Vol. 35 (1995)
REPORTS
ON MATHEMATICAL
PHYSICS
No. 1
PROJECTIVE REPRESENTATIONS OF THE LOOP GROUP
AND THE BOSON-FERMION CORRESPONDENCE
JOHNNY T. OTTESEN
Institute of Mathematics and Physics, Roskilde University,
Postbox 260, 4000 Roskilde, Denmark
(Received August 24, 1994)
We consider some particular projective representations of the restricted orthogonal and
symplectic groups. These representations are related to the second quantization. We derive
very simple and natural cocycles associated with these representations, as we get explicit and
simple expressions for the corresponding vacuum functionals. We apply our results to the
loop group LS’ and the diffeomorphism group Dif+(S’).
Finally, we derive the bosonfermion correspondence in a particular case in a quite transparent fashion.
1. Introduction
We shall consider some particular projective representations, the spin representation and the metaplectic representation, of the restricted orthogonal group and the
restricted symplectic group, respectively. Such a study occurs naturally in quantum
theory with infinitely many degrees of freedom, due to the fact that these representations are related to symmetries of the canonical commutation and anti-commutation
relations.
The study of these representations goes back to Friedrichs [16], van Hove [20],
I. Segal [41], Shale and Stinespring [43], and Berezin [4]. Lately there has been
a renewed interest in these representations
in connection with the study of KacMoody algebras, loop algebras, the Virasoro algebra and associated groups, partly
actualized by the books of Pressley and Segal [36], Kac and Raina [6], Vershik and
Zhenlobenko [25], and others.
This paper is expository, giving a coherent write up of some basic facts in this
field, but we also include some new results. Moreover, we use new elegant approaches
in studying the loop group LS’ and the diffeomorphism group D$+(S1). We derive
very simple and natural formulae for the cocycles of the projective representations
and the central extensions of the groups. This kind of analysis was first carried out
on a Lie algebra level for the restricted unitary group by L. E. Lundberg [28] and
was generalized by H. Araki [l] to the restricted orthogonal group. Furthermore,
we arrive at very simple expressions for the vacuum functionals for the projective
representations studied in this paper.
[391
40
J. T. OTTESEN
I would like to acknowledge
preparation of this paper.
2. The restricted
orthogonal
Lam-Erik
Lundberg
for constructive
advice in the
group
Let ‘Ft denote an infinite dimensional complex Hilbert space with an inner product
(., .) which is complex linear in the right hand argument. The orthogonal group
0(X) consists of real linear invertible transformations T on ‘Ft such that the real
part of the inner product is left invariant, i.e. .T(T~, Tg) = ~(f,g) for all f.g E R,
where r(f,g) = Re(.W.
Any real linear transformation T on ‘H can be split into a sum of a complex linear
transformation Tl and a complex antilinear transformation T2 as T = Tl + T2. In fact
the complex structure on ‘,Y is reflected in a real linear bounded operator J on the
corresponding real Hilbert space with inner product .T(., .). Then Tl = i(T - JTJ)
and Tz = i(T + JTJ), whereby it follows that Tl and J commute and T2 and J
anticommute. In the following the subscripts 1 and 2 refer to this splitting. Let T’
denote the transpose of T relative to 7-(., .), i.e. 7-(f! Tg) = r(TTf,g) for all f-g E IFt.
The adjoint TT of Tl is, as usual, given by (f, Tlg) = (TFf, g) for all f, g E 3-1,whereas
the adjoint T,* of T2 is given by (f, T2g) = (g, T;f) for all f,g E 3-1, due to the fact
that T2, as a transformation from X to the conjugated Hilbert space X*, is complex
linear. Then it follows by straightforward calculations that T-l = T’ = T[ + T; for
T E O(‘H), and hence
T;Tl + T;T2 = I,
T;T2 + T;Tl = 0.
This means that Tl and T2 are by definition the Bogoliubov transformations.
The restricted orthogonal group 02(R) is defined as
CJz(‘FI)= {T E C’(E) : Tz E Lz(‘FI)},
where L2('H)denotes the Hilbert-Schmidt operators on ‘H.
The group Q2(IFt) can be given the structure of a topological group in several
ways, which is typical for infinite dimensional groups. The strongest topology is given
by the uniform topology on the complex linear part and the Hilbert-Schmidt topology
on the complex antilinear part. However, in some applications one has to use a
weaker topology on the linear part, for example the strong topology. In [l] Araki
has shown that in both these topologies, there are two connected components of
02(3-t), each of which is simply connected.
The choice of topology on Oa(X) determines the Lie algebra of 0#-f).
Our
choice of “pre-Lie-algebra” 02(7-I) is
o&Y) = {A E La(R) : A’ = -A, A2 E L&V)},
where L~(7-l)
denotes the real linear bounded operators on ti. The phrase “preLie-algebra” means that in some applications we have to enlarge the “pre-Lie- algebra” to allow operators with unbounded linear part. The demand A’ = -A implies
PROJECTIVE
REPRESENTATIONS
OF THE
LOOP
GROUP
41
that both the linear and the antilinear part of A are skew-selfadjoint in their respective sense. In what follows we shall, in particular, consider a neighbourhood of the
identity in 0&V), generated from o&Q) by the exponential mapping. Notice that the
exponential mapping from an infinite dimensional Lie algebra to the corresponding
infinite dimensional Lie group, modelled on a general topological vector space, need
not be locally one-to-one nor locally onto, but in cases where the vector space is a
Banach space there is a well-developed theory, which is quite parallel to the theory
of finite dimensional Lie groups.
3. Existence
of the spin representation
Let FA(IFt) denote the antisymmetric Fock Hilbert space modelled over the complex Hilbert space 7-t, i.e. &,(IFt) is the Hilbert space completion of the exterior
algebra over ‘Ft, F,,(R) = @FTO~~ ‘FI for n > 2, A% = C, and nl?f = ‘FI. The
distinguished vector 0 E ?t, given by R = @rXOO, with tic, = 1 and QV2= 0 for
n > 1, is called the Fock vacuum.
An n-particle vector is the exterior product of n vectors f7 E ‘FI, i = 1,. . . , n,
defined by
where x(cr) is the sign of the permutation g in the permutation group S,,. If {ei}ZtN
is an orthonormal basis for the Hilbert space ‘?f, then {ei, A.. A ei7z}i,<.. il,, is an
orthonormal basis for the Hilbert space ~“3-t for R > 0. The inner product on r\“‘FI
turns out to be
here det((.f~,g,))i..j=l,....n denotes the determinant
of the matrix ((f3,gi))i,J,l,...,n.
Let
the dense subspace 2) in F,,(Y) consists of vectors F = @zEoFn with only finitely
many non-zero components F,,,i.e. V is the algebraic direct sum of the Hilbert
spaces A?.
