INSTRUCTIONS
The text between the lines BODY and ENDBODY is made of
1180 lines and 45448 bytes (not counting or )
In the following table this count is broken down by ASCII code;
immediately following the code is the corresponding character.
30115 lowercase letters
1965 uppercase letters
767 digits
4 ASCII characters 9
5955 ASCII characters 32
14 ASCII characters 33 !
23 ASCII characters 34 "
76 ASCII characters 35 #
1226 ASCII characters 36 $
156 ASCII characters 37 %
103 ASCII characters 38 &
41 ASCII characters 39 '
414 ASCII characters 40 (
414 ASCII characters 41 )
5 ASCII characters 42 *
16 ASCII characters 43 +
495 ASCII characters 44 ,
150 ASCII characters 45 -
526 ASCII characters 46 .
5 ASCII characters 47 /
39 ASCII characters 58 :
1 ASCII characters 59 ;
5 ASCII characters 60 <
92 ASCII characters 61 =
2 ASCII characters 62 >
4 ASCII characters 64 @
56 ASCII characters 91 [
1548 ASCII characters 92 \
55 ASCII characters 93 ]
108 ASCII characters 94 ^
244 ASCII characters 95 _
42 ASCII characters 96 `
381 ASCII characters 123 {
9 ASCII characters 124 |
381 ASCII characters 125 }
11 ASCII characters 126 ~
BODY
%%%%%% FORMAT %%%%%%%%%%%%
\magnification\magstep1
\hoffset=0.5truecm
\voffset=0.5truecm
\hsize=15.8truecm
\vsize=23.truecm
\baselineskip=14pt plus0.1pt minus0.1pt \parindent=19pt
\lineskip=4pt\lineskiplimit=0.1pt \parskip=0.1pt plus1pt
%%%%%% Extra Fonts %%%%%%%%%
\def\got#1{\bf#1}
%% If eufm10 fonts are available, you will get a much more
%% gothic effect for Lie algebras by replacing previous line
%% with the following:
%% \font\got=eufm10 scaled\magstep1
\def\G{\hbox{\got g}}
\def\P{\hbox{\got p}}
\def\V{\hbox{\got a}}
%%%%%% SYMBOLS %%%%%%%%%%%%%
\def\A{{\cal A}}
\def\tildA{\tilde{\cal A}}
\def\tildAnul{\tilde{\cal A}_0}
\def\a{\alpha}
\def\B{{\cal B}}
\def\b{\beta}
\def\C{{\cal C}}
\def\Co{{\bf C}}
\def\d{\delta}
\def\D{\Delta}
\def\E{{\cal E}}
\def\f{\varphi}
\def\g{\gamma}
\def\H{{\cal H}}
\def\K{{\cal K}}
\def\L{\Lambda}
\def\m{\mu}
\def\Na{{\bf N}}
\def\o{\omega}
\def\O{{\cal O}}
\def\p{\pi}
\def\Q{\Omega}
\def\r{\rho}
\def\R{{\cal R}}
\def\Re{{\bf R}}
\def\s{\sigma}
\def\t{\tau}
\def\T{\Theta}
\def\th{\theta}
\def\U{{\cal U}}
\def\W{{\cal W}}
\def\x{\xi}
\def\Z{{\cal Z}}
\def\Ze{{\bf Z}}
\def\ad{{\rm ad}}
\def\Lor{{\cal L}_+^\uparrow}
\def\Poi{{\cal P}_+^\uparrow}
\def\Spin{\widetilde{\cal P}}
\def\imply{\Rightarrow}
\def\np{\par\noindent}
\def\nu{\char'43}
\def\lar{\longrightarrow}
\def\ov{\overline}
\def\mappazza{\buildrel{\widetilde U}\over\lar}
\def\derteqzero{{d\over{dt}}\Bigr\vert_{t=0} \,\,}
\def\quot#1#2{\hbox{$\matrix{#1\cr\vphantom{#2}}$
\hskip-.8cm\vbox{\vfill/\vfill}\hskip-12.4cm
$\matrix{\vphantom{#1}\cr#2}$}}
%%%%%%%% Macros for theorems %%%%%%%%
\def\proof{\medskip\noindent{\bf Proof.}\quad}
\def\proofof#1{\medskip\noindent{\bf Proof of #1.}\quad}
\def\square{\hbox{$\sqcap\!\!\!\!\sqcup$}}
\def\endproof{\hskip-4mm\hfill\vbox{\vskip3.5mm\square\vskip-3.5mm}\bigskip}
\def\section #1\par{\vskip0pt plus.3\vsize\penalty-75
\vskip0pt plus -.3\vsize\bigskip\bigskip
\noindent{\sectionfont #1}\nobreak\smallskip\noindent}
\def\subs#1{\medskip\noindent{\it#1}\qquad}
\def\claim#1#2\par{\vskip.1in\medbreak\noindent{\bf #1.} {\sl #2}\par
\ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi}
\def\rmclaim#1#2\par{\vskip.1in\medbreak\noindent{\bf #1.} { #2}\par
\ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi}
%%%%%%%% AUTO REF. %%%%%%%%%%%%%
\font\sectionfont=cmbx10 scaled\magstep1
\newcount\REFcount \REFcount=1
\def\numref{\number\REFcount}
\def\addref{\global\advance\REFcount by 1}
\def\wdef#1#2{\expandafter\xdef\csname#1\endcsname{#2}}
\def\wdch#1#2#3{\ifundef{#1#2}\wdef{#1#2}{#3}
\else\write16{!!doubly defined#1,#2}\fi}
\def\wval#1{\csname#1\endcsname}
\def\ifundef#1{\expandafter\ifx\csname#1\endcsname\relax}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\ref(#1){\wdef{q#1}{yes}\ifundef{r#1}$\diamondsuit$#1
\write16{!!ref #1 was never defined!!}\else\wval{r#1}\fi}
\def\inputreferences{
\def\REF(##1)##2\endREF{\wdch{r}{##1}{\numref}\addref}
\REFERENCES}
\def\references{
\def\REF(##1)##2\endREF{
\ifundef{q##1}\write16{!!ref. [##1] was never quoted!!}\fi
\item{[\ref(##1)]}##2}
\section{References}\par\REFERENCES}
%%%%%%REFERENCES%%%%%%%%
\def\REFERENCES{
\REF(BiWi1)Bisognano J., Wichmann E. , ``{\it On the duality
condition for a Hermitian scalar field}", J. Math. Phys. {\bf
16} (1975), 985-1007.
