSFT}, the string bit basis:
\begin{equation}x^b(\sigma)=\sqrt{\frac{8}{\pi}}\sum_{n=1}^\infty x^D_{2n-1}
\cos(2n-1)\sigma\ \ \
p^b(\sigma)=\sqrt{\frac{8}{\pi}}\sum_{n=1}^\infty p_{2n-1}
\cos(2n-1)\sigma \label{eq:bit}\end{equation} for
$\sigma\in[0,\frac{\pi}{2}]$, satisfying,
$$ [x^b(\sigma),p^b(\sigma')]_\star=2i\delta(\sigma-\sigma')$$ While this
basis might be useful for studying tensionless
strings\cite{PP,MSFT}, it is probably the most problematic of all
the bases discussed in this paper, principally because it does not
unambiguously impose the correct open string boundary conditions.
Equation (\ref{eq:bit}) defines the boundary conditions
implicitly, but other mode expansions could be used and the basis
would have identical appearance. To see why this is a problem,
consider the the tachyon mass-shell constraint:
$$ L_0\left[x(\sigma),\fraction{\delta}{\delta
x(\sigma)}\right]\langle x(\sigma)|p\rangle=0 $$ This equation is
not strictly meaningful unless we consistently expand $x(\sigma)$
in terms of functions with a particular set of boundary conditions
appropriate for the BCFT in question. Otherwise, for example, we
could expand $x(\sigma)$ with Neumann boundary in $L_0$, but then
with periodic boundary conditions in $\langle x(\sigma)|p\rangle$;
the above equation will then clearly not work. Therefore the bit
basis can only work effectively when expanded in a set of
functions with specific boundary conditions, such as in \EQN{bit}.
But of course, in doing this we are not really working in the bit
basis anymore.

\section*{III. The continuous Moyal basis and the midpoint}
Having described some background, let us turn to the continuous
Moyal formalism. This approach was originally derived from the
zero momentum three string vertex $|V_3\rangle$ \cite{Vertices},
specifically by choosing a basis of oscillators which diagonalized
the quadratic form defining the squeezed state expression of the
vertex\cite{Spectroscopy}. In this form, the vertex factorized
into a continuous tensor product of vertices, each of which
described a Moyal product. This approach has since been extended
to include zero modes\cite{Zero-spectroscopy,Belov},
ghosts\cite{Erler,Belov-Konechny}, superstring
interactions\cite{Aref'eva}, and interactions in the presence of
NS-NS flux\cite{B-field}. In the original form of \cite{Moyal} the
the continuous basis is defined by the coordinates,
\begin{eqnarray} &\
&x(\k)=\sqrt{2}\sum_{n=1}^\infty v_{2n}(\k)\sqrt{2n}x_{2n}\nonumber\\
&\ &y(\k)=-\sqrt{2}\sum_{n=1}^\infty
\frac{v_{2n-1}(\k)}{\sqrt{2n-1}}p_{2n-1}\label{eq:continuous}
\end{eqnarray} where $\k\in[0,\infty)$ and the functions $v_n(\k)$
are defined implicitly through the generating
function\cite{Spectroscopy},
\begin{equation}\sum_{n=1}^\infty
\frac{z^n}{\sqrt{n}}v_n(\k)=\frac{1-e^{-\k\tan^{-1}z}}{\k N(\k)},\
\ \ N(\k)=\left(\frac{2}{\k}\sinh\frac{\pi
\k}{4}\right)^{1/2}\label{eq:gen-func}\end{equation} In
ref.\cite{Okuyama} the $v_n(\k)$'s were shown to be complete and
orthonormal on the interval $\k\in(-\infty,\infty)$. In fact,
noting that $v_n(-\k)=(-1)^{n+1}v_n(\k)$ we can write:
$$\sum_{m=1}^\infty v_{2m-1}(k)v_{2m-1}(k')=\half\delta(k-k')\ \ \
\ \int_0^\infty \ dk\ v_{2m-1}(k)v_{2n-1}(k)=\half\delta_{2m-1,
2n-1}$$
\begin{equation}\sum_{m=1}^\infty v_{2m}(k)v_{2m}(k')=\half\delta(k-k')\
\ \ \ \int_0^\infty \ dk\
v_{2m}(k)v_{2n}(k)=\half\delta_{2m,2n}\label{eq:vprop2}\end{equation}
These equations only strictly hold when $\k>0$. More information
about these very special functions can be found in the appendix.
Equation (\ref{eq:continuous}) is an orthogonal change of basis
with respect to the inner product,
$$\langle x_1,x_2 \rangle=\int_0^\pi d\sigma x_1(\sigma)
\sqrt{-\frac{\partial^2}{\partial\sigma^2}}x_2(\sigma)=\int_0^\infty
d\k [x_1(\k)x_2(\k)+y_1(\k)y_2(\k)]$$ After invoking the
formula\cite{Moyal},
\begin{equation}T_{2m-1,2n}= -2\sqrt{\frac{2m-1}{2n}}\int_0^\infty d\k\
\tanh \fraction{\pi \k}{4}
v_{2m-1}(\k)v_{2n}(\k)\label{eq:T-cont}\end{equation} it is simple
to show from \EQN{mixed} that the Moyal coordinates satisfy
commutation relations,
\begin{equation}[x(\k),y(\k')]_\star=2i\tanh\fraction{\pi \k}{4}\delta(\k-\k')
\label{eq:continuous-com}\end{equation} The noncommutativity
parameter\footnote{To avoid ugly factors of $\half$ we will define
the noncommutativity parameter so that $[x,p]=2i\theta$, so that
it differs by a factor of one half from the more standard
convention.} $\theta(\k)=\tanh\frac{\pi \k}{4}$ is diagonal, but
its magnitude varies as a function of $\k$. Most significantly,
$\theta(\k)$ vanishes continuously toward $\k=0$, so the
associated coordinates become commutative. At $\k=0$, $x(\k)=0$
while interestingly $y(\k)$ becomes proportional to the momentum
of a half-string, \EQN{P_L}. The fact that the half-string
momentum emerges naturally as the commutative coordinate has been
considered a powerful argument in favor of the continuous Moyal
formalism, since in a naive treatment of the discrete basis the
significance of this coordinate is easily missed.

Let us now take a moment to gain a little more insight into the
structure of the continuous Moyal formalism. Our approach will be
to recast the formalism in an (almost) equivalent split string
description, where the midpoint structure is more apparent. A much
more in depth treatment of the continuous half string formalism
will appear soon in ref.\cite{continuous-half}.

Often it is convenient to represent multiplication in the algebra
of functions on noncommutative $\Re^2$ as a nonlocal Moyal product
between otherwise commutative functions on $\Re^2$. However,
another useful perspective is the ``split string'' matrix-like
representation of the algebra. This is can be related to the Moyal
perspective as follows: Given a function $\psi(x,p)$ with
$[\hat{x},\hat{p}]=2i\theta$, define,
$$\hat{\psi}(\theta z+x,-\theta z+x)=\bar{\psi}(z+\fraction{x}{\theta},
-z+\fraction{x}{\theta})=\int_{-\infty}^\infty dp e^{ipz}
\psi(x,p)$$ A simple excercise\cite{Bars} shows that calculating
the ``matrix product'' of these objects,
\begin{eqnarray}\hat{A}\star\hat{B}(l,r)=\theta\int_{-\infty}^\infty dw
\hat{A}(l,w)\hat{B}(w,r) \nonumber\\
\bar{A}\star\bar{B}(l,r)=\int_{-\infty}^\infty dw
\bar{A}(l,w)\bar{B}(w,r) \end{eqnarray} is isomorphic to
calculating the Moyal product of $A(x,p)$ and $B(x,p)$. Note that
when $\theta\neq 1$ there are two ``natural'' split string
representations of the product. We distinguish between them by
writing fields alternatively with a hat or with a bar (the
significance of this distinction will be clear soon).

We can easily follow this recipe to construct a split string
representation of the continuous basis. Write,
\begin{eqnarray}\Psi[x_n]&=&\int\prod_{n=1}^\infty \left(\frac{dp_{2n-1}}{2\pi}
e^{ip_{2n-1}x_{2n-1}}\right)\Psi[x_{2n},p_{2n-1}]\nonumber\\
&=&\int [dy(\k)]_1 \exp\left(\int_0^\infty d\k
y(\k)z(\k)\right) \Psi^M[x(\k),y(\k)]\nonumber\\
&=&\hat{\Psi}[l(\k),r(\k)]=\bar{\Psi}[\bar{l}(\k),\bar{r}(\k)]\nonumber
\end{eqnarray}
with \begin{eqnarray} &\ &l(\k)=\theta(\k)z(\k)+x(\k),\ \ \ \ \
r(\k)=-\theta(\k)z(\k)+x(\k)\nonumber\\
&\ &\bar{l}(\k)=z(\k)+\frac{x(\k)}{\theta(\k)}\ \ \ \ \ \ \ \ \ \
\bar{r}(\k)=-z(\k)+\frac{x(\k)}{\theta(\k)}\label{eq:half}\end{eqnarray}
We can then calculate the open string star product as,
\begin{eqnarray}\hat{A}*\hat{B}[l(\k),r(\k)]=
\mathcal{N}\int [dw(\k)]_2
\hat{A}[l(\k),w(\k)]\hat{B}[w(\k),r(\k)] \nonumber\\
\bar{A}*\bar{B}[\bar{l}(\k),\bar{r}(\k)]= \mathcal{N}\int
[dw(\k)]_3 \bar{A}[\bar{l}(\k),w(\k)]\bar{B}[w(\k),\bar{r}(\k)]
\end{eqnarray} The measures in the previous formulae are
normalized so that,
\begin{eqnarray}1&=&\int[dw(\k)]_1 \exp\left[-\frac{1}{2\pi}\int_0^\infty
dk dk'K_{-1}(\k,\k')w(\k)w(\k')\right]\nonumber\\
&=&\int[dw(\k)]_2 \exp\left[-\frac{\pi}{2}\int_0^\infty dk
dk'K_{-1}(\k,\k')w(\k)w(\k')\right]\nonumber\\ &=&\int[dw(\k)]_3
\exp\left[-\frac{\pi}{2}\int_0^\infty dk
dk'\frac{K_{-1}(\k,\k')}{\theta(\k)\theta(\k')}w(\k)w(\k')\right]
\end{eqnarray} These expressions can all be derived by
translating from the corresponding measures in the mode basis. The
kernel $K_{-1}(k,k')$ is given by the sum $\sum_{n=1}^\infty
(2n-1)^{-1}v_{2n-1}(\k)v_{2n-1}(\k')$. An explicit formula for
$K_{-1}$ was found in \cite{Virasoro,Small}, but we will not need
it.

It is instructive to relate \EQN{half} to other half-string bases.
A simple computation shows,
\begin{eqnarray}l(\k)&=&\sqrt{2}\sum_{n=1}^\infty
\left(\sqrt{2n}v_{2n}(\k)x_{2n}-\sqrt{2n-1}v_{2n-1}(\k)\tanh\frac{\pi
\k}{4} x_{2n-1}\right)\nonumber\\
&=&\sqrt{2}\sum_{n=1}^\infty
\sqrt{2n}v_{2n}(\k)\left(x_{2n}+\sum_{m=1}^\infty
T_{2n,2m-1}x_{2n-1}\right)\nonumber\\ &=&\sqrt{2}\sum_{n=1}^\infty
\sqrt{2n}v_{2n}(\k)l_{2n}=-\tanh\frac{\pi \k}{4}
\sqrt{2}\sum_{n=1}^\infty \sqrt{2n-1}v_{2n-1}(\k)l_{2n-1}\nonumber
\end{eqnarray}
where $l_{2n}$ denotes even cosine half-string Fourier modes and
$l_{2n-1}$ denotes odd cosine half-string Fourier modes. Another
simple computation can relate $l(\k)$ to the left half of the
string, $l(\sigma)=x(\sigma),\ \sigma\in[0,\frac{\pi}{2})$:
\begin{eqnarray}&\ &l(\k)=-\frac{2}{\sqrt{\pi}}\frac{\sinh\frac{\pi
\k}{4}}{N(\k)}\int_0^{\pi/2}d\sigma \sec\sigma
M(\k,\sigma)l(\sigma)\nonumber\\
&\ &\bar{l}(\k)=-\frac{2}{\sqrt{\pi}}\frac{\cosh\frac{\pi
\k}{4}}{N(\k)}\int_0^{\pi/2}d\sigma \sec\sigma
M(\k,\sigma)l(\sigma)\label{eq:halfsigma}\end{eqnarray} where
$M(\k,\sigma)$ can be derived by using the generating function
\EQN{gen-func} to evaluate the sum,
\begin{eqnarray}M(\k,\sigma)&=&\frac{2N(\k)\cos\sigma}{\sqrt{\pi}\cosh\frac{\pi
\k}{4}}\sum_{n=1}^\infty \sqrt{2n-1}v_{2n-1}(\k)\cos
(2n-1)\sigma\nonumber\\&=&\frac{1}{\sqrt{\pi}}\cos\left(\frac{\k}{2}\tanh^{-1}
\sin\sigma\right)\label{eq:M}\end{eqnarray} The $M$'s obey
orthogonality relations:
\begin{eqnarray}\int_0^{\pi/2} d\sigma\sec\sigma M(\k,\sigma)M(\k',\sigma)
=\delta(\k-\k')\nonumber\\ \int_0^\infty d\k
M(\k,\sigma)M(\k,\sigma')=\cos\sigma\delta(\sigma-\sigma')\end{eqnarray}
These functions $M$ are a little perplexing. It seems reasonable
to expand a half string in a basis of cosines, as one does in the
discrete Moyal formalism, but
\begin{center}\resizebox{2.5in}{1.5in}{\includegraphics{M.eps}}
\end{center}
Figure1: An example, $M(10,\sigma)$.
\bigskip

\noindent why would anyone expand with these functions? Actually,
the $M$'s are somewhat special; they are eigenfunctions of the
midpoint preserving reparameterization generator,
$$ \mathcal{K}_1=L_1+L_{-1}=-2i\cos\sigma\der{\sigma}$$
Specifically, $$(\mathcal{K}_1)^2 M(\k,\sigma)=-\k^2 M(\k,\sigma)
$$ This is not surprising, since the continuous basis
is defined by the eigenvectors of the Neumann coefficients, which
in turn were derived in ref.\cite{Spectroscopy} from the
observation that a matrix $\mathrm{K}_1$, describing the action of
$\mathcal{K}_1$ on $|V_3\rangle$, possessed the same eigenvectors
as the Neumann coefficients.