Then F,,(Y) is the completion of 2) with respect to the norm arising
from the inner product on F,,(X). For further details on the Fock Hilbert spaces
see for example [8].
There is a canonical antilinear mapping S + CL(~), unique up to the *-isomorphism, of E into bounded operators on F,,(K) fulfilling
[4f), &)‘I+ = 4f>4>* + 4g>*4f) = (f: 9)1,
[4f>l4g>l+ = 0.
J. T. O’ITESEN
42
for all f,g E 7-1.This is called the Fock representation of the canonical anticommutation
relations (CAR), and it is irreducible.
The operators a(f)*, f E ‘H, the are given explicitly on the product vectors by
c(f)‘.n
= f,
a(f)*(fl
A
...
A fn)
=
f
A
fl
A
...
A fn,
where the product vectors are identified with vectors in F,,(X) in the obvious way.
Define r(f) by r(f) = &,(a(f) + a(f)*). It follows from the CAR that [r(f), r(g)]+
= ~(f, g)I, where r(., .) is the previously defined positive symmetric real bilinear form
on ‘FI. Then, r(f), f E X, generate a complex Clifford algebra over l-t considered
as a real Hilbert space. For each T E c?(H), define ?~~(f) by 7r~(f) = r(T-‘f),
then
[7~(f), &g)]+
= ~(f, g)I. Thus, the mapping r(f) + or
defines an automorphism
of the Clifford algebra, and these automorphisms form an automorphism group.
It is of interest to know for which T E O(‘Ft) the automorphism n(f) + XT(~)
is unitarily implementable, i.e. for which T does there exist a unitary operator U(T)
on F,,(X) such that rrT(f) = U(T)-h(f)U(T).
In fact this question has already been
answered by Shale and Stinespring in [43], as stated in the following theorem.
THEOREM 1. A unitary operator U(T) which implements the automorphism x(f)
-+ I
exists if and only if T E 02(V). Moreover, the operator U(T) is unique up to
a phase of modulus one.
Then there is a cocycle c(T, S) of modulus one such that
U(T)U(S)
= c(T, S)U(TS)
for all T, S E 02(R). This means that the mapping T + U(T) gives a projective representation of the restricted orthogonal group 112(E).
The group cocycle c(T, S) depends on the choice of the arbitrary phase in U(T).
In the following we will give an explicit formula for the cocycle c (T, S), by choosing
U(T) such that c(T, S) is smooth and that U(T) lifts one-parameter
subgroups into
one-parameter subgroups, for T and S close to the identity. To this end, we give a
constructive proof of the if-part of the above theorem of Shale and Stinespring in the
case of T in a neighbourhood of the identity in 02(‘FI). This is done by constructing the
spin representation on the Lie algebra level, i.e. we construct U(esA) for A E 02(x) and
s in a neighbourhood of 0 E R, by constructing its skew-selfadjoint generator O(A),
hence U(esA) is given by esdUcA).
4. Construction of Spin2(71FI)
As stated above we will now construct U(esA), for A E 02(E) and s E R, by first
constructing its skew-selfadjoint generator dU(A), i.e. U(esA) = esdUcA).
PROJECTIVE
REPRESENTATIONS
OF THE LOOP GROUP
43
First we consider the complex linear part, Al, of A E 02(7-l). Al is skew-selfadjoint. In this case the construction of U(esA1) is simple and was given by Cook
in [lo] as follows. Acting on product vectors it gives
U(e SAl)Q = 0,
U(esA1)(fr A . . . A fn) = eSA1fr A . . A eSA1f,,
and dU(A1) is obtained by differentiation of the appropriate vectors. The subspace
;I) is a dense set of analytic vectors for dU(A1).
Let us now turn to the antilinear part, A 2, of A E oz(‘H). Since A2 is a HilbertSchmidt operator, it has the representation
Azf =
XV, d~ir
are two orthogonal
where {K)SI and {uu,)iE~
AZ. The skew-selfadjointness
sets in 1-I, both spanning the range of
of A2 then implies that
CQ,Ui)
zEI
= --C~i(fGJ),
iEI
for all f E 1-I, and this means that
AZ = ~TJAZL~
$I
= t/i~v,@ui
iEI
defines a vector in A~%, i.e. we have identified A2 with a vector A2 E A~X.
We now generalize the mapping f + a(f)*, f E 7-t, in such a way that A2
is mapped into an operator a(Az)* in F,,(N). The operator a(A2)* is defined on
product vectors as
a(A2)‘f-I
4A2)*
f1 A . . .A fn
= A2.
=A~A fr A...A
This formula defines in general an unbounded operator,
analytic vectors. This is easily seen from the fact that
fn.
with V as a dense set of
for F E D such that F, = 0 for n > N. We denote the adjoint of u(A2)* by u(A2).
The following commutation relation holds on 22
]4f),
for all f E ‘FI.
442)*1
= WAzf)*
44
J.T. OTTESEN
THEOREM
2. The operator dU(A),
dU(A)
A E o#-t),
defined on D by
= dU(A1) + $(u(Az) - u(Az)*)
is essentially skew-selfadjoint and [dU(A),r(f)]
(fi,dU(A)R)
= 0 and
(0, dU(A)dU(B)R)
= ?r(Af) for all f E 3-1. Furthermore,
= -$(Az,
B,) = iTr(B2A2).
Proof It can be shown that the operator dU(A) has 2) as a dense set of analytic vectors, which implies the essential skew-selfadjointness. The rest follows by direct
calculations.
n
We now define a unitary one-parameter group U(e’“) by U(eSA) = &C’(A). At this
point we are rather close to having proved the if-part of Theorem 1 in a neighbourhood
of the identity, but the fact that U(esA) creates infinitely many particles is the reason
why we first have to prove the following technical lemma.
LEMMA 3.Let G be an essentially skew-selfadjoint operator with 2) as a dense set of
analytical vectors and let B be any bounded operator leaving 2) invariant, both defined
on the same Hilbert space. Then
e~GBe-“”
9 $[G, B]cn)
=
n=O
on D, for s suficiently small, where [G: B](“) = B, [G, B](l)
= [G, [,](“-l)J
inductively for n E N.
= [G, B] and [G, B](“)
Proof: Let f E 23, then ePsGf is well-defined for s sufficiently small, and so is
Belief.
