projective representations of the loop group and the boson-fermion correspondence

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 boson- fermion 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 Kac- Moody 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 Lam-Erik Lundberg for constructive advice in the preparation of this paper.

2. The restricted orthogonal 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 “pre- Lie-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 respec- tive 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 d enotes 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.

42 J. T. O’ITESEN

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).


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 Hilbert- Schmidt operator, it has the representation

Azf = XV, d~ir

where {K)SI and {uu,)iE~ are two orthogonal sets in 1-I, both spanning the range of AZ. The skew-selfadjointness of A2 then implies that

CQ,Ui) = --C~i(fGJ), zEI iEI

for all f E 1-I, and this means that

AZ = ~TJAZL~ = t/i~v,@ui $I 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 = A2.

4A2)* f1 A . . .A fn =A~A fr A...A fn.

This formula defines in general an unbounded operator, with V as a dense set of analytic vectors. This is easily seen from the fact that

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), 442)*1 = WAzf)*

for all f E ‘FI.

44 J. T. OTTESEN

THEOREM 2. The operator dU(A), A E o#-t), defined on D by

dU(A) = dU(A1) + $(u(Az) - u(Az)*)

is essentially skew-selfadjoint and [dU(A),r(f)] = ?r(Af) for all f E 3-1. Furthermore, (fi,dU(A)R) = 0 and

(0, dU(A)dU(B)R) = -$(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, B] and [G, B](“) = [G, [,](“-l)J inductively for n E N.

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 t D((e-SG)*) and e sG = C,“=, %[G, B]cn) on 2). 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 REPRESENTATIONS OF THE LOOP GROUP 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-T lTr(A1)U(eA), we get the cocycle c(eA, @) = (det(eAleBle-C1 ))3, where C is given explicitly by the Campbell-Baker- Hausdoff 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 of zero, given by

c(s) = (det(V_,Y))i,

where I;_, = ~?~l(e+~)~ = I - $ etAlA~(e-tA)2&.

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 invertible, we get

Vs-l$V3 = (e”A)11A2(e”A)2 = 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 operator. n

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.


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) + 2 TnF, 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),

and f E C1(S1) . IS real = 0. Fe(O) = nF8 + fo, 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.

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.


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

= -Tr(Pf(l - P)gP) + Tr(Pg(1- 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) = & 7 f’@)g(Q)dQ. 0

Notice that wp(., .) defines a non-degenerated symplectic form on slS1 x ~15’~. Fur- thermore. 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

Then ~p(if, ig) = -2iIm(f, g) 4. So islS1 becomes a real pre-Hilbert space with re-

spect to Re(., .) 4. The associated Hilbert space we denote by Xi and the correspond-

(.f > g) + = c kf,w

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:

(L?, Up(ezf)f2) = e -+llfll:_

2

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 for j = l,..., n. It follows that S is unitary. On 2) the following commutation relation

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


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’~~)-’ = up(e’Foh), as it should be. Moreover, we get

UP(e”~~)UP(eG) = ,~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 a projective unitary representation of the abelian charge group C.

10. The spin representation 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 value of the equation we get

c(f) = -(eo,$(f)e0) = -(ea,feo) = 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) = @,“=a V’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):

and

]a(.0 &>*I = (f> 9)11 [4f),4s>l = 0,

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.


THEOREM 5. A unitary operator U(S), implementing the automorphism r(f)

--j ns(f) = 7W1f) exists if and only if S E Sp$l-f). Moreover, the operator U(S) 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) = e sAlfi v . . v eSAlfn.

This defines a strongly continuous one-parameter unitary group in .&(7-Q and its skew-selfadjoint generator is denoted by dU(A1).

Let us now consider AZ, the antilinear part of A E spz(‘H). Due to the fact that AZ is Hilbert-Schmidt and self-adjoint, we can, in analogy with the orthogonal case, identify A2 = CzEl (., v,)ui with a vector A z = Xi,1 v, v ui E v2’H, and define a(A2)* on the product vectors by

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) = -iTr(&Aa).

54 J. T. OITESEN

By analogous considerations as those in Section 4 we can construct the global cocycle c(eA, e’) = (det(eAleBle-cl ))i, where C is given by the Campbell-Baker -Hausdorff formula. This cocycle formula also makes sense in the general case. We also get the following simple and explicit 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 diffeomor- phisms 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’.

The associated Lie algebra d@(S’) on ‘R is given by the real span of the basis 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


difl+(Sl), 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 are unbounded and fulfil d: = -d-k, and have the commutation relations

[dk,dk’] = -i(k - k’)dk+k’

on the (common) maximal domain given by DD,,, = {f E 3-1 : dif E 7-L). Let uZ(‘FI, P)c denote the complexification of u2(7f, P). It follows that dk belongs

to the enlarged Lie algebra of u2(XFI, P)c, allowing unbounded operators such that [P, &] 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 IC c on their common maximal (d ) 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, [H, Q] = 0 on D H. Hence H has the following decomposition as a direct sum:

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 charac-

terized 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 sec-

tor 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 com- ponent 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);

= $ CkG_ k(f,gk + fkg,). Since f is real, f,, = f-k. Note that tit has a one- 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 IF1: of ‘Mi is a Hilbert space with respect to the complex inner product

(f,g).J = Kg)+ + i(Jf>di,

We emphasize that this complex structure is not the usual one. Furthermore, we define

a bilinear non-degenerate symplectic form a(., .) on 3-t! by

The natural action of Di#+(S’) on ‘7f$ is given by

(saf)(Q) = f(&(@)),

with 4 given by @(et’) = ez4(s) for T+!J E Dif+(S’). 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,


where do is defined as earlier. Of course, the operators dk act in the ordinary comple-

xification $ of ti,$. Notice that these operators are unbounded, but with a common

maximal domain DD,,, given by f E 3-1, t such that C nEZ n2 lfi212 < co. One should 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;),

where d; = cos(M)do = $(dk - d;) and d; = sin(A = &(dk + d;). Then we 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)) = c I.il - $&? - 1) = ma s=l ,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 > i-2 > ‘a.) card(l+) - card(F) = q, i_, = q - s for s larger than some so E N, 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.

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projective representations of the loop group and the boson-fermion correspondence

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