On Fermion Gauge Groups, Current Algebras and Kac-Moody Algebras
A. L. C A R E Y Department of Mathematics, IAS, The Australian National Unioersity, Canberra ACT, Australia
and
S. N. M. R U I J S E N A A R S * Mathematics Department, Tiibingen University, Tiibingen, Federal Republic of Germany
(Received: 14 November 1986; revised: 3 December 1986)
Abstract. Representations of groups of loops in U(N), SO(N) and various subgroups are studied. The representations are defined on fermion Fock spaces, and may be regarded as local gauge groups in the context of the two-dimensional many-particle Dirac theory for charged or neutral particles with rest mass m/> 0. For m = 0, the representations are shown to give rise to type I® factors, while for m > 0 hyperfinite, type III~ factors arise. A key point in the structure analysis is a convergence result: We prove that suitably rescaled representers of certain nonzero winding number loops converge to the free Dirac fields. We also present applications to cyclicity and irreducibility questions concerning the Dirac currents, and to the representation theory of a class of Kac-Moody Lie algebras.
2. Operators on the Fermion Fock Space 5 2.1. The CAR, I', dF and other Wick monomials 5 2.2. The operations f' and dI" 10 2.3. Charge shifts 13 2.4. Schwinger terms and phase factors 16 2.5. Factors of type I~ and type 1111 17 2.6. The neutral case 19 2.7. Cyclicity + positivity ~ irreducibility 24
3. Implementable Gauge Groups in the One-Particle Dirac Theory 25 3.1 Preliminaries 25 3.2 The group U(1) 28 3.3. The groups U(N), SU(N) and SO(N) 31 3.4. The standard kinks for m = 0 32 3.5. The standard kinks for m > 0 36
* Present address: Centre for Mathematics and Computer Science Amsterdam, The Netherlands.
2
4. The Fock Space Gauge Groups II s and II,, o 4,1. The group U(N) for m = 0 40 4.2. The group SO(N) for m = 0 44 4,3. The group U(N) for m > 0 47 4.4. The group SO(N) for m > 0 48 4.5. Convergence proofs 49
5. Further Developments 54 5.1. Weyl algebras and projective multipliers 54 5.2. U(N) versus SO(2N) 58 5.3. Subgroups of Ils, o 59
6. Applications to Field Theory 62 6.1. Current algebra 62 6.2. The smeared Dirac currents 65 6.3. Boson-fermion correspondence 68
7. Applications to Kac-Moody Algebras 72 7.1. Preliminaries 72 7.2. The massless case 73 7.3. The massive case 76
8. Concluding Remarks 79 8.1. Wiener-Hopfoperators 79 8.2. Uniqueness of standard kinks 81 8.3. Vertex and Virasoro operators 82
Acknowledgements 83
References 83
A. L. CAREY AND S. N. M. RUIJSENAARS
40
1. Introduction
This paper concerns a study of local gauge groups arising in the context of the
two-dimensional Dirac theory for charged or neutral fermions with rest mass
m >i 0. Alternatively, it may be viewed as a study of au tomorphism groups of
C A R algebras in some special quasi-free representations, or as a study of representations of certain loop groups and K a c - M o o d y Lie algebras. ( C A R =
canonical ant icommutat ion relations, cf. (2.4) below.) Our results are also con-
nected with topics in operator theory, in particular with the theory of Wiener -
Hopf operators and Fredholm theory.
Thus, this work pertains to several subjects: relativistic quantum field theory, C*- and W*-algebra theory, the theory of loop groups and loop algebras, and opera tor theory. The viewpoint and language of this paper are principally
inspired by the first-mentioned subject. However , we have made a point of
explaining some physical terminology and standard field-theoretic concepts used below, so as to render this paper accessible to mathematicians f rom the latter
fields, who might not be familiar with quantum field theory. The cases m = 0 and m > 0 have a quite different character , not only f rom a
physical and mathematical viewpoint, but also as regards previous literature. In
the m = 0 case our results overlap with a plethora of papers in the field theory
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 3
context (to which we shall return below), and also with a considerable amount of work in the latter contexts, of which we mention specifically [1-11]. When m = 0, the Dirac theory can be avoided with no expense, and this is certainly more natural from the standpoint of the other fields involved.
For m > 0 nearly all of our results are new. In this case the Dirac theory appears essential, in the sense that there seem to be no reasons inside the field of, say, Kac-Moody algebras, to suspect the existence of the new representations we have found and studied in this paper.
Our main results can be described roughly as follows. Both for m = 0 and for m > 0 the representations of the local gauge groups/automorphism groups/loop groups/current algebras/Kac-Moody algebras defined below are primary. For m = 0 they give rise to type I~ factors, while for m > 0 they lead to hyperfinite, type III1 factors. The representation spaces can be concretely described as fermion Fock spaces or quite precisely delineated subspaces thereof. The key to the structure analysis of the representation is a convergence result that is of considerable interest in itselL especially in the context of field theory. Crudely speaking, we prove that the free Dirac fields can be reached as limits of unitary operators representing certain nonzero winding number loops, provided these representers are multiplied by suitably chosen scalars. This fact enables us to exploit well-known properties of CAR algebra representations, obtained by smearing the time-zero Dirac fields. Our convergence proofs hinge on the explicit formulas for such representers, obtained some time ago by one of us [12, 13], and on the fact that we can get sufficient control on the operator kernels occurring in'these formulas.
Connections between nonzero winding number gauge transformations (often called 'kinks') and fermion quantum fields have been found and discussed with varying degrees of explicitness and rigor for more than two decades. The realization that massless fermion currents are related to boson fields is even older. These are two aspects of what is nowadays often called boson-fermion cor- respondence. A selection from the vast literature that deals with this subject and/or is related to it and to the present work is [14-42]. We shall return to the boson-fermion correspondence in Subsection 6.3.
Here, we mention specifically papers by G. Segal [4, 5], who (among other things) presented convergence arguments in a CCR (boson) context. (CCR = canonical commutation relations, cf. (5.7) below.) His work and papers by Carey, Hurst and O'Brien [39, 41], Frenkel [10] and Lundberg [1, 31] provided much inspiration in the initial stages of our work. In particular, Frenkel's paper made us aware of the close connection between current algebras and Kac-Moody algebras.
In the massless case (m = 0) our work simplifies and extends the results of [39, 41, 1]. In as much as our m = 0 results overlap with [4, 5, 10], we feel this holds true to a certain extent as well. In particular, our methods avoid certain unresolved technical difficulties with the convergence arguments of [4, 5], and
4 A.L. CAREY AND S. N. M. RUIJSENAARS
our discussion of the quantum field theoretic aspects is mathematically rigorous, in contrast to the account in the second part of [10]. (We shall elaborate on this below (2.30).) More importantly, our novel approach renders possible a unified treatment of both the massless and the massive case which is largely self- contained.
We proceed by sketching the plan of the paper. Section 2 contains preparatory material concerning fermion Fock spaces, CAR algebras, and unitary im- plementers of Bogoliubov automorphisms. Some of the results are new and have independent interest. The reader might skip Section 2 at first reading, and refer back to it when needed.
Section 3 concerns the single-particle Dirac theory. It contains the definitions of the loop groups whose Fock space representations we study in later chapters. Moreover, we assemble information on operators connected with certain kink loops, which is needed as an ingredient of the convergence proofs mentioned above.
Section 4 contains our main results, viz., the convergence of the kink im- plementers to the free Dirac fields and the ensuing elucidation of the structure of the Fock space gauge groups.
In Section 5 we complement the general picture of Section 4 with additional structural information. We introduce and discuss Weyl algebras, which played a crucial role in previous literature, determine projective multipliers, clarify the relation between the U(N) ('charged') and SO(2N) ('neutral') cases, and consider subgroups. Of particular interest is Theorem 5.1, which concerns the structure of the group of SU(N) gauge transformations for m = 0.
In Section 6 we discuss the connection of our work with current algebra and the boson-fermion correspondence. The results on cyclicity and irreducibility of the smeared Dirac currents (Theorems (6.1)-(6.3)) may be viewed as corollaries of the theorems in Section 4.
Section 7 deals with representations of afline Lie algebras, which form a prominent class of Kac-Moody algebras. We show that for m = 0 the suitably smeared chiral Dirac currents give rise to the basic representation of these algebras (or tensor products thereof), and present new representations in the massive case. In contrast to the basic representation, which is irreducible, the massive representations we consider generate hyperfinite type 1Ill factors (Theorem 7.3).
The paper is concluded with Section 8, where we collect some observations and questions on connections with operator theory, on the kink loops we employ, and on the vertex and Virasoro operators, which have been employed in the mathematical literature pertaining to our m = 0 case.
We refrain from a more precise description of our results. Instead, we have included a table of contents, which, we hope, will make clear where the details on various topics mentioned cursorily above can be found.
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS
2. Operators on the Fermion Fock Space
2.1. THE CAR, F, dF AND OTHER WICK MONOMIALS
In this subsection we set up notation and summarize some essentials of second quantization that will be used throughout the paper. Let ~ be a separable Hilbert space (over C). The fermion Fock space ~A(~) is the completion of the vector space
co ~at---- (~ ^ n ~ (2.1)
n=O
of antisymmetric algebraic tensors in the obvious inner product. The vector (1, 0, 0 . . . . ) is denoted by [l and called the vacuum vector in physical language. The creation operator c * ( f ) , f e 8 , is defined inductively by
c * ( f l ) . - - c*(fn)~'~ ~(71!) -112 2 (--)°'f~r(1) ( ~ ' " " Qfo'(n) or E S.
(2.2)
so that
c*(~)II =det(fi,~-). (2.3)
(Here and henceforth, products are in the natural order of the indices.) It extends to a bounded operator of norm Ilfll; its adjoint is the annihilation operator c ( f ) . The c(*) ( f ) satisfy the canonical anticommutation relations
{c(f), c(g)} = {c*(f), c*(g)} = 0} Vf, g c 8 . (2.4) {c(f), c*(g)} = (f, g)
(Here and in the sequel, { } denotes the anticommutator, [ ] the commutator.) These relations determine a C*-algebra (the CAR-algebra) and the above Fock-Cook representation is the only irreducible one for which a vector l l :~ 0 exists such that
c(DO = 0, 8 . (2.5)
If U is a bounded operator on ~ , the product operator F(U) is by definition equal to U Q " • • (~) U on ^"Y(. Thus,
F( U ) c * ( f l ) . . . c*(f,,)l-I = c*( Uf l ) . . . c*( Uf,)l-I (2.6)
and
F(U,)F(U2) = F(U1U2). (2.7)
When U is unitary, the transformation c*(f)---> c * ( U f ) gives rise to an automor- phism of the CAR, which is unitarily implemented by F(U) in the Fock
6 A . L . C A R E Y AND S. N. M. RUIJSENAARS
representation. Indeed, (2.6) implies that
F(U)c*(f) = c*( Uf)F( U). (2.8)
If A is a bounded operator on ~, the sum operator dF(A) is by definition equal t o A @ . . . ( ~ ) i + . - - + D @ . . . @ A o n A"~.Thus,
dF(A)c*(f l ) . . , c*(f,)12 = ~ c*( f l ) . . . c*(Af , ) . . . c* ( f , ) fL (2.9) i=1
From this one easily obtains the important relations
dF(A)* = dF(A*), (2.10)
[dF(A), dF(B)] = dF([A, B]), (2.11)
exp[it dF(A)] = F(e~'A), (2.12)
[dF(A), c*(f)] = c*(Af ) . (2.13)
We proceed by noting the obvious bound
IIdF(A)Ptll ~< lllA]l. (2.14)
Here, Pt is the spectral projection of tlae particle number operator dF(B) on [0, l]. Thus,
l P,~a(~It0 = 0 ^ n~. (2.15)
n = 0
Moreover, we observe that for bounded U and A the operators F(U) and dF(A) are unbounded in general, but well defined on the subspace ~ of finite particle vectors,
~------{F~ ~ a l F = PIF for some I~N}. (2.16)
The above operator equations all hold on 9. Now let A be in the trace class J-1. Then one has
N
A f = ~, A,g,( f , , f ) , N < ~ , (2.17) n = l
where AI>~A2~ > . . - > 0 are the singular values of A and {fn} and {g~} are orthonormal. We set
N
A c * c = ~. A,c*(g,)c(f~). (2.18) n = l
This is well defined, since the r.h.s, does not depend on the choice of the {gn} and {f,}. Also, if N=oo, the series is norm-convergent, since IIc¢* (f)ll=llfll and EA, = Tr[A I < o~. Furthermore, one easily verifies that
A c * c = dF(A). (2.19)
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 7
Hence, dF(A) and therefore (by (2.12)) also F(ei'a), belong to the CAR-algebra for A e ~-1. (In contrast, A ~ ~-~ implies dF(A) is unbounded.) Note that the map A---) Ac*c is (complex-) linear, since A--* dF(A) is linear.
To guarantee that similar maps A---~Acc*, Acc, Ac*c* from ffl into the CAR-algebra are well defined and complex-linear, a conjugation J on Z is needed. We shall write f, A and A r instead of ~f, J A ~ and ~A*~, in keeping with the fact that in applications below ~ is a function space and J complex conjugation. (However, in this section ~ is assumed to be fixed, but otherwise arbitrary.) For A E ~-1 given by (2.17) we now set
Acc* =- ~)tnc(~,)c*(f~) = - -A Tc* c + Tr A,
Acc =- Y~AnC(g,,)C(f,,),
Ac* c* = ~,A,,c*(gn)c*( fn) = ( A * cc)*.
(2.20)
(2.21)
(2.22)
Since the r.h.s, of (2.20) is linear in A, the map A--~ Acc* is linear. To show linearity of A--~ Acc, we first note that
[Acc, c*(f)] = c ( ( A - A*)f) , (2.23)
which readily follows from the CAR. Thus, if A, B c ~-1, the operator D = (aA + [3B)cc - ot(Acc) - [3(Bcc) commutes with the c(*)(f) and hence D = c l . Since (l-l, D O ) = 0 it follows that.~c =0. The linearity of A---~Ac*c* follows similarly. Note that (2.23) implies that the operators Acc and Ac*c* are zero when A = A T.
The maps A ~ Acc, Ac*c* have a continuous extension on @ to the Hilbert- Schmidt (henceforth HS) class ~-2. More precisely, for any A ~ ~-2 there are uniquely determined operators Acc, Ac*c* from ~ into ~ such that Anc(*)c(*)F--~ Ac(*)c(*)F, VF ~ 9 , whenever ~'-1 ~) An --~ A in gr2. This is obvious from the following bounds, which contain much more information, needed in the next subsection:
II Acc P, ]1 ~< 1113112, (2.24)
[IAc* c* P, II <~ ( l + 2) 11AII2. (2.25)
Clearly, (2.25)follows from (2.24), since IlAc*c*P~ll = IIP,+2Ac*c*ll = IIA*ccP~+2ll. The proof of (2.24) will be postponed, cf. (2.32) below.
When A and B are rank-one operators, there is no difficulty in verifying that
[ Acc, dF(B)] = ( A - A T) Bcc, (2.26)
[Acc, Bc*c*] = ~Tr(B - B T ) (A - A T ) - dF((B - B T) (A - AT)). (2.27)
By linearity and continuity we can now conclude that (2.26)/(2.27) actually hold on ~ for any A ~ ~-2 and any B that is bounded/HS; this will be needed below.
So far, we have proceeded in a coordinate-free fashion, and we shall continue
8 A.L. CAREY AND S. N. M. RUIJSENAARS
to do so in this section whenever the coordinatization we are about to introduce can be avoided with little expense. However, the ensuing unsmeared creators and annihilators are needed to make contact with the physics literature and are in fact very convenient mathematically, as exemplified by the very simple proof of (2.24) they enable and by other developments below. Accordingly, we identify for the rest of this section ~ and J with L2(R, dp) and complex conjugation. (There is no difficulty in extending the following considerations to the various other L2-spaces occurring in later sections and, therefore, we shall use there cor- responding notation without further comment.) Denote by ~ the subspace of consisting of vectors F whose n-particle components F ~n) are in C~(R~). Define an operator c(p): ~ ~o by
Indeed, this is obvious for elementary tensors in ~3, so that the general case follows by linearity and continuity. Similarly, denoting the quadratic form adjoint on ~3 of c(p) by c*(p), one has
I dpf(p)c*(p) = c*(f), V f c ~f, (2.30)
in the sense that the form integral on the left equals the (form of the) operator
c*(f). At this point it is appropriate to insert some comments on the use (and abuse)
of the objects c(p) and c*(p). As we have seen, c(p) may be viewed as a densely defined operator. However, the domain of the operator adjoint of c(p) equals the zero vector, so that c*(p) must be interpreted as a quadratic form. (From (2.4) one obtains the formal CAR {c(p), c*(q)} = 3 ( p - q), which preclude a definition of c(p) and c*(p) as endomorphisms on a vector space larger than ~ . ) Thus, products and functions of unsmeared creators and annihilators do not generally have an analytical meaning.
Nevertheless, the physics literature abounds with such expressions. By proxy, this lack of rigor is shared by the second part of [10]. Specifically, the limits in (II.2.8) and (II.2.27) are not proved to exist in any topology. The resulting formal objects are treated as operators, although it follows from their vacuum expec- tation values that they cannot be. Similar criticism applies to formulas like (II.2.32).
We proceed by considering some cases in which products of unsmeared creators and annihilators can be made sense of. Let A be a trace class operator. Then A is an integral operator with square-integrable kernel A(p, q), and one
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS
has
I dp dq A(p, q)(F, c(p)c(q)G) ~ ~ , (2.31) (v, AccG), VF, G
since this clearly holds if F and G are elementary tensors and A is rank one. As a consequence,
/(F (/-2), AccGt')) t = [ l( l - 1)]1/21 I dp dqA(p, q) X
<~l I dpdqlA(p,q)l[I dpl . . . . . dp/-2lF(/-2)(pl . . . . . p1-2)12 x
I x dql . . . . . dql-glGm(q, p, ql , . - . , qt-z)l 2
-</llel12 IIf"-2qlll (2.32) where we used Schwarz's inequality twice. This entails the bound (2.24) we promised to prove. Thus, we are henceforth entitled to use the operators Acc and Ac*c* for A ~ if2; they are well defined on 9 , but are readily seen to be unbounded when A ¢ ~-1- Note that the linearity of the maps A---~ Acc, Ac*c* also directly follows from (2.31) and its analog
For bounded A there is a similar representation of dF(A), which will naturally arise in later sections: The map S(R)x S(R)-*C, (f, g)-+(f, Ag) is bilinear and continuous, so that A has a unique kernel A(p, q) ~ S'(R 2) by Schwartz's nuclear theorem. Then dF(A) corresponds to ~ dp dqA(p, q)c*(p)c(q) in the sense that
(F, dF(A)G) = I dp dqA(p, q)(c(p)F, c(q)G), VF, G c ~ . (2.34)
Here, the integral denotes distributional evaluation. Again, this follows by linearity and continuity from the rank-one case.