Let g E V, then also (e-SGg, BepSGf) is well-defined, and induction gives
&
(e-“Gg, BepsGf)
= (e-“g,
[G, B](")e-"Gf)
for n E N. From the Taylor formula we get
(eCsGg, BeesGf)
=
(9,F f$[G,B](“)f)
n=O
$;h
_f, g E 23, which is dense in the Hilbert space. Hence, BepsGf
e sG = C,“=, %[G, B]cn) on 2).
t D((e-SG)*)
and
n
It follows from the above lemma that
U(eSA)*7r(f)U(eSA)
= 7r(eCSAf)
on D, for all f E K. Hence we get the desired formula
U(zy7r(f)U(T)
= 7QTlf)
= 7rT(f)
(1)
PROJECTIVE
OF THE LOOP GROUP
REPRESENTATIONS
4.5
on IH, for all T = esA, A E 02(3t), f E 7-l, and s in a neighbourhood of zero in R,
where U(T) has been explicitly constructed, such that the arbitrary phase of U(T) has
been fixed on all one-parameter subgroups of Q&4) of the form T = esA, A E o~(7-l).
We emphasize that U(T) is indeed well defined on a neighbourhood of the identity
in c?*(?f), since the exponential mapping is one-to-one from a neighbourhood of zero
in oz(‘7f) onto a neighbourhood of the identity in 02(X). We call U: T + U(T) the
spin representation of the restricted orthogonal group and we define the spin group
5’pin2(7Q to be the group of all the unitary implementers U(T), T E c?~(Fl), from
Theorem 1.
The elements dU(A), A E 02(7-l), form a Lie algebra on 2) with a bracket
[dU(A), dU(B)]
= dU([A, B]) + w(A, B)I
and the Lie algebra cocycle is given by
w(A>B) = -$Tr([A2,&])
= -iiIm(Aa,B2)
This infinite dimensional Lie algebra is denoted as Spin,(%). This cocycle has also
been studied in the book by Vershik and Zhenlobenko [44].
Consider now the case where Al and Br are the trace-class operators. If we put
duo(A) = dU(A) - iTr(Ai)I,
then we get the following commutation relation:
[duo(A), duo(B)] = dUo([A, BI),
where the cocycle has been transformed away. This observation allows us to construct
the global cocycle c(e A,eD) in a neighbourhood of the identity. Let Uo(e”) = edc’o(A),
i.e. Uo(eA)Uo(e”) = Uo(eAe”), and since Uo(eA) = e-TlTr(A1)U(eA), we get the cocycle
c(eA, @) = (det(eAleBle-C1 ))3, where C is given explicitly by the Campbell-BakerHausdoff formula, i.e. es’ = esAesB. The cocycle formula does also make sense in
the general case.
5. The vacuum
functional
c(s) = (Q, U(esA)L’)
In this section we will calculate an explicit formula for the vacuum functional,
given by C(S) = (Q,U(eSA)R), for A E o#-f), and s in a neighbourhood of zero. We
notice that c(s) is analytic at s = 0, since R is an analytic vector for the generator
dU(A).
THEOREM
4. The vacuum functional
c(s)
is, in a neighbourhood
by
c(s) = (det(V_,Y))i,
where I;_, = ~?~l(e+~)~
= I - $ etAlA~(e-tA)2&.
of zero, given
46
J. T. OTIESEN
Proof Let Q(s) = V(esA)f2, for A E 02(7-l), and s in a neighbourhood of zero, and
put T = esA. Then, by formula (1) we have (a(T~f) + a(T~f)*)fi(s)
= U(T)a(f)f2
= 0, for all f E ti. Since Tl is invertible for s sufficiently small, we can define K
= TzTlpl, which is an antilinear skew-selfadjoint Hilbert- Schmidt operator. Hence,
(a(g) + a(Kg)*)R(s)
= 0 for all g E X. Put Q(s) = @~&Q,(s), where 0,(s) E r\“li
and 6’,(s) = c(s)n. It then follows by induction that
62(s) = c(s)e-+a(K)*R.
This formula allows us to get a differential equation for c(s) as follows:
c’(s) = (0, dU(A)0(s))
= -(dU(A)L?,
n(s))
= iTr(KAz)c(s).
We emphasize that K depends on s.
Put V, = e-SAl(esA)l, then
d
%Vs = e-SA1A2(eSA)2r
and for s so small that V, is close to the identity, and therefore
Vs-l$V3
= (e”A)11A2(e”A)2
invertible, we get
= Tlm1A2T2.
Hence
-$Tr(logV8) = Tr(KAs),
where we have used the fact that If-l and $Vs commute (only) under the trace symbol,
since both are complex linear operators. Then we may write the differential equation as
c’(s) = f
$T4~g(cI)))
44
which has the solution
c(s) = efrTr(iog(“*)) = (det(V_,))f ,
because c(-s) = c(s). Above we have used c(0) = 1 and the fact that the determinant of V, exists, due to the fact that V, -1 = Jo”eptAIAz(etA)&
is a trace-class opern
ator.
6. The spin representation
of the restricted
unitary
group
Since the (restricted) unitary group can be realized as a subgroup of the (restricted) orthogonal group, we may study the restriction of the spin representation to the
restricted unitary group. We will arrive at a nice explicit expression for the Lie algebra
cocycle which we will use later on.
PROJECTIVE
REPRESENTATIONS
OF THE
LOOP
GROUP
47
Let P be an orthogonal projection on the Hilbert space 7-f and let U(7-I) denote
a unitary group acting on ‘FI. We define the restricted unitary group Ua(‘FI,P) on
7-t by
U&Y P) = {V E U(X) : [P, V] E I&-q}.
Our choice of “pre-Lie
algebra” is
uZ(%, P) = {A E L(N) : A* = -A?
[P, A] E Lz(‘Ft)},
where L denotes the bounded linear operators on ‘7f; later on we may want to enlarge
our choice of “pre-Lie algebra” to unbounded operators.
Let r be an involution which commutes with P and put IP = I - P + TP, so
that (I~)~ = I and IP E c?(X). The restricted unitary group &(IH, P) can be realized
as a subgroup of 02(X) as follows. For V E &(‘H,P),
we put Vp = IpVIp,
then
V 4 VP defines a representation
of U&I-I, P) in 02(3-I). We notice that (VP)I =
(I - P)V(I - P) + PlVrP
and (Vp)2 = YP[P, V] - [P, V]TP. This allows us to
construct a spin representation of the restricted unitary group, and the corresponding
subgroup of S~i72~(7I) will be denoted by Spin,(lFI, P).
Define Up(V) = U(Vp) for any V E L&(‘H, P), where U(.) is the earlier defined
spin representation. Let
up(f)
= a((~-
p)f)
+ .(rpf)*
for all f E ‘Ft. This evidently gives a representation of the CAR labelled by P, since
(I - P) and TP are Bogoliubov transformations. Then rr(l~f) = up(f) + ap(f)*, for
all f E ‘FI, and
ap(Vf)
= &(V)aIJ(f)&(v)-l
for f E X and V E Zd2(‘7-l,P). Of course, we also have the analogous formula for
UP(.)*.
Let A E u2(‘HH!P) define dUp(A) as a generator of the unitary one-parameter
group Up(esA). For bounded operators A it follows that dUp(A) = dU(Ap).