$\mathcal{K}_1$ acts like a generator of translations, but
translations whose length becomes increasingly shrunk towards the
midpoint; so in a sense the midpoint appears infinitely ``far
away.'' Correspondingly, the eigenfunctions $M(\k,\sigma)$ look
locally like sine waves, but their frequency is position-dependent
and blows up towards $\sigma=\frac{\pi}{2}$. (See figure 1a). The
situation is therefore quite different from the discrete Moyal
formalism, which imposes very specific boundary conditions at the
midpoint; the value of $M$ and its derivative at the midpoint is
completely undefined. Actually, in the continuous Moyal formalism
we should really think of the midpoint boundary condition in the
sense of a singular Sturm-Liouville system\footnote{Perhaps it's
amusing to mention that $M(\k,\sigma)$ would describe the
vibrational modes of a string with linear mass density
$\sec\sigma$ and stiffness $\cos\sigma$. Of course, the strings in
string theory have mass density and stiffness equal to $1$, in
suitable units.}; one should require that the eigenfunctions
remain bounded as $\sigma$ goes to $\frac{\pi}{2}$. This
effectively fixes $\k$ to be real.

It's also worth noticing is that \EQN{M} implies that a pure
cosine wave in $\k$ space corresponds to a position eigenstate on
the half-string. The frequency of the cosine wave and the position
on the string are related: $\tanh\omega = \sin\sigma$. In
particular, the ultraviolet in $\k$ space describes phenomena
close to the string midpoint. Indeed, most singularities in the
continuous Moyal formalism manifest themselves in the apparent
need to invoke functions whose frequency is formally infinite. We
will see more of this in the following sections.

A simple but profound question is how the variables $l(\k)$ and
$\bar{l}(\k)$ represent the half string configuration
$l(\sigma)=1$ (or, equivalently, how $r(\k)$ and $\bar{r}(\k)$
represent $r(\sigma)=x(\pi-\sigma)=1$). The relevance of this
question should be explained. First, note that since we are
working at zero momentum, we can for definiteness fix the string
midpoint to be at zero: $x(\frac{\pi}{2})=0$. Then a string
configuration with a kink discontinuity at the midpoint is,
$$x(\sigma)=\left\{\matrix{+1 \ \ \ \ \mathrm{for}\
\sigma\in[0,\frac{\pi}{2})\cr \!\!\!\!0\ \ \ \ \mathrm{for}\
\sigma=\frac{\pi}{2}\cr -1 \ \ \ \ \mathrm{for}\
\sigma\in(\frac{\pi}{2},\pi]}\right. $$ Obviously, to describe
this we need to use a basis capable of describing the function
$l(\sigma)=1$. There is reason to believe that such a
configuration may be useful in string field theory, even though it
looks somewhat singular; for instance it was shown in
ref.\cite{Singular} that the sliver functional, which is the
canonical projector solution of VSFT, is independent of
transformations which break a string at its midpoint. These
considerations were further studied in
ref.\cite{Spectroscopy,GMS}. The discrete even Moyal basis, as
mentioned before, cannot describe such phenomena. However, if we
choose a half-string basis which {\it could} describe
$l(\sigma)=1$ we run into another ugly problem: the basis
accommodates configurations like:
\begin{equation}x(\sigma)=\left\{\matrix{+1 \ \ \ \ \mathrm{for}\
\sigma\in[0,\frac{\pi}{2})\cr \!\!\!\!0\ \ \ \ \mathrm{for}\
\sigma=\frac{\pi}{2}\cr +1 \ \ \ \ \mathrm{for}\
\sigma\in(\frac{\pi}{2},\pi]}\right.\label{eq:discontinuity}\end{equation}
This pathological situation describes a string which has become
estranged from its midpoint. There is no evidence that such
configurations have any realization in string field theory, in
fact there is some evidence to the contrary\cite{Spectroscopy}.
But now we seem to be stuck in a catch-22: if we cannot describe
$l(\sigma)=1$ we have too little freedom, but if we can we have
too much\footnote{Another interpretation of \EQN{discontinuity}
advocated in ref.\cite{Gross-Taylor} is that it really represents
a string which has been translated to $x(\sigma)=1$, and
$x(\pi/2)=0$ is a residual artifact. So in this perspective the
unwanted degree of freedom can be used to describe strings at
nonzero momentum. This is an interesting point of view, but it is
probably better to represent nonzero momentum directly with string
zero modes, rather than somewhat dubiously as a singular limit of
the higher mode excitations.}. This issue is at the heart of the
difficulty with the operator/Moyal approach to string field
theory.

So back to the question: how does the continuous Moyal formalism
deal with this? It is simple to plug $l(\sigma)=1$ into
\EQN{halfsigma},
\begin{eqnarray}&\ &\bar{l}(\k)=-\frac{2}{\sqrt{\pi}}\frac{\cosh\frac{\pi
\k}{4}}{N(\k)}\int_0^{\pi/2}d\sigma \sec\sigma
M(\k,\sigma)=-\frac{2}{\pi}\delta(\k)\nonumber\\
&\ &l(\k)=-\frac{2}{\sqrt{\pi}}\frac{\sinh\frac{\pi
\k}{4}}{N(\k)}\int_0^{\pi/2}d\sigma \sec\sigma
M(\k,\sigma)=-\frac{2}{\pi}\sinh\frac{\pi\k}{4}\delta(\k)=0\nonumber\end{eqnarray}
The situation is similar to the discrete Moyal formalism: In one
basis you can represent $l(\sigma)=1$ but in the other you can't.
Are we just back to the same problem we had with the discrete
basis? The answer is no, because neither of the half string bases
in \EQN{half} is really equivalent to the continuous Moyal
formalism as it was first conceived. The reason is that, in order
to construct a split string description, we implicitly had to make
a change of basis which normalized the noncommutativity parameter
to one. There are two ways to do this, corresponding to the two
half string formalisms:
\begin{eqnarray} \tilde{x}(\k)=\frac{x(\k)}{\theta(\k)}\ \ \ \ \ \
[\tilde{x}(\k),y(\k')]_\star=2i\delta(\k-\k')\Longrightarrow
\bar{l}(\k),\bar{r}(\k)\nonumber\\
\tilde{y}(\k)=\frac{y(\k)}{\theta(\k)}\ \ \ \ \ \
[x(\k),\tilde{y}(\k')]_\star=2i\delta(\k-\k')\Longrightarrow
l(\k),r(\k)\nonumber \end{eqnarray} The problem is that
$\theta(\k)$ possesses a zero mode, $\theta(\k)\delta(\k)=0$, so
we cannot really invert it to make this change of basis. The zero
mode of $\theta(\k)$ is the twin brother of the zero mode of $T$,
and as we can see it causes the same problems.

This clearly explains why the noncommutativity parameter in the
continuous Moyal formalism vanishes continuously towards $\k=0$:
If it did not, the formalism would either have too many or too few
degrees of freedom to describe a reasonable string. This also
sheds some light on why the Moyal formulation of the star algebra
is slightly preferable to the matrix/split string approach. With a
Moyal product, we have a deformation parameter $\theta$ which can
be taken to zero in a meaningful way to recover a purely
commutative algebra. This is not true for a matrix algebra. While
the matrix and Moyal descriptions are indeed isomorphic for
$\theta\neq 0$, they are not equivalent when $\theta=0$ and this
is why the split string formalism in any basis always has too many
or too few degrees of freedom\footnote{It should be noted that the
discrete Moyal formalism does not make use of this subtle
difference between matrix and Moyal representations. As far as the
author can tell, the discrete Moyal formalism is completely
equivalent to the split string formalism. The major breakthrough
in the approach of ref.\cite{Bars-Matsuo} is not so much the use
of the Moyal product, but the construction of a consistent
regulator which resolves the type of midpoint ambiguities we have
been discussing.}.

It was shown in ref.\cite{Spectroscopy} that the Neumann matrix
$M=CV^{11}$ (using standard notation) has a doubly degenerate
spectrum of eigenvectors, $v_{2n}(\k)$ and $v_{2n-1}(\k)$, for
$\k$ strictly greater than zero; however for $\k=0$ they argued
that it has only {\it one} twist odd eigenvector with eigenvalue
$-\frac{1}{3}$:
$$v_{2n-1}(0)=\frac{1}{\sqrt{\pi}} \frac{(-1)^{n+1}}{\sqrt{2n-1}}$$
which in the continuous basis \EQN{continuous} gives us the
half-string momentum. The twist even counterpart of this should
be, after normalizing by a vanishing factor of $\k$,
\begin{equation}
v_{2n}'(0)=\frac{1}{\sqrt{\pi}}\frac{(-1)^{n+1}}{\sqrt{2n}}
\left(1+\frac{1}{3}+\frac{1}{5}+
...+\frac{1}{2n-1}\right)\label{eq:v'}\end{equation} However, the
authors of ref.\cite{Spectroscopy} gave several reasons for
believing that this vector was in fact not an eigenvector of $M$.
If true, this implies that the corresponding Moyal coordinate,
\begin{eqnarray} x'(0)&=&\lim_{\k\to
0}\frac{x(\k)}{\k}=\sqrt{2}\sum_{n=1}^\infty
\sqrt{2n}v_{2n}'(0)x_{2n}\nonumber\\
&=&-\frac{1}{4\sqrt{\pi}}\int_0^\pi d\sigma
|\sec\sigma|x(\sigma)\end{eqnarray} should somehow be physically
pathological. Our discussion shows why this is the case, since we
can easily see that $x'(0)$ is responsible for the isolated
midpoint singularity \EQN{discontinuity},
$$\frac{x(\k)}{\theta(\k)}=\half\left
[\bar{l}(\k)+\bar{r}(\k)\right]=
-\frac{2}{\pi}\delta(\k)=\frac{4}{\pi}x'(\k)$$ This gives a
concrete justification of why the twist even vector should be
excluded from the spectrum of $M$ at $-\frac{1}{3}$.

However, this brings up a thorny issue. In a certain sense, it is
not really meaningful to simply ``remove'' $v_{2n}'(0)$ from the
spectrum, since it can always be recovered as a suitable limit of
vectors which {\it are} in the spectrum. Though you can never
``get'' to $v_{2n}'(0)$, you can get as close as you want, and
this is enough to cause trouble. Such limits, as we will see, are
the direct cause of the Horowitz-Strominger associativity anomaly.

\section*{IV. Philosophy of associativity anomalies}
Consider the following two Moyal coordinates: $P_L$, the momentum
of the half string, and $\bar{x}$, the ``center of mass'' position
relative to $x(\frac{\pi}{2})\equiv 0$. These are explicitly,
\begin{eqnarray}&\ &P_L=\frac{\sqrt{2}}{\pi}\sum_{n=1}^\infty
\frac{(-1)^{n+1}}{2n-1}p_{2n-1}=\int_0^{\pi/2}d\sigma
p(\sigma)=-\frac{1}{\sqrt{\pi}}y(0)\nonumber\\
&\ &\bar{x}=\sqrt{2}\sum_{n=1}^\infty
(-1)^{n+1}x_{2n}=\frac{1}{\pi}\int_0^\pi d\sigma
x(\sigma)=-2\int_0^\infty \frac{d\k}{\k N(\k)}x(\k)\ \ \ \ \
\label{eq:Px} \end{eqnarray} It would seem simple enough to
calculate the Moyal-star commutator, $$[\bar{x},P_L]_\star$$ but
if we attempt to do this, say in the mixed basis, we immediately
run into a problematic double sum, \begin{equation} i
\frac{4}{\pi}\sum_{m,n=1}^\infty (-1)^{m+1}T_{2m,2n-1}
\frac{(-1)^{n+1}}{2n-1}\label{eq:ambsum}\end{equation} As observed
numerous times\cite{Gross-Taylor,Associative,Bars-Matsuo}, the
value of this sum depends on the order in which the summation is
carried out. We can see the ambiguity in the continuous basis
too\cite{MSFT}:
$$[\bar{x},P_L]_\star=\frac{2}{\sqrt{\pi}}\int_0^\infty\frac{d\k}{\k
N(\k)}2i\tanh\frac{\pi \k}{4}\delta(\k)$$ Again, this integral is
difficult to define, since the integrand has delta function
support on the boundary of the region of integration. One could
simply declare that the commutator is zero by definition, since we
have explicitly removed the unphysical coordinate $x'(0)$ from our
theory. However, it would have been difficult to see this just by
inspecting the integral or the sum.