Finally, we recall the procedure of normal ord~ing, indicated by double dots. This puts the c*'s to the left of the c's in a, product with a minus sign for each transposition needed. Thus, e.g., the normal ordering of the expression c(p)c*(q) gives rise to the well-defined quadratic form -c*(q)c(p) on ~3. For A e ff~ one has
:Acc*:= --A T c* c, (2.35)
so that the normal ordering of Acc* amounts to the subtraction of Tr A (cf. (2.20)).
10 A.L. CAREY AND S. N. M. RUIJSENAARS
2.2. THE OPERATIONS F AND dF.
In the concrete situations below, ~ is the direct sum of two copies ~(+ and ~_ of an L2-space L. Thus, there is a natural conjugation J on 9~, defined as complex conjugation on ~÷-~ L and ~_-~ L. As in Subsection 2.1, we shall use bars to denote the action of J . The projections on ~ will be denoted P~. We set
A88, =-- P~APs,, 8, 8' = +, - , (2.36)
if A is an operator on ~ and correspondingly use the matrix notation
[A++ AA+-) (10 ~) A = \A_+ so that, e.g., P+ = .
Clearly, the entries give rise to operators from L to L, for which we shall use the same symbol, the meaning being clear from context. We also define a second conjugation
which obviously satisfies
CP~ = P_~C. (2.38)
The above decomposition of ~ corresponds to the fact that the free Dirac Hamiltonian H has spectrum (-0% - m ] U [m, oo), where m/> 0 is the particle rest mass; the projections P+ and P_ are the spectral projections of H on [m, oo) and ( - ~ , - m ] , resp. The second quantization A---> dF(A) is adequate and useful in a nonrelativistic context, but it is inappropriate for relativistic particles, since it does not get rid of the unphysical negative energies. From the mathematical point of view, the physicist's solution to this problem amounts to the choice of another irreducible representation of the CAR, defined by
~(f) - c(P+f) + c*(P_f). (2.39)
If U is a unitary operator on ~ that commutes with P~, the automorphism ~([) ~ 5(Uf) can be unitarily implemented by
F(U) = F( U++)F(/]__), (2.40)
as is clear from the definitions. (The notation here is an obvious shorthand: F(U÷+) stands for F(U++(~)0) e.g.) In particular, the automorphism 6(f)---> 6(¢,Hf) is implemented by F(e i'H) = F(ei~n++)F(ei'H--). Since H acts as multi- plication by a real-valued function, the complex conjugation ensures that the generator d[ ' (H)= dF(H++)- dF(H__) of F(e i'H) is positive. The replacement of c(P_f) by c*(P_f) thus leads to the desired positive spectrum of the 'second quantized' energy operator.
Physically, the states created from the vacuum by c*(g) with g e ~+(~_) are
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 11
regarded as states of the particle (antiparticle). The fact that ~+ and ~_ are both copies of the same space L reflects the fact that the antiparticles can only be distinguished from the particles by their charge and not by their wave functions. We shall follow the physicist's convention of expressing this state of affairs. This convention consists in using the symbols a(*)/bt*) to denote particle/antiparticle creators and annihilators. Thus we write, e.g., when f ~ ~ and g ~ L, a(P÷f) and a(g) instead of c(P÷f) and c(Lg) (where i+: L--+ ~+ is the identification map). As a rule, we shall also employ the symbol ~ instead of ?. With these conventions, (2.39) reads
dp(f) = a(P+f) + b*(P_f). (2.41)
Here, ~ ( f ) may be regarded as the smeared time-zero Dirac field, cf. (4.2) below. The automorphism d~(f)----~(Cf)* can be unitarily implemented by the so-
called charge conjugation operator,
g - - - r (~ ~), (2.42)
as a simple calculation shows. (It is traditional to refer to the anti-unitary operator C on ~ as charge conjugation, too.) Thus ~ satisfies
CCa(f)~ = b(q~f), Vf~ ~+. (2.43)
With the abuse of notation explained above, this can also be read as ~a(g)C¢ = b(g), Vg ~ L, which more clearly shows why ~ is called charge conjugation.
When U is a unitary operator on ~ , a unitary operator F(U) on Fock space such that
¢b(Uf) = F(U)cb(f)['(U)*, V f ~ ~ (2.44)
does not always exist. The well-known necessary and sufficient condition for the existence of the unitary implementer is, that the off-diagonal parts U~_~ be HS. As regards uniqueness, recall that the {$(f)(*)} are irreducible, so that existence implies uniqueness up to a phase factor. In view of the relation
(UV) ~ _~ = Us -~ V_~_~ + U~ V~ -~ (2.45)
implementable unitaries on ~ form a group, denoted by ~ henceforth. We denote the set of bounded operators on ~ whose off-diagonal parts are HS by g2, in keeping with the fact that g2 may be regarded as the (complex) Lie algebra corresponding to ~2. Indeed, if i A e 82 is skew-adjoint, then e in' is a norm- continuous 1-parameter subgroup of ~2. (To see this, expand the exponential and use the well-known estimates IIBCII2 IIBIIII CII2, IInCIl= Ilnll lt fit.) We intend to show next that one can choose the phases in the corresponding unitary im- plementers such that one obtains a strongly continuous l-parameter group. Hence
F(e uA) -- exp[it dI'(A)], VA = A* e 82 (2.46)
12 A.L. CAREY AND S. N. M. RUIJSENAARS
where dF(A) is the self-adjoint generator; the arbitrary additive constant in dF(A) is fixed by requiring that the vacuum expectation value of dF(A) vanish.
It is easy to see that this can be done if A is trace class. We shall consider this simple case first (cf. also [1]). Recall that dF(A) = Ac*c and l~(e irA) = e itAe*c are in the CAR-algebra when A c ffl. Since e"Ac*cc*(f)= c*(ei 'a / )e itac*c, it then follows tha t eitA~*~c*(f) = ~*(eitAf)e itAe*~, where
A~*~.=(a*b)(A+_ + A+_-_)(ba ) (2.47)
denotes the representant of Ac*c in the 'twisted' CAR-representat ion (2.39). But then
dF(A) ~ :A~*~: = A~*~ - Tr A__ (2.48)
has vanishing vacuum expectation value and also implements the automorphism 6(f) ~ ~.(eitAf), which is what we wanted to show.
We now note that (2.48) can be written
drY(A) = dF(A++) - d F ( A L ) + A+_a*b* + A_+ba. (2.49)
As we have seen before, the r.h.s, makes sense for any A e ~2, provided it acts on a finite particle vector. Thus, (2.49) leads to a well-defined operator on @ for any A ~ ~2, which leaves ~ invariant and satisfies
dF(A)* = dF(A*), VA ~ ~2 (2.50)
on ~ . We are now prepared for the following result.
P R O P O S I T I O N 2.1. Let A ~ ~2 be self-adjoint. Then dF(A), defined by (2.49), is essentially self-adjoint on 9. Denoting its closure by the same symbol, one has, moreover,
so that by the ratio test the power series Y,~=o zk(k[) -~ d['(A) kF converges for
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 13
Izl < 8 - (2111AIII)-'. Since dI'(A) is symmetric on 9 , it is essentially self-adjoint on ~ by Nelson's analytic vector theorem.
To prove (2.51), we note first that on
[d['(A), (1)([)*] = (I)(A/)*, (2.56)
which readily follows from the commutation relations (2.13) and (2.23) and their adjoints. Now consider
Fl(z) - exp[i d[~(zA)]~(f)* F (2.57)
and
FE(Z) ~- ~(eiZAf) * exp[i d['(zA)]F. (2.58)
We know already that F~(z) is strongly analytic in the circle Iz I < 8, and this also holds for F2(z), since (P(e~zA/) * is norm-entire in z. (To see this, recall that IlcP(f)*ll = Ilfll.) But using (2.56) one has
F~n)(0) = i" d[ ' (A)n~( f )*F
~-i" ~ (k)(l)(Akf) * d[ '(A)"-kF k=0
(iA) k , ( i d['(A)) n-k ~-n!k~_-o(1)(-~. f ) ( n - k ) , F
= F~")(0), Vn c N (2.59)
so that Fl(Z) = F2(z) [or Izl< ~. Thus, (2.51) follows for t e (-~i, ~) and by the group property for any t e R. [ ]
We remark that Lundberg [1] considered similar issues in a slightly more general context. His main results easily follow from the above proposition and, in fact, have been considerably extended by it. He proceeds in a much less direct way, and close inspection reveals that his arguments only apply directly to the case where the off-diagonal parts are trace class. The fact that Ac*c* is unbounded for A ~ 0-1 leads to additional complications in the proofs of Lemmas 3.7, 3.8, 4.1 and Theorem 2 of [1] that are not mentioned there. We should like to add that our stronger result and simpler proof hinges on the estimate (2.53). This bound is quite nonobvious in the coordinate free approach of [1], but it will have caused no surprise to readers familiar with the number operator estimates of [43]. Finally, we note that a similar bound appears in [39, p. 2216]. However, the proof turned out to have a gap and, in fact, the latter bound is in error.
2.3. CHARGE SHIFTS
Under the action of the charge operator
O =-- dr(8 ) = dr(P+) - dr(P_) (2.60)
14 A.L. CAREY AND S. N. M. RUIJSENAARS
ffa(~t °) splits up into a direct sum of charge sectors,
o%,(Ih'3 = (~)' ~ , . (2.61) n E Z
(That is, F ~ ,~,, ¢~ OF = nF.) It is easily seen that any implementer f '(U), U 6 ~ , must map ~ , onto ~,+q~U), where q(U)c Z is uniquely determined by U. Indeed, since e"°F(U) and f ' (U)e "° both implement the transformation generated by e"U = U e", one must have
e"°[ ' (U) = ei~")I'(U) e "° (2.62)
Using the group property and the relation exp(27riQ) = D, it follows that q~(t)= q(U)t with q(U)e Z, which proves the assertion. Similarly, it follows that
q( U1 U2) = q( U0 + q(Uz). (2.63)
Thus, if q(U) = 1, one can reach the charge-n sector ~ , from the vacuum sector ~0 by applying [ '(U)". It is also important to note that
q (e i'A) = 0, VA = A* ~ ~2. (2.64)
Indeed, the above implies
exp[iq(e"a)] = (exp[it df'(A)]O, e ' ° exp[it df'(A)]O) (2.65)
so that q(e "A) = q(n) = 0 by continuity in t. Thus, the operators ['(e "A) leave the charge sectors invariant. Finally, we note that (2.62) with t = 7r can be written as
V(- g)[~(U) = ( - )o(u)[,( U)F( - D) (2.66)
In fact, the integer q(U) equals the Fredholm index of U__. That is,
q(U) = dim Ker U__ - dim Ker U_*_ (2.67)
This was shown in [2] and can also be read off from the explicit formula for [ '(U) derived in [12, Section 5]. Below we shall need the latter formula for the special case dim Ker U__ = l, dim Ker U'3__ = 0, and we continue by stating this result in a form that is most appropriate for our later requirements.
Thus, let U e ~2 be such that
Ker U__ = {Ae { A ~ C},
Ker U*_ = {0},
where e_ is a unit vector. Setting
e+=-- Ue_,
it follows from the unitarity of U that
Ker U*+ = {Ae+ ] A ~ C},
Ker U++ = {0}.
(2.68)
(2.69)
(2.70)
(2.71)
(2.72)
F E R M I O N G A U G E G R O U P S , C U R R E N T A L G E B R A S , K A C - M O O D Y A L G E B R A S 15
Since the off diagonal parts are HS by assumption, these relations imply that the positive operators
E_ = U__ U*_ = P_ - U_+ U*+_, (2.73)
E+ = U*+ U++ = P+ - U*+_ U_+ (2.74)
have bounded inverses (considered as operators on ~(8). Thus we can define a bounded operator Z with off-diagonal HS parts by
Z + + ~ - U + + E + 1 , Z + _ ~ - U + + E + 1 U _ + * ,
(2.75) Z_+ - - U*-_E -1 U_+, Z - - -~ - U*--E -1.
We shall call Z the conjugate of U. (The conjugate is related to the associate A used in [12] by Z = 1 + ( P + - P_)A. Cf. also [13, Section 7].) It now follows from Theorem 5.1 in [12] that
I '(U) = det(l + Z+_*Z+_)-m[a*(e+)Ec(Z) + Ec(Z)b(&)] (2.76)
Put more precisely, in [12] it is proved that the r.h.s, of (2.76) implements the transformation generated by U, so that (2.76) fixes the arbitrary phase factor in F(U), Moreover, these formulas hold on ~at in the sense that the expansion of the exponential strongly converges on algebraic tensors.
For our purposes it will be more convenient to write Ec(Z) as
Ec( Z) = exp( Z+_a* b *)F( Z++)F( ZY_) e x p ( - Z_+ba). (2.78)
This is a well-defined operator equality on @,t, since the first three factors leave @at invariant and since exp(Z+_a*b*) is well defined on @a, [12]. (To see that it holds true, one needs only verify it holds in the sense of a form equality on ~at
and this is easy.) In the applications below U has the further property
C U C = U*. (2.79)
By the uniqueness of implementers up to a phase this entails
c¢[-(U) ~ = e'~I'( U)*. (2.80)
Using (2.43) and (2.76) one can evaluate both operators w.r.t. (l-l, . a*(e+)l)), which yields
(~_, Cg+) = e '~. (2.81)
In the case of interest to us the phases are such that
C e - ~ - e + ~ (2.82)
16 A.L. CAREY AND S. N. M. RUIJSENAARS
sO that e ~ = - 1 . Hence, using (2.66) and (2.67),
~ r (U) rg = F(- |)F( U)*F(- g). (2.83)
This consequence of (2.79) and (2.82) will be needed below.
2.4. SCHWINGER TERMS AND PHASE FACTORS
The currents occurring in Section 6 are normal-ordered quadratic expressions in un,;meared creators and annihilators. As such they have a well-defined meaning as quadratic forms on (analogs of) ~ . Smearing these forms with appropriate test functions a gives rise to (the form of) the operator dF(A), where A ~ ~'2 is uniquely determined by a. If (~ ~ 0, A__ is not compact, so that the normal ordering can be viewed as a renormalization, viz., the subtraction of the 'infinite constant Tr A__'.
The notion of 'current algebra' arises from the fact that the formal commutator of two currents is another current plus a constant term, which is called the Schwinger term in the physics literature. It seems that Lundberg [1, 31] was the first to realize that this is a special case of the more general state of affairs recorded in the next proposition, and we shall follow him in referring to the constant C(A, B) below as the Schwinger term.
PROPOSITION 2.2. For any A, B c ~2 one has on
[dI'(A), d['(B)] = dF([A, B]) + C(A, B) l, (2.84)
where
C(A, B) -~ Tr[A_ +B+_ - B_+A+ _]. (2.85)
Proof. This can be directly verified by using the commutation relations (2.26) and (2.27) and their adjoints. However, it is more illuminating (and less time- consuming) to argue as follows. By virtue of the bound (2.53) it suffices to prove (2.84) for A, B s ~-~, the general case following by continuity. In that case Ac*c and Bc*c are in (the unphysical representation of) the CAR-algebra, so
[A~*~, B~*6] = [A, B]~*~ (2.86)
by virtue of (2.19) and (2.11). Using (2.48) this can be written
[dF(A), dF(B)] = dF([A, B]) + Tr[A, B]__. (2.87)
Now one has
[A, B]_ _ = (A_ . B . _ - B_ . A . ) + [A_ _, B__], (2.88)
so that Tr[A, B]__ = C(A, B) by the cyclicity of the trace. Thus, (2.84) follows. []
It is important to further consider the special case where A and B commute. Then (2.88) entails that [A__, B__] is trace class. However, the trace of this commutator need not vanish when neither A__ nor B__ is trace class. This is
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 17
exemplified by the smeared currents below, where C(A, B) turns out to be nonzero in general, even if [A, B] = 0. The next result concerns this situation. Recall that we have defined the phase of the implementer F(e iA) such that it equals exp[i dr'(A)].
PROPOSITION 2.3. Let A , B e g2 be self-adjoint and let [A, B] = 0. Then
~(e/A)~(ein) = e-tl/2>C(A.B)~(e,(A+m). (2.~9)
Proof. In view of Proposition 2.2, the commutator of d['(A) and df'(B) equals C(A, B) on 9 . Proceeding as in the proof of Proposition 2.1, it follows from this by standard arguments that on
exp[i dI'(zlA)] exp[i d[~(z2B)]
= e x p [ - z l z 2 C ( A , B)/2] x
× exp[i dI ' (z lA + 7.2B)];
{zl I +lz21 < [2 max(Ill m Ill, Ill B liD].-' (2.90) Thus, for n large enough, (2.89) follows with A, B replaced by A/n , B/n. This implies the relations
F( U)* e i, d~'(A)~(U) = e i~(/) e it dr(A), Vt C R. (2.94)
Since the 1.h.s. is a strongly continuous 1-parameter group, one must have q~(t) = 3,t and
F( U)* dF(A)r'(U) = dF(A) + T. (2.95)
Taking vacuum expectation values and putting t -- 1 in (2.94), one obtains (2.93). []
2.5. FACTORS OF TYPE I~ AND TYPE III,
This section is devoted to some algebraic aspects. For more information on the algebraic concepts occurring in this section we refer to the books by Bratteli and Robinson [44].