\endREF
\REF(Borc1)Borchers H.J. , ``{\it The CPT theorem in
two-dimensional theories of local observables}", Commun. Math.
Phys. {\bf 143} (1992), 315.
\endREF
\REF(Brow1)Brown K.S., ``{\it Cohomology of groups}" Springer,
New York Heidelberg Berlin, 1982.
\endREF
\REF(BGL)Brunetti R., Guido D., Longo R. , ``{\it Modular
structure and duality in conformal quantum field theory}",
Commun. Math. Phys., {\bf 156} (1993), 201-219.
\endREF
\REF(BuSu1)Buchholz, D. Summers, S.J. ``{\it An algebraic
characterization of vacuum states in Minkowski space}", preprint
DESY 92-119
\endREF
\REF(DoLo1)Doplicher S., Longo R., ``{\it Standard and split
inclusions of von Neumann algebras}", Invent. Math. {\bf 73}
(1984), 493-536.
\endREF
\REF(FGL)In preparation.
\endREF
\REF(FrGa1)Gabbiani F., Fr\"ohlich J., ``{\it Operator algebras
and Conformal Field Theory}", pre-print ETH Z\"urich, 1992.
\endREF
\REF(GL1)Guido, D. Longo, R., ``{\it Relativistic invariance
and charge conjugation in quantum field theory}", Commun. Math.
Phys. {\bf 148} 521-551.
\endREF
\REF(Haag1)Haag R., {\it Local Quantum Physics}, Springer
Verlag, Berlin Heidelberg 1992.
\endREF
\REF(HiLo1)Hislop P., Longo R., ``{\it Modular structure of the
von~Neumann algebras associated with the free massless scalar
field theory}", Commun. Math. Phys. {\bf 84} (1982), 84.
\endREF
\REF(McLa1)Mc Lane S. ``{\it Homology}", Springer, Berlin, 1963.
\endREF
\REF(Mich1) Michel L., ``{\it Sur les extensions centrales du
groupe de Lorentz inhomog\`ene connexe}'', Nucl. Phys. 57 (1965)
356-385.
\endREF
\REF(Miln1)Milnor J., ``{\it Introduction to algebraic
K-theory}", Ann. of Math Studies {\bf 72}, Princeton University
Press, Princeton, 1971.
\endREF
\REF(Moor1)Moore C.C., ``{\it Group extensions and cohomology
for locally compact groups. III}'', Trans. Amer. Math. Soc. {\bf
221} (1976) 1.
\endREF
\REF(Moor2)Moore C.C., ``{\it Group extensions and cohomology
for locally compact groups. IV}'', Trans. Amer. Math. Soc. {\bf
221} (1976) 35.
\endREF
\REF(Simm) Simms D.J., ``{\it Lie groups and quantum
mechanics}" Lecture Notes in Math., Vol. 52, Springer, Berlin,
1968.
\endREF
\REF(StraZs1)Str\u atil\u a S., Zsido L., {\it Lectures on von
Neumann algebras}, Abacus press, England 1979.
\endREF
\REF(Wies1)Wiesbrock H.V., ``{\it A comment on a recent work of
Borchers}", to appear in Lett. Math. Phys.
\endREF
\REF(Wies2)Wiesbrock H.V., ``{\it Conformal quantum field
theory and half-sided modular inclusions of von~Neumann
algebras}", to appear in Commun. Math. Phys.
\endREF
\REF(Yngv1)Yngvason J. ``{\it A note on essential
duality}", preprint.
% available via anonymous ftp in mp_arc@math.utexas.edu
\endREF
}
\inputreferences
\font\ftitle=cmbx10 scaled\magstep1
\font\fabs=cmbx10 scaled800
\vskip2truecm
\centerline{\ftitle GROUP COHOMOLOGY, }
\medskip\centerline{\ftitle MODULAR THEORY}
\medskip\centerline{\ftitle AND SPACE-TIME SYMMETRIES}
\bigskip\bigskip
\centerline{R. Brunetti$^1$\footnote{$^*$}{ Supported in
part by MURST and
CNR-GNAFA.}
\footnote{$^\bullet$}{Supported in part by INFN, sez.
Napoli and by Fondazione ``A. della Riccia"},
D. Guido$^{2*}$ and R. Longo$^{23*}$}
\footnote{}{E-mail\ brunetti@na.infn.it,
guido@mat.utovrm.it, longo@mat.utovrm.it
}
\vskip1.truecm
\item{$(^1)$} Department of Physics, Syracuse University
\par
201 Physics Building, Syracuse, NY 13244-1130.
\par
\item{$(^2)$} Dipartimento di Matematica, Universit\`a
di Roma ``Tor Vergata''
\par
Via della Ricerca Scientifica, I--00133 Roma,
Italia.
\item{$(^3)$} Accademia Nazionale dei Lincei,
\par via della Lungara 10, I--00165 Roma, Italia
\vskip 3cm\noindent
{\bf Abstract. } The Bisognano-Wichmann property on
the geometric behavior of the modular group of the von Neumann
algebras of local observables associated to wedge regions in
Quantum Field Theory is shown to provide an intrinsic sufficient
criterion for the existence of a covariant action of the
(universal covering of) the Poincar\'e group. In particular
this gives, together with our previous results, an intrinsic
characterization of positive-energy conformal pre-cosheaves of
von Neumann algebras. To this end we adapt to our use Moore
theory of central extensions of locally compact groups by
polish groups, selecting and making an analysis of a wider class
of extensions with natural measurable properties and showing
henceforth that the universal covering of the Poincar\'e group
has only trivial central extensions (vanishing of the first and
second order cohomology) within our class.