It is probably meaningless to talk about the value of this
commutator without specifying a prescription for regulating and
resolving these ambiguous expressions. One approach would be to
impose a mode number cutoff $N$ to $P_L$ and $N'$ to $\bar{x}$,
and then take $N,\ N'$ to infinity in a way which specifies an
order for evaluating the double sum:
\begin{eqnarray}&\ &\lim_{N'\to\infty}\lim_{N\to\infty}[\bar{x}(N'),P_L(N)]_\star=i
wTv=0\nonumber\\
&\
&\lim_{N\to\infty}\lim_{N'\to\infty}[\bar{x}(N'),P_L(N)]_\star=i
v\bar{T}w =i vv=i\ \ \ \ \ \label{eq:ff}\end{eqnarray} Here, in
the notation of ref.\cite{Associative}, we introduced an even
moded vector $w_{2n}=\sqrt{2}(-1)^{n+1}\in\mathcal{H}^*_{even}$
satisfying $\bar{T}w=v$. The ambiguity above comes about because
$w$ and $v$ are not in $\mathcal{H}'_{even}$ and
$\mathcal{H}_{odd}$ respectively.  This mode number regulator
plays a central role in regulating and defining the discrete Moyal
formalism, though it is equally well-suited for the mixed basis.

We could try to impose the mode number cutoff in the continuous
basis, as suggested in ref.\cite{MSFT}, but this would be
unpleasantly inelegant. To make sense of $[\bar{x},P_L]_\star$ in
this context we should take a more refined point of view. The key
point is to notice that coordinates of the form
$$C=\int_0^\infty d\k\left[f(\k)y(\k)+g(\k)x(\k)\right]$$
will only have a well-defined Moyal-star commutator if $f(\k)$ and
$g(\k)$ live in a sufficiently smooth space of test functions,
since calculating the commutator inevitably involves integrating
$f$ and $g$ against a delta function. The problem with calculating
$[P_L,\bar{x}]$ is that $f$ and $g$ in this case are {\it not}
smooth functions; they are distributions. We can see this clearly
by regulating the formulas for $P_L$ and $\bar{x}$ as follows:
\begin{eqnarray}&\ &P_L(\omega)=\frac{\sqrt{2}}{\pi i}\sum_{n=1}^\infty
\frac{(i
\tanh\omega)^{2n-1}}{2n-1}p_{2n-1}=-\frac{1}{\pi}\int_{-\infty}^{\infty}
\frac{d\k}{\k N(\k)}\sin\omega\k y(\k)
\nonumber\\
&\ &\bar{x}(\omega)=-\sqrt{2}\sum_{n=1}^\infty
(i\tanh\omega)^{2n}x_{2n}=-2\int_0^\infty \frac{d\k}{\k N(\k)}
[1-\cos\omega\k]x(\k)\ \ \ \ \ \ \ \label{eq:regPx}
\end{eqnarray} Taking $\omega$ to infinity we should recover
$P_L$ and $\bar{x}$, but the coefficients in the integrand
oscillate with infinite frequency and diverge. What should we make
of this? The wild oscillation of $\sin\omega\k$ and
$1-\cos\omega\k$ is not necessarily a problem, if these singular
objects always appear integrated against a nicely behaved test
function; in this case the regulated formulas \EQN{regPx} converge
to \EQN{Px} in the sense of distributions. Unfortunately, when
calculating $[\bar{x},P_L]_\star$ we must integrate $\sin\omega\k$
against $1-\cos\omega\k$, and neither of them is smooth as
$\omega\to\infty$; formally, we are trying to integrate over a
product of distributions, which is mathematical nonsense. Still,
one can use the regulated formulas to calculate
$[\bar{x}(\omega),P_L(\omega')]_\star$, and take $\omega, \omega'$
to infinity in different orders,
\begin{eqnarray}\lim_{\omega'\to\infty}\lim_{\omega\to\infty}
[\bar{x}(\omega'),P_L(\omega)]_\star&=&2i \lim_{\omega'\to\infty}
\frac{2}{\sqrt{\pi}}\int_0^\infty d\k\frac{\tanh\frac{\pi
\k}{4}}{\k N(\k)}[1-\cos\omega'\k]\delta(\k)=0
\nonumber\\
\lim_{\omega\to\infty}\lim_{\omega'\to\infty}[\bar{x}(\omega'),
P_L(\omega)]_\star&=& 2i
\lim_{\omega\to\infty}\frac{2}{\pi}\int_{-\infty}^\infty d\k
\frac{\tanh\frac{\pi \k}{4}}{2\k \sinh \frac{\pi
\k}{2}}\sin\omega\k =i\nonumber\end{eqnarray} Consistently, we
find the same answers as we did in the mixed basis \EQN{ff}. This
is a concrete realization of our earlier observation that infinite
frequencies in $\k$ space are associated with midpoint
singularities.

Notice that, even though our calculation did not explicitly invoke
the unphysical coordinate $x'(0)$, we were still able to construct
an apparently reasonable limit where the value of
$[\bar{x},P_L]_\star$ was nonzero. Essentially, this limit allowed
us to get ``close enough'' to $x'(0)$ to effectively reintroduce
it into the theory. This might rightfully seem dubious, but it is
not {\it a priori} preventable. Strangely enough, in the regulated
discrete Moyal formalism\cite{Bars-Matsuo} the commutator
$[\bar{x},P_L]_\star$ always evaluates to $i$, not zero. This is
probably not really correct, at least not under all circumstances.
The difficulty is that usually when imposing the mode number
cutoff it is assumed that $T$ should be replaced by it's $N\times
N$ truncated version, $T_{N\times N}$. In fact it could equally
well be replaced by\footnote{In the truncated
theory\cite{Associative}, $T_{N\times N} v_N=v_N/(1+w_N^2)$.},
$$T_{N\times N}^{\mathrm{zero}}=T_{N\times N} -\frac{w_N v_N}{1+w_N^2}$$
which has a zero mode even at finite $N$. In this regularization,
$[\bar{x},P_L]_\star=0$. The fact that the space of Moyal
coordinates is infinite dimensional is important, and one should
be careful not to ``overregulate'' and gain false confidence into
the meaning of expressions which are intrinsically ill-defined.

The lesson we have learned is that the coordinates $\bar{x}$ and
$P_L$ do not really exist in the continuous/discrete Moyal
formalisms, unless multiplied with a sufficiently ``well-behaved''
string functional. This motivates the
definition,\begin{eqnarray}\Psi*P_L&\equiv &
\lim_{\rho\to\infty}\Psi*P_L(\rho)\nonumber\\\bar{x}*\Psi&\equiv &
\lim_{\rho\to\infty}\bar{x}(\rho)*\Psi\label{eq:dert}\end{eqnarray}
where $\rho$ is a regulator appropriate for the basis in question.
An important consequence of this definition is that star
multiplication of fields involving both $\bar{x}$ and $P_L$ is
generically {\it non-associative}. This phenomenon was observed
many years ago in a paper by Horowitz and
Strominger\cite{Horowitz}, and has long been a source of
puzzlement and speculation\cite{Associative, Strominger, Qiu,
Rakowski, Schnabl}. The anomaly in the guise we will consider
concerns the product $\bar{x}*\Psi*P_L$ where $\Psi$ is a
sufficiently well-behaved string functional, such as one in the
perturbative Fock space:
\begin{eqnarray}
(\bar{x}*\Psi)*P_L&= &
\lim_{\rho'\to\infty}\left[\lim_{\rho\to\infty}
[\bar{x}(\rho)*\Psi]*P_L(\rho')\right]\nonumber\\
\bar{x}*(\Psi*P_L)&= &\lim_{\rho\to\infty}
\left[\bar{x}(\rho)*\lim_{\rho'\to\infty}[\Psi*P_L(\rho')]\right]\
\ \ \ \label{eq:assoc}
\end{eqnarray} The product $\bar{x}*\Psi*P_L$ fails to be
associative because these two limits are not the same\footnote{The
limits differ if we calculate in the Moyal formalism, but they are
the same if we calculate in position space. See next paragraph.}.
Let's see how this comes about explicitly. The calculation is
simplified by noting,
\begin{eqnarray}\bar{x}(\rho)\star A &=& \bar{x}(\rho) A +\half[\bar{x}(\rho), A]_\star\nonumber\\
A\star P_L(\rho) &=& A P_L(\rho)
+\half[A,P_L(\rho)]_\star\nonumber
\end{eqnarray} Multiplying and writing out the terms, \begin{eqnarray}
\bar{x}(\rho)\star \Psi \star P_L(\rho') &=& \bar{x}(\rho)\Psi
P_L(\rho') + \half [\bar{x}(\rho),\Psi]_\star
P_L(\rho')\nonumber\\
&\ &+\half \bar{x}(\rho)[\Psi,
P_L(\rho')]_\star+\fraction{1}{4}[\bar{x}(\rho),[\Psi,
P_L(\rho')]_\star]_\star\nonumber\\
&\ &+\half[\bar{x}(\rho), P_L(\rho')]_\star\Psi
\label{eq:reg_prod}
\end{eqnarray}
We can now calculate the associator, cancelling terms which are
manifestly equal regardless of the order of limits:
\begin{eqnarray}
&\ &(\bar{x}\star\Psi)\star P_L-\bar{x}\star(\Psi\star
P_L)=\nonumber\\ &\ &\half
\left[\lim_{\rho'\to\infty}\lim_{\rho\to\infty}
-\lim_{\rho\to\infty}\lim_{\rho'\to\infty}\right]\left(\half[\bar{x}(\rho),[\Psi,
P_L(\rho')]_\star]_\star +[\bar{x}(\rho), P_L(\rho')]_\star\Psi
\right)\nonumber
\end{eqnarray}
For a well behaved string functional we can argue that the first
term on the right hand side vanishes. For example, in the mixed
basis this term looks like,
\begin{eqnarray}&\ &\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
[\bar{x}(N),[\Psi,
P_L(N')]_\star]_\star\nonumber\\
&\ &=\frac{2}{\pi}\sum_{n=1}^N\sum_{n'=1}^{N'}
\sum_{k,l=1}^\infty\frac{(-1)^{n'+1}(-1)^{n+1}}{2n'-1}T_{2n,2k-1}T_{2l,2n'-1}
\frac{\partial^2 \Psi}{\partial p_{2k-1}\partial
x_{2l}}\label{eq:double-com}
\end{eqnarray} Depending on how the mixed partial derivative evaluates,
this expression might have an ambiguous $N,N'\to\infty$ limit.
However, for the ground state wavefunctional,
\begin{equation}\frac{\partial^2
\Psi_{|0\rangle}}{\partial p_{2k-1}\partial x_{2l}}=4 \frac{
2l}{2k-1} p_{2k-1}x_{2l}\Psi_{|0\rangle}\nonumber
\end{equation}
and noting that $\Psi_{|0\rangle}$ only has support on
configurations satisfying
$$\sum_{n=1}^\infty \frac{p_{2k-1}^2}{2k-1}<\infty\ \
\ \ \sum_{n=1}^\infty 2l x_{2l}^2<\infty $$ we can see that the
$N,N'\to\infty$ of \EQN{double-com} limit is unambiguous (in fact,
it is zero). Therefore, the ambiguity of $[\bar{x},P_L]_\star$
gives us the only contribution to the associator,
\begin{equation}(\bar{x}\star\Psi)\star P_L-\bar{x}\star(\Psi\star
P_L)=\fraction{i}{2}\Psi \label{eq:anomaly}
\end{equation} Hence associativity is anomalous.

