%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% (plain tex, 8 pages)
% K-H REHREN
% BOUNDED BOSE FIELDS (may 1996)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\magnification=\magstep 1
\hsize=134mm
\vsize=200mm
\hoffset=0mm
\voffset=-2mm
\parindent=0mm \parskip=7pt plus1pt minus.5pt % 5 + 1 - 0.5
%
\global\newcount\secno \global\secno=0
\def\newsec#1{\global\advance\secno by1 \bigbreak\mgap
{\bbf\the\secno. #1} $\hbox{\vrule width0pt height0pt depth13pt}$
\nobreak\newline \message{(\the\secno. #1)}}
\global\newcount\subsecno \global\subsecno=0
\def\newsubsec#1{\global\advance\subsecno by1
\bigbreak\sgap {\bf\the\secno.\the\subsecno. #1} $\hbox{\vrule
width0pt height0pt depth8pt}$ \nobreak\newline
\message{(\the\secno.\the\subsecno. #1)}}
\global\newcount\equation \global\equation=0
\def\nummer{\global\advance\equation by1 \eqno(\the\equation) }
\font\large=cmr12
\font\eightpoint=cmr8
\font\ninepoint=cmr9
\font\fivepoint=cmr5
\font\hbf=cmbx10 scaled\magstep2
\font\bbf=cmbx10 scaled\magstep1
\font\sbf=cmbx9
\font\srm=cmr9
\font\bsl=cmsl10 scaled\magstep1
\font\caps=cmcsc10
\font\tenit=cmti10
\font\tenbf=cmbx10
\font\bbl=msbm10
\def\newline{\hfil\break\noindent}
\def\newpage{\vfill\eject\noindent}
\def\sgap{\par\vskip 3mm\noindent} %2
\def\mgap{\par\vskip 5mm\noindent} %4
\def\bgap{\par\vskip 8mm\noindent} %7
\def\hh{{\cal H}} \def\a{\alpha} \def\b{\beta} \def\udl{\underline}
\def\frac#1#2{{#1\over#2}} \def\supp{{\rm supp}\,} \def\id{{\rm id}}
\def\double#1{\,{}\colon #1 \colon\,}
\def\trip{\vcenter{\vbox{\hbox{$.$}\hrule width.0ex height.45ex
\hbox{$.$}\hrule width.0ex height.45ex \hbox{$.$}}}}
\def\triple#1{\,\trip #1 \trip\,}
\def\norm#1{\vert\!\vert #1 \vert\!\vert}
\def\be{$$} \def\ee{\nummer $$}
%
\def\RR{{\rm I\!R}} \def\NN{{\rm I\!N}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% reference no.s %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\cite#1{[#1]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bgap
\centerline{\hbf Bounded Bose fields}
\mgap
\centerline{\caps K.-H.\ Rehren} \sgap
\centerline{II.\ Institut f\"ur Theoretische Physik}
\centerline{Universit\"at Hamburg ({\caps Germany})}
\centerline{{\srm email: rehren@x4u2.desy.de}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bgap
\baselineskip12pt
{\sbf Abstract.}
{\srm Examples for bounded Bose fields in two dimensions are presented.}
%\sgap
\baselineskip15pt % 13
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newsec{Introduction}
%
Scalar free fields give rise to unbounded smeared field operators
$\varphi(f) = \int f(\udl x)\varphi(\udl x)\, d^dx$. In contrast \cite{1},
free Fermi
fields give rise to bounded smeared field operators $\psi(f) = \int f(\udl x)
\psi(\udl x)\, d^dx$. These familiar facts are due to the unlimited occupation
number in every excitation mode of the symmetric Fock space, and to the Pauli
exclusion principle in the anti-symmetric Fock space, respectively.
The boundedness or unboundedness being a probe for the structure of the state
space (in a sense suggested by the above examples), the question is of
interest whether unboundedness of smeared field operators is a necessary
feature of Bose fields.