We
define the so-called charge operator Q on 23 by Q = --idUp
= i&7(1 - 2P). The
spectrum of Q is Z.
The charge operator Q is selfadjoint and commutes with every dUp(A) for A E
u2(lFI, P), but it does not commute with’all operators Up(V) for V E &(E, P). This
means that the antisymmetric Fock Hilbert space F,,,(R) has the following decomposition:
FA(IFI) = @-fq>
qEz
called the charge gradation of F*(E), where ‘& denotes the eigenspace of Q corresponding to the eigenvalue q E Z. The operator Up(eA) for A E u2(lH, P) maps XFt,
into 7-&, i.e. it conserves the charge, but we emphasize that not all operators Up(A)
do leave 3-1, invariant.
48
J. T. O’ITESEN
By a direct calculation we get
(Q, dUp(A)dUp(B)Q)
= Tr(PA(1 - P)BP)
for A, B E u2(‘H, P). Hence we arrive at the following explicit expression for the Lie
algebra cocycle:
up(A, B) = Tr(PA(I-
P)BP)
- Tr(PB(I-
P)AP),
(2)
which will be applied in the next section.
7. The loop group LS1
In this section we shall consider a particular abelian subgroup of the restricted unitary
group, the loop-circle LS l. This group has been studied in the book by Pressley and
Segal [36], but we shall approach the subject slightly differently.
The loop group is realized in terms of multiplication operators on the Hilbert space
7-1 = L2(S1) = ‘II+ 63 7-_, where S1 denotes the unit circle, ‘Ft+ = span{ek, k 2 0},
ek(d) = eircQ,and ‘Ft- = span{ek, k < 0). We denote the projection onto ‘H_ by P. The
inner product on X is given by
The elements of LS1 are of the form eiF, where F is a smooth function from S1 into
R such that F(B + 27r) = F(B) + 2TnF, for some integer nF, the winding number of eiF.
In the following analysis it appears that it is sufficient to demand that F E C1(S1) and
not necessarily smooth. Below this demand guarantees that the multiplication operator
e iF is in 24z(X, P). We have the following splitting of F:
F(B) = nF0 + fo + f(e),
= 0.
and f E C1(S1) IS
. real
then F = F. + f and eiF E LS1 can be factored as eiFoezf. The
subgroup of LS1 consisting of elements of the form e if will be called the special loop
group, SLS1, and the subgroup generated by elements of the form eiFo will be called
the charge group, C.
Fe(O)
= nF8 + fo,
8. The spin representation
of the special
loop group SLS’
First we notice that SLS’ is a subgroup of the restricted unitary group 242(X, P).
Moreover, SLS’ is generated through the exponential mapping from the Lie algebra,
called the special loop algebra, slS1, given by the skew-selfadjoint multiplication operators if, such that [P,if] is Hilbert-Schmidt (this requires that f E C1(S1)). In what
follows we show that the spin representation of SLSl gives a projective representation
of positive energy.
PROJECTIVE
REPRESENTATIONS
OF THE
LOOP
GROUP
49
The skew-selfadjoint generator da for rotations in 3-t = Lz(S1), do = $ with domain
D(da) = {f E ‘FI : dof E ‘H}, is unbounded, but its commutator with P still makes
sense and vanishes. Then H = -idUp
defines an unbounded selfadjoint operator
in F,,(E) on its maximal domain. The operator H is called the energy operator. The
spectrum of H is N {0}, then
and Hf2
0.
us put
= --idUp
for if E slS1, i.e. 4(f) is
we get
L@(f)>
4(9)1 = --Wp(if,G/v
- P)gP) + Tr(Pg(1-
= -Tr(Pf(l
P)fP),
for all if,ig E slS’. This means that we have constructed a representation of the canonical commutation relations (CCR) in the antisymmetric Fock Hilbert space F,,(X),
indicating the so-called boson-fermion correspondence to which we shall return later
on. This representation is highly reducible because &4(f) commutes with the charge
operator Q on D.
If we complex@ the mapping f + 4(f), it is no longer selfadjoint, but it is still
a *-quantization mapping, i.e. 4(f)* = 4(f). M oreover, the splitting f = f+ $ f- E
7-t+ 63 ‘FI- gives 4(f) = 4(f+) + 4(f-), d(f+) = 4(f-)* and 4(f-)fi
= 0, for all
if E slS1.
The cocycle wr(if, ig) can be computed explicitly and we get
dP(if, ig) = c
k(.G,
- f&3
kEN
where f~,. = (ek,f),
i.e.
LJP(if, ig)
=
& 7f’@)g(Q)dQ.
0
Notice that wp(., .) defines a non-degenerated
symplectic form on slS1 x ~15’~. Furthermore. the so-called two-point function becomes
Let us consider islS’
functions are determined
f. g E is1.Y’ let us define
i.e. real functions on S1 whose integrals vanish. These
by their Fourier components fr;, k E N. For each pair
(.f>g)+ =
Then ~p(if, ig) = -2iIm(f,
c kf,w
g) 4. So islS1 becomes
a real pre-Hilbert
spect to Re(., .) 4. The associated Hilbert space we denote by Xi
space with re-
and the correspond-
50
.I.
T. OTTESEN
ing complex Hilbert space, given by introducing of the ordinary complex structure on
the Fourier components, by ‘Ft 4.
The spin representation, Up(eif) = ei$(f), of the special loop group, SLS1, leaves
each charge sector XFt, invariant and fulfils
Up(ezf)Up(eLS)
= e~WF(if.is)Up(el(f+9))
= eWP(if.is)Up(elg)Ur(eif)l
recognized as the Weyl form of the canonical commutation relations. We shall see later
on that all these representations are unitarily equivalent.
By a straightforward calculation we arrive at the following simple formula for the
vacuum functional:
-+llfll:_
2
(L?,Up(ezf)f2) = e
In the next section we turn to the charge subgroup C.
9. The spin representation
of the charge group C
The charge group C has infinitely many disconnected components, as a subgroup
of LS1, and these components are labelled by the winding number no. Let s denotes
the usual shift operator on N, (s/z)(O) = eieh(6’), for h E ‘Ft. Evidently s is unitary
and it is easily verified that [P, s] is Hilbert-Schmidt,
so s belongs to the restricted
unitary group. However, s is not generated by an element in the Lie algebra fulfilling
the Hilbert-Schmidt
condition. Hence, we can not use previously developed theory.
However,
in this case we can still construct
explicitly
a unitary
operator
Up(s)
such that
for all h E ‘If. This equation is equivalent
to Up(s)a(e~)U~(s)-’
= a(ek+l), for k # -1,
and Up(s)u(e_l)Up(s)-’
= a(eo)*.