18 A. L. CAREY AND S. N. M. RUIJSENAARS
In Section 3 we shall encounter two subgroups ~r~, s = + , - , of the group ~d2. These subgroups correspond to a decomposit ion of 9( through projections q,, s = +, - , in the sense that, putting
X ~ ---- q~g, s = +, - , (2.96)
~r~ acts like the identity on ;¢8-~. This entails that the implementing groups II~ = f'(~-s) commute with (the representation of) the C A R algebra over ~ - ' , generated by the ~(f)(*) with f c )F -~. To exploit this, we introduce several algebras related to the latter, viz.,
~¢s = {¢(f)(*)l f • ~}" ,
~ ~- { ( r ( - O)¢(f))(*) I f • 9(~}",
(2.97)
(2,98)
e(~ ") - {(I)(f)*eP(g)l f, g • 9(~}" I ~--, n ~ Z. (2.99)
Thus, the algebras ~(~") are the strong closures of the representations of the gauge-invariant algebra over 9(s on the charge sectors ~ , . They arise in connection with the identity component IIs,0 of II~.
We shall now discuss those general features of the above algebras that are relevant to the study of the groups IIs and Hs,0 undertaken in later sections. These features depend on how the pairs of projections P~ and q~ are related. Two cases occur that we shall consider separately. In both cases the projections qs satisfy
dim(q~Y0 = oo, q~ = qs, Cq~ = q~C. (2.100)
Case 1. The projections qs and P8 commute and dim(q~Ps~ e) = oo, s, ~ = +, - . To study this case, it is convenient and illuminating to introduce the Fock
spaces ~a(Y(s) with their corresponding a's, b's and field operators
• (f)-~ a(P+f) + b*(P-f), f e 9(~. (2.101)
Here and below we abuse notation in an obvious way. Clearly, the ~(f)(*), f • ~ s , form an irreducible representation on ~ , ( W ) of the C A R over ~s . Moreover, it can be seen that
~" ) = ~(~, ,~), Vn • Z (2.102)
where ~,,~ denotes the charge-n sector of ~a(9(s). In Section 4 we shall first study II~ on ~a(9(~). The results can then easily be transferred to ~ , (9 t ) by using the isomorphism
obtained through the identification lq = l-I (~)lq and
• ( f ) = ~ ( q s f ) @ l + F ( - i ) @ F ( - I ) ~ ( q _ , / ) , V f e 9(. (2.104)
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 19
From this isomorphism it is clear that the W*-algebras M, and ~, (regarded again as acting on ~ ( : ~ ) ) are factors of type L , related by M's = ~_, .
To study II,.o on ,ff'~(X a) it is convenient to introduce the 'chiral charges'
Q, -- OF(q,) = dr(P+q,) - dr(P_qs), (2.105)
and decompose ~-~(~) into their eigenspaces, so that
~(Y()= @ .~. ,.+. (2.106) rt , n ~ E Z
Thus the charge sectors are decomposed into chiral charge sectors:
.~ .= @ o%._,... (2.107) /1 + n + = r l
Evidently, the algebras (?~") are reduced by this decomposition. Restricted to ~,_,,+ (with n_+ n+ = n), they are factors of type I®, related by (?~")'= ~TL"~. Indeed, this follows from (2.102) and the isomorphism (2.103). (In fact, (2.102) will be clear from our later work. We have included it here primarily for expository reasons.)
Case 2. The operators P, qsP8 have purely continuous spectrum [0, 1 ] (restricted to ~,).
This implies in particular that qsP8 is injective on ~ , so that P89~ '~ is dense in ;~8. Hence, Ms, ~s and 0~ °) are cyclic on ~ in ~ ( ~ ) and ;~0, resp. The spectrum condition entails that the modular operator corresponding to (M~, ~) and ( ~ , ~) has purely continuous spectrum on ~1. Thus, M~ and ~ are (hyperfinite) factors of type III1. Moreover, they are related by
M's = ~_, . (2.108)
Similarly, ~7~ ) are hyperfinite factors of type III1 on .,~o, related by
~o)' = ~ ) . (2.109)
(These arguments may be a bit brief for readers who are not familiar with Tomita-Takesaki theory and/or quasi-free states. For a more leisurely exposition of similar issues in a somewhat different context, of. a forthcoming paper by one of us [45].)
2.6. THE NEUTRAL CASE
Till now we have been dealing with the fermion Fock space ~:a(X")-~ ~a(~+) ~ ) ~ ( ~ _ ) . As explained in Subsection 2.2, from a physical viewpoint this amounts to considering the case where the antifermion can be distinguished from the fermion. If the particle is neutral, it is by definition equal to its own antiparticle, and the relevant state space for arbitrarily many such so-called
20 A. L. CAREY AND S. N. M. RUIJSENAARS
Majorana fermions is ~a(Y(+), and not ~ ( ~ ) . (To avoid creating misunder- standing among nonphysicist readers, we should add that the charge concept at issue here does not coincide with electric charge.)
The physicist's prescription for dealing with this is to identify a's and b's and use c's instead. Following this convention, the analog of (2.41) reads
B(f) = c(P+f)+ c*(CP_f), V f c ~, (2.110)
where B(f) now may be regarded as the smeared time-zero Dirac-Majorana field, cf. (4.30) below. Note that B(f) is a well-defined operator on ~a(~+), since not only P+f, but also CP_f belongs to ~+ in view of (2.37). One also easily verifies that the B(f) (*) satisfy the algebraic relations
{B(f), B(g)*} = (f, g), (2.111)
B(f)* = B(Cf). (2.112)
Note this implies the B (*) do not represent the CAR algebra over ~. We mention that Araki [46, 47] first introduced the algebra defined by (2.111) and (2.112) for mathematical reasons. We shall follow him in referring to it as the self-dual CAR algebra over ~.
Most of the developments of the preceding subsections can now be adapted to the neutral context, and we proceed by doing this.
First, Subsection 2.1 applies in toto, provided one changes the symbol ~ to Y(÷ throughout. Correspondingly, we shall use in the sequel the symbols @at, 9 , ~ , F, dF and Ac(*)c (*) as defined with this change.
Secondly, let us discuss the changes needed in Subsection 2.2. Here, a unitary operator U on Y( only generates an automorphism B(f)---> B([)= B(U[) of the algebra (2.111) and (2.112), when it satisfies
CU= UC. (2.113)
Indeed, this is necessitated by the requirement/~(f)* = B(Cf). As will be seen in Section 3, the restriction (2.113) leads to the allowed gauge group being O(N) in the neutral case, as opposed to U(N) in the charged case. (In physical terms, gauge transformations on a Majorana field must be real so as to preserve its Majorana character.) The second necessary condition for the existence of a unitary operator f'(U) on ,~a(Y(+) such that
B(Uf) = I'(U)B(f)['(U)*, Vf ~ ~(, (2.114)
is that the off-diagonal parts of U be HS. These two conditions are also sufficient (see [13] and references given there) and it is again clear that the implementer is unique up to a phase. We denote the subgroup of unitaries in % commuting with charge conjugation by <g~. Similarly, we denote the Lie subalgebra of consisting of those operators satisfying A * = - CAC by y~. Thus, if A ~ y~ is self-adjoint, then eUAE ~ .
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 21
To state and prove the analog of Proposition 2.1, we first recall that ~(+ and ~_ are copies of the same space L, so that the matrix elements As~, of an operator A on ~ may be regarded as operators on ~+ - L, cf. Section 2.2. Thus, if A ~ y~, we can express the fact that CAC equals - A * through the equations
on 9. Also, if A c y~ is self-adjoint and has vanishing off-diagonal parts, one gets dF(A) = dF(A++) in view of (2.115). Hence, exp[itdF(A)] = F(e~ A) implements the transformation B(f)*--> B(eitAf) *. The following proposition shows that this holds more generally, rendering dF(A) the neutral analog of the charged object dF(A).
PROPOSITION 2.5. Let A ~ g~ be sel[-adjoint. Then dF(A) is essentially sel[- adjoint on 9. Denoting its closure by the same symbol one also has
Proof. since A is self-adjoint, it follows from (2.117) that dF(A) is symmetric on 9. Using the commutation relations (2.13), (2.23) and their adjoints, and then the relations (2.115) between the matrix elements it moreover follows that
[dF(A), B( f )*]= B(Af)*, V f~ ~. (2.119)
The proof now proceeds along the same lines as the proof of Proposition 2.1 and will therefore be omitted. [ ]
Note that (2.116) makes sense for any A e g2, but that (2.119) only holds for A c ,~. In fact it is clear from its definition that dF(A) vanishes for every A e satisfying A * = CAC. We also point out that if A c y~ is a trace class operator given by (2.17) one has
N
dF(A) = ½ )-'. A~B(gn)*B(f,)-½Tr A__. (2.120) n = l
(Indeed, using (2.111) and (2.112) it follows that the r.h.s, has the derivation property (2.119). Furthermore, its vacuum expectation value vanishes. Hence, it equals dF(A).)
Let us now turn to aspects paralleling Section 2.3. Of course, in this case the 'charge operator' d]~(I) vanishes. The neutral analogs of the charge sectors of Section 2.3 are the sectors with even and odd particle numbers, which we shall denote by .~o and ~ , resp. Equivalently, we can define ~ o / ~ as the eigenspace
22 A.L. CAREY AND S. N. M. RUIJSENAARS
of the self-adjoint unitary F(-g) corresponding to the eigenvalue 1/-1. Hence,
~,(Y(+)= @ ,~,,. (2.121) r~EZ2
Now, any implementer F(U), U e ~d~, must map • onto o%,+a(ta), where d(U) e Z2 is uniquely determined by U. (Also, here and below addition is rood 2.) To prove this assertion, we note first that the transformation generated by -1 is implementable; in fact we may and shall take ['(-0) = F(- I ) (where, of course, t denotes the identity on Y(/~+ at the left/right). As in Section 2.3 it now follows that F(-0)f '(U) = ei~f'(U)F(-0). This entails e i~ = +1, since F(-6) 2 = D. As F(- I ) has eigenvalue 1/-1 on ~o /&, the assertion follows. As a consequence we also have
F(-~)I '(U) = (_)a(v}[,(U)F(-t). (2.122)
By the same argument,
d( U1U2) = d( U1) + d(U2) (2.123)
and, moreover, arguing as in Section 2.3,
d(e ~'A) = O, VA , A* ~ ~ . (2.124)
Since ~ is a subgroup of ~ , one's first guess is that d(U) equals q(U) (mod 2). However, this is wrong. In fact, q(U)= O, Y U e q2~. Indeed, [C, U] = 0 and unitarity implies that the kernels of the four operators ,.,rr(*)s8 all have the same dimension. From the explicit formula for the implementer established in [13, Section 6], it follows that d(U) is equal to this common dimension (rood 2). We shall need this formula in the special case where the dimension equals 1 and we proceed by stating it.
To this end, let U E ~ satisfy
Ker U*+ = {a.e 1,~ e C}, (2.125)
where e is a unit vector in ~+. Setting
e' -- CU* e
it follows that
Ker U++ = {ae'l ,x • C}. (2.127)
In this case, the positive operators E8 defined by (2.73) and (2.74) have a one-dimensional kernel Ks, but they are again boundedly invertible on the ortho-complement K} (w.r.t. Y(s) in view of the HS condition. We denote by E~ 1 the bounded operator that equals 0 on Ks and equals the inverse of Es on K~. The conjugate Z of U is then defined again by (2.75). It now follows from [13, Theorem 6.3] that
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 23
where
E~(Z) - :exp(½ [Z+_c*c* + (Z+ + - P+)c* c - (Z__ - P-)cc* - Z - ÷cc]):.
(2.129)
These formulas hold in the same sense as in Subsection 2.3, and as the analog of (2.78) one obtains
E,(Z) = exp(½ Z+_c*c*)F(Z+ +)exp(- ½ Z_ +cc). (2.130)
(Indeed, it is not hard to see that the assumption UC = C U implies Z_ T_ = Z÷+.) We continue by adapting Subsection 2.4 to the neutral context.
PROPOSITION 2.6. For any A, B ~ g~ one has on ~)
[dI'(A), dF(B)] = dF([A, B]) + 1 C(A, B) Q, (2.131)
where the Schwinger term C ( A , B) is given by (2.85). Proof. This follows from (2.116) and (2.115) by a straightforward computation,
using the relations (2.26) and (2.27) and their adjoints. [ ]
As in the case of Proposition 2.2, one can also prove (2.131) in a more algebraic way, taking this time (2.120) as a starting point, and using (2.111) and (2.112). Let us now consider the analog of Proposition 2.3.
PROPOSITION 2.7. Let A, B c y~ be self-adjoint and let [A, B] = O. Then
['( e iA ) ['( e iS) = e -(1/4)C( A'B)['( e i( A + B)). (2.132)
Proof. This follows from Proposition 2.5 and Proposition 2.6 by arguing as in the proof of Proposition 2.3. [ ]
Finally, we present the analog of the projective multiplier formula (2.93).
PROPOSITION 2.8. Let [ U , A ] = 0 where U c ~3~ and A c ~ is self-adjoint. Then
It remains to discuss the changes needed in Subsection 2.5. In the neutral case one is dealing with two subgroups ar~ of the groups ~3~. The projections qs are the same as in the charged case. Here too, 7r~ acts like the identity on ~e-s, cf. (2.96). The relevant W*-algebras are now
sgs =- { B ( f ) J f e ~q",
~ =- { r ( - 1 ) S ( f ) J f e ~}",
~5~")=-{n(f)n(g)Jf, g e ~ q " t ~;,, n c Z 2 .
(2.134)
(2.135)
(2.136)
24 A.L . CAREY AND S. N. M. RUIJSENAARS
Note that the relation
Cy(S = ~s, (2.137)
which follows from (2.100), is essential for (2.134)-(2.136) to be *-algebras, cf. (2.112).
In Case 1 we introduce the fermion Fock space ff~(~_) over
N$ _ p+ ygs, (2.138)
which carries the irreducible representation
B(f) = c(P÷f)+ c*(CP_f), f e Ns (2.139)
of the self-dual CAR over ~s. Then one has
~n) = "~(~n,s), V?I E Z 2 . (2,140)
where ~0,~/,~,~ are the even/odd sectors of ,~,(~-) . The analogs of (2.103) and (2.104) are then
~, (~+) - ~ , ( ~ $ ) @ ff~(~7~ s) (2.141)
and
B(f) = B(qf f )@l + F( -O®F(-OB(q-s f ) , Y f e ~. (2.142)
From this one reads off that M, and ~s are factors of type I~ satisfying 91'~ = ~_, . In view of (2.100), the 'chiral charges' dF(q~) vanish. Their role is played by
L--F(U - 2q~), (2.143)
in the sense that
~,,(~+)~- @ ,~.,,~ (2.144) n - - ~ n + E Z 2
where F e ~ . . . . . ¢~/~F = n~F. (Note that Is is well defined, since qs~+ C ~÷. Also, note that Is = F ( - I ) @ I under the isomorphism (2.141).) Then (2.107) holds again (with addition mod 2) and the algebras ~7~ ") are factors of type I~ on ~ , . . . . such that ~")' = ~7(2, ).
Finally, let us consider Case 2. Here, essentially no changes are needed. Indeed, it is clear that all that has been said in Subsection 2.6 holds true here, too, the sole difference being that o~.(~) should be replaced by 8~,(~÷).
2.7. CYCLICITY + POSITIV1TY =), IRREDUCIBILITY
We close this section with a lemma that will be very useful to use in the case where qs and P8 commute. This concerns a result that dates back to work of Borchers [48] and Ruelle [49] in the context of axiomatic relativistic quantum field theory.
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 25
Let ~ be a Hilbert space and M a set of (possibly unbounded, but closable) operators on ~ such that polynomials in operators from M are well defined and cyclic on a vector f~ e ,~. Moreover, let M be self-adjoint (i.e., O e M implies O* e M). Finally, let U(t) = e "~ be a strongly continuous one-parameter group of unitary operators with f~ as its only invariant vector and such that
U(t)MU(t)* = M. (2.145)
This relation is assumed to hold on the dense subspace
~ ~ PoI(M)fL (2.146)
We can now state the lemma. For completeness we include the proof, which is adapted from [50].
L E M M A 2.9. In addition to the above assumptions, suppose that the generator G of U(t) is positive. Then the set M acts irreducibly on ~ , in the sense that if A ~ ~ ( ~ ) is such that
(F, A O G ) = ( O ' F , AG) , V O e M, VF, G s ~ , (2.147)
then it follows that A = cL Proof. Since U(t) leaves 1] and M invariant, (2.147) implies
(fL A ei 'GO1.. . One) = ( O * . . . O*f~, e-"CAfl). (2.148)
Integrating this with the Fourier transform of the characteristic function of ( -b , - a ) C (-w, 0), one obtains
0 = ( 0 " . . . 0"1~, P(o,b)Al~), (2.149)
where PI is the spectral projection of G on the interval I. Taking a ---> 0, b ~ 0% it follows from the cyclicity assumption that Pto,o~)A~ = 0. Since f~ is the only eigenvector with eigenvalue 0 of the positive operator G, it follows that AI~ = c~. Thus,
(F, a 0 1 . . . OnlY) = ( O*n . . . O ' f , Af~)
= c(F, 0 1 . . . O,I~), V F ~ ~a~,
so that A = cO.
(2.150)
[]
3. Implementable Gauge Groups in the One-Particle Dirac Theory
3.1. PRELIMINARIES
In this subsection we introduce various operators in the context of the Dirac theory for a one-dimensional particle of mass m ~> 0. To distinguish the cases m = 0 and m > 0, we shall sometimes append a subscript (0) and (m), resp. The
26 A.L. CAREY AND S. N. M, RUIJSENAARS
first operator we consider is the Dirac Hamiltonian, which we choose to be the differential operator
) n ~ I d x m
dx
on the Hilbert space LZ(R, dx) 2. This amounts to the choice
,,-,°,,-- (; ;), -;),
(3.1)
(3.2)
for the Dirac T-algebra. Here, the domain of t7/is the Sobolev space Hi(R) 2 and its action the obvious one. We diagonalize /2/by means of the unitary operator
where O is the Heaviside function. We shall use the convention
A=- W-l fi, W (3.9)
if ~i, is an operator on ~ and vice-versa. Using this convention one verifies that
( n f )dp) = 6E,,f~(p), (3.10)
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 27
while the momentum, charge conjugation and parity operators
. d (["g)(x)--,(-d--~x g)(x), (3.11)
((~g) (x) - V5g(x), (3.12)
(Pg) (x) = y° g(-x) (3.13)
transform into
(P'f)8(p) = 6pfs(p), (3.14)
( Cf)n(p) = f-8(p), (3.15)
(Pf)8(p) = 6fs(-p). (3.16)
We denote the spectral projections of H on Ira, ~) and (-o~, - m ] by P÷ and P_, resp., so that in view of ( 3 . 1 0 ) one has
~(~=--Ps~-L2(R, dp), 6 = + , -
Note that (3.14)-(3.16) imply
[p1, p~] __ 0,
CP~= P-sC,
[P, P~] = O.