Indeed, Baumann \cite{2} has proven that the spectrum condition entails that
{\it chiral}\/ Bose fields in two dimensions are necessarily unbounded. On the
other hand, Buchholz \cite{3} has observed that a very simple Bose field
in two dimensions which is the tensor product of two chiral free Fermi fields,
that is
\be \varphi(f) = \int d^2x \, f(t,x) \, \psi(t+x) \otimes \psi(t-x) \; ,
\ee % 1
is bounded. Clearly, this field commutes with itself not only at space-like
distance but also at time-like distance. In view of Huygens' principle, this
feature is usually interpreted as the absence of interaction for massless
fields.
In the present letter we construct bounded Bose fields in two dimensions which
commute with themselves at space-like distance but not at time-like distance.
All our examples are based on various factorizations of chiral Fermi
fields into chiral vertex operators.
\newsec{The construction}
%
The starting point is the bosonization formula \cite{4}, which represents
chiral free Fermi fields as vertex operators of unit charge. Using the
boundedness of smeared free Fermi fields, we shall infer the boundedness of
smeared chiral vertex operators with charge below unity. In a second step, we
pass to two-dimensional local fields.
\bigbreak
We briefly recall the relevant definitions. Chiral vertex operators
$E_\a(x)$ are defined by
\be
E_\a(x) = \mu^{\frac 12 \a^2} \triple{e^{i\a\phi(x)}} \equiv
\mu^{\frac 12 \a^2} e^{i\a\phi_+(x)}e^{i\a\phi_-(x)} = E_{-\a}(x)^*
\ee % 2
where $\phi = \phi_+ + \phi_-$ with $\phi_+ = (\phi_-)^*$, satisfying the
commutation relations
\be
[\phi_-(x),\phi_+(y)] = -\log i\mu(x-y-i\epsilon)
\ee % 3
while $[\phi_-,\phi_-] = [\phi_+,\phi_+] = 0$. The variable $x \in \RR$ is a
chiral light-cone coordinate. The regulating parameter $\mu > 0$ for the
highly singular field $\phi$ will ultimately be sent to zero. Expressions
involving $i\epsilon$ are always understood as boundary values as
$\epsilon\searrow 0$.
The vertex operators (2) satisfy commutation relations
\be
E_\a(x) E_\b(y) = e^{\pm i\pi\a\b} E_\b(y)E_\a(x)
\ee % 4
with the $+$ sign if $x>y$ and the $-$ sign if $x \supp g_2$ resp.\ $\supp g_2 > \supp g_1$.
In order to obtain Bose fields, we proceed to the tensor product of two
non-local chiral fields $\psi_\tau$
\be
\varphi_\tau(g \otimes h) := \psi_\tau(g) \otimes \psi_\tau(h)
\ee % 22
for test functions $(g \otimes h)(t,x) = g(t+x)h(t-x)$ on $\RR^2$. This
tensor product is understood to extend by linearity, as in eq.\ (1), to the
two-dimensional field
\be \varphi_\tau(f) =
\int d^2x \, f(t,x) \, \psi_\tau(t+x) \otimes \psi_\tau(t-x) \; .
\ee % 23
\bigbreak
If two test functions $f_i = g_i \otimes h_i$ on $\RR^2$ ($i = 1,2$) have
supports at space-like distance, then either $\supp g_1 < \supp g_2$ and
$\supp h_1 > \supp h_2$, or $\supp g_1 > \supp g_2$ and $\supp h_1 <
\supp h_2$. In either case, the commutation relations (21) produce two
opposite phase factors, and consequently $\varphi_\tau(f_1)$ and
$\varphi_\tau(f_2)$ commute, whenever $f_i$ have supports at space-like
distance. On the other hand, if $f_i$ have supports at time-like distance,
then the phase factors add rather than cancel, and consequently
$\varphi_\tau$ does not satisfy Huygens' principle. Choosing $\cos^2\tau =
\frac 12$ one may even have time-like anti-commutation. (This possibility
was also previously discovered by Buchholz \cite{3}.)