This calculation makes the associativity anomaly appear
deceptively straightforward. For example, if we repeat the
argument in the coordinate space basis $x(\sigma)$, the anomaly
disappears altogether. To see this consider,
\begin{eqnarray}\bar{x}(\rho)\star A[x(\sigma)]&=&
\frac{2}{\pi}\int_0^{\pi/2}d\sigma
f_\rho(\sigma)x(\sigma)A[x(\sigma)]\nonumber\\
A[x(\sigma)]\star P_L(\rho)&=& i\int_{\pi/2}^{\pi}d\sigma\
g_\rho(\sigma)\frac{\delta}{\delta x(\sigma)}A[x(\sigma)]
\label{eq:35}\end{eqnarray} where
$\lim_{\rho\to\infty}f_\rho=\lim_{\rho\to\infty}g_\rho=1$. For
finite values of $\rho$, these formulas involve integrals of the
position $x(\sigma)$ and the momentum
$p(\sigma)=-i\frac{\delta}{\delta x(\sigma)}$ over opposite halves
of the string. This is no accident, otherwise multiplication would
not be associative. On the other hand, when $\rho$ is strictly
infinite, $A\star P_L$ involves $p(\sigma)$ integrated over {\it
both} halves of the string, with opposite sign:
\begin{eqnarray}
A[x(\sigma)]\star P_L&=& i\int_{\pi/2}^{\pi}d\sigma\
\frac{\delta}{\delta x(\sigma)}A[x(\sigma)]\nonumber\\
&=&-\frac{i}{2}\left[\int_{0}^{\pi/2}d\sigma\ \frac{\delta}{\delta
x(\sigma)}-\int_{\pi/2}^{\pi}d\sigma\ \frac{\delta}{\delta
x(\sigma)}\right]A[x(\sigma)]\nonumber
\end{eqnarray}
This is because when $\rho\to\infty$, $A \star P_L(\rho)$
describes a differential where the right half of the string is
broken away from the midpoint (see figure 2). This is the same as
breaking the left and right halves symmetrically away from each
other, since the difference involves the unphysical string
configuration \EQN{discontinuity}, or equivalently the Moyal
coordinate $x'(0)$, which apparently reasonable string functionals
can't see. Keeping this in mind, we can calculate,
\begin{eqnarray}\bar{x}\star (\Psi\star P_L)&=&\lim_{\rho\to\infty}
\frac{2i}{\pi}\int_0^{\pi/2}d\sigma
f_\rho(\sigma)x(\sigma)\left[\int_{0}^{\pi/2}d\sigma\
\frac{\delta}{\delta x(\sigma)}-\int_{\pi/2}^{\pi}d\sigma\
\frac{\delta}{\delta x(\sigma)}\right]\Psi[x(\sigma)]\nonumber\\
&=&-\frac{i}{\pi}\int_0^{\pi/2}d\sigma
x(\sigma)\left[\int_{0}^{\pi/2}d\sigma\ \frac{\delta}{\delta
x(\sigma)}-\int_{\pi/2}^{\pi}d\sigma\ \frac{\delta}{\delta
x(\sigma)}\right]\Psi[x(\sigma)]\nonumber
\end{eqnarray}
Now let's consider $(\bar{x}\star\Psi)\star P_L$. According to
\EQN{35}, $\bar{x}(\rho)\star A$ multiplies $A$ by the average
value of $x(\sigma)$ on the left half of the string, weighted by
$f_\rho(\sigma)$ (see
\begin{center}a) \resizebox{2in}{1.3in}{\includegraphics{J.eps}}
\ \ \ \ \ \ \ \ b)
\resizebox{2in}{1.3in}{\includegraphics{K.eps}}\end{center} Figure
2: Graphs of the magnitude of the action of $p(\sigma)$ on the
string when calculating $A\star P_L(\rho)$, as a function of the
regulator in a) the discrete/Mixed basis and b) in the continuous
basis. a) shows values for the mode number cutoff $N=1,3,5,7,9,11$
and b) shows $4\omega= 1,2,3,4,5,6$ with $\omega$ as in
\EQN{regPx}.
\bigskip

\begin{center} \resizebox{2in}{1.3in}{\includegraphics{L.eps}}\end{center}
\noindent Figure 3: $f_\rho$ in the continuous basis, for
$\rho=2\omega=1,...,9$ with $\omega$ as in \EQN{regPx}. For
$\omega<\infty$ the area under the curve vanishes.
\bigskip



\noindent figure 3). For finite values of $\rho$,
$\bar{x}(\rho)\star A$ has no functional dependence on $x'(0)$ or
\EQN{discontinuity}, because $\int_0^{\pi/2}d\sigma f_\rho=0$. In
fact, $f_\rho$ approaches $1$ as $\rho$ gets large, but in such a
way that there is an increasingly narrow and deep ``well'' near
$\sigma=\frac{\pi}{2}$ which cancels off the area under the curve.
When $\rho$ is strictly infinite, the ``well'' becomes a delta
function at $\frac{\pi}{2}$, which can be ignored since
$x(\frac{\pi}{2})=0$ by definition. But then in this limit
$\bar{x}\star A$ has a nontrivial dependence on $x'(0)$, as is
easy to see; therefore we cannot argue that star multiplication by
$P_L$ will multiply $\bar{x}\star \Psi$ by $-i\frac{\delta}{\delta
x(\sigma)}$ integrated over both halves of the string. In fact,
\begin{eqnarray}(\bar{x}\star \Psi)\star P_L
&=&\lim_{\rho\to\infty} \frac{2i}{\pi}\int_{\pi/2}^{\pi}d\sigma\
g_\rho(\sigma)\frac{\delta}{\delta x(\sigma)}\int_0^{\pi/2}d\sigma
x(\sigma)\Psi[x(\sigma)]\nonumber\\
&=&-\frac{i}{\pi}\int_0^{\pi/2}d\sigma
x(\sigma)\left[\int_{0}^{\pi/2}d\sigma\ \frac{\delta}{\delta
x(\sigma)}-\int_{\pi/2}^{\pi}d\sigma\ \frac{\delta}{\delta
x(\sigma)}\right]\Psi[x(\sigma)]\nonumber
\end{eqnarray} Therefore, $(\bar{x}\star\Psi)\star P_L=\bar{x}\star(\Psi\star
P_L)$ and the associator vanishes!

How do we reconcile this with \EQN{anomaly}? The problem is that
the limit $\rho\to\infty$ does not commute with the the operation
of taking the Fourier transform. Translating \EQN{reg_prod} into
position space, we see that the offending term is, $$\half
[\bar{x}(\rho),\Psi]_\star P_L(\rho')=-\frac{i}{2\pi} \int_0^\pi
g_{\rho'}(\sigma) \frac{\delta}{\delta x(\sigma)} \left[\int_0^\pi
d\sigma f_\rho(\sigma)x(\sigma) \right] \Psi$$ where in our
notation $f_\rho$ and $g_\rho$ are odd functions around
$\sigma=\frac{\pi}{2}$. In the phase space of even coordinates and
odd momenta, the limits $\rho,\rho'\to\infty$ commute because,
$$\lim_{\rho\to\infty}\lim_{\rho'\to\infty}\half
[\bar{x}(\rho),\Psi]_\star P_L(\rho')=\half\left(
\lim_{\rho\to\infty}[\bar{x}(\rho),\Psi]_\star\right)
\left(\lim_{\rho'\to\infty}P_L(\rho')\right)$$ However, in
coordinate space, $P_L$ is a differential operator whose action on
$[\bar{x}(\rho),\Psi]_\star$ needs to be evaluated before we take
can take $\rho\to\infty$. Taking the derivative gives us an extra
term, $$-\frac{2i}{\pi}\int_0^{\pi/2} d\sigma f_\rho(\sigma)
g_{\rho'}(\sigma)\Psi $$ which has an ambiguous $\rho,\rho'\to
\infty$ limit, cancelling the contribution to the associator from
$[\bar{x},P_L]_\star \Psi$.

To make matters more complicated, we should note that the products
$\bar{x}\star\Psi$ and $\Psi\star P_L$ need not be defined in
regulated form through \EQN{dert}; $\bar{x}$ and $P_L$ are not
singular in every basis. For instance in the mixed basis, they
exist directly $N\to\infty$ limit, which means we can construct
these operators explicitly {\it before} multiplying them with a
state. Calculating in this order, we find:
\begin{eqnarray}\Psi\star P_L &=& \Psi P_L\nonumber\\
\bar{x}\star\Psi &=& \bar{x}\Psi+i\frac{2\sqrt{2}}{\pi}
\sum_{m=1}^\infty \frac{(-1)^{m+1}}{2m-1}
\frac{\partial\Psi}{\partial p_{2m-1}}\nonumber
\end{eqnarray} Then, \begin{eqnarray}
\bar{x}\star(\Psi\star P_L)& =& \bar{x}\Psi P_L
+i\frac{2\sqrt{2}}{\pi} \sum_{m=1}^\infty \frac{(-1)^{m+1}}{2m-1}
\frac{\partial\Psi}{\partial p_{2m-1}}+\frac{i}{2}\Psi\nonumber\\
(\bar{x}\star\Psi)\star P_L & =& \bar{x}\Psi P_L
+i\frac{2\sqrt{2}}{\pi} \sum_{m=1}^\infty \frac{(-1)^{m+1}}{2m-1}
\frac{\partial\Psi}{\partial p_{2m-1}}P_L\nonumber \end{eqnarray}
So,
$$(\bar{x}\star\Psi)\star P_L-\bar{x}\star(\Psi\star P_L)=-\fraction{i}{2}\Psi$$
Now the associator has {\it opposite} sign from \EQN{anomaly}.
Apparently the limit $\rho\to\infty$ does not commute with star
multiplication either.

It has been suggested in the literature many
times\cite{Horowitz,Strominger,Qiu,Associative} that the failure
of associativity in the star algebra is a necessary feature
indicating the emergence of closed string physics from open string
field theory. Bars and Matsuo have recently argued that the
anomaly is needed in VSFT too\cite{Bars-Matsuo}, in order to
ensure that D-brane vacua have a nontrivial spectrum of
fluctuations. These ideas are interesting, but they are based on
the assumption that when associativity fails, it does so in a
consistent and physically meaningful way. This seems not to be the
case, at least from the perspective of our analysis. A more sober
point of view is that associativity anomalies indicate that we are
attempting to multiply objects which are outside the algebra of
string fields, and really the star product of such objects should
not be defined. It might be possible that one day physical
considerations will dictate an unambiguous procedure for
regulating and calculating the star product when associativity no
longer holds, but at this point the definition of products like
$\bar{x}\star\Psi\star P_L$ is rather a question for philosophy,
not physics.

\section*{V. Ghosts\footnote{This discussion is borrowed
and improved from and earlier discussion in ref.\cite{Erler}}}
Sofar, our analysis has dealt with the matter sector exclusively.
In bosonized form\cite{Belov-Konechny}, the star algebra of ghosts
is identical\footnote{Restricting to the subalgebra of fields with
ghost momentum $-3/2$ is equivalent to studying zero momentum in
the matter sector} to the algebra in the matter sector,
\EQN{mixed}. However, in the $bc$ ghost language, the algebra of
string fields looks quite different from the matter sector because
the ghost coordinates, $\partial_\sigma b(\sigma)$ and $c(\sigma)$
are identified using half-string {\it anti}overlap conditions
(overlap with a sign)\cite{Vertices,Erler,Bars_amplitudes}. This
leads to some novel structures beyond what we've seen. Expand the
worldsheet ghosts $b_{zz},c^z$ on the unit semicircle $|z|=1,
\Im(z)>0$ in terms of Fourier modes as follows:
\begin{eqnarray}b(\sigma)&=&i\sqrt{2}\sum_{n=1}^\infty x_{n}\sin
n\sigma\ \ \ \ \ \ \ \ \ \spaces x_n=\frac{i}{\sqrt{2}}(b_n-b_{-n})\nonumber\\
\pi_c(\sigma)&=&\frac{1}{\pi}\left(b_0+\sqrt{2}\sum_{n=1}^\infty
q_{n}\cos n\sigma\right)
\ \ \ \ \ \ q_n=\frac{1}{\sqrt{2}}(b_n+b_{-n})\nonumber\\
c(\sigma)&=&c_0+\sqrt{2}\sum_{n=1}^\infty y_{n}\cos n\sigma
\ \ \ \ \spaces y_n=\frac{1}{\sqrt{2}}(c_n+c_{-n})\nonumber\\
\pi_b(\sigma)&=&\frac{i\sqrt{2}}{\pi}\sum_{n=1}^\infty p_{n}\sin
n\sigma \ \ \ \ \ \ \ \ \ \ \spaces
p_n=\frac{1}{\sqrt{2}}(c_n-c_{-n}) \nonumber\end{eqnarray} where
$b_n$ and $c_n$ are the anti-commuting $bc$ ghost oscillators
satisfying $\{b_n,c_{-m}\}=\delta_{m,n}$ (note that our use of the
$x_n$ and $p_n$ differs from previous sections). Restricting
ourselves to Siegel gauge\cite{Erler}, we will think of the string
field in the ghost sector as a functional of the eigenvalues of
$x_{2n-1},p_{2n},y_{2n-1},q_{2n}$ for $n\geq 1$, in which case we
can calculate the reduced star product\cite{Okuyama2} ($=b_0$
times the full star product) as: $$
\Psi*_{b_0}\Phi[x_{2n-1},p_{2n},y_{2n-1},q_{2n}]=\mathcal{N}_{ghost}
\Psi\star\Phi[x_{2n-1},p_{2n},y_{2n-1},q_{2n}]$$ where
$\mathcal{N}_{ghost}$ is a normalization\cite{Erler} and $\star$
is a non-anticommutative ``Moyal product''\cite{Ferrara} (i.e. a
product of elements in a Clifford Algebra) satisfying,
\begin{equation}\{y_{2m-1},q_{2n}\}_\star=2 R_{2m-1,2n}\ \ \
\{x_{2m-1},p_{2n}\}_\star=2\frac{2m-1}{2n}R_{2m-1,2n}
\label{eq:mixed_ghost} \end{equation} with $R$ the matrix in
\EQN{T-inv}. The appearance of $R$ here is interesting, since $R$
has a much more restricted domain than $T$; this is a reflection
of the fact that the anti-overlap is more singular than the simple
matrix-like overlap in the matter sector. We can make a change of
basis into $\kappa$ space:
\begin{eqnarray}x_\e(\k)&=&-\sqrt{2}\sum_{n=1}^\infty \frac{1}{\sqrt{2n}}v_{2n}(\kappa)q_{2n}\nonumber\\
x_\o(\k)&=&\sqrt{2}\sum_{n=1}^\infty \frac{1}{\sqrt{2n-1}}v_{2n-1}(\kappa)x_{2n-1}\nonumber\\
y_\e(\k)&=&\sqrt{2}\sum_{n=1}^\infty \sqrt{2n}v_{2n}(\kappa)p_{2n}\nonumber\\
y_\o(\k)&=&\sqrt{2}\sum_{n=1}^\infty
\sqrt{2n-1}v_{2n-1}(\kappa)y_{2n}\label{eq:k-ghost}
\end{eqnarray} Writing $\vec{x}=(x_\e,x_\o)$ and similarly for
$\vec{y}$,
\begin{equation}\{\vec{x}(\k),\vec{y}(\k')\}_\star=2i\sigma_y \coth\fraction{\pi \k}{4}
\left[\delta(\k-\k')+\delta(\k+\k')\right]\label{eq:continuous-gcom}\end{equation}
with $\sigma_y$ the $y$th Pauli matrix. The extra $\delta(\k+\k')$
is put there since we will be letting $\k$ range over the whole
real line (even though negative $\k$s don't give independent
coordinates) for reasons that will be clear in a minute.