From the above formulas one can read off
(3.17)
what we have
(3.18)
(3.19)
(3.20)
anticipated in Subsection 2.2 as regards the one-particle theory. In particular, (2.37) and (2.38) correspond to (3.15) and (3.19). We finish this subsection by making explicit the projections q~ of Subsection 2.5. On ~ these are simply the spectral projections of 7 5 . That is,
Setting
(Rf)(p) = f(-p), f e L2(R, dp) (3.22)
one readily verifies that
1 (Ep+sp smR~ qs = ~ p \staR Ee + sp/' s = +, - (3.23)
(The matrix notation used here has been explained in Subsection 2.2.) In particular,
q(o) = (Oo(SP) O) O(sp) " (3.24)
28 A.L. CAREY AND S. N. M. RUIJSENAARS
The properties mentioned in Subsection 2.5 are obvious from this: Case 1 corresponds to m = 0, Case 2 to m > 0.
3.2. THE GROUP U(1)
At the one-particle level considered m this section, gauge transformations are simply unitary matrix multiplication operators on ~ . However, since we are interested in 'transporting' these unitaries to the (positive energy) Fock space level, we should single out those gauge transformations defining unitaries in the group ~ of ~ , i.e., the group of unitaries whose off-diagonal parts are HS, cf. Subsection 2.2. Since ~ C ~ 2 , the Lie algebra of bounded operators with off-diagonal HS parts, let us first consider bounded multiplication operators on
Specifically, let a(x) e C~(R). Then a gives rise to two multiplication operators
00/' o -x t on 9~ and a calculation shows that the kernels of As, s = +, - , are given by
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 29
For m > 0 the analog of (3.32) is
I dp, m)l
= ~ dO1 dOz l&(Sm[sinh O1 + sinh O2])l z e °'+°~
= 4--~ d~b dOi6(2~m sinh ~0 cosh O)12 e 2~
= 2--~ d k I la(~k)l 2 (3.33)
where
I(Y) ~ ~ I cosh 2dO (~ [2y + 2(cosh20 + y2)1/2 _ cosh 20(cosh 2 O + y2)-(lm].
(3.34)
Now one easily sees that
0 < I ( y ) - < C ( l y l + l ) , Vy~R, (3.35)
so that the integral (3.33) converges, too. Hence, the off-diagonal parts of A~,,)s are also HS for m > 0.
As regards the multiplication operators
CI 7,")oo (0o,, one readily verifies they are not in y2. Thus we restrict ourselves henceforth to multiplication operators commuting with 3, 5 . Clearly, one need not require a c C~) for the corresponding operators A to be in y2. We have assumed this at first to fix our thoughts, but the above shows one needs only require a to be in L~(R) and to have a distributional Fourier transform such that the integrals (3.32) and (3.33) make sense and converge. However, to avoid unnecessary technical complications, we shall settle for a slightly smaller class of ot's henceforth, viz., those that are in the Sobolev space H1(R). Recall that Hi(R) consists of all absolutely continuous L2-functions with L2-derivatives. Since such functions vanish at infinity, they are bounded and the HS condition is satisfied since S dk( k2 + 1)Ic~(k)l 2 < oo implies S dk I k[ Ic~(k)[ 2 < oo.
Henceforth we denote the space of real-valued H,(R)-functions by V. Clearly, for any a e V, exp[ia(x)] is a continuous path in the group U(1), converging to the identity for I xl---~ o0; furthermore, such paths form a group under pointwise multiplication, since V is a linear space. Thus, one obtains a subgroup of the group of loops in U(1). We shall denote this subgroup by LeU(1)o. Here, the subscript e indicates that the loops 'start' at the identity e, while the subscript 0
30 A.L. CAREY AND S. N. M. RUIJSENAARS
denotes the winding number. We shall now introduce a larger group of arbitrary winding number loops through e. Specifically, we set
(To see that this is a group under pointwise multiplication, note that f c H~ and g - 1 ~/-/1 implies fg ~ H1 .) Defining
~/(x) ~- ~r+ 2 Arctan x (3.37)
and
o'(x) -= exp[i'q(x)] = (x - i)/(x + i) (3.38)
one readily verifies that any u(x )EL~U(1) can be uniquely written as exp[ia(x)]tT"(x) with a e V and n e Z. Thus, LeU(1) is generated by its identity component L~U(1)0 and the winding number 1 loop or.
Let us now return to ~ . On )~ we define two faithful unitary representations -k± of LeU(1) by setting
(x, (339,
and
(3.40)
From the above it is then clear that ~rs(u)e ~2. As a result, we obtain two corresponding projective unitary representations F(~-±) on ~ a ( ~ ) , whose study is deferred to later sections. We shall conclude this subsection with some comments on the definition of the loop group LeU(1) and its representations "~'s.
Let us begin by emphasizing that the products of the operators ~r.(u) and "~'.(u') do not give rise to all implementable gauge transformations. For one thing, as already indicated above, we could have chosen a larger space than H1 in the definition of LeU(1) and still get continuous loops leading to unitaries in ~d2. However, HI turns out to be large enough for our purposes, in particular as concerns answering questions of cyclieity and irreducibility and as regards establishing the connection with Kac-Moody algebras. Nevertheless, we feel a precise description of the 'largest' space would be of interest. (This seems to be tied up with subtle questions in Fourier analysis.)
A more conspicuous omission is formed by the global gauge transformations
ei ~ , ~p±~(0,2~').
It will be convenient to treat these on a separate footing in the sequel, one reason being that for m > 0 one needs to require in addition that ~p+ = tp_. Indeed, the HS condition is violated for tp+ ~ ~p_, as follows from (3.23).
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 31
Finally, let us comment on our choice of a minus sign in (3.25) and (3.40). The point is that with this choice one can set up a unitary equivalence between or÷ and w_ and their Lie algebras through a unitary that commutes with P , , viz., the parity operator (3.16). (Indeed, it is obvious from these relations and (3.13) that
P , ~ k = A_~, /5"fi'~(u)P = "h'_~(u) (3.41)
so that the assertion follows from (3.20).) Thus, we get a unitary equivalence of the corresponding Fock space representations through the Fock space parity operator. This will enable us to restrict ourselves in many respects to considering only one of the two representations.
3.3. THE GROUPS U(N), SU(N) AND SO(N)
Let us now pass to the case where an internal symmetry space CN(N> 1) is tensored on to ~ and ~ ; the resulting Hilbert spaces will be denoted again by and ~ . Similarly, we denote (e.g.) the operator/-:/@1 by/2/. With this convention all formulas and relations from Subsection 3.1 also hold in this case; in (3.1) (e.g.), ~ should be regarded as L2(R, dx)N(~ L2(R, dx) N. AS the analog of (3.36) we define
LeG(N) =- {u(.) ~ Map(R, G(N))[ u0(- ) - 6 o c H1(R),
i, j = I . . . . . N}, G -- U, SU, SO. (3.42)
As before, it easily follows this is a subgroup of the loop group of G(N). We define the two representations of interest again by (3.39) and (3.40), where u(±x) now stands for an N × N matrix in G(N). It also follows that the operators ~'s(u) belong to the group ~ of ~. Indeed, their off-diagonal parts w.r.t, the ~ / ~ _ - decomposition of ~ can be regarded as N x N matrices of HS operators on L2(R, dp) by virtue of Subsection 3.2.
The comments at the end of Subsection 3.2 apply here, too and, therefore, will not be repeated. Instead, let us note that the (complex) Lie algebra of LeG(N) is given by
Leg(N) = {M(.) ~ Map(R, g(N)) ] M,j(.) c HI(R),
i , j = 1 . . . . . N}, g = u, su, so. (3.43)
We shall presently show that any loop in LeSU(N) can be written as a finite product of loops of the form exp(ia), where a = a* c Lesu(N). This follows from the well-known fact that SU(N) is simply-connected. The fundamental group of U(N) is Z; To generate LeU(N) one therefore needs not only loops of the form exp(ia) with ¢¢ = cr* c L~u(N), but also one of the loops
where o-(.) (cf. (3.38)) stands in slot j. Since w~(SO(N)) = Z2 for N > 2, one needs in this case also one of the 'winding number 1' element~ ~rjk('), j ¢: k, to generate
32
LeSO(N). Here, e.g.,
//'cos n(') or12(, ) ~__- f-sin ~7(')
\ cos r/(.) 1.
0 1 '
A. L. C A R E Y A N D S. N. M. R U I J S E N A A R S
~r2,(') =-- tr,2(')* (3.45)
where 19(') is defined by (3.37); the definition of o'sk(.) will be obvious from this. Note that (3.4'5) implies that the representers ,h-~(o-sk ) commute with ~7. Thus, the operators 7r~(o'jk) belong to the group q3~, cf. Subsection 2.6.
Let us now keep our promise made below (3.43). As a preliminary, we introduce the set N of loops v in LeG(N) such that v - I has matrix elements with supremum norm smaller than N -I. Then the eigenvalues of v stay at a distance smaller than 1 from 1, so that In v is well defined for v ~ N. Moreover, it is not difficult to see that In v ~ Leg(N). Indeed, invoking the power series for In(1 - z), one needs only verify that the matrix elements of ~ = 1 / 3 " / n are in Hi(R) if/3 has matrix elements in Hi(R) with supremum norm smaller than N -~, and this is routine.
We are now prepared to prove the finite product property. To this end, let u c LeSU(N). Since u is a continuous loop in SU(N), it can be continuously deformed to the identity loop. It easily follows from this that u can be written as e"~, . . . e~'~,, where &~ , . . . , &t are continuous functions on R vanishing at infinity with values in the self-adjoint traceless N × N matrices. Approximating &s by a self-adjoint a j e Lesu(N) (in supremum norm), we can ensure that e - " ~ , . . . e - i " , u c N . Hence, u = e i " , . . . e i ' ~ , e i',~, with a~=a*cLesu (N) , j = 1 . . . . . l + 1, completing the proof.
Finally, we note that one may view LeG(N) as a Banach Lie group with Banach Lie algebra Leg(N); one can take e.g.
II MII -= sup II Miill..tm and d(u. v) = supl Iuii - v~ill H,<.) i ,i i ,/
as norm and metric on Leg(N) and LeG(N), resp. In this picture N is a neighborhood of the identity in LeG(N) (since the Hi(R) norm dominates the L~(R) norm) and the finite product property translates into LeSU(N) being connected.
3.4. T H E S T A N D A R D K I N K S F O R m = 0
We introduce
~r~(x) ~ exp[i~9~'(x)] = 1
X - - / " rc,an(-v), 2ie
X - - r + ie. '
r c R, e > 0, (3.46)
(3.47)
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 33
and define cr;,j(.) and o'~,~k(') correspondingly, cf. (3.44), (3.45). We shall refer to the unitary representers
U s , j , e ( r ) ~ "/rs(Orr,j) , j = 1 , . . . , N, (3.48)
U~,jk,~(r) =-- 7rs(o'~,ik), j, k = 1 . . . . . N (3.49)
and their adjoints as standard kinks. Roughly speaking, they will be shown to give rise to Fock space implementers that converge to components of the free Dirac field when e-~ 0. In this section and the next one we prepare the ground for the convergence proof by deriving the relevant properties of the standard kinks. We shall first consider the U(N) case (3.48).
It suffices to take N = 1, as it will be apparent from what follows that the case N > 1 can then be dealt with by making obvious notational changes. Moreover, we shall restrict our attention to U+,,,,(r) since an appropriate use of charge conjugation and parity will enable us to avoid deriving similar properties for the adjoint and the case s = - . Thus, no confusion should be caused by suppressing all indices in the sequel. In particular, U henceforth stands for U+.L,(r).
We begin by recalling that the functions
1 Fn(x ) ~ ,/./.-1/2 O'(X) n, n C Z (3.50)
x + i
form an orthonormal base of L2(R, dx). One way to see this is to note that they are the transforms of the orthonormal base {(21r) -1/2 ei"°}7=_~ of L2([0, 2~'], dO) under the unitary transformation
(Tf)(x) =- ~+ i f('rl(x))" (3.5 l)
It is also readily seen that
supp F, = [[0, oo) n/> 0, (3.52) t(-o% 0] n < 0 .
(In fact, the F, are essentially the well-known Laguerre functions.) Setting now
e , - W - l ~ , , n ~ Z + ½ (3.53)
where
e n ( x ) ~ - e -112 " (3.54)
it follows from (3.5), (3.8) and (3.52) that the set {e8.}~=,/2 forms an orthonormal base for q+~n. Moreover, one has the crucial relation
Uen = e , , + l , VnEZ +½. (3.55)
3 4 A.L . CAREY AND S. N. M. RUIJSENAARS
Indeed, the corresponding relation on ~ is obvious from the above definitions. We are now in the position to state and prove the following lemma, which
ensures that U has all the properties assumed in Subsection 2.3.
L E M M A 3.1. One has U ~ ~2 ; moreover,
Ker U__ = {)re_ I )t ~ C}, (3.56)
Ker U*__ = {0}, (3.57)
Ue_ = e+, (3.58)
where en(p) ~ L2(R, dp) = ~ are the unit vectors
es(p) =- 6(2e)l/20(p) e -ispr e -*p. (3.59)
Finally,
C U C = U*,
Ce_ = -e+.
(3.60)
(3.61)
Proof. A straightforward calculation shows that e~ equals ie~/2, cf. (3.54). Thus, (3.56)-(3.58) follow from the shift relation (3.55). Furthermore, it is clear that (~0(~ = U*, so that (3.60) holds true, and (3.61) follows from (3.15). [ ]
In view of this lemma we can introduce the conjugate Z of U by (2.75). In the next lemma we collect the properties of U and Z needed to control the e ~ 0 limit.
L E M M A 3.2. The operators
D~ - U~ - P~ (3.62)
are integral operators on L2(R, dp) -~ ~8 with kernels given by
D~(p, q) = - 2 e O ( p ) O ( q ) O ( 6 p - 8q) e -~ap-q)" e -'8(p-q). (3.63)
They satisfy the bounds
IID+fH = O(e 1/2), E ""~ 0 , (3.64)
119411= o(,), , 4 0 (3.65)
for any f c L2(R, dp) with compact support. Furthermore, the conjugate of U is given by
Thus, (3.63) follows from (3.29). To prove the bounds, let f c L 2 have support in
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS
]Pl ~< N. Then we have, using the Schwarz inequality,
35
2 , ee'NNIIfll z, (3.68)
which entails (3.64). Similarly,
{ID-fll2 = 4,2 IoNdp e2"p lloNdq
<~ 4,2 N eZ'NNIIfll z,
e -~q~ e-'qf(q) [ 2
(3.69)
implying (3.65). The shift relation (3.55) entails U_+ = 0, so that Z~.-8 = 0 and moreover E8 = Ps, cf. (2.73), (2.74). Thus, (3.66) follows from the defining relations (2.75). [ ]
Let us now consider the SO(N) kinks Us,ik,,(r) (*). In this case it suffices to take N = 2 and study U÷.lz,,(r). To ease the notation we shall denote this operator again by U; the context will prevent confusion with the operator U of Lemmas 3.1 and 3.2. The relevant properties of the former operator can be readily established by using the above results on the latter one and the observation that
0/~/= Diag(~r~(. ), ~ ' ( . ), 1, 1), (3.70)
where M is the unitary matrix multiplication operator
M - ( N ON), N----2-112(1i l i ) (3.71)
on ~ ~ LZ(R, dp) 2 (~) C 2. Since M is of the form Q I~) N, one has
[M, Psi -- 0. (3.72)
We are now prepared for the analog of Lemma 3.1.
LEMMA 3.3. One has U • q3~ ; moreover,
Ker U*+ = {Ae [ A • C}, (3.73)
e' = CU* e, (3.74)
where e(')(p) • L2(R, d p ) Q C 2 ~ ~o+ are the unit vectors
Here, e+(p) is defined by (3.59). Proof. In view of (3.72), we can determine Ker U*+ by considering the
++-kernel of Diag ( ~ , o'i, 1, 1), transformed to ~. By virtue of Lemma 3.1 the
36 A.L. CAREY AND S. N. M. RUIJSENAARS
latter kernel is spanned by e+(p)(10). Thus, Ker U*+ is spanned by e+(p)N(~)=
e(p). Similarly, (3.74) follows by making suitable use of (3.58) and (3.61). [ ]
This lemma shows that we are in the situation described in Subsection 2.6, cf. (2.125) and (2.126). Thus, we can define the conjugate by (2.75), where the meaning of the symbols 'E~ -1' is explained below (2.127). The proof of the convergence lemma of Subsection 4.2 (Lemma 4.4) hinges on the following properties of U and Z.
LEMMA 3.4. The operator
D+ = U++ - P+ (3.76)
is an integral operator on L2(R, dp)~)C 2= ~+ with kernel given by
for any f e L2(R, dp)(~)C 2 with compact support. Furthermore, the conjugate of U is given by
Z = - U++ - U'~__. (3.79)
Proof. These claims follow by exploiting the relations (3.70) and (3.72): Using Lemma 2.2 this yields (3.77) and (3.78), while (3.79) follows by using.the shift property (3.55). Indeed, this property entails that E¥1/E -1 acts like the identity on Ker U++±/Ker U*_ I. [ ]
3.5. THE STANDARD KINKS FOR m > 0
For m > 0 the standard kinks are again defined by (3.48) and (3.49). Proceeding as in Subsection 3.4, the shift relation (3.55) still holds. However, e, now has nonvanishing components both in ~÷ and in ~_, so that there seems to be no way to exploit this shift property in determining properties of the operators U~,. Also, note (3.26) and (3.27) imply that a multiplication operator fi,~,,~s vanishes if one of its off-diagonal parts vanishes. Thus, conjugates with nonzero off-diagonal parts cannot be avoided. These are the main reasons why the massive case is considerably more involved, both at the single-particle and at the Fock space level. It is a remarkable fact that with our choice of kink function many properties of the massless case do persist. The first example of this is Lemma 3.5, which is the exact analog of Lemma 3.1.