By linearity, the local commutativity extends to tensor product fields
smeared with arbitrary test functions on $\RR^2$.
\newsubsec{Boundedness of tensor products}
% 2.4
By Schwartz' nuclear theorem, the extension of the tensor product of
two (operator-valued) distributions in one variable to a functional in two
variables is again a distribution, and the chiral bound (19) implies bounds
for the two-dimensional fields (23) in terms of some Schwartz norms of the
test function $f$. The following estimates due to Buchholz \cite{3} improve
the bounds due to the theorem.
We rewrite eq.\ (23) in the form
\be \varphi_\tau(f) = \int dp\,dq\;P(i\partial_p)P(i\partial_q)\tilde f(p,q)
\; \int d^2x \frac{e^{ipx_+}}{P(x_+)} \psi_\tau(x_+) \otimes
\frac{e^{iqx_-}}{P(x_-)} \psi_\tau(x_-) \; ,
\ee % 24
where $\tilde f(p,q)$ is the Fourier transform of $f$ with respect to the
chiral coordinates $x_\pm = t \pm x$, and $P$ is a polynomial in one chiral
coordinate such that $1/P$ is in the domain of $m_\omega$. The
operator-valued integral $\int d^2x \ldots$ in eq.\ (24) is of the form (22),
and is hence of norm less than $const. \cdot \norm{m_\omega/P}_2^2$. Thus
\be \norm{\varphi_\tau(f)} \leq const. \cdot
\norm{P(i\partial_p)P(i\partial_q)\tilde f}_1 \; .
\ee % 25
Estimating the $L^1$-norm by the Cauchy-Schwarz inequality
$\norm f_1 \leq \norm{1/Q}_2\norm{Qf}_2$ provided $1/Q$ is square integrable,
and using the Plancherel equality of the $L^2$-norms of a function and its
Fourier transform, we also have
\be \norm{\varphi_\tau(f)} \leq const. \cdot
\norm{Q \cdot P(i\partial_p)P(i\partial_q)\tilde f}_2 = const. \cdot
\norm{Q(-i\partial_+,-i\partial_-) (P \otimes P) f}_2
\ee % 26
for appropriate polynomials $P(x_\pm)$ and $Q(p,q)$ as
qualified before, e.g., $P(x) = 1+x^2$ and $Q(p,q) = (1-ip)(1-iq)$. The
bound (26) holds for all test functions $f$ which are sufficiently smooth
and of sufficiently rapid decay, and in particular for all Schwartz functions.
The constant depends on $\tau$, on the state $\omega$, and on $P$ and $Q$.
We conclude that $\varphi_\tau$ are bounded Bose fields with non-trivial
time-like commutators. As the parameter $\tau$ varies between $0$ and
$\frac \pi 2$, the fields $\varphi_\tau$ interpolate between Buchholz'
example (1) and the identity operator.
\newsubsec{Further examples}
% 2.5.
The above construction has a non-abelian generalization. It was shown by
Wassermann and by Loke, with essentially the same method as ours, that --
when passing from level $k$ to level $k+1$ -- the coset model factorization
of chiral free $SU(N)$ Fermi fields produces $L^2$-{\it bounded}\/ chiral
exchange fields (non-abelian vertex operators)
which are primary for the level $k+1$ current algebra \cite{5}, and for the
coset Virasoro algebra (for $N=2$) \cite{6}, respectively. The $SU(2)$ coset
Virasoro algebra has central charge in the discrete series $c<1$, and the
primary chiral fields thus obtained are, in the standard nomenclature,
the fields $\phi_{12}$ and $\phi_{22}$ with chiral scaling dimension
$h_{12} = \frac{k-1}{4(k+2)}$ and $h_{22} = \frac 3{4(k+1)(k+2)}$.
The primary fields for the current algebra are those corresponding to the
vector and conjugate vector representations of $SU(N)$.