Define an operator S on F,,(R) by its action on the product basis vectors SR = eo,
Se-1 = R, S(ek,, A .. A ek,,) = ekI+l A ... A ek,,+l A eo, provided each rE_j# -1 for
j = l,..., n, and S(ek, A ... A ek, A e-1) = ekl+l A .. A ek,+l, where each k, # -1
relation
for j = l,..., n. It follows that S is unitary. On 2) the following commutation
holds: [Q, S] = S, or equivalently
QS = S(Q + I).
By a direct calculation
it follows that S satisfies our demand on Up(s) and we may put
UP(s) = S. Then we have handled the first term nF0 in Fa. We thus turn to the second
term f0 in Fc.
Consider eifo, with JO E R, which trivially belongs to the restricted unitary group
Z&(X, P). Then the unitary operator Up(eifo) on F,,(ti) is explicitly given by Up(eifo)
= edu~(ifo) = ,ifoQ
PROJECTIVE
REPRESENTATIONS
OF THE
LOOP
GROUP
51
Combining the above discussion of the two terms in Fa and the fact that 5’ and Q
do not commute, we are led to define Up(eiFO) by
t&(e’FO) = ,~~foQS~F&.f”Q,
Then Up(eiFo) is unitary and Up(eiF~)a,(h)Up(e’~~)-’
Moreover, we get
UP(e”~~)UP(eG)
= up(e’Foh), as it should be.
= ,~z(fo~~o--~~~~~),~~(fo+so)QS(~~r+nr;),~~(fo+~o)Q
= c(eiFu,eiGo)lip(ei(Fo+co)),
whereof we see that the cocycle is given by
c(el’~l~e’Go) = e2‘i(.f”no-_g”liF)
Thus we have constructed
group C.
10. The spin representation
a projective
unitary representation
of the abelian charge
of the loop group LS1
We will now show that Up(e”f) and Up(ezFo) commute for all if E slS1 and eiF”
E C. This is a consequence of the fact that the unitary operators Seid(f) and er6(f)S
implement the same automorphism. From the irreducibility of the representation of
the CAR, labelled as P, it follows:
SKOS-l
= 4(f) + 4.0~7
where c(f) is a constant which vanishes, because for the vacuum expectation
of the equation we get
c(f) = -(eo,$(f)e0)
= -(ea,feo)
value
= 0.
This also implies that ail representations of SLS1, in different charge sectors, are
unitarily equivalent.
Let us finally put Up(eiF) = Up(e iFo)Up(eif), whereby we get a projective representation, i.e. the spin representation, of the loop group LS1.
11. The restricted
symplectic
group
The restricted symplectic group was first studied in detail by Shale [43] in 1962.
We will treat it completely analogously to the restricted orthogonal group, introduced
in Section 2, and use similar notation. Let Sp(ti) denotes a symplectic group consisting of all continuous real linear invertible operators S on E such that a(Sf, Sg) =
a(f,g), and for all f.g E 3-1, and a(., .) = Im(., .). By S~~(7-i) we denote the restricted
symplectic group given as
S&Y)
= {S E S&H); s2 E L,(X)}.
52
J. T. O’ITESEN
The transpose of S E Sp(‘N) relative to the form o(., .) is denoted by S”, and we
have S-l = S” = S; -S& i.e. S1 and Sz are Bogoliubov transformations in the sense
that
s;sr - s;s:! = I. s;sz - S,*SI = 0.
The group s~~(17-I) can be given the structure of a topological group in several
different ways, as was the case for the orthogonal group. The simplest choice of
“pre-Lie algebra” spa(‘M) is
s&X)
= {A E La(%) : A” = -A,
A2 E I&)},
where A” = -A means that A; = -Al and Aa = AZ. In some applications, however,
we have to enlarge the “pre-Lie algebra” to allow for unbounded operators.
12. The metaplectic
representation
In this section we construct the metaplectic representation in close analogy with the
spin representation.
Let 3”(X) denote the symmetric Fock Hilbert space modelled over a complex
Hilbert space ‘Ft, i.e. 3”(E) is the Hilbert space completion of the symmetric tensor
algebra over ‘Ft, 3”(X) = @,“=aV’H, where v stands for the symmetric tensor product.
An n-particle vector fr v I_. v fn in 344,
fi E 7-l, i = 1,. . . , n, is given by
By 27 we denote the finitely many particle vectors, i.e. those vectors F = cE~=~F,, in
3,,(X) for which only finitely many F,, are non-zero. The unbounded creation operators
a(f)*, f E fi, in 3~(7-0 are defined on product vectors by
Q)*.n
= f,
a(f)”
(fl
v
. . . Vfn)
=
f
Vfl
v...v.fn.
Hence V is a dense set of analytic vectors for u(f)*, for all f E IH. We define the
annihilation operators a(f),
f E K, on 2) as the adjoint of n(f)*. Then D is also a
dense set of analytic vectors for u(f), for all f E 7l. The creation and annihilation
operators fulfil the canonical commutation relations (CCR) on 2):
]a(.0
&>*I = (f>9)11
[4f),4s>l = 0,
and
a(f)Q
= 0,
for all f, g E ‘FI.
Put n(f) = &(u(f) + u(f)*), then [x(f), n(g)] = ia(f,g)l.
These commutation
relations are invariant under the action of Sp(‘Ft). It is interresting to know for which
S E Sp(3-I) there exists a unitary operator U(S) such that n(S-‘f)
= U(S)-lr(f)U(S),
for f E X. The answer was given by Shale [42] in 1962.
PROJECTIVE REPRESENTATIONS OF THE LOOP GROUP
53
THEOREM
5. A unitary operator U(S),
implementing the automorphism r(f)
exists if and only if S E Sp$l-f). Moreover, the operator U(S)
--j ns(f)
= 7W1f)
is unique up to a phase of modulus one.
Shale also gave an explicit formula for the cocycle when he normalized U(S) so that
(Q, U(S)Q) > 0. We shall choose a different phase in the definition of U(S), in analogy
with the orthogonal case. In fact, we will construct the unitary operator U(S) for S in
a neighbourhood of the identity in Sp#t)
in such a way that U(.) lifts one-parameter
subgroups into one-parameter subgroups, whereby the phase is determined.
We construct U(esA) for A E ~p~(3-I) by constructing its skew-adjoint generator
&J(A). First we consider A 1, the complex linear part of A.
The construction of U(esAl) was given by Cook [lo] in 1953. On product vectors
we have
U(e”A’).n = R,
U(&l
)(fl V . . V .fn) = esAlfi
v . . v eSAlfn.
This defines a strongly continuous one-parameter
skew-selfadjoint generator is denoted by dU(A1).