\section{Introduction}
\par
In this paper we discuss the problem of characterizing
the existence of a Poincar\'e covariant action for a net of
local observable algebras in terms of net-intrinsic algebraic
properties.
In other words, given a state with a Reeh-Schlieder cyclicity
property, we look for a condition that ensures it to be a
relativistic vacuum.
The work done by Bisognano and Wichmann [\ref(BiWi1)]
during the seventies about the essential duality has shown the
geometrical character of the Tomita-Takesaki modular operators
associated with von Neumann algebras generated by Wightman
fields with support based on wedge regions and the vacuum
vector.
While an algebraic version of the Bisognano-Wichmann theorem in
the Poin\-ca\-r\'e covariant case is not yet available, there
are general results in this direction given by the recent work
of Borchers [\ref(Borc1)], see also
[\ref(BuSu1),\ref(Wies1),\ref(Yngv1)] for related results. Based
on the result of Borchers, there is a complete algebraic
analysis in [\ref(BGL)] and [\ref(FrGa1)] showing that, for a
conformally invariant theory, the modular groups associated with
the von~Neumann algebras of double cones are associated with
special one-parameter subgroups of the conformal group leaving
invariant the given double cone (cf. [\ref(HiLo1)]).
Applications of the Bisognano-Wichmann theorem to the analysis
of the relations between the space-time symmetries and
particle-antiparticle symmetry in quantum field theory are
contained in [\ref(GL1)].
Here we show how to reconstruct the Poincar\'e group
representation from the net-intrinsic property of {\it
modular covariance} (assumption 2.1), i.e. the modular operators
associated with the algebras of the wedge regions act
geometrically as boosts.
In our previous paper [\ref(BGL)] we showed in particular that,
for a Poincar\'e covariant net, the modular unitaries associated
with wedge regions should necessarily coincide with the boost
unitaries in the given representation if the
modular covariance and the split property are
assumed. Here, we do not have any given unitary representation
of the Poincar\'e group from the start. Instead we recover a
canonical representation of the Poincar\'e group, which is
unique if the split property holds, just by using the
modular unitaries corresponding to all wedge algebras.
This step forward makes two significant changes. The first
concerns physics, since our result shows that the net itself
contains all the space-time symmetry informations, and that the
modular covariance property gives rise to a
canonical representation even when (non-split case) it is
not unique.
The second concerns mathematics, providing a new application of
the group cohomology describing central extensions of groups.
As is well known, central extensions of groups appear naturally
in physics, as shown for example by the Wigner theorems
[\ref(Simm)] on the (anti-)unitary realization of the
symmetries in quantum physics.
The class of results in this context which is relevant
for our purposes concerns the triviality of (suitable classes of)
central extensions of the universal covering of the Poincar\'e
group.
One of the first results of this type is due to Michel
[\ref(Mich1)], but his hypotheses seems unapplicable to our
problem.
Another result in this direction is a particular case of the
theory of Moore [\ref(Moor1),\ref(Moor2)] on central
extensions of locally compact groups via polish groups.
The extensions described by Moore are continuous and open, and
his theorem on the universal extension concerns extensions by
closed subgroups of $\U(\H)$, the group of unitary operators on a
separable Hilbert space $\H$.
As a main step in our analysis, we study a class of extensions
with natural measurable properties, which we call weak Lie
central extensions. These extensions are given by a (not
necessarily closed) subgroup $G$ of $\U(\H)$, and we assume that
there are continuous one-parameter subgroups in $G$ which
correspond to a set of Lie-algebra generators (see section 2 for
the precise definition). We show that weak Lie central
extensions of a simply connected perfect Lie group are trivial,
and this applies to the universal covering of the Poincar\'e
group.
Our proof strongly relies on Moore theory. In fact, even though
the continuity properties of the central extensions described by
Moore are too restrictive for our purposes, we prove that his
theorem on the structure of the first and second order
(measurable) cohomology of a connected perfect Lie group
(see [\ref(Moor2)] or theorem 1.3) can be generalized to the weak
Lie central extensions (see theorem 1.7).
The cases of the Poincar\'e group and of the conformal
group find application in section 2.
Finally we mention that our results, together with theorem 2.3
in our previous paper [\ref(BGL)], give a complete
characterization of positive-energy conformal pre-cosheaves as
those where the modular groups associated with
double cones act geometrically as expected (see corollary 2.9).
The plan of the paper is the following: in the first section we
review some known facts about cohomology and extensions of
groups, recall elements of Moore theory and provide a
generalization of results needed in the following.
In the second section we present our results about the
generation of the unitary representation of the group of
space-time symmetries by modular groups, in case of local
algebras on a separable Hilbert space.
The last section contains an outlook.
\section 1. Group extensions and cohomology
\par
We begin by shortly reviewing basic elements of central
extensions of groups (see [\ref(McLa1),\ref(Brow1)]). We first
deal with {\it algebraic central extensions}, i.e. with short
exact sequences of groups
$$
1\to A\buildrel i\over\to G\buildrel\pi\over\to P\to
1\eqno(1.1)
$$
where $P$ is the group to extend and $A$ is a central
subgroup of $G$. The pair $(G,\pi)$ determines the
extension, and we often refer to it as a central
extension of $P$ (by $A = ker(\pi)$).
\par
Two extensions $(G_1,\pi_1)$, $(G_2,\pi_2)$ are called
{\it equivalent} if there is an isomorphism between $G_1$
and $G_2$ such that the following diagram commutes:
$$
\matrix{
& & & & G_1 & & & & \cr
& & &\nearrow& &\searrow& & & \cr
1&\to&A& &\updownarrow & &P&\to&1\cr
& & &\searrow& &\nearrow& & & \cr
& & & & G_2 & & & & \cr}
\eqno(1.2)
$$
An extension of $P$ via $A$ is {\it trivial} if it
is equivalent to the direct product $(A\times P,\pi)$ where
$\pi$ is the projection onto $P$, i.e. sequence (1.1) splits.