On the other hand, it is well known (see, e.g., \cite{7}) that tensor
products of chiral exchange fields in rational theories yield {\it local}\/
two-dimensional fields of the form
\be
\varphi(t,x) = \sum_{e\bar e} \zeta_{e\bar e} \; \phi_e(t+x) \otimes
\phi_{\bar e}(t-x)
\ee % 27
with appropriate numerical coefficients $\zeta_{e\bar e}$; the labels $e$
and $\bar e$ run over the pairs of initial and final sectors connected by
the primary field $\phi$ and its conjugate, respectively. This result relies
only on a `CPT' symmetry due to pentagon and hexagon identities of fusion and
braiding matrices.
Taken together, these two results show that $\varphi$ in eq.\ (27) are bounded
two-dimensional Bose fields if the primary field $\phi$ is one of the fields
produced by the coset factorization of chiral free Fermi fields.
These fields are the fields $\varphi_{12}$ and $\varphi_{22}$ in the minimal
models \cite{8}, and the basic matrix fields $g_{ab}$ of the two-dimensional
$SU(N)$ Wess-Zumino-Witten models \cite{9}, respectively.
\newsec{Conclusion}
% 3.
We have presented a variety of bounded Bose fields in two dimensions. The
family of fields (23) for various parameters $\tau$ are actually defined
on a common Hilbert space and are relatively local among each other.
All the bounded Bose fields we have constructed here are conformally covariant
fields with scaling dimension $d \equiv h_+ + h_- \leq 1$. To remove this
upper bound for the scaling dimension, one may consider (non-primary)
derivative fields
\be
\partial_{\mu_r} \ldots \partial_{\mu_1}\varphi(f) = (-1)^r
\varphi(\partial_{\mu_r} \ldots \partial_{\mu_1}f)
\ee % 27
which are again bounded Bose fields but with scaling dimensions
$r \leq d \leq r+1$.
We conclude that in two dimensions, bounded Bose fields are quite abundant.
We still do not know whether bounded Bose fields exist in more than
two dimensions.
%
\mgap
\bigbreak
{\bf Acknowledgments.} I thank J. Yngvason for encouraging me to write
down these notes, and D. Buchholz and K. Fredenhagen for helpful comments.
\mgap
\bigbreak
{\bbf References} \vskip 3mm
\def\ref#1{\par \noindent \hangafter=1 \hangindent 14pt \cite{#1}}
\parskip 4pt
\baselineskip=2.5ex\smallskip
\def\CMP#1{Com\-mun.\ Math.\ Phys.\ {\bf #1}}
\def\PR#1{Phys.\ Rev.\ {\bf #1}}
%
\ref{1} H. Araki, W. Wyss: {\it Representations of canonical
anticommutation relations}, Helv.\ Phys.\ Acta {\bf 37}, 136--159 (1964).
%
\ref{2} K. Baumann: Talk at the Symposium on Algebraic Quantum Field Theory
and Constructive Field Theory, G\"ottingen, 1995, and preprint in
preparation (1996).
%
\ref{3} D. Buchholz: unpublished.
%
\ref{4} S. Mandelstam: {\it Soliton operators for the quantized sine-Gordon
equation}, \PR{D 11}, 3026--3030 (1975).
%
\ref{5} A. Wassermann: {\it Operator algebras and conformal field theory,
III}, preprint Cambridge (UK), 1995.
%
\ref{6} T. Loke: {\it Operator algebras and conformal field theory of the
discrete series representations of {\rm Diff}$(S^1)$}, PhD thesis Cambridge
(UK), 1994.
%
\ref{7} K.-H. Rehren: {\it Space-time fields and exchange fields}, \CMP{132},
461--483 (1990).
%
\ref{8} A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov: {\it Infinite
dimensional symmetries in two-dimensional quantum field theory},
Nucl.\ Phys.\ {\bf B 241}, 333--380 (1984).
%
\ref{9} V.G. Knizhnik, A.B. Zamolodchikov: {\it Current algebra and
Wess-Zumino model in two dimensions}, Nucl.\ Phys.\ {\bf B 247}, 83--103
(1984).
\bye