LEMMA 3.5. One has U c ~2 ; moreover, the relations (3.56)-(3.58) and (3.60),
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 37
Proof. By (3.18) translations commute with Ps, so that we need only consider the case r - 0. Let g c Ker Us~. Then one has from (3.67) and (3.26)
1/2 + 8k'~ 1/2 g(P)=ee-8"P(a+~p) I-n2 d k e'k( 1 E--kk! g(~k) (3.82)
Hence, it follows that g is C 1 and that the C 1 function f(p) = (1 + p/Ep)-l/2g(p) satisfies the ODE f' = (6Ep/Ep)f. Thus, f(p)= c exp[8~Ep], which implies that Ker U__ = {Ae_} and Ker U÷+ = {0}. By unitarity, the latter equation implies Ker U*_ = {0}. The equations CUC = U* and Ce_ = - e ÷ follow as in the proof of Lemma 3.1. Together with the kernel properties they entail Ue_ = ei~e+, and checking that e i~ = 1 is routine. [ ]
Thus, we can again define the conjugate by (2.75). The following information on U and Z will suffice for proving the convergence Lemma 4.7 below.
LEMMA 3:6. The operators Ds, defined by (3.62), are integral operators with kernels
D~(p, q ) = - e ( l +~v) U2(l +~q) l/20(Sp-6q)e-'~P-q)" e -'~'p-q' (3.83)
and satisfy the bounds (3.64), (3.65). The conjugate of U satisfies
IIz.,- ll= = (3.84)
IIz++ + u++ll-- ( 3 . 8 5 )
I I z - -+ u*--II = o(d/2). (3.86)
Proof. The kernel formulas follow from (3.67) and (3.26). Proceeding as in the proof of Lemma 3.2, one has
N N q \ 1/2 e -~" e_,qf(q)12 ][O_fll2=,ZI_ dp ( l+~)e -2"P l l p dq(1 +~q)
= (3.88)
38 A.L. CAREY AND S. N. M. RUIJSENAARS
since 1 + (p/Ep) is integrable for p ~ -0o. It remains to prove the bounds on the conjugate.
To this end, we first note that
Io o II o-+ll~ = 4mZE 2 dq~ dO exp[-4em sinh q~ cosh O] e -2~, (3.89)
where we used (3.33). Changing variables, one gets
dO f[ e Ilu-+l122<~cl~jo c o s h O _ _ dke-nmkcosh~p(k,O)
f f ~ G ~ dk e -4"k dO e-°[1 + (k e-°/E)2] -3/2
C2E2Iod~e-4mk~k/" = dy[1 + y2]-3/2
Io = C2e 2 dk e-4'~k[E 2 + k2] --1/2
= O(E2 In 1), (3.90)
so that, afortiori,
II u-+l12 = 0(E3/4) • (3.91)
This is readily seen to imply
E~ 1 = P~ + R8 (3.92)
where
II ~112 = o( ,3 '=) . (3.93)
Hence, the desired bounds follow from the defining relations (2.75) and well- known properties of Hilbert-Schmidt norms. [ ]
As noted above, we shall only need the estimates (3.84)-(3.86) on the matrix elements of Z. It is of considerable interest, however, that Z can be actually explicitly determined. Using the polar decomposition of U__ one sees that Z__ is the bounded operator that satisfies
U - _ Z _ _ = - P _ , Z _ _ U - - = e _ @ e _ - P _ . (3.94)
Similarly, Z++ is uniquely determined by
U*+Z++ = - P + , Z++ U*+ = e+ @ e+ - e+. (3.95)
Thus, determining Zsn is tantamount to solving integral equations, cf. (3.83). Since the result may be useful in further developments and is of interest in itself, we state it here: The matrix elements ( Z + I)a~, are integral operators whose
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 39
kernels are given by
p ~ 1/2(1 + q~,,2 (Z+U)++(p,q)=E\I \ -~q! e-i(p-q)r e ~(E,,-Ep) X
× [T(q) - @(q - p)], (3.96)
( Z + D--(P, q) = (Z+g)++(p, q), (3.97) [ p ~ 1/2[ + q ~ 112
Z+_(p, q ) = e t l +-~p ) ~1 -EqqJ e-i(P+q)' e~(E"-E')[T(q)--O(q+ P)]'
(3.98)
Z_+(p, q) = Z÷_(p, q) (3.99)
where
T(q) =- C~ 1 + k e_2,~k" (3.100)
The proof consists of the following steps, whose details we shall skip, however.
(1) Proving that the kernel at the r.h.s, of (3.96) defines a bounded integral operator by using the well-known inequality
IIK~]2 <~sup I dqlK(P' q)lSqUpa I dplK(P' q)l. (3.101)
(2) Verifying that the relations (3.94), (3.95) hold true, when Z~ is defined through (3.96), (3.97) (the integrals that arise can be done explicitly).
(3) Verifying that Z s - , , defined by (3.98), (3.99), satisfies Z+_ = Z++ U_+* and Z_+ = Z__ U_+ (again, the integrals can be evaluated).
Let us now proceed with the analogs of Lemmas 3.3 and 3.4, using the notation introduced there.
LEMMA 3.7. One has U • q3~ ; moreover, (3.73), (3.74) hold true, where et')(p) is defined by (3.75) and e+(p) by (3.80), (3.81).
Proof. Arguing as in the proof of Lemma 3.3, these relations follow from Lemma 3.5. [ ]
Defining the conjugate by (2.75) we can now close this section with the massive analog of Lemma 3.4.
LEMMA 3.8. The operator D+, defined by (3.76), is an integral operator with kernel
D+(p, q ) = - , ( 1 +~p)1/2(1 +~q) 1/2 x
-i(p-q), N[O(P - q) e-'(P-q) 0 × IV* e (3. 1 O2) 0 19(q - p) e -'(q-P)/
40 A. L. CAREY AND S. N. M. RUIJSENAARS
and satisfies the eslimate (3.78). The conjugate of U satisfies the bounds (3.84)- (3.86).
Proof. Again, diagonalization of U through M reduces this to the claims made in Lemma 3.6. [ ]
4. The Fock Space Gauge Groups Hs and 1-ls,o
4.1. THE GROUP U(N) FOR m = 0
In this Subsection and Subsection 4.3 we study the local gauge groups IIs and IIs,o for charged Dirac fermions with internal symmetry space C N. Thus, the relevant Fock space is ~ ( ~ ) , with ~ ~ L2(R, dp) 2 (~)C N, cf. Subsections 2.2 and 3.3. The free massless charged Dirac field is then a C2N-valued quadratic form on the dense subspace @~ of ~a(~) , whose components are defined by
Smearing • leads to the field operator (I) of Section 2 in the sense that
I dx~(x) • ~(t , x) = (I)(ei'HW-lg)
= exp[it dF(H)]~(W- 'g) exp[-i t d['(H)], Vg c ~. (4.2)
(To verify this, use (3.5) and (3.8).) Thus, the time dependence of ~ is governed by the positive 'second-quantized' Dirac Hamiltonian
d~(H) = dr(H++) - dr(H__), (4.3)
which is usually written
dI'(H) = f dpEp[a*(p)" a(p) + b*(p)" b(p)] (4.4)
in the physics literature. For later reference we also note the relation
~¢~tq.j(0, x)~¢ = sXt,*j(0, x),
where ~* denotes the form adjoint of • and c¢ is conjugation operator, cf. (2.42). Moreover,
~ s , j ( 0 , x )~ = ,It_~,j(0, -x) ,
where ~ denotes the Fock space parity operator
~ F(P) = r (e ) ,
of. (3.16), (3.20) and (2.40).
(4.5)
the Fock space charge
(4.6)
(4.7)
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 41
A further comment concerning the precise meaning of the above equations may be in order to assist readers who are not accustomed to unsmeared creators and annihilators. As explained at the end of Subsection 2.1, these objects are well-defined quadratic forms on ~ . Evaluation of a normal-ordered product on F, G ~ ~ leads to a C~-function, on which a given distribution (in particular, an L~oc-function) can then be evaluated. The result of this is a new form on ~ , which in some cases, e.g., (4.2) and (4.4), is the form of an operator whose domain includes ~ . However, ~ , j ( t , x) is not the form of an operator, so that (4.5), e.g., denotes equality of forms on ~ .
The representations IIs of LeU(N) on ~ ( ~ ) are by definition given by
II~(u) -- ['(Try(u)), u • L,U(N). (4.8)
In view of (3.41) we can and shall fix the arbitrary phase on H_ in terms of that on II+. That is, we set
II_(u)-= ~I I+(u)~ . (4.9)
The phase choice on II+ will be discussed shortly. By construction, we now have
H~(u)~(f) (*) = ~P(~r~(u)f)(*)IIs(u), (4.10)
so that
[IIs, ~(f)(*)] = O, V f c W -s - q _ ~ . (4.11)
(To see this, recall that ~'s(u) acts like the identity on ~_s~, cf. (3.21), (3.39) and (3.40).) In view of (3.24) it follows that we are in the situation described abstractly in Subsection 2.5, Case 1.
As we have seen in Subsection 3.3, LeU(N)o is generated by loops of the form e i~(x), where a is a self-adjoint matrix with entries in HI(R). Thus, the Fock space representations II~,o are generated by
exp[i dF(A~)] = IIs(e'~), a = a* c Leu(N) (4.12)
This relation fixes the phase unambiguously for loops in the identity neighbor- hood W; note that (4.12) is consistent with (4.9), since d['(A_) = ~ dF(A+)~. For loops not in W there appears to be no natural phase choice. In fact, it is not even clear whether the phase can be fixed for loops in the range of the exponential mapping: (4.12) may be ambiguous, since e ~ = e it3 does not entail a =/3 when the loop eigenvalues do not stay away from -1 . However, this state of affairs plays no role in the sequel.
The key result enabling us to elucidate the structure of the representations Hs and IIs,o is Lemma 4.1 below, for which we shall now prepare the ground.
First, we introduce the charge shifts
~/s,i,e(r)-IIs(o'~,j), j = 1 . . . . , N (4,13)
cf. (3.47). Combining Lemma 3.1 with Subsection 2.3 it follows that we may
For brevity we have suppressed several indices at the r.h.s. (as in Subsection 3.4). Of course, the arguments are assumed to correspond to the first slot of ~ - L2(R, d p ) 2 0 ' ' ' (~)L2(R, dp) 2. We can then fix the phase of °ll +.j,,( r) by using (4.14) and the obvious transposition on C N. Finally, for s = - the phase is fixed through (4.9), so that
~_j . , (r ) = ~°R+,i,,(r)~. (4.15)
It is important to note that °lls4,~(r) is obtained from ~ls.j.,(O) through translation:
Indeed, if s = +, (2,83) may be used, since then (2.79) and (2.82) are satisfied, el. (3.60) and (3.61). If s = - , this follows by using (4.18). (Note that c¢~= ~CgF(-a).) Now let y(x) be a function whose Fourier transform is in C~(R). Then the form integral
I dxy(x)~]*)(O, x) - ~(*)(- ~ (4.2O) = s , i ~ Y !
defines a bounded operator in view of (4.2). Also, the expression
~ drT(r)W~ ,(0, r) (*) = ~(*) t^ ~ (4.21) • j , - - s , j , E , , Y !
is well defined as a strong improper Riemann integral, since ~ j , , ( 0 , r) (*) is strongly continuous in r and has n o r m (4"tre) -O/2). Hence,
11 s,j,,('Y)ll ~ (4'rrE) -('/2) d r i y ( r ) l . (4.22)
We shall call the operators ~. j , , (0 , r) (*) approximate Dirac fields. The
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS
justification for this terminology now follows
L E M M A 4.1. For any F E ~ t ,o one has
43
s ' lim $~.~)~(?)F - $(*)t a F - , - s,ixrt , s = + , j = I , . . . , N . (4.23) E--*0
This lemma will be proved in Subsection 4.5. Here we only remark it easily follows from Lemmas 3.1 and 3.2 that ~+. j , j0 , r ) * - ~ $ 4 ( 0 , r) for e ~ 0 in the sense of forms on 5~a,,o. Thus, ~+,i,,(0, r ) - - ~ + # ( 0 , r), too, and using parity, $ _ i , j 0 , r) (*)~$(*)~0-,i, , r). However , this topology would be far too weak to justify the conclusions that we shall draw from (4.23). The convergence (4.23) will be 'just strong enough' for these conclusions, and we believe it is in essence
xlt (.) , best possible: It appears likely that ,.i,,,L'Y) gets unbounded as E ~ 0. If this is true, strong convergence cannot hold for all F E ~ a ( ~ ) . Moreover, (4.23) may well be false, when ~ does not have compact support. (The C~-restriction is inessential, however.)
Using notation explained in Subsection 2.5, we shall first consider IIs on ~a(~s) . From Subsection 2.3 we know that IIs,o leaves the sectors -~,,s invariant. Thus we get a sequence of representations i-I(-)~1s.o of L~U(N)o. In Subsection 5,1 we shall see that these representations are not unitarily equivalent. Here, we restrict ourselves to proving that the representations IIs and II~"2 are irreducible.
T H E O R E M 4.2. The representation IIs of L,U(N) on ~ (~(~) is irreducible. The representations r~(.) xxO,s of LeU(N)o on ~n,s are irreducible, and
II '')'= ~'(~") 5f(,~.,s), Vn E Z. (4.24) O,s
Proof. To prove the first assertion, let A E II'~. Then A also commutes with F ( - 0 ) ~ s 4 , j y ) and its adjoint. Now consider vectors of the form
f i (F ( - 0)~ ~,j,.,, (3,~))(*), fl, (4.25) i = 1
where (*)~ signifies that either the operator or its adjoint occurs, depending on i. By Lemma 4.1 we can successively send e , , , . . . , ex to 0, which clearly leads to a set of vectors that is total in ~(Ygs). Hence, vectors of the form (4.25) are total as well. Putting
U(t) - exp[it d [ ' (n ) ] (4.26)
( , ) • _ ~ , / - {F(-I) ' t 's.j , j~,) I 1 - 1 . . . . . N, e > 0, ~ e Co(R)}, (4.27)
one readily verifies that the assumptions of Lemma 2.9 are satisfied. (To see that M is invariant under U(t), note that H equals sP ~ on Y(~ and recall (4.16).) Thus it follows that A = cO, proving irreducibility of [Is.
If we now take m = 2n and n *'s in (4.25), we get a total set in ,~0,s • Another r r (0) application of Lemma 2.9 then proves that Us,o acts irreducibly on ~o,s. Since
4 4 A . L . CAREY AND S. N. M. RUIJSENAARS
operators of the form dp(f)*dp(g),f, g e ~s, are left invariant by U(t), a third application of Lemma 2.9 entails ~o )= ~(,~o.s). Thus, (4.24) follows for n = 0. Moreover, by the uniqueness of implementers up to phase factors, the operators in q/~,l,l(0)*"II~"~q/~,m(0)" can differ from the operators in rl(m_~,o only by phases. Thus, II~"d is irreducible. Finally, if A ~ ~(ff,,~) commutes with ~(f)*~(g), then 0//~,l,l(0).-Aa//~,la(0), c ~(ffo,~) commutes with ~(U~aa(0)*"f)*~(U~.m(0)*~g), V f, g ~ ~ . Hence, A -- d , so that (4.24) follows. [ ]
Let us now return to ~ , (N) , Again, the notation used in the next theorem is explained in Subsection 2.5.
THEOREM 4.3. The representations IIs of L~U(N) on ~,(YO are primary and generate type I~ factors. Specifically, one has
P t I I , - ~ , (4.28)
where ~ is given by (2.98). The representations II~"~ of LeU(N)o on the chiral charge sectors ,~ . . . . . with n_ + n+ = n are primary and generate type I~ factors. Specifically, one has
II~"~" = II~_"~)io = tg~ ") (4.29)
on ~ . . . . . where 6~ ") is given by (2.99).
Proof. These assertions are straightforward consequences of the previous theorem and the isomorphism (2.103). [ ]
4.2. THE GROUP SO(N) FOR m = 0
This subsection and Subsection 4,4 are devoted to a study of the gauge groups II~ =-F(Trs) and their identity components IIs,0 for neutral Dirac fermions with state space o%a(N+), where Y(+= L2(R, d p ) @ C N and N > 2 , cf. Subsections 2.6 and 3.3. (The SO(2) case is unitarily equivalent to the U(1) case. This will be shown in Subsection 5.2, where more generally, the connection between the U(N) case and the SO(2N) case is discussed.) The treatment proceeds along the same lines as in Subsections 4.1 and 4.3.
First, the free massless neutral Dirac field is given by (4.1) with a---> c and b*---> c*. The analogs of (4.2)-(4.4) are then
l dxg(x). ~(t, x) = B(ei'HW-lg)
= exp[it dI'(H)]B( W -1 g) exp[- i t dF(H)], Vge ~ , (4.30)
a, here
d['(H) = dF(H÷+) = I dpEpc*(p)" c(p). (4.31)
For neutral particles, the Fock space charge conjugation operator equals the
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 45
identity, so that there is no analog of (4.5). As regards parity, the F-operation is only defined for unitaries commuting with C (cf. Subsection 2.6), so that it is not defined for P, cf. (3.15) and (3.16). However, iP does commute with C, so that we may set
=- f'( ie) = F(iP++). (4.32)
(We could have defined parity this way in the charged case as well. However, for charged Dirac particles it is possible and customary to choose ~9 self-adjoint, as we have done. In contrast, no self-adjoint parity operator exists for Majorana particles.) With this definition one obtains
Nq~s.j(0,x)~* = - i~-s , i (0 , -x) , (4.33)
which should be compared with (4.6). The analogs of (4.8), (4.9) and (4.11) are
II,(u) ~- F(Tr~(u)), u c LeSO(N), (4.34)
II_(u) = 3~II+(u)~ *, (4.35)
[II~, B(/)] = 0, Vfc yg-s, (4.36)
resp. Here, the identity component H~.o is generated by the unitaries exp[i dF(A~)] = II~(ei'~), where a = a* E Leso(N), cf. Subsection 3.3.