A {\it section} of an extension $(G,\pi)$, i.e. a
map $s:P\to G$ such that $\pi\cdot s=id_P$, determines an
identification of $G$ with $A\times P$ as sets given by
$g\to(s(\pi(g))^{-1}g,\pi(g))$. Then, the
multiplication rule on $A\times P$ is given by
$$
(a,p)\cdot(b,q)=(ab\o(pq)^{-1},pq),\eqno(1.3)
$$
where $\o(p,q) = s(q)s(pq)^{-1}s(p)$ satisfies the 2-{\it
cocycle} condition $\d_2\o=0$, where $\d_2$ is defined in
$(1.4)$.
Conversely a 2-cocycle gives an associative multiplication on
$A\times P$, thus defining an extension.
Two cocycles $\o_1$, $\o_2$ give rise to
equivalent extensions {\it iff} there is a group
automorphism of $A\times P$ which is compatible with
diagram (1.2) and intertwines the products given by $\o_1$
and $\o_2$. This means that there is a map $\f:P\to A$
such that the morphism between $(A\times P,\o_1)$ and
$(A\times P,\o_2)$ is given by $(a,p)\mapsto(a\f(p),p)$,
and $\o_1=\o_2\cdot\d_1\f$, with $\d_1$ defined in (1.4),
i.e. $\o_2$ differs by $\o_1$ of a coboundary, showing
that equivalence classes of extensions are indeed
cohomology classes.
\par
The $n$-{\it cochains} $C^n(P,A)$ of $P$ with values in $A$
are maps from $P^n$ to $A$, and the
$n$-th {\it coboundary map} $\d_n:C^n(P,A)\to
C^{n+1}(P,A)$ is given by
$$
\eqalign{
\d_n f(p_1,\dots,p_{n+1})
&=f(p_2,\dots,p_{n+1})-f(p_1p_2,p_3,\dots,p_{n+1})\cr
&+f(p_1,p_2p_3,\dots,p_{n+1})+\dots\cr
&+(-1)^nf(p_1,\dots,p_np_{n+1})-(-1)^nf(p_1,\dots,p_n)}
\eqno(1.4)
$$
and verifies $\d_{n+1}\d_n=1$.
\par
The range of $\d_{n-1}$ is denoted by
$B^n(P,A)$, the kernel of $\d_n$
by $Z^n(P,A)$, and their quotient by
$H^n(P,A)$. Since $C^n(P,A)$ is an abelian group with respect to
the pointwise multiplication, $H^n(P,A)$ is a group too.
\par
The above discussion shows that equivalence classes of
central extensions of $P$ via $A$ are in $1-1$
correspondence with elements of $H^2(P,A)$, and therefore
form a group. Note that the cocycle equation for
1-cochains means exactly that $Z^1(P,A)\equiv H^1(P,A)$
is the group of homomorphisms from $P$ to $A$, therefore
the vanishing of the first cohomology group for all
abelian $A$ is equivalent to the fact that $P$ coincides
with its commutator $[P,P]$, i.e. $P$ is {\it perfect}.
\par
The groups $G$ for which all central extensions
split in a unique way, namely $H^n(P,A) =0, n=1,2$ for all A as
above, are called {\it algebraically simply
connected} (see [\ref(Miln1),\ref(Brow1)]).
\par
A {\it universal central extension} $(E,\s)$ of $P$ is a
central extension such that for each central extension $(G,\pi)$
of $P$ there is a homomorphism $U:E\to G$ such that the
following diagram commutes:
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt }
\def\mapse#1{\searrow\hskip-.3cm {}^#1}
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow
\rlap{$\vcenter{\hbox{\hskip-.1cm $\scriptstyle#1\/$}}$}}
\matrix{
E &\mapright U&G\cr
&\mapse{\s}&\mapdown{\pi}\cr
&&P\cr}\ .\eqno(1.5)
$$
If a universal central extension exists, it is unique (up to
equivalence) and the extensions of $P$ via $A$ may be easily
described.
Indeed consider the following commutative diagram:
$$
\def\normalbaselines{\baselineskip20pt
\lineskip3pt \lineskiplimit3pt }
\def\mapright#1{\smash{
\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdownu#1{\Big\downarrow
\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapdownd#1{\Big\Vert
\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\matrix{
&1&\mapright{}&A&\mapright{}&E\times
A&\mapright{id_E\cdot1_A}&E&\mapright{}&1&\cr
&&&\mapdownd{}&&\mapdownu{U\cdot id_A}&&\mapdownu\sigma
&&&\cr &1&\mapright{}&A&\mapright{}&G&\mapright\p
&P&\mapright{}&1&.\cr}\eqno(1.6)
$$
By a classical diagram chasing argument, it is easy to
see that $U\cdot id_A$ is surjective, and that the
restriction of the projection $id_E\cdot1_A$ to
$\ker(U\cdot id_A)$ is an isomorphism onto $S:=\ker(\s)$.
Moreover the embedding of $S$ in $E\times A$ via the
preceding isomorphism has the form $id_S\times\psi$, where
$\psi$ is a morphism from $S$ to $A$.
As a consequence, there is an isomorphism
$$
\quot{E\times A}S \rightarrow G\eqno(1.7)
$$
where $S$ is embedded into $E\times A$ as described,
therefore the extension $(G,\pi)$ is described in terms
of this embedding, i.e. in terms of the morphism
$\psi$, which implies that $H^1(S,A)\simeq H^2(P,A)$.
\par
A central extension $(E,\s)$ of $P$ where $E$ is
algebraically simply connected is the universal central
extension of $P$. Indeed fix an extension $(G,\pi)$
of $P$, choose a section $s$ of $\pi$ and consider the
2-cocycle $\o$ defined in equation (1.3). Then the map
$$
\matrix{
\tilde\o&:E\times E&\to&A\cr
&(g,h)&\mapsto&\o(\s(g)\s(h))}\eqno(1.8)
$$
is indeed a cocycle in $Z^2(E,A)$. Since $H^2(E,A)$ is
trivial, there is a 1-cochain $\f$ s.t.