In contrast to the matter sector, where the continuous basis gives
a smooth noncommutativity parameter and Moyal coordinates with a
nice physical interpretation, the algebra \EQN{continuous-gcom} is
perplexing. First, the metric/non-anticommutativity parameter
$g(\k)=\coth\fraction{\pi \k}{4}$ {\it diverges} towards
$k=0$---in fact, the quantity $\coth\fraction{\pi
\k}{4}[\delta(\k-\k')+\delta(\k+\k')]$ is not a well defined
Schwartz distribution, since its inner product with a $C^\infty$
function generically diverges. Even more confusing is the fact
that the even coordinates vanish at $\k=0$,
\begin{eqnarray}x_\o(0)&=&\sqrt{\frac{2}{\pi}}\left(\hat{x}_1-
\fraction{1}{3}\hat{x}_3+\fraction{1}{5}\hat{x}_5-... \right)\ \ \
\ y_\o(0)=\sqrt{\frac{2}{\pi}}\left(\hat{y}_1-
\hat{y}_3+\hat{y}_5-... \right) \nonumber\\
y_\e(0)&=&0\spaces\spaces\spaces\ \ \ \ \ \ \ \ x_\e(0)=0
\end{eqnarray}
yet they are somehow supposed to have {\it infinite}
non-anticommutativity with the odd coordinates! Furthermore, the
odd coordinates at $\k=0$ don't seem to have any obvious physical
interpretation.

The problems of the vanishing coordinates and the diverging metric
are actually connected. Note that the odd coordinates at $\k=0$
generically have finite and nonzero non-anticommutativity:
$$\{y_\o(0),q_{2n}\}_\star=-2\sqrt{\frac{2}{\pi}}\sum_{m=1}^\infty(-1)^m
R_{2m-1,2n}=-\frac{\pi\sqrt{2}}{2}\sqrt{2n}\ v_{2n}'(0)$$ which is
not zero, in contrast to what we found in the matter
sector\footnote{This result may be derived independently from the
continuous basis from the manipulation,
\begin{eqnarray}\{y_\o(0),\sum_{n=1}^\infty z^{2n}q_{2n}\}_\star
&=& -2\sqrt{\frac{2}{\pi}}\sum_{m,n=1}^\infty (-1)^m z^{2n}
R_{2m-1,2n}\nonumber\\ &=& \frac{4}{\pi}\sqrt{\frac{2}{\pi}}
\int_0^{\pi/2}d\sigma
(\sec\sigma)\left(\frac{2z^2}{(1+z^2)(1-z^2)}\frac{\cos^2\sigma}{
1+(\fraction{2z}{1-z^2}\sin\sigma)^2} \right)\nonumber\\
&=&\frac{4}{\pi}\sqrt{\frac{2}{\pi}}
\frac{z}{1+z^2}\int_0^{\fraction{2z}{1-z^2}}\frac{du}{1+u^2}\nonumber\\&=&
\frac{8}{\pi}\sqrt{\frac{2}{\pi}}\frac{z
\tan^{-1}z}{1+z^2}\nonumber
\end{eqnarray} Taylor expanding in $z$ and recalling \EQN{v'} we get the answer.}. Translating this
equation into the continuous basis,
\begin{eqnarray}\{y_\o(0),q_{2n}\}_\star&=
&-\frac{1}{\sqrt{2}}\sqrt{2n}\int_{-\infty}^\infty dk
v_{2n}(k) \{y_\o(0),x_\e(k)\}_\star\nonumber\\
&=&-2\sqrt{2}\sqrt{2n}\int_{-\infty}^\infty dk v_{2n}(k)
\coth(\fraction{\pi k}{4})\delta(k)\nonumber\\
&=&-\frac{\pi\sqrt{2}}{2}\sqrt{2n}\ v_{2n}'(0)
\nonumber\end{eqnarray} The divergence of $\coth\frac{\pi \k}{4}$
cancels against the vanishing of $v_{2n}(\k)$, giving a finite
answer. Apparently, the diverging metric serves to compensate for
the disappearance of the even coordinates, so that $x_\o(0)$ and
$y_\o(0)$ can still have finite non-anticommutativity. Note that
$\k$ was assumed to range over the whole real line, otherwise our
calculation would have been ambiguous.

Though $\coth\fraction{\pi \k}{4}[\delta(\k-\k')+\delta(\k+\k')]$
is not a Schwartz distribution, it is a valid distribution in the
topological dual of the vector space of $C^\infty$ functions which
vanish at $\k=0$. We should really think of \EQN{continuous-gcom}
as being defined in this sense. However, this means we need to be
careful, since many products we would not have suspected to be
problematic are actually singular---for instance, $x_\e(\k)\star
y_{2n-1}$ is undefined near $\k=0$.

The remaining question is how the the $\k=0$ coordinates should be
interpreted physically. Let's write $x_\o(\k)$ and $y_\o(\k)$
explicitly in terms of $c(\sigma)$ and $b(\sigma)$:
\begin{eqnarray}y_\o(\k)=\frac{\cosh\frac{\pi \k}{4}}{\sqrt{\pi}N(\k)}
\int_0^\pi d\sigma \sec\sigma M(\k,\sigma)\ c(\sigma)\nonumber\\
x_\o(\k)=\frac{\cosh\frac{\pi \k}{4}}{\sqrt{\pi}N(\k)}\int_0^\pi
d\sigma \sec\sigma M(\k,\sigma)\ \tilde{b}(\sigma)\nonumber
\end{eqnarray} where we define,
$$\tilde{b}(\sigma)=\int_{\pi/2}^\sigma d\sigma' b(\sigma')$$
With this formula it is simple to see that $x_\o(0)$ and $y_\o(0)$
are responsible for describing string configurations where
$c(\sigma)$ and $\tilde{b}(\sigma)$ have a kink discontinuity at
the midpoint\footnote{The derivatives of the even coordinates at
$\k=0$
are,\begin{eqnarray}x_\e'(0)&=&\frac{1}{\sqrt{\pi}}\int_0^\pi
d\sigma
\left[\left(\frac{\pi}{4}\right)^2+(\half\tanh^{-1}\sin\sigma)^2\right]\pi_c(\sigma)\nonumber\\
y_\e'(0)&=&\frac{i}{2\sqrt{\pi}}\int_0^\pi d\sigma \sec\sigma
\tanh^{-1}\sin\sigma\ \pi_b(\sigma)
\end{eqnarray} The author has not succeeded in finding
a particularly compelling interpretation for these coordinates.}.
Suppose for example that $c(\sigma)$ is a step function which
jumps from $1$ to $-1$ at the midpoint. Then,
\begin{eqnarray}y_\o(\k)=\frac{2\cosh\frac{\pi
\k}{4}}{\sqrt{\pi}N(\k)} \int_0^{\pi/2} d\sigma \sec\sigma
M(\k,\sigma)=\frac{2}{\sqrt{\pi}}\delta(\k)\nonumber
\end{eqnarray} so the configuration is described solely with the
$y_\o(0)$. We may also write,
\begin{eqnarray}\frac{\delta}{\delta y_\o(0)}&=&\frac{\sqrt{\pi}}{2}
\left(\int_0^{\pi/2} d\sigma\pi_c(\sigma)- \int_{\pi/2}^\pi
d\sigma\pi_c(\sigma)\right) \nonumber\\ \frac{\delta}{\delta
x_\o(0)}&=&\frac{\sqrt{\pi}}{2}\pi_b(\fraction{\pi}{2})\end{eqnarray}
In the last equation we noted that a kink in $\tilde{b}(\sigma)$
translates into a delta function in $b(\sigma)$.

The $\k=0$ coordinates in the ghost sector are probably not as
physically important as they were in the matter sector, since they
have no particularly special algebraic properties (the fact that
$x_\e'(0)$ and $y_\e'(0)$ have ``infinite'' non-anticommutativity,
for instance, is a property shared by uncountably many other
coordinates). Much more interesting are,
\begin{eqnarray} b(\fraction{\pi}{2})&=&i\sqrt{2}\sum_{n=1}^\infty
(-1)^{n+1}
x_{2n-1}\nonumber\\
c'(\fraction{\pi}{2})&=&\sqrt{2}\sum_{n=1}^\infty (2n-1)(-1)^{n+1}
y_{2n-1}\nonumber \\
A &=& -\frac{i\sqrt{2}}{\pi}\sum_{n=1}^\infty
\frac{(-1)^{n+1}}{2n} p_{2n}\nonumber\\ B &=&
\frac{\sqrt{2}}{\pi}\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n)^2}
q_{2n}
\end{eqnarray} The first two of these are especially important
since they are purely {\it anticommutative} under the open string
product:
\begin{equation} \{b(\fraction{\pi}{2}),\ p_{2n}
\}_\star=0\ \ \  \{c'(\fraction{\pi}{2}),\ q_{2n}
\}_\star=0\nonumber
\end{equation} since $\bar{R}$ has a zero mode, \EQN{R-zero}. The
second two coordinates, $A$ and $B$, are analogous to $\bar{x}$ in
the matter sector since they have ambiguous star-anticommutator
with  $b(\fraction{\pi}{2})$ and $c'(\fraction{\pi}{2})$
\begin{eqnarray}\{b(\fraction{\pi}{2}),\ A
\}_\star=\{c'(\fraction{\pi}{2}),\ B\}_\star=\left\{\matrix{
w'\cdot \bar{R}v'=w\cdot Tv=0 \cr v'\cdot Rw'=v\cdot \bar{T}w=1
}\right.\label{eq:amb-gcom}\end{eqnarray} where
$w'_{2n}=\sqrt{2}\frac{(-1)^{n+1}}{(2n)^2}\in
\mathcal{H}_{even}'^*$. Apparently associativity anomalies occur
in the ghost sector just as they do in the matter sector.

From the perspective of the continuous basis, the existence
anticommutative coordinates is startling. Indeed,
$\coth\fraction{\pi \k}{4}$ has no zeros, so it seems impossible
that any linear combination of $x_\o(\k)$ and $y_\o(\k)$ could
anticommute with {\it all} $x_\e(\k)$ and $y_\e(\k)$. The
unexpected twist here, however, is that $b(\p2)$ and $c(\p2)$ are
actually described by Moyal coordinates with an {\it imaginary
argument}, and for imaginary $\k$, $\coth\fraction{\pi \k}{4}$ has
zeros. To explain this, it is convenient to define,
$$X_\o(\k)=N(\k)x_\o(\k)\ \ \ Y_\o(\k)=N(\k)y_\o(\k)$$
This scaling is useful since $X_\o$ and $Y_\o$ are analytic
everywhere on the complex plane, unlike $x_\o$ and $y_\o$; in
particular, the change of basis \EQN{k-ghost} becomes defined in
terms of functions $N(\k)v_n(\k)=\Tch_n(\k)/\sqrt{n}$, which as
discussed in the appendix are just polynomials. So write $b(\p2)$
in this basis: $$b(\p2)=2i\int_0^\infty d\k
\left(\sum_{n=1}^\infty(-1)^{n+1}\Tch_{2n-1}(\k)\right)\frac{X_\o(\k)}{\frac{2}{\k}\sinh\frac{\pi
\k}{2}}$$ We can immediately see that something strange is going
on, since the above series doesn't converge. However, the authors
of ref.\cite{Virasoro} showed how this type of divergence could be
handled, at least formally: One writes,
\begin{eqnarray}b(\p2)&=&2i\int_0^\infty d\k
\half\sin\left(2\frac{d}{d\k}\right)\k
\left[\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n-1}\Tch_{2n-1}(\k)\right]\frac{X_\o(\k)}{\frac{2}{\k}\sinh\frac{\pi
\k}{2}}\nonumber\\ &=&2i\sqrt{\pi}\int_0^\infty d\k
\left[\half\sin\left(2\frac{d}{d\k}\right)\k
N(\k)\delta(\k)\right]\frac{X_\o(\k)}{\frac{2}{\k}\sinh\frac{\pi
\k}{2}}\label{eq:dist}\end{eqnarray} The strange object in the
brackets is some type of distribution---let us hold off on this a
moment, and assume this makes sense. Integrating by parts,
\begin{eqnarray}b(\p2)&=&-\frac{i\sqrt{\pi}}{2}\int_{-\infty}^\infty d\k
\delta(\k)N(\k)\k\Im\left[\frac{X_\o(\k+2i)}{\frac{2}{\k+2i}\sinh\frac{\pi
(\k+2i)}{2}}\right]\nonumber\end{eqnarray} Cancelling the pole at
$\k=0$ zero gives, \begin{equation}b(\p2)=i X_\o(2i)\ \ \ c'(\p2)=
Y_\o(2i)\label{eq:z-bc}\end{equation} An identical calculation
follows for $c'(\p2)$. Note that the argument of $X_\o$ and $Y_\o$
is evaluated precisely where $\coth\fraction{\pi \k}{4}$ has a
zero! Yet, it is still not clear that these coordinates are
anticommutative, since when plugging in to \EQN{continuous-gcom}
we have to evaluate a delta function with an imaginary argument,
which makes no sense. The way to see the anticommutative nature of
$b(\p2)$ and $c'(\p2)$ is to represent them as in \EQN{dist}, and
calculate the anticommutator this way:
\begin{eqnarray}\{b(\p2),p_{2n}\}_\star&=&-\frac{4i\sqrt{2\pi}}{2n}
\int_{-\infty}^\infty d\k
\left[\half\sin\left(2\frac{d}{d\k}\right)\k
N(\k)\delta(\k)\right]\frac{\Tch_{2n}(\k)\coth\frac{\pi
\k}{4}}{\frac{4}{\k}\sinh\frac{\pi \k}{2}}\nonumber\\
&=&-\frac{4i\sqrt{2\pi}}{2n} \int_{-\infty}^\infty d\k
\delta(\k)N(\k)\k \Im\left[\frac{\Tch_{2n}(\k+2i)\coth\frac{\pi
(\k+2i)}{4}}{\frac{4}{\k+2i}\sinh\frac{\pi
(\k+2i)}{2}}\right]\nonumber\\ &=&0\nonumber\end{eqnarray} This
vanishes, as claimed, because $\coth\fraction{\pi \k}{4}$ has a
zero at $\k=2i$.