The unitaries
~lls,ik, ,(r) =- 1Is (o~4k) (4.37)
will play the role of the charge shifts (4.13). Lemma 3.3 and Subsection 2.6 imply that we can take
~s.~k,,(O, r)~* = - i~_, , jk . , (0 , - r ) . (4.41)
Finally, we define the bounded operators
~,*j(3') = ~ dx 34x)~*j(O, x) (4.42)
46
and
A. L. CAREY AND S. N. M. RUIJSENAARS
* I qr~4k.,( y) ----- dry(r)~. ik . , (0 , r)*, (4.43)
where ~ c C~(R). The following lemma exhibits the relation between these field operators and plays the role of Lemma 4.1 in the neutral context. It will be proved in Subsection 4.5.
Let us first consider II~ on ~ ( ~ - ) , cf. Subsection 2.6. We denote by II~,~d the restriction of l-ls,o to the sectors ~, ,s , n c Z2.
T H E O R E M 4.5. The representation IIs of LeSO(N), N ~ > 2, on ~(g(~_) is irre- ducible. For N > 2 the representations ~O.sWrl(n)"¢ LeSO(N)o o n g'~,t.s are irreducible and
are total in f fa (~- ) . Indeed, for e . . . . . . el--> 0 the vector (4.46) has the strong limit 2"( ) , . i , . - l) /21-ILl * - ~,n-,(Y~)O, and such vectors are obviously total. Thus, setting
Lemma 2.9 applies and irreducibility of [Is follows Taking m even in (4.46) leads to a total set in ~ ~o,s. Also, for m even and
,, n(o) is N > 2 , the operator acting on ~ in (4.46) belongs to IIs.o. Hence . . . . . o 072 irreducible on 5%,s for N > 2 by Lemma 2.9. Using Lemma 2.9 once more with
_ { B ( f ) B ( g ) I f , g~ :~s}, one gets tF] °) = ~(~o,~) and hence (4.45) for n = 0. Employing the shift q/s,12,1(0) as in the proof of Theorem 4.2, the remaining assertions follow. [ ]
Finally, let us present the analog of Theorem 4.3.
T H E O R E M 4.6. The representations I]s of LeSO(N), N ~ > 2, on ~,(~(÷), generate type I® factors. The representations II~.d of LeSO(N)o, N > 2, on the sectors ~ . . . .
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 47
with n_ + n+ = n (rood 2) generate type L factors. Moreover, (4.28) and (4.29) hold true, where ~ and 6~ ") are given by (2.135) and (2.136), resp.
Proof. This follows from Theorem 4.5 and the isomorphism (2.141). [ ]
4.3. THE GROUP U(N) FOR rn > 0
The study of the massive case undertaken in this section and the next one runs parallel to the treatment of the m = 0 case presented in the previous subsections. The massive analog of (4.1) reads
--- x-(l/2) sp 11/2 *s.i(t,x)=-(27r) I dp[~+2Epj [a,(p)eiPX-'E,/ + sb*(p)e-ipx+iE,/],
s = + , - , j = l . . . . . N. (4.49)
Using Subsection 3.1 it then follows that (4.2)-(4.7) hold true again. We define II~ through (4.8) and ensure unitary equivalence of II_ and II+ through (4.9). Then (4.10)-(4.12) follow, too. However, here we are dealing with Case 2 of Sub- section 2.5 in view of (3.23).
Using the notation (4.13) we now have as the analog of (4.14) the formula
Indeed, this follows from Lemma 3.5 and Subsection 2.3. We continue by taking over the phase fixing for °-tts,i,,(r) and the definitions (4.17), (4.20) and (4.21) from the massless case. Then it is readily verified that (4.16), (4.18), (4.19) and (4.22) hold again. The next lemma is the analog of Lemma 4.1 and shows that the operators ~, j , , (0 , r) ~*) may be regarded as approximate massive Dirac fields.
LEMMA 4.7. For any F ~ ~at,o the relations (4.23) hold.
We shall prove this lemma in Subsection 4.5. The remark made below Lemma 4.1 applies here, too. However, for m > 0 no chiral charges exist. Indeed, it is clear from (3.23) that the off-diagonal parts of q~ are not even compact. Thus, there is no analog of Theorem 4.2. In fact, we shall presently see that II(,,)~1) is total in ~,(Y(), in contrast to II(o)~II. We again denote the representations of L~U(N)o on the charge-n sector ft , by rw,),,,,o. We are now ready for the main result of this section.
T H E O R E M 4.8. The representations IIs of LeU(N) on ~a( ~ a) are primary. Their bicommutants are hyperfinite type III1 factors satisfying (4.28), where ~s is given by (2.98). The representations nt,) 11s,o of LeU(N)o on ~n are unitarily equivalent and primary. Their bicommutants are hyperfinite type IIIl factors satisfying (4.29), where ~7~") is given by (2.99).
48
Proof. Let A c II's. Then one has
(F, AF(-O)~s.i,,(7)G) = (~*,i.,(~/)F(-U)F, AG), VF, G e ~at,o. (4.51)
Using (4.2) it follows by linearity and continuity that
[ a , F(-I)@(f)] = 0, Vfe Y(s. (4.53)
Similarly, it follows that A commutes with @(f)*F(-0), VfE ~ . Thus, A ~ N'~, cf. (2.98). Using (2.108) we infer II'~ C ~_~. But (4.11) implies s¢_~ C II'~, so that II', = sC-s. Hence, II~ = s~'_~ = N~ by (2.108), proving (4.28).
Now from (4.28) one concludes by a standard density argument that any II~(u) is a strong limit of a sequence of polynomials in @(/~)*F(-n) and F(-0)@(gi), f~, gi e N~. When u e L~U(N)o, II,(u) leaves the charge sectors invariant, so that one may take polynomials in @(f~)*~(gi). This implies II~"dcff~ "), and, moreover,
0//_~,,,t(0)*"I-l~(u)0//_~,l,l(0)" = II~(U), VU E L~U(N)o, (4.54)
since U_s.~,~(0)* acts like the identity on ~ . Conversely, since the operator qb(f)*~(g) is in ~ when f, g c ~s, it follows from (4.28) that it is a strong limit of a sequence of polynomials in elements from II~. But ~,orT~") may be written as F, FI,F,, where F, denotes the projection on ,~.. Hence, one also has (~(n) ,-- ~(n)" rl(~)" -- ~(n) ~ ~,,.o , so that 1 . 1 S , O - -
From (4.54) the asserted unitary equivalence of n~,) rT~o) • ,~,0 and ~,~.o can be read off, and the remaining assertions then follow from Section 2E. [ ]
A. L. C A R E Y AND S. N. M. RUIJSENAARS
4.4. THE GROUP SO(N) FOR m > 0
The massive Dirac-Majorana field is given by (4.49) with a ~ c and b*---~ c*. Then (4.30) and (4.31) hold true again. Taking over the definitions in Subsection 4.2, all relations preceding Lemma 4.4 follow as before, except (4.38), whose analog is
°Y+.lZ,~(r ) = det(1 + Z+_*Z+_) -1/4 x
x [c*(e) exp(½Z+_c*c*)F(Z++) exp(-½Z_+cc) +
+ exp(½Z+_c*c*)F(Z++) exp(-1Z_+cc)c(e')], (4.55)
cf. Lemma 3.7 and Subsection 2.6. We are now prepared for the massive analog of Lemma 4.4.
LEMMA 4.9. For any F ~ ~at.O the relations (4.44) hold.
This lemma is proved in Subsection 4.5. Denoting the representations of
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 49
L~SO(N)o, N > 2, on the two sectors ~ , by '~t") ~ , o , we are in a position to state and
prove the following theorem.
T H E O R E M 4.10. The representations II~ of L¢SO(N), N ~ 2 , on ~ ( ~ + ) are primary. Their bicommutants are hyperfinite type llI~ factors satisfying (4.28), where ~ is given by (2.135). The representations ,.~,orl~ of L~SO(N)o, N > 2, on ~ , are unitarily equivalent and primary. Their bicommutants are hyperfinite type III~ factors satisfying (4.29), where ~") is given by (2.136).
Proof. This follows in the same way as Theorem 4.8, Lemma 4.9 playing the role of Lemma 4.7. The asserted unitary equivalence follows from
~-~,t2,t(0)*l-ls(u)°//-s,12,1(0) = Hs(u), VU ~ L~SO(N)o, (4.56)
which is the analog of (4.54). [ ]
4 .5 . C O N V E R G E N C E P R O O F S
This Subsection contains the proofs of Lemmas 4.1, 4.4, 4.7, and 4.9.
Proof of Lemma 4.1. To ease the notation we take N = 1 and correspondingly omit the index j; the general case can be handled similarly, as will be obvious from what follows.
First, let us note that we need only show that
s - l im ~ * ~ ( T ) F - - ~ * ( T ) F , V F ~ ~at.,, (4.57)
for any y with ~, ~ C~. Indeed, the charge conjugation and parity operators leave @at,O invariant, so that the three remaining relations follow from (4.57) by using (4.5), (4.6) and (4.18), (4.19). By linearity we can also restrict ourselves to proving (4.57) for a vector of the form
M N
F = I-I a*(g,) [ I b*(hj)lI, g, ~ Y(+, h~ c ~_, (4.58) i = t i = t
where the supports of g~ and hj are compact. Consider first the contribution of the annihilation part
~*.,(T) .... F = (47re) -'/2 I drT(r)F(- U++)F(- O__)b(~_)F(-I)F. (4.59)
Using the C A R to get rid of the b(~_)-term, one obtains M terms, a typical one being the first one:
I M N (-)M+l(4¢r)-'~2 drT(r)(E-'~2e_, h,) [I a*(U++g,) l-I b*(O__hj)ll. (4.60) i = l i = 2
From the definition (3.59) of e_ one infers that the inner product term is bounded uniformly in r and e, since h~ has compact SUlal~O. ft. Writing U~ as P~ + D~ and
5 0 A . L . C A R E Y AND S. N. M. RUIJSENAARS
expanding, it follows from the bounds (3.64), (3.65) that all terms containing at least one D8 converge strongly to 0 for ,--*0. (Recall Ila*tg)ll--Ilgl[, e.g.) The remaining term is the first contribution to
(4"n'E) -1/2 ~ dr'r(r)b(-~_)F -- b(P_f,)F, (4.61)
where
cf. (3.59), (3.5) and (3.8). Clearly,
s-limf~ = W-t (0 ), (4.63) E~'>0
so that the strong limit of (4.61) exists. Now from (4.2) it follows that
~*(T) =@( W-I (0))* , (4.64)
so that this limit equals ~*(7)a,, .F in view of (2.41). Secondly, consider
I M N ~*,~(7)cr.f= (4w,) -~/2 drT(r)a*(e+) I-[ a*(U++g,) 11 b*(Cl__hj)lL (4.65) i=1 j=l
Telescoping U~8 and expanding, it follows from the bounds (3.64), (3.65) that all terms with at least one D_ or more than one D+ strongly converge to zero. Let us therefore consider the M terms with one D+ and no D_, e.g., the term
M M
(47r')-1/2 1 dr'y(r)a*(e+)a*(D+gO [I a*(gi) I I b*(hi)fL (4.66) i = 2 j=1
Using notation explained in Subsection 2.1, the operator in front of the products can be written K,a*a*, where, using (3.59) and (3.63),
Since ~, and gl have compact support, H(p, q, e) has compact (p, q)-support not depending on , , and satisfies ]H] ~< C on R 2 x [0, l] (say). Hence,
I dp dqlK,(p, q)12 = 0(,2). (4.69)
Since K,a*a* acts on a finite particle vector, the bound (2.25) now implies that
F E R M I O N G A U G E G R O U P S , C U R R E N T A L G E B R A S , K A C - M O O D Y A L G E B R A S 51
the term (4.66) strongly converges to zero. Using the C A R to write the remaining M - 1 terms with only one D+ and no D_ into the form (4.66), it follows that they go to zero, too.
The only term remaining equals
(4'n'E)-'/2 I dry(r)a*(e+)F = a*(P+f,)F, (4.70)
cf. (4.62). In view of (4.63), (4.64) and (2.41) it strongly converges to ~*('r)cr F for E--* O, concluding the proof. [ ]
Proof of Lemma 4.4. We take N = 2, the general case being clear from this special case. Also, it suffices to prove
s . lim ~*,,z.,(3,) F = [xI,*,,(3,) + ixP'* 2(3,)] F E---~O
(4.71)
for a vector of the form
M
F : I-I c*(g~)l), g~ c x+, (4.72) i=1
since the corresponding results for ij = 21 and s = - then follow by using transposition and parity.
Treating the annihilation part
'l'I't~+,12,~('~) . . . . F = (47rE) -1/2 1 dry( r )F( - U++)c(e')F(-O)F (4.73)
as in the previous proof by using Lemma 3.4, one infers that this vector has the same limit as
f (47rE) -I/2 J dry(r)c(-e')F = c(CP_f,)F
where
(4.74)
f , ( p ) = l W - t l i i l l ( p ) e ,Ipl.
Since
xP'+,l('y) + i~*,2(~y) = B
(4.75)
(4.76)
52 A . L . CAREY AND S. N. M. RUIJSENAARS
by virtue of (4.30), it follows from (2.110) that (4.74) has the strong limit [XI?+, l (Y) + ixI**,2( Y)] . . . . F .
Using Lemma 3.4 once more, we can handle
xIf~+,12,e(Y)cr.F = (4"?TE) I/2 I dry(r)c*(e)F(- U++)F(-~)F (4.77)
as in the proof of Lemma 4.1, too. The analog of (4.67) is then
K~(p, q ) = 2e e ~p [ ; dkO(p)®(q)~,(p+ q - k)2-'/2(li)@
@ N(O(q - k) e -e (q-k ) 0 ) N* (gl,l(k)~ (4.78)
0 O(k - q) e ,(k-q)j \gl,2(k)]"
Due to the support restrictions the four integrals define functions with the same properties as the function H of (4.68), so that IlK, If2 = O(e). This reduces us to considering
(4~'e) 1/2 1 dry(r)c*(e)F = c*(P+fe)F. (4.79)
Using (4.76) once more, the claim now follows. [ ]
Proof of Lemma 4.7. Arguing as in the proof of Lemma 4.1 we conclude that it is again enough to prove (4.57) for a vector of the form (4.58).
The analog of (4.59) is
XI/'~,~(Y) . . . . F = ( 4 r y E ) - I / 2 I dry(r) det( l + Z+ *Z+ )--1/2 X
x exp(Z+_a* b*)F(Z++)F(Z r ) exp( - Z_+ba)b(&)F(- U) F. (4.80)
We begin by proving that the pure creation part may be telescoped away. This is a consequence of the bound
[l(exp(Z+_a* b*) -U)~II = O(E3/4), (4.8 l)
which we shall presently prove. To see this, note that the five operators preceding the pure creation part give rise to a finite sum of algebraic tensors, each of which has a norm that remains bounded as E--+0. Thus, pushing the creation part through the finitely many creators of each term till it acts on the vacuum, and then using (4.81) and the fact that the determinantal factor is bounded above by one, it follows that we may replace the creation part by D.
Let us now prove (4.81). The square of the l.h.s, equals
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 53
from which (4.81) follows. Here, we used (3.84) in the last step and the estimate det (l + T)~<exp(lITll~)in the second step; for the first one, cf., e.g., [12, Eq. (3.47)].
Now that the (unbounded) creation part is replaced by g, we may use the bound (3.84) twice more to replace the determinantal factor by 1 (the difference being O(E 3/2) in view of the estimate l - d e t ( l + T)-l/2<~ 1 -exp(-~l[TIl0), and to replace the annihilation part by n. Also, the bounds (3.85), (3.86) imply we may place Z++ by -U++ and Z_ T_ by -U__.
After these replacements we are in the same situation as in the proof of Lemma 4.1. Since the hounds (3.64), (3.65) again hold, we can argue in the same fashion and conclude that the only term in (4.80) remaining for e--->0 equals (4.61), where now
cf. (3.8(I) and (3.5), (3.6). Now it is easy to verify that
lim C,E - in = 1, (4.84)
so that (4.63) holds again, implying the strong limit of (4.80) exists and equals
**(3/) .... F. Proceeding in the same manner for the creation part, we can again follow the
proof of Lemma 4.1. As the analog of (4.67) we obtain, using (3.80) and (3.83),
Comparison with (3.26) and inspection of (3.33) now leads to the conclusion that the kernel at the r.h.s, defines a HS operator. Using (4.84), it then follows that IlK& -- 0(~). Thus, we are left with the term (4.70), where f, is defined by (4.83). Using once more (4.63) and (4.84), the lemma follows. [ ]
Proof of Lemma 4.9. The proof is similar to the previous one, Lemmas 3.7 and 3.8 playing the role of Lemmas 3.5 and 3.6. We therefore restrict ourselves to mentioning the analog of (4.82), as this may not be obvious. It reads
Here, the second and last step follow in the same way as for (4.82); for the first one, cf. [13, Eq(4.45)]. [ ]
5. Further Developments
5.1. WEYL ALGEBRAS AND PROJECTIVE MULTIPLIERS
As mentioned before, the m = 0 representations of the previous chapter are intimately related to the loop group representations of [4-6] and the Kac-Moody algebra representations of [7-11]. (The connection can easily be made via the Cayley transform, cf. also Subsection 7.2.) In this work a prominent role is played by Weyl algebras (or, in another dialect, Heisenberg groups). In this Subsection we make contact with this picture. This will also enable us to determine cocycles and resolve some further issues. We restrict our attention to the U(N) case, since the relevant information for the neutral case can then be obtained by invoking the isomorphism sketched in the next Subsection.
Let us first take N = 1. Then from Proposition 2.2 and Subsection 3.2 it follows that
8ss' f dkk&(-k)l~(k), Va,/3c Hi(R). (5.1) [d[~(As), dI'(Bs.)] =
(To verify this for m > 0, use the transformations indicated in (3.33).) Thus, taking a,/3 real, Proposition 2.3 implies
II~(ei~)l-Is(e 't3) = e(-i/2)P(~'a)II,(ei(~+a)), Vot,/3 e V (5.2)
where
p(c~,/3)-----~ I dxot'(x)/3(x). (5.3)
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 55
Since p is a nondegenerate symplectic form on the real vector space V, it follows that II~,o gives rise to a representation of the Weyl algebra (equivalently: represents the Heisenberg group) corresponding to (V, p).