$\d_1\f=\tilde\o$. Then, setting
$$
U(g)=s(\s(g))\f(g)^{-1},\qquad g\in E\eqno(1.9)
$$
we easily get that $U$ is a homomorphism from
$E$ to $G$ and makes diagram (1.5) commutative.
We note that since $\s$ is surjective and $E$ is
perfect, necessarily $P$ is perfect too.
\par
If $P$ is a topological group, extensions with given
topological properties are of interest. A theory in this sense
has been developed by Moore in a series of papers
[\ref(Moor1),\ref(Moor2)]. We recall that a topological group
$A$ is called {\it polish} if its topology may be obtained by
a separable complete metric (that can be chosen compatible with
the uniformities of $A$).
Moore considers {\it topological central extensions} of a
locally compact group $P$ via a polish group $A$, i.e. exact
sequences (1.1) such that $i$ is a homeomorphism into its image
and $\pi$ is continuous and open. He denotes by $Ext(P,A)$ the
equivalence classes of topological extensions of $P$ via $A$,
where the isomorphism in diagram (1.2) is asked to be a
homeomorphism. In analogy with the algebraic case, topological
cohomology groups are to be defined in such a way that the
identifications of $Ext(P,A)$ with $H^2_{top}(P,A)$, and of
$H^1_{top}(P,A)$ with the continuous homomorphisms from $P$ to
$A$ hold. Moore proves this to be the case if cochains are
Borel measurable. With this hypothesis, the previously mentioned
identifications hold, cohomology groups are polish groups and
all usual functorial properties are satisfied. From now on we
consider topological cohomology groups only, therefore we drop
the subscript $_{top}$.
\par
Let us call {\it unitary group} a
subgroup of $\U(\H)$, the group of the unitary operators on a
separable Hilbert space $\H$ equipped with the weak topology.
It turns out that closed (i.e. complete) unitary groups are
polish. Concerning universal extensions, Moore deals with {\it
unitary topological extensions}, namely $A$ in (1.1) is a closed
unitary group.
We shall say that $E$ is $\U$-{\it simply connected} if all
central unitary extensions split in a unique way, that is to say
$H^1(E,A)$ and $H^2(E,A)$ vanish for all closed abelian unitary
groups $A$.
\claim{1.1 Theorem [\ref(Moor2)]} Let the locally compact
group $P$ have a universal topological central extension $(E,\s)$
with $E$ a polish $\U$-simply connected group. Then
$(E,\s)$ is a universal central extension for the class of all
unitary central extensions, namely for
every topological central extension $(G,\pi)$ via a closed
unitary group $A$ there exists a continuous map $U:E\to G$ s.t.
the diagram (1.5) commutes. As a consequence, the isomorphism
(1.7) is a homeomorphism and $H^1(S,A)$ is isomorphic to
$H^2(P,A)$, where $S:=\ker(\s)$.
\par
According to theorem 1.1, we call the $\U$-covering of $P$ the
(unique) central extension of $P$ via a simply connected group
(if it exists). We need however a generalization of theorem 2 to
a larger class of extensions.
\claim{1.2 Theorem} Let $P$ be a locally compact group which
admits a $\U$-covering $(E,\s)$. Let $(G,\pi)$ be an algebraic
central extension of $P$ with $G$ a unitary group and
$A:=\ker(\pi)$ a topological subroup of $G$, closed as unitary
group. If $\pi$ has a Borel section $s$, then $\pi$ splits on
$E$, i.e there exists a continuous map $U:E\to G$ s.t. diagram
(1.5) commutes.
\np
As a consequence, the group $\quot{E\times A}S$ is
polish and isomorphism (1.7) is continuous.
\np
If, in addition, the map $\pi$ is continuous, then $G$
is a closed unitary group and the extension $(G,\pi)$ is
a topological extension in the sense of definition 1.1.
\par
The last statement of the theorem shows that the
difference between the extensions in theorems 1.1 and 1.2 lies
in the continuity of the map $\pi$.
\proof
Since $s$ is a measurable section, the 2-cocycle defined
in (1.3) is measurable too, and it is an element of
$Z^2(P,A)$. Then we define $\tilde\o$ as in (1.8), and
since $\s$ is continuous, $\tilde\o$ turns out to be an
element of $Z^2(E,A)$. Since $A$ is a closed unitary group
and $E$ is $\U$-simply connected, $\tilde\o$ is indeed a
coboundary, i.e. there exists $\f\in C^1(E,A)$ s.t.
$\tilde\o=\d_1\f$. Then $U$ defined by equation (1.9)
is a homomorphism from $E$ to $G$, it is measurable by
construction, and makes diagram (1.5) commute. By
proposition $5(a)$ in [\ref(Moor1)], $U$ is indeed
continuous.
\np
Now we come back to diagram (1.6). Surjectivity of
$U\cdot id_A$ and the existence of the isomorphism
$j:\ker(U\cdot id_A)\to S$ follow by algebraic reasons.
Then we observe that $j$ is simply the restriction to
$\ker(U\cdot id_A)$ of the projection $id_E\cdot1_A$,
which is continuous and open, therefore $j$ is a
homeomorphism. Then $\quot{E\times A}S$ is a polish
group and the isomorphism (1.7) is continuous.
\np
If the map $\pi$ is continuous, then
$$
g\in G\to\left(\s^{-1}\circ\pi(g)\ ,\
\left[U\circ\s^{-1}\circ\pi(g)\right]^{-1}\
g\right)\in E\times A
$$
gives a map from the subsets of $G$ to the subsets of
$E\times A$ for which the preimage of an open set is
open. On the other hand, composing such a map with the
projection on $\quot{E\times A}S$, we get a map of
points which is the inverse of the isomorphism
(1.7). Such isomorphism is therefore a homeomorphism,
and all other properties follow.
\endproof
Note that, in the topological setting, the only
property that $[P,P]$ is dense in $P$ is needed for the
existence of a universal central extension. A remarkable
result of Moore shows that a perfect connected group $P$
admits the $\U$-covering.
Moreover, if $P$ is a Lie group, central extensions may be
completely described in terms of the Lie algebra cohomology and
the fundamental group $\pi_1(P)$.