But what sort of horrible thing is the distribution in \EQN{dist}?
It is certainly not a Schwartz distribution, since for a general
$C^\infty$ function the infinite rank differential operator
$\sin(2\frac{d}{dk})$ does not converge. The convergence of
$\sin(2\frac{d}{dk})$ on a function $f(\k)$ is determined by the
sum,
$$ \sum_{n=0}^\infty \frac{f^{(2n+1)}(\k)}{(2n+1)!} 2^{2n+1}< \infty $$
This says that a Taylor expansion of $f$ around any particular
$\k$ must converge at least within a radius of $2$; thinking of
$f$ as a function in the complex plane, this means that $f$ must
be analytic inside a band outside the real axis extending from
$-2<\Im(\k)<2$. On this restricted space of test functions (much
more restricted than $C^\infty$ functions), \EQN{dist} is a
perfectly well defined distribution, and our formal calculations
have a rigorous meaning. Actually, \EQN{dist} is in a class of
generalized functions which have been studied by mathematicians,
called ``ultradistributions''\cite{text}.

For those who are uncomfortable with these formal arguments, we
can explicitly regulate and show that everything works fine.
Define,
\begin{equation} b_\omega(\p2)=i\sqrt{2}\sum_{n=1}^\infty
\fraction{1}{i}(i\tanh\omega)^{2n-1}x_{2n-1}\sim i
\int_{-\infty}^\infty d\k
\frac{e^{2\omega}\cos\omega\k}{4N(\k)}x_\o(\k)
\end{equation} We recover $b(\p2)$ in the limit $\omega\to \infty$. (The $\sim$ here means that this formula
gives only the leading order correction for large $\omega$, but
this is all we need). This is similar to what we saw earlier in
\EQN{regPx}; when regulating we encounter an infinitely
oscillatory divergence in the integrand. Here the divergence seems
much worse, since the {\it amplitude} of the oscillations diverge
as well as the frequency. A very similar class of divergences
emerge when formulating the Virasoro generators in the continuous
basis\cite{Virasoro,Moyal}. In order to understand how $b(\p2)$
functions in the continuous basis, we need to see how
$e^{2\omega}\cos\omega\k$ behaves under an integral as $\omega$ is
taken to infinity. Consider a function $f(\k)$ which has simple
poles at $\k_n=a_n+i b_n$ (no branch points, for the sake of
argument) and for which we can close the contour in the upper half
plane. Then,
$$\int_{-\infty}^\infty d\k e^{i\omega(\k-2i)} f(\k)=2\pi i\sum_{n=1}^{N}
e^{-\omega(b_n-2)}e^{i\omega a_n}\mathrm{Res}(f,\k_n)$$ Taking the
large $\omega$ limit, we have the following possibilities: If $0<
b_n <2$ the answer will diverge; if $2<b_n<\infty$ the answer
vanishes; if $b_n=2$ but $a_n\neq 0$ the integral also vanishes
because it is washed out by the oscillating exponential; only when
$b_n=2$ and $a_n=0$ do we get a nonzero and finite answer. A
simple extension of this argument allows us to include branch
points too, and we see
\begin{equation}\lim_{\omega\to \infty}\int_{-\infty}^\infty d\k
e^{i\omega(\k-2i)} f(\k)=\left\{ \matrix{\infty \mathrm{\ if\ } f
\mathrm{\ has\ poles\ or\ branch\ cuts\ on\ or\ between\ }\cr \ \
\ \mathrm{ the\ real\ line\ and\ the\ line\ } \Im(\k)=2i \cr 2\pi
i\mathrm{Res}(f,2) \mathrm{\ otherwise}\spaces\spaces\spaces\ \ \
\ }\right.\label{eq:Kint}\end{equation} This thankfully agrees
with our earlier more abstract discussion. We can use this
integral to evaluate the anticommutator,
\begin{eqnarray}\{b(\p2),p_{2n}\}_\star=\frac{i\sqrt{2}}{2n}
\lim_{\omega\to\infty}\int_{-\infty}^\infty d\k
\frac{\Tch_{2n}(\k)\coth\frac{\pi
\k}{4}}{\frac{4}{\k}\sinh\frac{\pi \k}{2}}e^{i\omega(k-2i)}
\end{eqnarray} The integrand is analytic for $0\leq \Im(\k)\leq
2$; the divergence of $\coth\fraction{\pi \k}{4}$ is cancelled by
the vanishing of $\Tch_{2n}(\k)$, and the pole from the
$\sinh\fraction{\pi \k}{2}$ in the denominator at $\k=2i$ is
cancelled by the vanishing of $\coth\fraction{\pi \k}{4}$.
Therefore, by \EQN{Kint} the anticommutator is zero as claimed. An
identical reasoning applies to $c'(\p2)$.

However, as mentioned earlier, the anticommutator
$\{b(\p2),A\}_\star$ is ambiguous. How does this come about in the
continuous basis? Again, we regulate\footnote{This equation
follows from the following asymptotic formulas for large $\omega$,
which we list for reference:
\begin{eqnarray}\sum_{n=1}^\infty\frac{(i\tanh
\omega)^{2n}}{(2n)^{3/2}}v_{2n}(\k)&\sim & \frac{1}{\k
N(\k)}\left[\half \psi(\half+\fraction{i \k}{4})+\half
\psi(\half-\fraction{i \k}{4})+\gamma+2\ln
2-2e^{-2\omega}\frac{1+(\fraction{\k}{2})^2-\cos\omega\k
+\fraction{\k}{2}\sin\omega\k}{1+(\fraction{\k}{2})^2}\right]\nonumber\\
\fraction{1}{i}\sum_{n=1}^\infty\frac{(i\tanh
\omega)^{2n-1}}{(2n-1)^{3/2}}v_{2n-1}(\k)&\sim& \frac{2}{\k
N(\k)}\left[\fraction{\pi}{2} \tanh\fraction{\pi
\k}{4}+e^{-2\omega}\frac{\sin\omega\k+\fraction{\k}{2}\cos
\omega\k}{1+(\frac{\k}{2})^2}\right]\nonumber\end{eqnarray}The
exact formulas are transcendental integrals which we will not
need.}:
\begin{eqnarray}A_\omega &=& \frac{i\sqrt{2}}{\pi}\sum_{n=1}^\infty
\frac{(i\tanh\omega)^{2n}}{2n}p_{2n}\nonumber\\
&\sim& \frac{i}{\pi}\int_{-\infty}^\infty \frac{d\k}{\k
N(\k)}\left[\half \psi(\half+\fraction{i \k}{4})+\half
\psi(\half-\fraction{i \k}{4})+\gamma+2\ln 2\right. \nonumber\\
&\ & \spaces\spaces
\left.-2e^{-2\omega}\frac{1+(\fraction{\k}{2})^2-\cos\omega\k
+\fraction{\k}{2}\sin\omega\k}{1+(\fraction{\k}{2})^2}\right]y_\e(\k)
\nonumber \end{eqnarray} where $\psi(z)=\frac{d}{dz}\ln\Gamma(z)$
denotes the digamma function and $\gamma$ is Euler's constant.
Here too we have an oscillating ``divergence,'' bit curiously it
actually disappears as $\omega\to\infty$. Calculating the
anticommutator with $b_\omega(\p2)$ we are lead to consider the
integral,\begin{eqnarray}\{b_\omega(\p2), A_{\omega'}\}_\star&\sim
&\frac{1}{2\pi}\int_{-\infty}^\infty d\k \frac{\coth\frac{\pi
\k}{4}}{\sinh \frac{\pi \k}{4}}e^{i\omega(\k-2i)}\Big[\half
\psi(\half+\fraction{i \k}{4})+\half
\psi(\half-\fraction{i \k}{4}) \nonumber\\
&\ & \spaces \left.+\gamma+2\ln
2-2e^{-2\omega'}\frac{1+(\fraction{\k}{2})^2-\cos\omega'\k
+\fraction{\k}{2}\sin\omega'\k}{1+(\fraction{\k}{2})^2}\right]
\nonumber\end{eqnarray} Taking $\omega'\to\infty$ first, we must
evaluate ,\begin{eqnarray}\{b_\omega(\p2), A\}_\star&\sim
&\frac{1}{2\pi}\int_{-\infty}^\infty d\k \frac{\coth\frac{\pi
\k}{4}}{\sinh \frac{\pi
\k}{4}}e^{i\omega(\k-2i)}\left[\psi(\half-\fraction{i \k}{4})-
\psi(\half)+\frac{\pi i}{4}\tanh\fraction{\pi \k}{4}\right]
\nonumber\end{eqnarray} Closing the contour in the upper half
plane, and keeping in mind \EQN{Kint}, we see two possible
relevant singularities: a double pole from $\frac{\coth\frac{\pi
\k}{4}}{\sinh \frac{\pi \k}{4}}$ at zero and a simple pole from
$\frac{\pi i}{4}\tanh\fraction{\pi \k}{4}$ at $2i$ (the
$\psi(\half-\fraction{i \k}{4})$ is analytic in the upper half
plane). The double pole at $\k=0$ is cancelled by a second order
zero in $\psi(\half+\fraction{i \k}{4})+ \psi(\half-\fraction{i
\k}{4})-\psi(\half)$. The pole in the integrand at $2i$ gives the
only nonzero contribution to the $\omega\to\infty$ limit,
$$\lim_{\omega\to\infty}\lim_{\omega'\to\infty}
\{b_\omega(\p2), A_{\omega'}\}_\star=\frac{1}{2\pi}(2\pi
i)(-\half)(2i)=1$$ in agreement with \EQN{amb-gcom}. Let's now
evaluate the limits in the opposite order:
\begin{eqnarray}&\ &\lim_{\omega'\to\infty}\lim_{\omega\to\infty}
\{b_\omega(\p2), A_{\omega'}\}_\star=\nonumber\\
&\ &\ \ \ \ \ \ 1-
\frac{1}{\pi}\lim_{\omega'\to\infty}\lim_{\omega\to\infty}
\int_{-\infty}^\infty d\k \frac{\coth\frac{\pi \k}{4}}{\sinh
\frac{\pi \k}{4}}e^{i\omega(\k-2i)}e^{-2\omega'}\left[1-
\frac{\cos\omega'\k
-\fraction{\k}{2}\sin\omega'\k}{1+(\fraction{\k}{2})^2}\right]
\nonumber\end{eqnarray} The integrand has a pole at $2i$ which
contributes as $\omega\to\infty$:
\begin{eqnarray}\lim_{\omega'\to\infty}\lim_{\omega\to\infty}
\{b_\omega(\p2), A_{\omega'}\}_\star&=&1-
\frac{1}{\pi}\lim_{\omega'\to\infty}2\pi
i\left[e^{-2\omega'}(-\half)(-1)\fraction{1}{i}(\cos 2i\omega'-i
\sin 2i\omega')\right]\nonumber\\ &=&1-\lim_{\omega'\to
\infty}e^{-2\omega'}(\cosh 2\omega'+\sinh
2\omega')=0\end{eqnarray} The answer is again in agreement with
\EQN{amb-gcom}.