As pointed out and exploited by Carey, Hurst and O'Brien [39] (who were studying the m = 0 case using a subspace of V), there is a simple way to turn V into a complex pre-Hilbert space (V, ( . , .)), such that the above representations may be viewed as C C R representations (in this connection, cf. [44, especially p. 19 of Vol. II]). Specifically, the required complex structure is given by oL---> a c, where o~ c is the conjugate function of a • V. That is, a ~ is the real-valued Hj(R) function given by
i&(p), p > O, &C(p) = (- i&(P), P < 0. (5.4)
The corresponding inner product is then
(or,/3) = p (a , /3c) + ip(ot, fl)
I; 1 dkk&(k)13(k), Va, /3 • V, (5.5) 71"
and the corresponding boson creation and annihilation operators are
c,(a) = 2-'/2[d[~(A~) + i df'(A~)],
c * ( a ) 2-1/2[d[ '(As)- i dF(A§)]. (5.6)
As promised, the c~ *) satisfy the C C R over (V, (-, .)),
= [c*, = o} [c~(a), c* (/3)] = (a, /3) _ Va, /3 • V. (5.7)
In the case m = 0, the relation
supp & c ( -~ , 0] ¢:> dF(A(0)~)12 = 0, (5.8)
which follows from (3.31) and (2.49), implies that
C(o)s(a)12 = O, Y a • V. (5.9)
Thus, on the vacuum sector one is dealing with the Fock representation of the CCR, so that the generating functional
E(~) --- (ft, IL(e'~)ft) (5.10)
satisfies
E(o)(a) = exp(-¼(a, a)). (5.1 I)
In the massive case, no nontrivial linear combination of c's and c*'s can annihilate 12, as is clear from (3.26) and (3.27). In fact, it is not difficult to verify that the C C R representation on ~0 is not quasi-free. It would be of considerable
56 A.L. CAREY AND S. N. M. RUIJSENAARS
interest to obtain more information on /5(,,) (a); cf. in this connection the first question below Theorem 5.1 in Subsection 5.3, and the discussion below Pro- position 8.3 in Subsection 8.1.
To study other sectors, let us use the notation
°Us = q&,,,,(O), (5 .12)
cf. (4.14) and (4.50), and let us put
E(")(a) -- (a//~l), I]~(ei'~)q/~l)), n c Z, a E V. (5.13)
By virtue of Proposition 2.4 one then has
E(")(a) = e x p [ i n ( ~ l ) , dF(As) ~ l ) ) ]E(°) (a)
=- exp[in~(a)]E(°)(a). (5.14)
(Recall II+ and II_ are unitarily equivalent.) Using (4.14), (2.49) and (3.59), one gets
Similarly, (4.50) leads to an expression for sr(,,)(a). One can argue heuristically that this expression should be equal to the r.h.s, of (5.15), i.e., that ~" should be mass-independent, cf. (6.9), (6,10) below. However, it appears not easy to confirm or refute this.
Let us now complete the picture as regards the projective multipliers arising for the LeU(1) representations IIs with m >/0. First, the cocycle on Hs can be read off from the formula
k lot 1 i/3 a//~IIs(e )°//~l-I~(e ) (5.16)
= exP[2 p(a,/3) + il~(a)]°ll~+'IL(ei(C'+t3)),
which follows on combining (5.2), Proposition 2.4 and the definition (5.14) of ~'. Secondly, a simple algebraic argument yields the formula
where w denotes the winding number. In particular, 6//+ and o-//_ anticommute. Indeed, w(u) equals the charge change q(m(u)) effected by H~(u), cf. Subsection 2.3. Thus, since 1-Is(u)~ ~¢'~ = ~ (cf. Subsection 2.5), it can be reached as a strong limit of a sequence of polynomials in F(-U)~(~) (*), j~ c ygs, which are even/odd for w(u) even/odd. Hence, (5.17) follows.
Before turning to the U(N) case, let us make one more remark on the massless U(I) case. Since the function e-k/k is not in L2([0, oo), k dk), there exists no oq in
FERMION GAUGE GROUPS, CURRENT AtGEBRAS, KAC-MOODY ALGEBRAS 57
the completion of V w.r.t. ( - , - ) such that K(o)(a)= Im(a~, a), Va • V. By well-known arguments, this implies that the displaced Fock representations
El~))( a ) = e xp[ in~(o)( a ) ] exp(-¼(a, a)) (5.18)
are mutually unitarily inequivalent. Thus the representations -s,orT(") of LeU(1)o considered in Subsection 4.1 are unitarily inequivalent.
As mentioned in passing before Theorem 4.2, this holds true as well when N > 1. This fact will be obvious from the discussion of the U(N) case on which we now embark.
First of all, if we replace LeU(N) by its maximal torus
L e T N =- {u • LeU(N) [ uq = 0, i ~ j} (5.19)
in Theorems 4.2, 4.3 and 4.8, then they still hold true. Indeed, the charge shifts represent elements of LeT N, and an inspection of the proofs of these theorems shows that only products of shifts are involved. Put differently, the above L,U(N) representations belong to the strong closure of the linear hull of LeT N represen- tations. (In Subsection 8.3 we shall explain this extension property in a heuristic fashion.) Similarly, the representations II(~"~ of LeU(N)o may be viewed as extensions of representations of
LeTo N =- LET" ("1 LeU(N)o
= {u • L~TN[ w(det u) = 0}. (5.20)
We shall now determine the multipliers on these LeT N representations. To this end, let us first present the arbitrary N generalization of (5.1). It reads
[d['(A~), dF(~s,)]
= ~ , d['([A~, B~]) +
~ " ~ d k k 6 ( - k ) f l ( k ) , ot~j, + ~--~ Tr /3ij c H~(R). (5.21)
This formula follows in the same way as (5.1) from Proposition 2.2. From this one infers as before that the group
leads to representations of the CCR over V N with the obvious scalar product derived from (5.5). Setting
¢//~,i -= ~,j,,(O), j = 1 . . . . . N, (5.23)
(5.16) follows for each slot separately. Moreover, (5.17) holds again, by the same argument. A similar argument implies that °/t~,j and °g~,k anticommute for j ~ k. One may now fix the phase on the LeT N representations by writing each representer in the form
N
l-I °//~;iH~(Diag(e'~' . . . . . e'~'O), n j • Z , oq• V. (5.24) j=l
58 A. L. CAREY AND S. N. M. RUIJSENAARS
Then the above remarks suffice to obtain the generalization of (5.16). However, since the resulting formula is rather unwieldy, we shall omit it.
The Fock character of the CCR representation for m = 0 can also be exploited to elucidate the structure of the L~SU(N) representations for m = 0. For reasons of exposition we shall postpone this till Subsection 5.3, however (cf. Theorem 5.1).
5.2. U(N) VERSUS SO(2N)
In this subsection we shall clarify the relation between the U(N) and SO(2N) cases. Combined with the previous section, this correspondence leads to a 'Weyl algebra picture' for the latter case, which we shall not detail, however.
From the point of view of quantum field theory, this relation can be easily described. For N = 1 it amounts to the well-known fact that a charged field can be traded against two neutral ones. In terms of the Dirac fields of Section 4 this boils down to making the identification
in terms of charged and neutral creators. In smeared form, (5.26) leads to an isomorphism between the charged Fock
space ,~a(~) of the U(1) case and the neutral Fock space ,~a(~+) of the SO(2) case: The space ~ = L2(R, alp) 2 of the former case is identified with the space Y(+ = L2(R, dp)@C 2 of the latter case through the map
( ~ ) ~ N ( f g ) , N-= 2-°/2)(: 1 i ) , (5.27)
which leads to an identification f f~ (~ )= ffa(~+) through the isometric product operator F(N).
This isomorphism is such that
= e x P [ 4 ( 1 - s)w(eiV)] ×
~[,_, [ cos 7(') sin 3'(')]] ×l~llS~--siny(') cos 'y( ' ) / / ' e'~'CLeU(1)" (5.28)
To prove this, let us first note that (the inverse of) (5.26) entails
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS
Thus, one has
(I)(( UII - iU21) C f l -[- ( U22 -[- i U~ 2) Ci f 2) +
+ $(( U,, - iU2~) f~ + ( U22 + iU~ 2) ( - i[2))*
59
(5.30)
provided U commutes with charge conjugation. If, in addition, the operators U~I - iU2~ and U22 + iU~2 are equal and unitary, then this relation shows that the self-dual CAR automorphism generated by U amounts to a CAR automorphism generated by the unitary U ~ - i U 2 , . For the SO(2) gauge transformations considered by us, all provisos are clearly satisfied, so that the implementers correspond up to phase factors. Now it is straightforward to check that our conventions in Section 4 are such that these phases equal one when s = +. Since the neutral parity operator ~ , = f '(iP) corresponds to F(iP) = f'(i)~c, one gets a phase
[o ] exp i-~ w(e 'v) for s = - .
Thus, (5.28) follows. More generally, we may set
a *(p) ~- 2-'l/2)( c *j_,(p) + ic *~(p)) ] b*i~x~2-t l /2)~c* l ~ i c * l - ~ J j = 1 . . . . . N (5.31)
j ~ .VI-- \ 2 j - I \ F ] - - 2j\p)2
in the U(N) case. Then the analog of (5.30) implies that the representers of loops in U(N) are identified with the representers of loops in SO(2N), the cor- respondence between U(N) and SO(2N) amounting to a realification of C N. Specifically, the SO(2N) image of a given U(N) matrix is obtained by replacing each entry in the latter one by the 2 × 2 matrix
R e z I m p ) .
- Im z Re
5.3. SUBGROUPS OF II,,o
SO far, we have restricted our attention to groups of loops in U(N) and SO(N), and their identity components. However, the corresponding representations IIs and II~,o by restriction also lead to representations of loops in subgroups, and it is of interest to obtain more information on such restrictions. This subsection concludes the section with some results, questions and conjectures in this direction.
We shall begin by clarifying the structure of the LeSU(N) representations provided by (the restriction to SU(N) of) Hs,0 when m -- 0. More specifically, we restrict ourselves to the vacuum sector ~o,~ of ~a(~s); the state of affairs on other sectors and on ~ ( ~ ) readily follows from this.
60
First, let us note the equality
Hs.,, = I I s ( L ~ S U ( N ) ) I I d L e D ( N ) )
where
LeD(N)-= {u ~ GU(N)I u(.)= e~"(~, a c V}.
Secondly, the commutation relations (5.21) are easily seen to entail
[[l,(u3, l-Uug] = 0, u~ e LeSU(N),
whence one has
II~(LeSU(N))" C II,(LeD(N))'.
A. L. CAREY AND S. N. M. RUIJSENAARS
(5.32)
(5.33)
THEOREM 5.1. The mapping
l:lls(ul)IL(u2)~ IIs(U0~ ® IL(u2)~,
ul ~ L e T N-I, u2 c LeD(N)
extends to an isomorphism
such that
nAGSU(N))" = ~(~ '~) ® D,
II~(LeD(N)) " ~ ~ @ ~0,(~7)).
Proof. We begin by noting Lemma 2.9 implies
I-Is(LeTN-I) "= ~97(.~ 1)) (on o~1)),
II.dL, D(N))"= ~(~2) ) (on ,~2)).
To prove that
(5.39)
(5.40)
(5.41)
(5.42)
(5.43)
(5.44)
I may be extended by linearity and continuity to an isometric
The result we are about to present implies that C may be replaced by = in (5.35). It provides, however, much more information. Let us introduce the maximal torus
L e T N-' =- { u c L~SU(N) I u o = O, i 7 ~ j}, (5.36)
and recall from Subsection 5.1 that the representation of L e T N - I L e D ( N ) = LeTo N
(cf. (5.20)) is already irreducible in ffo.s. Next, we set
~ t ~ = L H I I s ( L e T N - 1 ) f k (5.37)
~ - - - LHII~(LeD(N))fL (5.38)
where LH stands for the closure of the linear hull. Then we are in the position to state and prove the following theorem, which reveals the simple structure of the LeSU(N) representation in the massless case.
(5.35)
u2 ~ LeD(N), (5.34)
FERMION G A U G E GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 61
operator from az , ~ @ ,~2), ,~o,:~ onto ~{') it is therefore enough to show
(IL(u,)IL(u2)~, IL(vOIL(v2)f~)
= (II~( Ul)11, II , (vl)~) (Ils (u2)11, II,(V2)11),
VUl, el ~ LeT N-l, VU2, V2 C LeD(N). (5.45)
Now observe that operators in [Is(LeT N-~) are of the form (5.24) with
N N
Z n , = 0 and Y. ~ j (x )=0 . j=l j=l
If the slot winding numbers n i of u~ and v~ differ, both sides of (5.45) vanish, since one is then dealing with two vectors having different slot charges (=- eigenvalues w.r.t, df'(Pj) with Pj the projection on slot j). If they are all equal, we can cancel the shifts on both sides, so that it remains to prove (5.45) for the case where u~, v~ have zero winding number in each slot. But then the four operators in (5.45) belong to the Weyl algebra, and since one is dealing with the Fock representation of the C C R over (V N, (., .)), one can explicitly evaluate the inner products, cf. (5.2) and (5.11). The factorization property then follows from the fact that the subspace of V ~ consisting of vectors of the form (a(x) . . . . . o~(x)) is orthogonal to the subspace of vectors of the form (al(x) . . . . . aN(x)) with ~=~ oq(x)= 0. Thus, (5.45) holds true.
It remains to prove (5.41), (5.42). But (5.42) is clearly from (5.44) and the definition of the isomorphism, so that (5.41) follows from (5.35) and (5.43). [ ]
For m > 0 the situation as regards SU(N) is much less clear to us. Of course, the inclusion (5.35) follows as before and from this we are able to conclude that the W*-algebras
"/g'~- II~(LeSU(N))" ]" ,~o (5.46)
are factors. Indeed, any central element of ~ commutes with II_,,o and also, due to (5.35) and (5.32), with II,,o. Hence, the factor property is a consequence of
Theorem 4.8.
However, we have not answered the following natural questions:
(1) Does the generating functional (5.111) of the massive C C R representation also have the factorization property (5.45), entailing a tensor product structure for :~o?
(2) If not, is it perhaps true that the subspaces
' ~ ) -= LHIls(LeSU(N))F~ (5.47)
are equal to ~o? (3) If these subspaces are smaller than ~o, is it at least true that they are
equal? (4) Assuming . ~ ) = ~ 9 ) ~ . ~ ( u is it true that the restrictions of °W+ and
to .~(J) are each other's commutants?
62 A.L. CAREY AND S. N. M. RUIJSENAARS
(5) Are the factors °tVs hyperfinite and type 1II1 ? (6) Does the modular automorphism group associated to (II",D) leave ~
invariant?
Of course, these questions are not independent. In particular, an affirmative answer to (6) would imply that the factors ~/,V~ are type 1111. We believe that the answer to questions (1) and (2) is no, and to the remaining questions yes.
Let us now briefly discuss subgroups of SU(N). Here, an important open problem appears to us to classify the subgroups G of SU(N) in relation to the 'size' of the subspaces
~( G)-=- LHIIs( L~G)IZ (5.48)
In particular, when is ,~s(G)= ?T~ ~--- ffs(SU(N))? We have one observation to offer in this connection: If G is a real subgroup, then ff*,(G) is a proper subspace of ff~ll. Indeed, I IdL , G) then commutes with charge conjugation, so that ~ ( G ) belongs to the fixed point space of ~, whereas ~!~) clearly does not.
Of course, similar questions can be asked in the neutral case concerning subgroups of SO(N). A natural one is, for instance: To ensure that ~ , (G) coincide with ~(~(~_) for m = 0 and with ~(gf+) for m > 0 , is it sufficient to require that G ~ SO(N) act irreducibly on cN? In this connection we note that the isomorphism described in the previous subsection shows that equality already holds for maximal tori in SO(N) with N even. Thus, an irreducible action on C N is not a necessary condition.
These size questions are particularly relevant in the massless case. As we shall see in Subsection 7.2, for m = 0 the irreducible projective representation of LeG on .~AG) constitutes, roughly speaking, the exponentiated version of the so- called basic representation of the affine Lie algebra corresponding to the Lie algebra of G (or tensor products thereof), provided G is simple and simply- connected. (if G is not simply-connected, one should replace L~G by its identity component.) Answers to the above questions might in particular yield rather precisely delineated representation spaces for the symplectic and exceptional affine Lie algebras, which we do not consider in Section 7B.
6. Applications to Field Theory
6.1. CURRENT ALGEBRA
In this subsection we establish the relation of the above to the picture of currents and gauge transformations that can be found in the physics literature. We shall begin by discussing the U(1) case. Then the components J~',/z = 0, 1, of the vector current corresponding to the free Dirac field are the quadratic forms
The relation of the latter to the j ' - o p e r a t i o n is given by
I dxct(sx)J~(t, x)
= d['(ei'HAs e -~,n)
= exp[it d[ '(H)] df~(As) exp [ - i t d[ '(H)], t r e H,(R). (6.3)
(As concerns the meaning of this formula, we refer to the discussion at the end of Subsection 2.1 and to the remark below (4.7). To verify it, use the definitions (4.1) and (2.49) of • and d[', and various results from Subsections 3.1 and 3.2.) By virtue of (6.3) one can rewrite the commutation relations (5.1) in unsmeared form as
[J ' ( t , x), J"( t , y)] = 0, /~ = 0, 1, (6.4)
[J°(t, x), j l ( t , y)] = 8'(x - y)/rri. (6.5)
If one does not worry about rigor, then a calculation using the unsmeared equal-time C A R
and the defining relations (6.1) leads to the conclusion that the l.h.s, of (6.5) vanishes. However , it was realized several decades ago that this conclusion contradicts some cherished principles in theoretical physics. The nonzero 'c- number' commutator was first pointed out by Goto and Imamura [51], and is generally known as the Schwinger term [52]. In the physics literature it is derived in a quite different fashion.