\par
We recall that when $\P$ is a Lie algebra, a central
extension of $\P$ via a vector space $\V$ is an exact
sequence $0\to\V\to\G\buildrel\pi\over\to\P\to0$ where
$\V$ is a central Lie subalgebra of $\G$ and $\pi$ is
a Lie algebra homomorphism. Extensions are still
described in terms of 2-cocycles, and equivalence
classes of extensions in terms of 2-cohomology classes. We
denote by $H^n(\P,\V)$ the cohomology groups of $\P$
with respect to $\V$.
\par
\claim{1.3 Theorem [\ref(Moor2)]} The $\U$-covering
$(E,\s)$ of a perfect connected Lie group $P$ is a perfect
Lie group, and
$$
\ker(\s)\simeq\pi_1(P)\times H^2(\P,\Re)
$$
where $\P$ is the Lie algebra of $P$.
If $H^2(\P,\Re)=0$, $E$ coincides with the universal covering of
$P$.
\par
In the case of Lie groups, the existence of a measurable
section in theorem 1.2 may be replaced by a natural
condition.
\claim{1.4 Definition} Let $(G,\pi)$ be an algebraic
extension of the Lie group $P$ where $G$ is a unitary
group. We say that $\pi$ is a weak Lie extension if there
exists a set $\{L_1\dots L_n\}$ of generators of $\P$ as
a Lie algebra ad a corresponding set
$\{V_1(t),\dots,V_n(t)\}$ of strongly continuous
1-parameter groups in $G$ such that
$$
\pi(V_i(t))=\exp(tL_i),\qquad i=1,\dots,n.
$$
where we have denoted by $\P$ the Lie algebra of $P$.
\par
\claim{1.5 Lemma} Let $(G,\pi)$ be a central weak Lie extension
of the Lie group $P$. Then there is a finite set of strongly
continuous 1-parameter groups in $G$ such that the generators of
their images in $P$ are a basis for $\P$ as a vector space.
\par
\proof Let us consider the map $v$ which associates to a
given set of one parameter groups in $P$ the (finite
dimensional) vector space spanned by their generators.
\np
We observe that given $A,B\in\P$, the commutator
$[A,B]$ belongs to
$$
X:=v(\{t\to\exp(sA)\ \exp(tB)\ \exp(-sA):s\in\Re\}).
$$
Indeed, $X$ is spanned, by definition, by the range of
the Lie algebra valued function $s\to\exp(sA)\ B\
\exp(-sA)$, whose derivative in 0 is $[A,B]$.
\np
As a consequence, by finite dimensionality, we may find
a finite set $\{s_i^j\}$ such that the vector space
$$
v\circ\pi\left\{t\to V_{h_1}(s_1^j)\cdot\dots\cdot
V_{h_p}(s_p^j) \cdot V_{h_0}(t)\cdot V_{h_p}(s_p^j)
\cdot\dots\cdot V_{h_1}(s_1^j):j=1,2,\dots\right\}
$$
contains the element
$[L_{h_1},[\dots[L_{h_p},L_{h_0}]\ ]$, where the $V_i$ are
described in definition 1.4. Then the thesis easily follows.
\endproof
\claim{1.6 Proposition} Let $(G,\pi)$ be a central weak
Lie extension of the connected Lie group $P$, with $G$ a
unitary group. Then there exists a Borel measurable
section $S:P\to G$ of the extension $\pi$.
\par
\proof According to the previous lemma, we may suppose
that $\{L_1,\dots L_n\}$ is a basis for $\P$ as a vector
space.
\np
The map $\a :\Re^n\to P$ given by
$
\a (t_1,\dots t_n)=\prod_{i=1}^n \exp(t_iL_i)
$
has non trivial Jacobian at the origin because
$\{L_1\dots L_n\}$ is a basis. Therefore there there exist
two open sets, $U\subset\Re^n$ and
$V\equiv \a (U)\subset P$, where $U$ is a neighborhood
of zero in $\Re^n$ and, consequently, $V$ is a
neighborhood of the identity element in $P$, such that
the map $\a \,:\, U\to V$ is a diffeomorphism.
\np
Now we define the (strongly) continuous map $\b:\Re^n\to
G$ given by $\b(t_1,\dots t_n)=\prod_{i=1}^n V_i(t_i)$,
and observe that the map $S_0:=\b\ \a^{-1}:V\to G$ is a
(strongly) continuous section of $\pi$ on $V$, i.e.
$\pi\cdot S_0\equiv id_V$.
\np
Since $V$ is an open neighborhood of the origin and $P$
is a connected group, then $P$ is algebraically generated
by $V$, i.e., each element of $P$ can be written as a
finite product of elements in $V$.
\np
Now we consider the open covering
$\bigcup_{g\in P} gV= P.$
Since $P$ is $\sigma$-compact, we may extract a
countable sub-covering,
$$
P =\bigcup_{k\in\Na}g_k V.
$$
Finally, define the measurable
partition of $ P $ given by,
$$
\eqalign{ A_1 &= g_1 V\cr A_{n+1}&=\, g_{n+1}
V\cap\left(\bigcup_{k=1}^{n} g_k V\right )^c \quad
n\in\Na.\cr }
$$
\np
For each $n\in\Na$, we may write $g_n =v_{n,1} \cdots
v_{n,m_n}$, $v_{n,k}\in V$, hence, since $A_n \subset g_n
V$, any element of $A_n$ may be written as $ v_{n,1}
\cdots v_{n,m_n} v$, when $v$ varies in $V$. Then, we
define
$$
\matrix{
S_n :&\quad A_n &\longrightarrow&G&\cr
&&&&\cr
&v_{n,1} \cdots v_{n,m_n} v &\longmapsto&S_0 (v_{n,1})
\cdots S_0 (v_{n,m_n})S_0 (v)&\quad.\cr
}
$$
As before, $S_n$ is a strongly continuous section of $\pi$
on $A_n$. Therefore it is easy to understand that the map
$S=\sum_{n\in\Na}\chi_{A_n}\,\,S_n$, where $\chi_B$ is
the characteristic function of $B$, is the desired
measurable section.