Let us now consider AZ, the antilinear part of
AZ is Hilbert-Schmidt
and self-adjoint, we can, in
identify A2 = CzEl (., v,)ui with a vector A z = Xi,1
on the product vectors by
unitary group in .&(7-Q
A E spz(‘H). Due to the fact that
analogy with the orthogonal case,
v, v ui E v2’H, and define a(A2)*
a(Az)*Q = dz,
442)*(f1
v . . v fn) =
A2 V .fl v . . v j-n
The
4.01
= 4A.f)
on V, for f E ‘H and all A E sp$f),
(R,dU(A)R)
= 0,
for all A E sp2(7-L), and
(R,dU(A)dU(B)R)
= -a(d2,232)
and its
= -iTr(&Aa).
54
J.T. OITESEN
By analogous considerations as those
cocycle c(eA, e’) = (det(eAleBle-cl ))i,
-Hausdorff formula. This cocycle formula
also get the following simple and explicit
in Section 4 we can construct the global
where C is given by the Campbell-Baker
also makes sense in the general case. We
formula for the vacuum functional.
THEOREM 7. The vacuum functional C(S) = (L?, U(esA)f2) for A E s&Y)
and s in
a neighbourhood of zero, where U(.) denotes the metaplectic representation, is simply
c(s) = (det(V_,s))-~,
where V-, = esAl(e-sA)l
= I - s; etAIAz(epfA)z
dt.
We call U: S + U(S) the metaplectic representation of the restricted symplectic
group. It turns out that the elements dU(A), A E s&J-f), form a Lie algebra mpz(3-t),
called the metaplectic Lie algebra, corresponding to the metaplectic group Mp&Y)
defined as the group of all unitary implementers U(S), S E Sp&),
from Theorem 5.
13. The diffeomorphism
group Difsf(S1)
as a unitary
group
In this section we will study the group consisting of orientation preserving diffeomorphisms of the unit circle S l, denoted Difs+(S1). We realize Difl+(S’) as a subgroup of
the restricted unitary group and then apply the spin representation, given earlier, on a
Lie algebra level. It turns out that we get representations of the Virasoro algebra in
terms of the spin representation. We emphasize that this is not the only possible realization of II@+(
in fact we will study another in the next section, using the metaplectic
representation.
An element Q of Difff(S1) is of the form $(e”‘) = e@(‘), where eiH E S1, 4 is a
smooth real function such that 4(Q + 27r) = 4(e) + 27r and b’(Q) > 0.
As mentioned, the diffeomorphism group Diff+(S’) can act on 8 = Lz(Sl), the
space of all square Lebesgue integrable function on S1, in more than one way. The
action becomes unitary if we choose it as follows:
(%fW) = f(dQ>>. lo’(s)l+
,
for any f E IFt. It turns out that ~4 E IA2(ti: P), where P is the natural projection
introduced when we considered the loop group LS’.
on ‘R is given by the real span of the basis
The associated Lie algebra d@(S’)
vectors
d, = cos(k0) ’ do - i’c sin(M):
lc E Z, and
d; = sin(M) . do + $kcos(kO),
k E Z \ {0}, where do = d/de, in generalized sense in R. This is a realization of the
Lie algebra Vect(Sl). However, it is easier to work with the complexified Lie algebra
PROJECTIVE
difl+(Sl),
REPRESENTATIONS
OF THE LOOP GROUP
55
given by the choice of basis vectors
dk = ek . do + $ikek = ek (do + fik) ,
where k E Z and do is as above. These operators
and have the commutation relations
are unbounded
and fulfil d: = -d-k,
[dk,dk’] = -i(k - k’)dk+k’
on the
Let
to the
[P, &]
(common) maximal domain given by DD,,, = {f E 3-1: dif E 7-L).
uZ(‘FI,P)c denote the complexification of u2(7f, P). It follows that dk belongs
enlarged Lie algebra of u2(XFI,P)c, allowing unbounded operators such that
is Hilbert-Schmidt. Moreover, complexify the mapping A -+ dUp(A) by putting
dUp(A)c = dUp(A-) + idUp(
where A = A- + +iA+ and A* E uz(‘H, P). Then
we may define the unbounded operators Dk = dU p (d IC) c on their common maximal
domain consisting of the finite energy vectors including VH, the algebraic direct sum
of & = span{ei, A . ’ A e& : cy=“=,ik = m, n E N}, m E N u (0). For these operators
0; = -D-k.
This provides us with a positive energy representation of the Virasoro
algebra. It is of the positive energy due to the fact that the energy operator H = -iDo
is non-negative. The associated Lie algebra cocycle becomes
w(dk, d,,)c
= Tr(Pdk(I
- P)d,P)
- Tr(Pd,(I
- P&P)
= h(k” + 2k)6k+m.
If we add a constant h to the energy operator H and put Hh = H + h, we get the
well-known Lie algebra cocycle of the Virasoro algebra by choosing h = i:
w@,d,)c
= &k(k2 - l)Sk+,,.
Hence our representation is labelled by the pair (h, c) = (i, l), where c denotes the
so-called central charge.
Since [H, S] = SQ on 2)~ c ;I), where S is the lifted shift operator and Q is the
charge operator introduced in Section 6, we get
[H-+Q(Q-I),s]=o
on 2)~. Moreover,
a direct sum:
[H, Q] = 0 on D H. Hence
H has the following decomposition
as
H = @Hli;
qtz
where H, = H )%, . Thus H 2 iQ(Q - I) = $q(q - 1) on the product vectors in ‘Rq.
Therefore the representation of the Virasoro algebra restricted to 3-1, is characterized by the label ($ + iq(q - l), l), where i + iq(q - 1) is the minimal energy
eigenvalue of the new energy operator Hi on 31, corresponding to the vacuum sector fiq. The earlier mentioned label (i, 1) then corresponds to q = 0 (L?, = L?).
56
J. T. O’ITESEN
Observe that the representations
corresponding to q and -q + 1 give rise to the
same label, they are therefore unitarily equivalent. Since DI, map H, into XFI, and
[H, Dk] = kDk, for k E Z\(O), by the earlier derived commutation relations, it follows
that HDI,Q, = ($q(q - 1) + k)DkR,. Hence D,Qq = 0, for any negative k E Z.
14. The diffeomorphism
group Difl+(S’)
as a symplectic
group
In this section we will study the diffeomorphism group as a symplectic group and
apply the metaplectic representation in parallel to what we did in Section 13.
Consider an infinite dimensional vector space ‘FI; of real functions on the unit
circle S1 such that CkEN k lfk12 < co, where fk = (ek,f)x is the k’th Fourier component of f with respect to the inner product in iFt = L2(S1). We introduce a
semi-inner product in ‘H? given in terms of the Fourier components as (f, g);
has a one= $ CkG_ k(f,gk + fkg,). Since f is real, f,, = f-k. Note that tit
dimensional null space with respect to the semi-norm arising from the semi-inner product. This null space consists of the constant functions f = fe. Hence the quotient space
7-$ = 7fi /{f : f = f. E R} is a Hilbert space.