\section*{VI. The Hamiltonian}

One of the most interesting aspects of operator/Moyal formulation
is that it provides a way to represent the Hamiltonian in Siegel
gauge algebraically in terms of the star
product\cite{Bars-Matsuo,Bars_amplitudes,Bars-SFE}. To see how
this is done, write the zeroth Virasoro generator in the mixed
basis $x_{2n}$ $p_{2n-1}$:
$$L_0=\half\sum_{n=1}^\infty \left[-\frac{\partial^2}{\partial
x_{2n}^2}+(2n)^2 x_{2n}^2\right]+\half\sum_{n=1}^\infty
\left[-(2n-1)^2\frac{\partial^2}{\partial p_{2n-1}^2}+
p_{2n-1}^2\right]$$ This is not normal ordered, so the action of
$L_0$ on the vacuum gives a sum of the zero point energies for
every string oscillator,
$$L_0\Psi_{|0\rangle}=\half\sum_{n=1}^\infty n\ \Psi_{|0\rangle}$$
where, $$\Psi_{|0\rangle}=\exp\left[-\half \sum_{n=1}^\infty 2n
x_{2n}^2-\half \sum_{n=1}^\infty \frac{p_{2n-1}^2}{2n-1}\right]$$
To represent $L_0$ using the star product, the trick is to define
a deformed Moyal star algebra satisfying,
\begin{equation}[x_{2n},p_{2m-1}]_\star = 2i T^{N\times
N}_{2n,2m-1}\label{eq:deformed}\end{equation} where $T^{N\times
N}$ is a matrix with nonzero entries only inside an $N\times N$
block, and $\lim_{N\to\infty} T^{N\times N}=T$, the matrix we know
from \EQN{T-matrix}. For now we will suppress the $N$ dependence
and write $T^{N\times N}$ as $T$. Bars and Matsuo showed that one
can define $N\times N$ matrices $R,n_\e,n_\o$\footnote{Bars and
Matsuo denote $n_\e$ and $n_\o$ by $\k_\e$ and $\k_\o$
respectively.} and $N$ component vectors $v,w$ satisfying the
properties (bar denotes transpose),
\begin{eqnarray}&\ &RT=TR=1\ \ \ T\bar{T}=
1-\frac{w\bar{w}}{1+w^2}\ \ \ R\bar{R}=1+(1+w^2)v\bar{v}\nonumber\\
&\ &\bar{T}T= 1-v\bar{v}\ \ \ \bar{R}R=1+w\bar{w}\ \ \ R=n_e^{-2}T
n_\o^2 \nonumber\\&\ &\bar{T}w=v\ \ \ Tv=\frac{w}{1+w^2}\ \ \
Rw=(1+w^2)v \ \ \ Tv=w \label{eq:Bars-relations}\end{eqnarray}
which, when $N\to\infty$ become equal to the $R$, $v$ and $w$
introduced before\footnote{Note that the entries of the deformed
matrices/vectors are not in general equal to the truncation of the
infinite $N$ matrices/vectors. However, we will for simplicity
define our regularization so that this is true for $n_\e$ and
$n_\o$.}, and $n_{\o,2m-1,2n-1}=(2m-1)\delta_{2m-1,2n-1}$,
$n_{\e,2m,2n}=2m\delta_{2m,2n}$. All the above equations have
analogues when $N=\infty$, except that the norm of $w$ diverges
for infinite $N$. If one imposes a mode number cutoff $N$ on
$L_0$, a short calculation reveals that one can represent $L_0(N)$
using the deformed algebra \EQN{deformed}:
\begin{eqnarray}L_0(N) \Psi = \{\mathcal{L}_0,\Psi\}_\star +\half
(1+w^2)[[\Psi,P_L]_\star,P_L]_\star\label{eq:L_0}\end{eqnarray}
with,
\begin{eqnarray}
\mathcal{L}_0=\fraction{1}{4}\sum_{n=1}^N (p_{2n-1}^2+(2n)^2
x_{2n}^2)\nonumber \\ P_L=\half \sum_{n=1}^N
v_{2n-1}p_{2n-1}\end{eqnarray} Taking $N\to\infty$, on the left
hand side we get $L_0$; on the right hand side, the deformed
products become identical to the open string star product. We have
therefore found an explicit representation of $L_0$ in terms of
the star algebra.

B8ut there's a problem: the second term on the right hand side of
\EQN{L_0} makes no sense at infinite $N$. The factor $(1+w^2)$
blows up, but it is multiplied by an expression which naively
vanishes because $P_L$ becomes commutative. The best way to
understand the role of this singular term is to apply the formula
in a concrete example, like the ground state functional:
\begin{eqnarray}\{\fraction{1}{4}\sum_{n=1}^N
(2n)^2 x_{2n}^2,\Psi_{|0\rangle}\}_\star &=& \half
\sum_{n=1}^N\left[(2n)^2x_{2n}^2-p_{2n-1}^2+(2n-1)\right]\Psi_{|0\rangle}
\nonumber\\ \{\fraction{1}{4}\sum_{n=1}^N
p_{2n-1}^2,\Psi_{|0\rangle}\}_\star &=& \half
\left[\sum_{n=1}^N(p_{2n-1}^2-(2n)^2x_{2n}^2)+\frac{\left(
\sum_{l=1}^N  2l w_{2l} x_{2l}\right)^2}{1+w^2}\right.\nonumber\\
&\ &\spaces \left.+\sum_{n=1}^N 2n
\left(1-\frac{1}{1+w^2}\right)\right]\Psi_{|0\rangle}\nonumber\\
\half (1+w^2)[[\Psi,P_L]_\star,P_L]_\star&=
&-\half\frac{1}{1+w^2}\left[\left(\sum_{l=1}^N 2l w_{2l}
x_{2l}\right)^2-\sum_{n=1}^N 2n \right]\Psi_{|0\rangle}\nonumber
\end{eqnarray} Adding these pieces up we get the sum of string zero
point energies. At $N=\infty$, the singular term makes an infinite
contribution to the answer, but the infinity is subleading to the
infinity of zero point energies.

What is the meaning of this? We will argue that the singular term
computes the midpoint contribution to the zero point energy.
Consider the following ``almost ground state'' functional:
$$\Psi=\exp\left[-\frac{1}{\pi}\int_0^\pi d\sigma
x(\sigma)\sqrt{-\frac{d^2}{d\sigma^2}}x(\sigma)+\frac{1}{\pi}
\int_{\pi/2-\epsilon}^{\pi/2+\epsilon} d\sigma
x(\sigma)\sqrt{-\frac{d^2}{d\sigma^2}}x(\sigma)\right]
$$ where $\epsilon<<1$. The second integral in the exponential
serves to cancel off the midpoint dependence of this functional.
Translating this to the mixed basis gives, to leading order in
$\epsilon$, $$\Psi=\exp\left[-\half \sum_{m,n=1}^\infty 2n
x_{2m}x_{2n}(\delta_{2m,2n}-4\epsilon w_{2n}w_{2m})-\half
\sum_{n=1}^\infty \frac{p_{2n-1}^2}{2n-1}\right]$$ Acting on this
with $L_0$, $$L_0\Psi=\half\sum_{n=1}^\infty
\left[2n+(2n-1)-4\epsilon 2n+ 4\epsilon\left(\sum_{l=1}^\infty 2l
x_{2l}w_{2l} \right)^2\right] \Psi$$ Comparing this to our
previous calculation, we see that this is precisely the answer we
would have gotten if we had ignored the singular term in
\EQN{L_0}. Apparently, $\{\mathcal{L}_0,\ \}_\star$ does not see
the energy of the midpoint.

\section{VII. Conclusion}
In this paper we have studied midpoint structure of the open
string star product, in both the matter and ghost sectors, and
have shown explicitly how the resulting singularities are
manifested in differing operator/Moyal formulations of the star
algebra. Many of the issues we've been talking about are
intricate, both conceptually and mathematically, but we believe
that they are of fundamental importance, since they give us a
clear view into the structure of what is arguably our best
candidate for a nonperturbative, background independent
formulation of string theory. However, we certainly do not claim
to have come to a definitive understanding; some other directions
of future research are as follows:

\bigskip \noindent$\bullet$ Little work has been devoted to
understanding the Moyal formulation of the open superstring star
algebra and it's relation to the midpoint. One approach was
developed in the continuous basis in ref.\cite{Aref'eva} but no
corresponding formulation has been codified in the discrete/mized
basis
\bigskip

\noindent$\bullet$ The other Virasoro generators can also be
formulated singularly in terms of the star product
\cite{Bars-Matsuo}, and it would be very interesting to understand
what these singularities mean physically and how they should be
properly regulated. A related question is that of midpoint anomaly
cancellation, either for the midpoint preserving
reparameterizations $\mathcal{K}_n$ or for the BRST operator.
\bigskip


\noindent$\bullet$ The work of ref.\cite{Belov-Konechny} it was
shown that, including ghost contributions, the open string star
product is related to a Moyal product only up to an infinite
normalization. It is important to understand the physical meaning
of this divergent normalization and how it should be properly
handled
\bigskip

\noindent$\bullet$ Finally, more thought needs to be devoted to
the definition of the algebra of string fields. The basic
criterion for this definition (if it exists) is that the algebra
is closed and that Witten's axioms hold. The associativity axiom,
for instance, essentially amounts to requiring that the matrix $T$
always acts in a suitably defined Hilbert space of even/odd moded
sequences. However, it is not yet clear how this observation can
be realized as a concrete restriction on the function space of
string functionals.

I would like to thank I.Bars, D.Belov, D.Gross, A.Konechny, and
D.Reynolds for many useful discussions. I would also like
especially M.Putinar for helping me gain a deeper appreciation of
the subject from the perspective of functional analysis.

\section*{Appendix: more about the $v_n$'s}
The harmonic oscillator Hamiltonian is $$H=a^+ a +\half{}.$$ This
operator has a discrete spectrum of eigenvectors $|n\rangle$ (in
our conventions $1\leq n<\infty$) corresponding to the $n-1$st
excited state of the harmonic oscillator. On the other hand, the
position operator,
$$X=\frac{i}{\sqrt{2}}(a-a^+)$$ has a continuous spectrum of
eigenvectors, $|x\rangle$ ($-\infty<x<\infty$). In quantum
mechanics we are often interested in the inner product,
$$\phi_{n}(x)=\langle x|n-1\rangle=H_n(x)\frac{e^{-x^2/2}}{(\pi
4^n(n!)^2)^{\frac{1}{4}}}.$$  The $\phi_n$'s are of course the
familiar Hermite functions and the $H_n$'s are the Hermite
polynomials. But in this paper we do not care so much about $X$
and its eigenvalues. More interesting is the
operator\cite{Spectroscopy},
$$\mathrm{K}_1=-\sqrt{aa^+}a^+ -a\sqrt{aa^+}$$
$\mathrm{K}_1$ has a continuous spectrum of eigenvectors
conventionally labelled by $\k$ for $-\infty<\k<\infty$. Consider
wavefunctions formed from the inner product,
$$v_n(\k)=\langle \k|n\rangle=\frac{\Tch_n(\k)}{
\sqrt{n}\sqrt{\frac{2}{\k}\sinh\frac{\pi\k}{2}}}$$ The functions
$\Tch_n$ are simply a set of polynomials, like the Hermite
polynomials, but orthogonal with respect to the weight
function\cite{Okuyama},
$$w(\k)=\left(\frac{2}{\k}\sinh\frac{\pi \k}{2}\right)^{-1}$$ The
$\Tch_n$'s are unfortunately not one of the classical orthogonal
polynomials we would easily find in a handbook. Therefore it's
worthwhile cataloging a few of their properties:

\bigskip
\noindent {\bf Differential equation}: The statement that the
$v_n$'s are eigenstates of $H$ translates to the differential
equation\cite{Virasoro,Small}: $$\sin\left(2\frac{d}{d\k}\right)\k
\Tch_n(\k)=2n\Tch_n(\k)$$

\bigskip
\noindent {\bf Generating function}\cite{Spectroscopy}:
$$\sum_{n=1}^\infty \frac{z^n}{n}\Tch_n(\k)=\frac{1}{\k}(1-e^{-\k\tan^{-1}z})
=f_{\k}(z) $$

\bigskip
\noindent {\bf Recurrence relations}: The polynomials can be
systematically generated from the relation
$$\Tch_{n+1}(\k)+\Tch_{n-1}(\k)=
-\frac{\k}{n}\Tch_{n}(\k);\ \ \ $$ after defining $\Tch_0(\k)=0$
and $\Tch_1(k)=1$. The first few polynomials are explicitly:
\begin{eqnarray}
&\ &\Tch_1(\k)=1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \Tch_4(\k)=
-\fraction{1}{6}\k^3 + \fraction{4}{3}\k\nonumber\\ &\
&\Tch_2(\k)=-\k\ \ \ \ \ \ \ \ \ \ \ \ \ \
\Tch_5(\k)=\fraction{1}{24}\k^4-\fraction{5}{6}\k^2+1\nonumber\\
&\ &\Tch_3(\k)=\half \k^2-1\ \ \ \ \ \ \ \
\Tch_6(\k)=-\fraction{1}{120}\k^5+\fraction{1}{3} \k^3
-\fraction{23}{15} \k  \nonumber
\end{eqnarray}
Here is a graph of the first five $v_n(\k)$'s:


\resizebox{4.5in}{1.5in}{\includegraphics{v.eps}}


\bigskip
\noindent{\bf Ladder operators}:
$$\cos^2\left(\frac{d}{d\k}\right) \k \Tch_n(\k)=-n\Tch_{n+1}(\k)$$
$$\sin^2\left(\frac{d}{d\k}\right) \k \Tch_n(\k)=-n\Tch_{n-1}(\k)$$