\endproof
Making use of theorems 1.2 and 1.3 and proposition 1.6,
we can prove the following theorem on weak Lie
extensions. We remark that no condition on the closure of
$A$ is needed.
\claim{1.7 Theorem} Let $P$ be a perfect connected Lie
group. Then all central weak Lie extensions $(G,\pi)$ of $P$,
with $G$ a unitary group, split on the $\U$-covering
$(E,\s)$, i.e. there exist a continuous homomorphism
$U:E\to G$ such that diagram (1.5) commutes.
\par
\proof Let $\tilde G$ be the group algebraically generated by
$G$ and $\ov A$, where $\ov A$ is the weak closure of $A$ in
$\U(H)$, and extend $\pi$ to a morphism $\tilde\pi$ in such a
way that $\tilde\pi|_{\ov A}=0$. Then, applying proposition~1.6
and theorem~1.2 to the extension $(\tilde G,\tilde\pi)$
we get a continuous map $U:E\to\tilde G$ such that
$\tilde\pi\cdot U=\s$.
\np
By theorem 1.3 $E$ is perfect, therefore
$U(E)=U([E,E])\subseteq[\tilde G,\tilde G]$.
Since $\ov A$ commutes with $G$ and $\tilde G$,
equality $[\tilde G,\tilde G]\equiv[G,G]$ holds. As a
consequence, $Rg(U)\subseteq G$, and the proof is
concluded.
\endproof
We recall that the (4-dimensional) Poincar\'e group
$\Poi$ is perfect and its Lie algebra $\wp$ satisfies
$H^2(\wp,\Re)=0$. Therefore, next corollary immediately
follows.
\claim{1.8 Corollary} Let $(G,\pi)$ be a central weak Lie
extension of the Poincar\'e group $\Poi$ where $G$ is a
unitary group.
Then there exists a strongly continuous unitary
representation $U$ of the universal covering $\Spin$ of
$\Poi$ such that $\pi\cdot U=\s$, where $\s$ is the
covering map, and there is a continuous isomorphism
$$
\quot{\Spin\times A}{\Ze_2}\to G
$$
where $\Ze_2$ is a suitable order two central subgroup of
$\Spin\times A$.
\par
\proof The corollary easily follows by theorems 1.2, 1.3 and
1.7, the previous observations on the Lie algebra $\wp$ and the
fact that the kernel of the covering map from $\Spin$ to $\Poi$
is $\Ze_2$.
\endproof
\section 2. Modular covariance and the
reconstruction of space-time symmetries.
\par
In this section we study how the unitary representation of
the group of the space-time symmetries of a Quantum Field
Theory [\ref(Haag1)] may be generated by the modular unitaries
associated with the algebras of a suitable class of regions.
Local Quantum Theories are described by a local pre-cosheaf of
von~Neumann algebras (see [\ref(GL1)]), i.e by a map
$$
\A:\O\to\A(\O),\qquad \O\in\K
$$
where $\K$ is the family of the double cones in the Minkowski
space $M$, such that
$$
\eqalign{ \O_1\subset\O_2&\imply\A(\O_1)\subset\A(\O_2)
\hskip1.5cm {\rm(isotony)}\cr
\A(\O)&\subset\A(\O')'\hskip3.cm {\rm(locality)}.\cr}
$$
\par
Local algebras are supposed to act on a {\it separable} Hilbert
space $\H$, and $\A_0$ denotes the $C^*$-algebra of quasi-local
observables generated by the local algebras. There is a vector
$\Q$ (vacuum) which is cyclic for all of them.
\par
The pre-cosheaf is extended by additivity to general open
regions of $M$.
\par
As a consequence of locality, $\Q$ turns out to be cyclic
and separating for the algebras associated with all non
empty open regions whose complement has non-empty
interior (Reeh-Schlieder property).
\par
We recall that a {\it wedge} region is any Poincar\'e
transformed of the region $W_1:=\{x\in\Re^n:|x_0|2$.
Then there exists a vacuum preserving, positive energy,
$\A$-covariant, unitary representation $U$ of $\Spin$ which is
canonically determined by the property
$$U(\tilde\L_W (t))=\D_W^{it},\qquad W\in\W$$
\par
In order to prove Theorem $2.3$ we need some results
about the structure of $\Poi$ (Lemma $2.5$), and about
the group generated by modular operators associated with
wedge regions.
\claim{2.4 Proposition} The generators of the boost
transformations generate the Lie algebra of the Poincar\'e
group. As a consequence, the boosts algebraically generate
$\Poi$.
\par
\proof The first statement is well known, and the second
follows by the connectedness of $\Poi$.
\endproof
\claim{2.5 Lemma} Let $\A$ be a local pre-cosheaf on
$M$ satisfying assumption $2.1$, and consider the set
$$
H=\{g\in\Poi:\A(g\O)=\A(\O)\quad\forall\O\in\K\}
$$
Then, H is a normal subgroup of $\Poi$ and either
\item{$(a)$} $H=\{1\}$
\np
or
\item{$(b)$} for each $\O\in\K$, $\A(\O)$ is equal to
$\A_0$ and is a maximal abelian subalgebra of $B(\H)$.
\par
\proof $H$ is clearly a group, and we prove that it is indeed
normal in $\Poi$. In fact, by modular covariance, $\forall
g\in H$ we get
$$
\A(\L_W (t) g\L_W (-t)\O)=Ad\D_W^{it}\cdot\A(g\L_W (-t)\O)=
Ad\D_W^{it}\cdot\A(\L_W (-t)\O)=
\A(\O)
$$
therefore, since the boosts generate $\Poi$
(proposition~2.4), $H$ is normal.