Define a complex unit operator
J on Hi by
J
J introduces a complex structure on the set 7-1:. The complexification
Hilbert space with respect to the complex inner product
(f,g).J = Kg)+
+
IF1: of ‘Mi is a
i(Jf>di,
We emphasize that this complex structure is not the usual one. Furthermore,
a bilinear non-degenerate
we define
symplectic form a(., .) on 3-t! by
The natural action of Di#+(S’)
on ‘7f$ is given by
(saf)(Q) = f(&(@)),
E Dif+(S’).
with 4 given by @(et’) = ez4(s) for T+!J
In fact, II, -+ s+ defines an anti-
representation of Difl’(S’). By a standard computation it follows that s++E sp(H,!).
This means that we can construct the metaplectic representation of D@(S1)
considered
as a symplectic group.
A basis for the Lie algebra of real vector fields acting on the Hilbert space H$ is
given by cos(kO)do and sin(ke
but it is more convenient to use the ordinary complex
structure and introduce the basis
dk = eik8do,
PROJECTIVE REPRESENTATIONS OF THE LOOP GROUP
where do is defined as earlier. Of course, the operators
xification $
of ti,$. Notice that these operators
57
dk act in the ordinary comple-
are unbounded,
but with a common
t such that C nEZ n2 lfi212 < co. One should
maximal domain DD,,, given by f E 3-1,
realize that we are now operating with two different complex structures. There will be
no trouble if the (ordinary) complex linear operators commute with J. However, this is
not quite the case for the basis elements d k, but the commutator is Hilbert-Schmidt, in
fact, it is of finite rank.
Complexify the mapping A --f dU(A) in such a way that
dU(dk)c
= dU(d’,) + idU(d;),
= &(dk + d;). Then we
where d; = cos(M)do = $(dk - d;) and d; = sin(A
put DA, = dU(dk)c. Hence 0; = -D-k: on the domain 2). Furthermore, the energy
operator H = -iDo is non-negative, i.e. we have a positive energy representation.
Moreover, the Lie algebra cocycle becomes
This positive energy representation is the so-called level one representation, i.e. (h, c) =
(0, l), where h = 0 is the minimal energy and c = 1 is the central charge. Finally we
notice that there are other symplectic actions of D@(S1)
than the one considered
here, see for example [33].
15. The boson-fermion
correspondence
In Section 8 we showed that eif -+ U(eif) gave a representation of the special
loop group SLS’, which fulfils the Weyl form of the canonical commutation relations,
or equivalently that the mapping f + 4(f) provides us with a representation
of ii
the canonical commutation relations in the antisymmetric Fock Hilbert space. This.
representation
is unitarily equivalent to the Fock representation
f ---f r(f) in the
symmetric Fock Hilbert space modelled over Xi. This remarkable equivalence is
well understood by the boson-fermion
correspondence.
In this section we will not
discuss the boson-fermion correspondence in general, but refer to [25]. However, we
will prove it in this particular case, where it turns out to be quite illustrative.
THEOREM 8. The “sector energy operator” H,, = H - iQ(Q - I) is unitarily equivalent to the boson enew operator, in each charge sector.
Proof: We notice that Uhlenbrock [45] considered similar correspondence. However,
the arguments for equal multiplicities are not immediately intelligible to the author,
even though the result is correct. Alternately one could compute Tr(e-tH) in each
case. However, we give another argument.
Since [do, f] = f’, by a direct calculation it follows that [H, 4(f)] = +(f’).
Using
[Q, 4(f>l = 0, we get [HA,4Wl = -Nf’), as expected. From earlier considerations
we have H,, 2 0 on each I-&,, and HA&$ = 0. So the spectrum of H/, is N
58
J. T. OITESEN
Now we show that the multiplicities of H,, and those of the boson energy operator
are the same, whereof the unitary equivalence follows.
In the boson case, any basis product vector ekl v . . . v ek,, E .&(N~),
with Icr
> .** > k, > 0 and energy C;“=, k, = m E N, corresponds uniquely to a particular partition of m E N into a sum of positive integers, i.e. a set {ICI,. . . , AT,,}, where
ICI + ... + k, = m and ICI 2 . . . 2 k,, > 0. Moreover, different partitions correspond
to orthogonal vectors. The eigenspace Bk of the energy operator corresponding to
energy eigenvalue m E N is spanned by ekl v . 1 . v ek,, where Cy__, k,i = m. Hence
the dimension dim(BK) is exactly the number P(m) of partitions of m E N into a
sum of positive integers. It can be shown that p(m) = G(flT=,(l
- x~~‘))~)j,r=O, but
we will not need this result here.
In the fermion case each product vector ej, A . . 1 A ej,, E F&, with jl > . . . > j,
and sector energy m = CIkl Ijl) - iq(q - l), is uniquely determined by the ordered
index set (jl,. . . , j,), with ji > . ‘. > jTL and jl E Z, 1 = 3,. . . ! n, such that card(J+) card(L) = q and Cy=, [$I - $q(q - 1) = m, where J+ = {j E Nu (0) : j E (jl,. . , ,j,l)}
and J_ = {j E -N : j E (jl, . . . ,.jn)}. That is, we have an isomorphism between
the set of orthogonal basis vectors in 7-& and the set of ordered integer tuples such
that the difference 4 between the number of non-negative and negative elements is
such that Cy=l IjlI = m + iq(q - 1). Notice that different index tuples are mapped
into orthonormal basis product vectors.
Define the mapping y from the set of such
index tuples, defined above, into the set of ordered integer sequences by r(j,, . . . . jTL)
= (i) E (Ll,L2,.
..), where i-t = j, if j, is non-negative, and the negative elements
i-l E (i) are the negative integers which do not occur in (jl, . . . , jn). The sequence (i)
is ordered in decreasing order iLl > iL2 > .. . and i_l_l = i_l - 1 from a certain step
(1 > n). We will briefly write this as y: J+ + I_+ Y J+ and 7: J_ + I_ F (-N)\ J_. We
emphasize that 4 = card(l+) - card(1’), where 15 = J-, and that there exists so E N
such that i_, = Q - s, for s > sO. Integer sequences fulfilling these demands will be
called semi-infinite integer sequences (of charge q). Then it follows by a straightforward
computation that
33
In i_,- (q s=l
s)) =
c
I.il - $&?- 1) = ma
,jE.I+U.T-
Hence y defines an isomorphism between the set of index tuples with charge q and
sector energy m and the set of semi-infinite integer sequences (i) such that i-r >
= q, i_, = q - s for s larger than some so E N,
i-2 > ‘a.) card(l+) - card(F)
and C,“,,(Ls
- (4 - s)) = m. Hence, the dimension of the eigenspace BcL(q) of
the sector energy operator HA IN,, corresponding to the energy eigenvalue ‘m f N,
is equal to the number of different ways in which one can choose semi-infinite integer
sequences fulfilling the above demands. This number of ways is equal to the number p(m) of partitions of m into a sum of positive integers (in non-decreasing order).