\bigskip\noindent {\bf Special values}:
\begin{eqnarray} &\ &\Tch_{2n}(0)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \
\Tch_{2n-1}(0)=(-1)^{n+1}\nonumber\\
&\ &\Tch'_{2n}(0) =(-1)^n \sum_{m=1}^n\frac{1}{2m-1}\ \ \ \ \
\Tch'_{2n-1}(0)=0\nonumber\\
&\ &\Tch_n(2im)=-\frac{n}{m}(-i)^{m+n}\Tch_m(-2in)
\nonumber\end{eqnarray}

\bigskip
\noindent {\bf Derivative}:
$$\Tch_{2n}'(\k)=-\frac{\Tch_{2n}(\k)}{\k}+2n
\sum_{m=1}^{n}\frac{(-1)^{m-n+1}}{2n-(2m-1)}\frac{\Tch_{2m-1}(\k)}{2m-1}$$
$$\Tch_{2n-1}'(k)=\frac{(-1)^{n+1}-\Tch_{2n-1}(\k)}{\k}+
(2n-1)\sum_{m=1}^{n-1}\frac{(-1)^{m-n}}{(2n-1)-2m}\frac{\Tch_{2m}(\k)}{2m}$$

\bigskip
\noindent {\bf Integral representations}\cite{Virasoro,squeezed}:

\begin{eqnarray} \Tch_{n}(\k)&=&\frac{n}{2\pi}i^{n-1}w(\k)\int_{-\infty}^{\infty}du\
e^{i\k u} \frac{\tanh^{n-1}u}{\cosh^2 u}\nonumber\\
&=&\frac{n}{2\pi}i^{n-1}w(\k)\left(\int_1^\infty+
\int_{-\infty}^{-1}\right)dx\
\frac{e^{i\k\coth^{-1}x}}{x^{n+1}}\nonumber\\ &=&\frac{n}{2\pi
i}\oint_{z=0} dz\ \frac{f_\k(z)}{z^{n+1}}
\end{eqnarray}

\bigskip\noindent {\bf Darboux-Christoffel Formula}:
$$ \sum_{n=1}^N\frac{1}{n}\Tch_n(\k)\Tch_n(\k')=-\frac{\Tch_{N+1}(\k)\Tch_N(\k')
-\Tch_{N+1}(\k')\Tch_N(\k)}{\k-\k'}$$


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\end{document}






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Whether and how open string field theory\cite{Witten} may be
formulated as a noncommutative field theory remains a compelling
open question\cite{Aref'eva_review}. At stake is a systematic
framework for understanding nonperturbative string physics and
ultimately defining string theory itself. The past two years have
seen a proliferation of alternative operator/Moyal approaches to
string field theory: split strings\cite{half-strings}, the
discrete Moyal formalism\cite{Bars,Bars-Matsuo,MSFT}, the mixed
basis \cite{Moyal,MSFT}, the string bit basis\cite{PP,MSFT}, the
continuous Moyal formalism\cite{Moyal,Belov,
various,Erler,Virasoro}, the continuous half-string
formalism\cite{Myself,continuous-half}. So many formalisms now
exist that one seriously questions whether any physical principle
distinguishes them or whether we are just playing games
translating between equivalent bases.

&\ &
\Tch_n(2iM)=\left\{\matrix{\frac{(-i)^{n-1}}{2}\frac{n}{M}\sum_{k=0}^M\pmatrix{M
\cr M-k}\pmatrix{k+n-1 \cr M-1}\ \ \ \mathrm{for}\ n\geq M \cr
\frac{(-i)^{n-1}}{2}\frac{n}{M}\sum_{k=0}^n\pmatrix{M \cr
n-k}\pmatrix{k+n-1 \cr n-1}\ \ \ \mathrm{for}\ n<M}
\right.\nonumber

One could try to explicitly remove $v_{2n}'(0)$ from the space,
but since it can be approximated to any accuracy by vectors
$v_{2n}(\k)/\k$ which {\it are} in the space, anyone sufficiently
determined could effectively recover $v_{2n}'(0)$ by taking an
appropriate limit. In some sense, this is analogous to considering
\EQN{discontinuity} to be the limit of a sequence of smooth
functions $f_n(\sigma)$ which vanish at the midpoint but approach
$1$ monotonically towards the endpoints.



 , like the ground state, the first term above does not
contribute Well behaved string functionals (like the ground state)
should satisfy,
$$P_L\star\Psi=\lim_{N\to\infty}P_L(N)\star\Psi=P_L\Psi$$
Then we can calculate,
\begin{eqnarray}
(P_L\star\Psi)\star x_{2n} &=& P_L\Psi x_{2n}-i\sum_{k=1}^\infty
T_{2n,2k-1}P_L\frac{\partial\Psi}{\partial p_{2k-1}}\nonumber\\
(P_L\star\Psi)\star \bar{x} &=&
\lim_{N\to\infty}(P_L\star\Psi)\star\bar{x}(N)=P_L\Psi
\bar{x}-i\frac{2\sqrt{2}}{\pi}\sum_{k=1}^\infty
\frac{(-1)^{k+1}}{2k-1} P_L\frac{\partial\Psi}{\partial
p_{2k-1}}\nonumber\end{eqnarray} On the other hand, we
have,\begin{eqnarray}p_{2n-1}\star(\Psi\star\bar{x})&=&
p_{2n-1}\star(\lim_{N\to\infty}\Psi\star\bar{x}(N))\nonumber\\
&=& p_{2n-1}\Psi\bar{x}\nonumber\\ &\
&-i\frac{2\sqrt{2}}{\pi}\sum_{k=1}^\infty\frac{(-1)^{k+1}}{2k-1}
p_{2n-1}\frac{\partial\Psi}{\partial p_{2k-1}} -i\sum_{k=1}^\infty
T_{2l,2n-1}\frac{\partial\Psi}{\partial{x_{2l}}}\bar{x}\nonumber\\
&\ &-\frac{2\sqrt{2}}{\pi}\sum_{k,l=1}^\infty
\frac{(-1)^{k+1}}{2k-1}T_{2l,2m-1}\frac{\partial^2\Psi}{\partial
p_{2k-1}\partial x_{2l}} \nonumber\\ &\
&-i\frac{2\sqrt{2}}{\pi}\frac{(-1)^{n+1}}{2n-1}\Psi\nonumber\\
P_L\star(\Psi\star\bar{x})&=&\lim_{N\to\infty}P_L(N)\star(\Psi\star\bar{x}) \nonumber\\
&=&P_L\Psi \bar{x}-i\frac{2\sqrt{2}}{\pi}\sum_{k=1}^\infty
\frac{(-1)^{k+1}}{2k-1} P_L\frac{\partial\Psi}{\partial
p_{2k-1}}-\frac{i}{2}\Psi\end{eqnarray} Bringing these two results
together,
 Note that the anomaly comes
directly from the ambiguous double sum discussed just a moment ago
in \EQN{ambsum}; the same sum appears {\it both} in
$(P_L*\Psi)*\bar{x}$ and $P_L*(\Psi*\bar{x})$, but evaluated in a
different order so that the difference is nonzero. The same
computation can be repeated in the continuous basis with the
$\omega$ regulator, with identical results.

 for instance in the mixed basis.

\begin{eqnarray}
&\ &\bar{x}(\rho)\Psi P_L(\rho')=-\frac{i}{2\pi} \int_0^\pi
d\sigma |f_\rho(\sigma)|x(\sigma)\ \left[\int_0^\pi
g_{\rho'}(\sigma) \frac{\delta}{\delta x(\sigma)}\right] \Psi
\nonumber\\
&\ & \half [\bar{x}(\rho),\Psi]_\star P_L(\rho')=-\frac{i}{2\pi}
\int_0^\pi g_{\rho'}(\sigma) \frac{\delta}{\delta x(\sigma)}
\left[\int_0^\pi d\sigma f_\rho(\sigma)x(\sigma)
\right] \Psi\nonumber\\
&\ &\half \bar{x}(\rho)[\Psi, P_L(\rho')]_\star=\frac{i}{2\pi}
\int_0^\pi d\sigma |f_\rho(\sigma)|x(\sigma)\left[\int_0^\pi
|g_{\rho'}(\sigma)| \frac{\delta}{\delta x(\sigma)}\right] \Psi
\nonumber\\
&\ &\fraction{1}{4}[\bar{x}(\rho),[\Psi,
P_L(\rho')]_\star]_\star=\frac{i}{2\pi} \int_0^\pi d\sigma
f_\rho(\sigma)x(\sigma)\ \left[\int_0^\pi
|g_{\rho'}(\sigma)| \frac{\delta}{\delta x(\sigma)}\right] \Psi\nonumber\\
&\ &\half[\bar{x}(\rho), P_L(\rho')]_\star\Psi=\frac{i}{2\pi}
\int_0^\pi d\sigma f_\rho(\sigma)g_{\rho'}(\sigma) \Psi\nonumber
\end{eqnarray}





\noindent
\begin{eqnarray} \bar{x}(\rho)\star(\Psi\star
P_L(\rho'))&=&\frac{2}{\pi}\int_0^{\pi/2}d\sigma
f_\rho(\sigma)x(\sigma)\ i\int_{\pi/2}^{\pi}d\sigma\
g_{\rho'}(\sigma)\frac{\delta}{\delta x(\sigma)}\Psi\nonumber\\
&=&i\int_{\pi/2}^{\pi}d\sigma\
g_{\rho'}(\sigma)\frac{\delta}{\delta x(\sigma)}\
\frac{2}{\pi}\int_0^{\pi/2}d\sigma
f_\rho(\sigma)x(\sigma)\Psi\nonumber\\&=&
(\bar{x}(\rho)\star\Psi)\star
P_L(\rho')\label{eq:36}\end{eqnarray} i.e. since the integrals
have no overlap they commute with each other, implying
associativity.  Multiplication with $P_L$ is still associative,
$$(\bar{x}(\rho)\star\Psi)\star
P_L=\bar{x}(\rho)\star(\Psi\star P_L)$$ because we can see that
the integrals involved commute,
\begin{equation}\int_{0}^{\pi/2}d\sigma\ \frac{\delta}{\delta
x(\sigma)}\int_0^{\pi/2}d\sigma
f_\rho(\sigma)x(\sigma)=\int_0^{\pi/2}d\sigma
f_\rho(\sigma)x(\sigma) \int_{0}^{\pi/2}d\sigma\
\frac{\delta}{\delta x(\sigma)}\label{com_int}\end{equation}

\noindent bearing in mind that $f_\rho(\sigma)$ has a Fourier
decomposition in terms of $\cos 2n\sigma$ for $n\geq 1$, so that,
$$\int_0^{\pi/2}d\sigma f_\rho(\sigma)=0$$ Now, however, we have
a small paradox. How is the above equation consistent with the
fact that $\lim_{\rho\to\infty} f_\rho=1$?




This in turn implies:
$$\int_{0}^{\pi/2}d\sigma\ \frac{\delta}{\delta
x(\sigma)}\int_0^{\pi/2}d\sigma f_\rho(\sigma)x(\sigma) \
\Psi[x(\sigma)]=\int_0^{\pi/2}d\sigma f_\rho(\sigma)x(\sigma)
\int_{0}^{\pi/2}d\sigma\ \frac{\delta}{\delta x(\sigma)}\
\Psi[x(\sigma)]$$ which by an argument similar to \EQN{36} implies
associativity. Note that we still have associativity:

However, in this particular case the fact that $A\star P_L$
involves an integral of $p(\sigma)$ over both halves of the string
does not in this particular case imply that associativity is lost,
since $P_L$ is a commutative coordinate\footnote{In particular,
note that $f_\rho(\sigma)=\sqrt{2}\sum_{n=1}^\infty
f_{2n}(\rho)\cos 2n\sigma$. Therefore $\int_0^{\pi/2}d\sigma
f_\rho(\sigma)=0$ and associativity follows along the lines of
\EQN{36}.} (assuming, again, that $x'(0)$ is excluded). However,
we have seen that $P_L$ may appear noncommutative if the proper
limit is taken, and in such situations associativity might fail.


no longer involves an integral over only half the string. However,
associativity is more delicate when we take the regulator to
infinity. In this limit, $f_\rho$ and $g_\rho$ approach $1$, and
\EQN{A} reduces to,

Apparently the anomaly is sensitive not only to the grouping of
parentheses, but also the way in which the multiplication is
carried out. A priori, there is no reason why we should favor
multiplying the fields either before or after taking the regulator
to infinity. In the previous paragraph we chose to multiply
before, because $P_L$ and $\bar{x}$ do not necessarily exist in
other bases (such as the continuous or discrete basis) before they
are multiplied with a sufficiently well-behaved state. However,
this just a matter of convenience, and is not an objectively valid
argument in favor of the prescription \EQN{assoc}.

It is interesting to try to translate the above computations into
position space, where they can be given a clearer visual
interpretation. Performing a fourier transform on the odd momenta,
$$p_{2n-1}\star\Psi\star x_{2m}=i\left(-\der{x_{2n-1}}-
\sum_{l=1}^\infty
T_{2l,2n-1}\der{x_{2l}}\right)\left(-\sum_{k=1}^\infty
T_{2m,2k-1}x_{2k-1}+x_{2m}\right)\Psi $$





 in other bases, such as the
continuous basis, we chose the former prescription we multiplied
This lends credence to the viewpoint that associativity anomalies
are really an indication that the formalism is being stretched
beyond its limits, and probably not and has become ambiguous.