\np
Now we prove that each non trivial normal subgroup $K$ of
$\Poi$ contains the translation subgroup
$T=\{T(a):a\in\Re^n\}$. Let $1\not=g\in K$. If $g$ is not
a translation then, by normality,
$$
K\ni g T(a) g^{-1} T(-a)=T(g(a))T(-a)\in T
$$
i.e. $K\cap T$ contains at least a non trivial element
$g$. Again by normality, $K\cap T$ is globally invariant
under the action of the Lorentz group. Since the (minimal)
Lorentz-invariant subsets of $\Re^n$ are the future light
cone, its surface, the past light cone, its surface, and
the complement of the preceding ones, and since any of
these generates $\Re^n$ as a group, then $K\supset T$. In
particular, we proved that if $H$ is non trivial it
contains $T$.
In this case, for each $\O\in\K$ we can find a
translation $T(a)$ such that $T(a)\O$ is spatially
separated by $\O$, and therefore, by locality, $\A(\O)$ is
abelian. Then it is easy to see that all local algebras
coincide, and they are maximal abelian sub-algebras
[\ref(StraZs1)] because the vacuum is cy\-clic.
\endproof
\rmclaim{Remark} If alternative $(b)$ of Lemma 2.5 holds, then
Theorem 2.3 is true by taking the trivial representation of the
universal covering of the Poincar\'e group.
In the rest of the paper we shall discuss alternative $(a)$,
i.e. we shall suppose $H=\{1\}$.
\par
\claim{2.6 Lemma} Let $G$ be the subgroup of $\U(\H)$ which is
algebraically generated by the modular groups of the algebras
$\A(W)$, $W\in\W$. Then $G$ is an algebraic central extension of
the Poincar\'e group $\Poi$.
\par \proof First we prove that the map $\pi$ which satisfies
$$
\p (\D^{it}_{W})=\L_{W}(t)\qquad W\in\W,\quad t\in\Re\eqno(2.1)
$$
extends to a well defined surjective group homomorphism
$\p :G\to\Poi$.
\np
Indeed let
$\D_1^{it_1}\cdot\dots\cdot \D_n^{it_n}=1$ be a non
trivial identity in $G$, where
$\D_i$ is the modular operator of $\A(W_i)$ and
$\{W_i , i=1,\dots,n\}$ is any collection of $n$ wedges
in $\cal W$. Then
$$
\A(\O)=
\D_1^{it_1}\cdot\dots\cdot\D_n^{it_n}\A(\O)
\D_n^{-it_n}\cdot\dots\cdot\D_1^{-it_1}=
\A(\p (\D_1^{it_1}\cdot\dots\cdot\D_n^{it_n})\O),
\quad\forall\O\in\K.
$$
From the Lemma 2.5 we get
$\p(\D_1^{it_1}\cdot\dots\cdot \D_n^{it_n})=1$.
As a consequence $\p$ is a well-defined homomorphism.
Since the boosts algebraically generate $\Poi$
(proposition 2.4), $\pi$ is surjective.
\np
Now we observe that if $U\in\ker(\p)$, then
$\ad U(\A(W))=\A(\p(U)W)=\A(W)$ for each wedge $W$, therefore
$U$ commutes with the modular group of any wedge algebra
[\ref(StraZs1)], and hence with any element in $G$. As a
consequence, the exact sequence $1\lar\ker(\p)\lar
G\lar\Poi\lar 1$ gives the announced central extension.
\endproof
\proofof{Theorem 2.3} Equation (2.1) and proposition
2.4 implies that the extension $\pi$ described in
lemma~2.6 is a weak Lie extension. Since dim$(M)>2$, the
Poincar\'e group is perfect and its Lie algebra has
trivial second cohomology. Therefore, by lemma 2.6 and
corollary 1.8, we get a strongly continuous unitary
representation $U$ of $\Spin$. Setting
$U_W(t):=U(\tilde\L_W(t))$, $W\in\W$, $t\in\Re$, the
unitary operator
$$
z(t)=\D_W^{it}U_W(-t)
$$
implements internal symmetries by modular covariance,
therefore is in the center of $G$.
Let us denote by $\th_g$, $g\in\Poi$, the action of
$\Poi$ on the central extension $G$.
It follows that $\th_g(\D_W^{it})=\D_{gW}^{it}$,
$\th_g(U_W(t))=U_{gW}(t)$. Since $\th_g$ acts trivially on
the center of $G$ and $\Poi$ acts transitively on the
family $\W$, then $z(t)$ does not depend on $W\in\W$.
Moreover
$$
z(s)z(t)=\D_W^{it}z(s)U_W(-t)=z(s+t)\qquad s,t\in\Re
$$
i.e. $z(t)$ is a one-parameter group. Finally, since
$\D_{W'}^{it}=\D_W^{-it}$ by essential duality (theorem
2.2), and $U_{W'}(t)=U_W(-t)$ because $\tilde\L_W (t)$ and
$\tilde\L_{W'}({-t})$ are the unique lifting of $\L_W
(t)=\L_{W'}({-t})$, we have $z(t)=z(-t)$, i.e.
$z(t)\equiv I$.
\np
In order to check the positivity of the energy-momentum, we
observe that the generator of any time-like translation
is a convex combination of generators of light-like
translations. Moreover each generator of a light-like
translation gives rise to a one-parameter semigroup of
endomorphisms of a suitable wedge. Therefore the result follows
by proposition 2.7.
\endproof
The following proposition is a partial
converse to Borchers theorem, in a slightly different form it
appeared also in Wiesbrock paper [\ref(Wies1)], with the
difference that we do not assume the commutation relations of the
modular conjugation with the one-parameter group.
\claim{2.7 Proposition} (Converse of Borchers theorem) Let $\R$
be a von Neumann algebra standard with respect to the vector
$\Q$. If $U(a)$ is a one-parameter semigroup ($a\in \Re_+ $) of
endomorphysms of $\R$ such that
\item{$(i)$} $U(a)\Q=\Q$
\item{$(ii)$} $ad\D_{\R}^{it} U(a)=U(e^{-2\pi t} a)$
\np
then $U(a)$ has positive generator.
\par
We give here an easy proof of the preceding Theorem.
\proof From Lemma II.$3$ in [\ref(Borc1)] we know that
the operator valued function $t\to U(e^{-2\pi t} a)$ can
be analitically extended on the strip $\{ z\in\Co ;
-1/2