Each semi-infinite integer sequence, for fix q E Z, can be uniquely written as (4 - 1 +
+kl,q-2+kz
,..., q-n+k,,q-n-l,q-n-2
,...) with kl 2 k2 2 ... 2 k, > 0, since
60
J. T. OITESEN
than 1 (especially c = i) by use of either the spin representation or the metaplectic
representation, in analogy with the construction made in this paper.
REFERENCES
H. Araki: Bogoliubov
automorphisms
and Fock representations
of the canonical anticommutation
relations, in: Operator Algebras and Mathematical Physics, P. E. T. Jorgensen
and P. S. Muhly (eds.);
Contemporary Mathematics 62, 23-141, Providence,
Rhode Island 1987.
PI H. Araki and W. Wyss: Helv. Phys. Acta 37 (1964) 136.
Springer,
[31 V. I. Arnold: Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics,
New York 1978.
[41 F. A. Berezin: The Method of Second Quantization, Academic Press, New York 1966.
Nuclear Phys. B241 (1984) 333.
I51 A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov:
J. Statist. Phys. 34 (1984) 763.
PI A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov:
[71 0. Bratteli and D. W. Robinson: Operator Algebras and Quantum Statistical Mechanics 1: C’- and
W*-Algebras, Symmetry Groups, Decomposition of States, Texts and Monographs
in Physics, Springer,
Harrisonburg,
Virginia 1987.
0. Bratteli and D. W. Robinson: Operator Algebras and Quantum Statistical Mechanics II: Equilibrium
PI
States Models in Quantum Statistical Mechanics, Texts and Monographs
in Physics, Springer, New York
1981.
I91 A. Chodos and C. B. Thorn: Nuclear Phys. B72 (1974) 509.
J. M. Cook: Trans. Amer. Math. Sot. 74 (1953), 222.
WI
P. A. M. Dirac: Proc. Roy. Sot. London 114 (1972) 248.
VI
J.
Dixmier: C*-algebras, North-Holland,
Amsterdam
1977.
WI
P31 B. L. Feigin and D. B. Fuchs: Funct. Anal. Appl. 17 (1983) 241.
1141 V. Fock: Z. Phys. 75 (1932), 622.
1151 D. Friedan, E. Martinee and S. Shenker: Nuclear Phys. B271 (1986) 93.
K. 0. Friedrichs:
Comm. Pure Appl. Math. 4 (1951), 161.; 5 (1952) 1, 349.; 6 (1953) 1.
WI
P71 J. Frohlich, G.‘t Hooft, A. Jaffe, G. Mack, P. K. Mitter and R. Stora: New Symmetry Principles
in Quantum Field Theory, NATO AS1 series, Series B: Physics, vol. 295, Plenum Press, New York
1992.
R. Goodman
and R. Wallach: .I. Reine angew. Math. 347 (19X3), 69.
WI
P91 R. Goodman and R. Wallach: J. Funct. Anal. 63 (1985) 29’)
PO1 L. van Hove: Mem. Acad. Roy. Belg. 26 (1951), 68.
J. E. Humphreys:
Introduction to Lie Algebras and Representation Theory, Graduate
Texts in MathePI
matics, Springer, New York 1972
P. Jordan and E. Wigner: Z. Phys. 47 (1928) 631.
WI
Operators and Representation Theory, Mathematics
Studies, 147, Notas de MaP31 P. E. T. Jorgensen:
tematica, North-Holland,
Elsevier Science Publishers B. V, Amsterdam
1988.
v41 V. G. Kac: Infinite Dimensional Lie Algebras, Cambridge University Press, Singapore 1987.
I251 V. G. Kac and A. K. Raina: Bombay Lectures on Highest Weight Representations of infinite Dimensional Lie Algebra, Advanced
Series in Mathematical
Physics, Vol. 2, World Scientific, Singapore
1987.
M. Kaku: Strings, Conformal Fields, and Topology: An Introduction, Springer, New York 1991.
WI
On Schwinger terms in (3 + l)-dimensions,
to appear (1991).
[271 E. Langmann:
L. E. Lundberg:
Commun. Math. Phys. 50 (1976), 103.
PI
v91 D. lust S. Theisen: Lectures on String Theory, Lecture Notes in Physics 346, Springer, New York 1989.
[301 J. Mickelsson: Current Algebras and Groups, Plenum Press, preprint 1991.
and quantum symmetry of the conformal
lsing models,
[311 G. Mack and V. Schomerus, Endomorphisms
to be published
in: Algebraic Theory of Superselection Sectors and Field Theory, World Scientific,
Singapore.
F. Murray and J. von Neumann: Ann. Math. 37 (1936), 116.
[321
PI
PROJECTIVE
REPRESENTATIONS
OF THE LOOP GROUP
61
Yu. A. Neretin: SW. Math. Dokl. 28 (1983), 411.
J. T. Ottesen: Infinite Dimensional Groups and Algebras in Quantum Physics, to be published in:
Lecture Notes in Physics, New Serie m: Monographs, Springer.
[351 J. Palmer: J. Funct. Anal. 27 (1978), 308.
[361 A. Pressley and G. Segal: Loop Groups, Clarendon Press, Oxford 1986.
1371 M. Reed and B. Simon: Methods of Modem Mathematical Physics, Vol. 1: Functional Analysis, Academic
Press, New York 1980.
[381 M. Reed and B. Simon: Methods of Modem Matematical Physics, Vol. II: Fourier Analysis, SelfAdjointness, Academic Press, Orlando,
1975.
1391 M. Reed and B. Simon: Methods of Modem Mathematical Physics, Vol. IV: Analysis of Operators,
[331
[341
[401
t411
[421
[431
[441
[451
[461
[471
Academic Press, New York 1978.
G. Segal: Commun. Math. Phys. 80 (1981) 301.
1. E. Segal: Mathematical Problems of Relativistic Physics, Amer. Math. Sot., Providence 1963
D. Shale: Trans. Amer. Math. Sot. 103 (1962), 149.
D. Shale and W. F. Stinespring: J. Math. and Mech. 14 (1965) 315.
A. M. Vershik and D. P. Zhenlobenko:
Representation of Lie Groups and Related Topics, Advanced
Studies in Contemporary
Mathematics,
Vol. 7, Gordon and Breach Science Publishers, New York 1990.
D. A. Uhlenbrock:
Commun. Math. Phys. 4 (1967) 64.
M. A. Virasoro: Phys. Rev. Dl (1970), 2933.
K. Yamabana: On the Algebra Generated by Canonical Commutation Relations, RIMS, Kyoto Univ.,