Applied to the case at hand, the relation
exp[it d[ ' (A)]~(f)* exp [ - i t d['(A)] = e0(e"af) * (6.7)
from Subsection 2.2 (cf. (2.51)) amounts to a precise version of the physicist's saying: 'The currents generate the local gauge transformations': This assertion is usually expressed through the formula
and is obtained by the same arguments that lead to the erroneous conclusion just mentioned. However , in this case (6.8) does hold true after smearing with appropriate test functions. Indeed, in view of (6.3) and (4.2) this leads to the infinitesimal form (2.56) of (6.7).
For loops e i'~ in LeU(l)0 one may write II~(e ~'~) as exp[i J dy a(sy)J,(O, y)] in view of (6.3). But the notation exp[ i fdy 'o ( sy)J , (0 , y)] for the charge shifts q/s = II~(e ~'7) is purely symbolical, since the form integral does not give rise to an
64 A.L. CAREY AND S. N. M. RUIJSENAARS
operator. (Indeed, neither for m = 0 nor for m > 0 the off-diagonal parts are HS.)
If one uses it nevertheless, and moreover formally applies the current algebra (6.4), (6.5) to it (as is customary in theoretical physics literature), one obtains, using (3.37),
IIAe~'~)°l/.~ = e~a~°//sIIs(e~) (?) (6.9)
where
~(a)=-l l dxa(x)/(l + x2). (6.10)
Comparison with (5.15) and (5.16) shows that (6.9) is correct for m = 0 even though its derivation is purely formal. As we have mentioned below (5.15), this leads to the conjecture that (6.9) holds true for m > 0, too.
In the U ( N ) case, the analog of (6.3) (with t = 0) reads
f dx a o E H~(R). (6.11) :xJYs*(() , X)OL(SX)X!Ys(O, x)" dF(As),
If one chooses a basis M 1 , . . . , MN2 and introduces the charged time-zero currents
JZk(0, x)--= :q~*(0, x)Mk~(O, x): k = 1 . . . . . N z (6.12)
one can write the commutation relations (5.21) in the symbolic form
[J.,C.,k(O, X), J's'.t(O, y)] N:
= a,,, ~, ckTJ~.,. (0, x) a ( x - y) + r e l = l
s + 2 7r--i a~, a ' ( x - y )Tr MkM~. (6.13)
Here and in the sequel, the structure constants corresponding to the Lie algebra basis are denoted by ck"], and the superscript c is used to avoid confusion with the neutral case, which we shall consider next.
which is the unsmeared form of (2.111), (2.112) (cf. (4.30)). Using (2.116) one obtains as the analog of (6.11)
I dx x)a(sx)qt~(O, x): a~ i e Hi(R). (6.16) :xlt.s*(() , 2 dF(A~),
At this point it is important to recall a property of the dF-operation that has played no explicit role so far. In Subsection 2.6 we have seen that dI'(A) is well
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 65
defined for any A e g 2 , just as is dI'(A). However, the map A ~ d F ( A ) is injective, whereas d F ( A ) = 0 whenever A * = CAC. For the above special case this can also be read off directly from the quadratic form q - : ~ * a ~ s " Indecd, using (6.14) and (6.15) one has .~ , i~ s .k : =--.W~,k~s,i SO that q = 0 when a = ot T. In particular, the neutral current (6.16) vanishes for N = l (in agreement with the physical picture). When N > 1 one may therefore just as well restrict a to be skew-symmetric, and we shall do this henceforth.
Let N1 . . . . . NNtN-~1)/2 be a basis for the skew-symmetric matrices in MN(C). We set
J~,k(0, x) = :~*(0, x)Nk~s(0, x):, k = 1 . . . . . N ( N - 1)/2 (6.17)
As before, this holds in the sense of operators on ~ if one smears appropriately. (Here, smearing leads to a special case of (2.119).) The commutation relations (2.131) then lead to the current algebra
[J," k(0, x), JL,(o, y)] N(N-1)/2
= 2a~s, Y~ c k"l,s",,,,,(o, x) a ( x - y)+ m=l
S +---: 6,,, 6'(x - y) Tr NkNi (6.19)
rn
which is the neutral analog of the charged current algebra (6.13). It is through formulas like (6.4), (6.5), (6.13) and (6.19) that current algebras
usually appear in the physics literature. For space-time dimension d > 2, smear- ing the free currents at time zero does not lead to operators (the HS condition is violated), so that there appears to be no obvious way to make analytical sense of such equal-time current commutation relations (in contrast to the commutation relations for the fields, since smearing the free time-zero Dirac fields gives rise to bounded operators for any d). Also, there seems to be no reason why the situation should be more favourable for interacting quantum field theories with d > 2. In spite of this, current algebras for interacting fermion fields have been used extensively and with great success in the physical case d = 4 (cf., e.g., [53] and references given there); The difference between the cases d = 2 and d > 2 just described is a precise version of the physicist's saying: 'The Schwinger term for free fermion currents is infinite for d > 2'.
6.2. THE SMEARED DIRAC CURRENTS
In this Subsection we prove three corollaries of our results in Section 4 that concern the time-zero currents smeared with arbitrary H~(R) functions. They
66 A . L . C A R E Y A N D S. N. M. R U I J S E N A A R S
correspond to Theorems 4.2, 4.3 and 4.8 in the charged case, and to Theorems 4.5, 4.6 and 4.10 in the neutral case. First, let us consider the massless case, restricting attention to the Fock spaces of right-moving (s = +) or left-moving ( s - - - ) particles. We denote by l-ln,s a fixed vector in the sector o~n,s of o~(~gs)/o~a(9~ s) in the charged/neutral case, which is obtained by acting on fl0,s -= ~ with a charge-n element of l'Is (one can take, e.g., lq,,s = °?/~,1,1(0)"~ and l~,.s = q/s,12,1(0)"l~, resp.).
T H E O R E M 6.1. Let m = O. Then the U(N) currents (6.12), smeared with Hi(R) functions, are well defined, cyclic and irreducible in ~n,s c ,~a(~(s) on II . . . . V n c Z . The smeared SO(N) currents (6.17) with N > 2 are well defined, cyclic and irreducible in o~,,s c ~ a ( ~ ) on ~ . . . . Vn ~Z2. Furthermore, the represen- tations on o~,,s of the smeared current algebras (6.13) and (6.19) are mutually unitarily inequivalent.
Proof. We only consider the charged case, since the proof for the neutral case is similar. Let us first take n = 0. Suppose F ~ ff0.s is orthogonal to the subspace of ~0.s obtained by acting on the vacuum with polynomials in the smeared currents. Then it follows from (6.11) that
N, N k ( i t l ) n ' . . ( i t k ) n ~ • .. ~ " (fL df~(As,~) ~' df~(A,,k)"~fl) = 0 (6.20)
. , = o . k = o n l ! n k ! " " "
where the smearing functions a~ giving rise to As,i (cf. (3.25)) are selfadjoint elements of Leu(N) (cf. (3.43)). If we now take 0 < tj < (2111A ,;III)-' we infer from the proof of Proposition 2.1 that we can send N ~ , . . . , Nk successively to 0% obtaining
SO that F = 0 by virtue of Theorem 4.2. Thus, cyclicity for n = 0 follows. Irreducibility in the vacuum sector is then a consequence of invariance under time translations and Lemma 2.9.
For n ~t 0 it is not immediate that ll,,s is in the domain of polynomials in the smeared currents. However, this is a straightforward consequence of Stone's theorem, cf. (2.95). The latter equation can also be used to reduce the case n ~ 0 to the case n = 0, as is easily seen.
The inequivalence of the representations follows from the fact that the dis- placement functions that arise are not in the completion of V N w.r.t. ( . , .), cf. Subsection 5.1. [ ]
Note that the smeared SO(2) current is not cyclic in the even and odd sectors. This follows from the isomorphism between the U(1) and SO(2) cases, cf, Subsection 5.2.
F E R M I O N G A U G E G R O U P S , C U R R E N T A L G E B R A S , K A C - M O O D Y A L G E B R A S 6 7
Secondly, we study the situation on the Fock spaces ffa(~i~) and .~a(~+) for charged and neutral particles of both chiralities, denoting by ~ . . . . a vector in
. . . . obtained by acting on f~o,o = ~ with a charge - n - element of H_ and a charge -n+ element of If+.
T H E O R E M 6.2. Let m = O. Then the smeared U(N) currents (6.12) with s = +
and s = - are cyclic and irreducible in ~ . . . . c ff%( ~() on ~ . . . . . V (n_ , n+) ~ Z 2. The smeared SO(N) currents (6.17) with N > 2 and s = + , - are cyclic and
irreducible in ~ . . . . c ~(Y(+) on ~ . . . . . V(n_,n+)~Z2. In both cases, the
smeared chiral currents with a .fixed value of s are not cyclic on gl . . . . in ~ . . . . . Proof. This readily follows from Theorem 6.1 by exploiting the isomorphisms
(2.103) and (2.141). [ ]
"Thirdly, we discuss the massive Dirac currents. In the following theorem, f~, denotes a vector in ~ , obtained from f~0-- f~ through a charge-n element of I L or H_.
T H E O R E M 6.3. Let m > O. Then the smeared U(N) currents (6.12) with s f ixed
are cyclic and reducible in ~ , c ~ ;o (~) on f l . , V n ~ Z . The smeared SO(N) cur-
rents (6.17) with N > 2 and s f ixed are cyclic and reducible in ~ . ~ ~a (~+) on
f~,, Vn ~ Z2. The representations on ~ , of the smeared current algebras (6.13) and
(6.19) are unitarily equivalent. Moreover, in both cases the vector currents
smeared with HI(R) functions, act irreducibly in the sectors ~ , .
Proof. The cyclicity claims are proved in the same way as in the prooi of Theorem 6.1; one needs only replace Theorems 4.2 and 4.5 by Theorems 4.8 and 4.10. Reducibility is obvious from the latter theorems.
To prove irreducibility of the time-zero currents J~ on ~o, assume that B ~ ~(,~o) commutes with J~,k, s -- +, - . Then it follows in particular that
N ~lit~ . , ~ (it)" F ~--o-~- (dF(A,) F, B G ) = ~,, ~ ( , B dF(A,)nG), (6.24)
n~O n .
where c~ c V, t~ = c~* ~ L,i u(N) and 0 < t < (2111A, III)-', and F, G are vectors obtained by acting on f~ with polynomials in the smeared currents. Taking N----> ~, it follows that [~3, H~(exp(ita))] = 0, s = +, - . Hence, B ~ (I1(+()i)o U ri<o)v • • - , 0 ! •
Thus it follows from Theorem 4.8 (specifically, from (4.29) and the factor • ~(o)~ that B = cl. Therefore the smeared time-zero currents J~ act property of ll~,o,
irreducibly on ~ in ~o. Irreducibility for the other sectors follows by exploiting (2.95) and its neutral
analog. [ ]
Finally, we wish to draw attention to a peculiar property of the chiral currents for m > 0. In contrast to the massless case, the massive time-zero currents are not left invariant under time translations (to see this, compare e.g. (6.3) and (3.26),
68 A.L. CAREY AND S. N. M. RUIJSENAARS
(3.27)). But, of course, the currents Js,k(t, X), t C R, are invariant. Thus we can use Lemma 2.9 to conclude that the smeared currents at arbitrary times with a fixed value for the chiral quantum number s are irreducible in the vacuum sector (and hence in the other sectors, too). This is a physically unexpected result that might be useful in other contexts.
6.3. BOSON-FERMION CORRESPONDENCE
From (5.6) and (6.3) it follows that the smeared fermion currents may be viewed as linear combinations of smeared boson creators and annihilators. This is one half of what is often called 'boson-fermion correspondence'. For m = 0 the simplest version of this correspondence dates back to a paper by Jordan on the neutrino theory of light [54]. His work was partly inspired by a physical picture: A massless fermion-antifermion pair in two spacetime dimensions, whose con- stituents have the same chirality, 'looks' like a neutral massless boson, since the constituents move along the line in the same direction and with the same speed (the speed of light).
Though the picture is physically appealing, it is in fact not borne out by the model at hand: The relation between the fermion and boson description of the states in the vacuum sector is rather more complicated. It is obvious that a one boson state corresponds to a fermion-antifermion pair state; more generally an N boson state clearly contains at most N pairs. However, a state describing one pair is in general a superposition of arbitrarily many bosons. For instance, the state
H - °R+,l,,.(r0~+A,~(r2)*~, (El, r0 ~ (Ez, 1"2) (6.25)
is on one hand a linear combination of fl and a fermion-antifermion pair state, by virtue of the explicit formula (4.14). On the other hand one has
H = e i¢ exp(i d['(A+H))I) (6.26)
where
a te (x) - -2 Arctan(~--~r~) - 2 A r c t a n ( X - r 2 ) \ E2 ,' (6.27)
and e i* is an irrelevant phase factor. Thus, in the boson picture H is a coherent state.
The fermion---~boson half just mentioned and (a generalization of) the 'boson'---~ fermion half we are about to discuss play an important role in some interacting two-dimensional field theories, in particular the massless Thirring model and the massive Thirring model alias sine-Gordon theory. For a concise review from a mathematical viewpoint we refer to the Appendix of [50]; for the massless Thirring model, cf. also the recent reference [42]. Our results in this paper have a bearing on the case where the coupling cons t an t ~2 of the sine-Gordon theory equals 4~r. This corresponds to vanishing coupling constant
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 69
in the Thirring model, so that the Thirring field reduces to the free Dirac field. We shall only consider the charged case with N = 1. Thus, the arena for what follows is the Fock space ~a(Y0 with ~ = L2(R, dp) ~.
It is routine to verify that the current J~(t, x), given by (6.1), is conserved (i.e., that do + j1, = 0). This fact is reflected in the existence of a field q~ such that
¢(t, x) = exp(it dF(H))q~(0, x) exp( - i t df'(H)). (6.30)
(In the literature one often adds a term (77/2/2)O or -(~?/2/2)O to the r.h.s, of (6.29).) Moreover, the current algebra (6.4), (6.5) is formally equivalent to
Thus, q~ may be regarded as a neutral pseudo-scalar boson field. The question whether q~ is a free field can be answered by evaluating I-lq~. This is straightfor- ward and the result can be written
0~o= 2m~/2:~,(0 i - i ) ~ : . 0 (6.33)
Thus, q~ is a free massless Klein-Gordon field when m = 0, but is not a free field when m > 0.
Before continuing, let us briefly digress to comment on the meaning of the above equations. First of all, recall our comments at the end of Subsection 2.1 and below (4.7). Secondly, note that evaluation of the form J°(t, y) on F, G c ~ leads to a function that is in 5¢(R) w.r.t, y, so that the form integral (6.29) leads to a well-defined form on ~S. Thirdly, the derivatives in the above are meant to be taken after evaluation on ~ . Finally, (6.31), (6.32) are well defined and valid on
after smearing with S°(R) functions whose Fourier transforms vanish at the origin. Indeed, for a c SO(R) with ~(0) = 0 one has, e.g.,
I dx ot(x)q~(O, x) = dI'(A,¢),
I dx o~(x)~b(O, x) = dI'(A~),
(6.34)
(6.35)
70 A.L. CAREY AND S. N. M. RUIJSENAARS
where
and & is the 5e(R) function
= 1 x ~)
(6.36)
(6.37)
Let us now return to boson-fermion correspondence as presented in the theoretical physics literature. Here, the fermion---~boson half is embodied in (6.28), showing how 9 may be obtained from W. The other direction, recovering
from q~, is usually expressed either through a formula due to Mandelstam [25] or through an earlier formula due to Coleman [24]. These formulas connect the massive Thirring model field with the exponentiated sine-Gordon field. For the special case g = 0, /3 = - 2 7 r m, Mandelstam's formula reads (compare [25, Eq. (2.8a,b)], and note that q ~ = ~0 + (,n'1/2/2)Q)
W~(', x )= Nexp[iTrl/2IX_ dy(~(t, y)-isTrl/2~(t,x)-is2~Q ]. (6.39,
(Here and in the sequel, N denotes some renormalization prescription, and also includes constants.) Mandelstam's work was inspired by Coleman's correspon- dence, which in our special case can be expressed through (6.28) and
• *(t, xNt_,(t, x) = Nexp[2ilrl/2s~o(t, x)] (on ~0). (6.40)
This correspondence consists in the equality of the n-point functions of the fields on both sides in formal perturbation theory.
Unfortunately, neither of the above formulas (6.39), (6.40) has a direct nonperturbative meaning, even in the free field context considered here. The point is that the exponential of a quadratic form is not defined; recall, moreover, that ~o and ~b leave the vacuum sector invariant, whereas W maps ~o in ~--1.
However, a way to give such exponentials a precise sense has been pointed out in the mathematical physics literature. In the massless case, the first result in this direction is due to Wightman [16]. By considering m - ~ 0 limits of n-point functions he arrived at a rigorous interpretation of the r.h.s, of (6.40), and this is the object whose n-point functions Coleman compared with those of the l.h.s. (See also [28] for the special case considered here.)
For m > 0 the status of the object Nexp(2i~rl/2q~) and of the formula (6.40) is much less clear. This holds a [ortiori for general values of g and /3, with one important exception: Fr6hlich [27, 29] rigorously constructed the sine-Gordon theory and its nonzero charge sectors for/32 < 16/7r, and in particular arrived at a
(6.38)
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 71
well-defined meaning for N exp(i/3q~(/3)). It appears to us, though, that his results shed little light on (the generalization of) (6.40). The point is, that his charge sector intertwiners can only be regarded as approximate Thirring fields, and even this is an interpretation by fiat, in the absence of any independent construction of the massive Thirring model. Similar remarks can be made concerning the work of Fr6hlich and Seller [28], who dealt with analogous correspondences. Moreover, the extraordinary difficulty of the constructive field theory program and the implicitness of the resulting Wightman field theory appear to prevent a com- parison with what is now widely believed to be the S-matrix of the massive Thirring model/sine-Gordon theory [55, 56].
It is therefore surprising that the much more modest problem of establishing the nonperturbative meaning of (6.40) is still open. In fact, the only reference we know that addresses this problem is [30], which impresses us as rather formal and not quite convincing.
As concerns the meaning of Mandelstam's formula, expressing the Dirac field in terms of the boson field ~: we feel this problem is solved for the first time by
our results. Indeed, recall that
o'tl~,~.,(sr)*"=" exp[-i l dy,o~r(sy)J~(O, y)] , (6.41)
since the r.h.s, is formally unitary and formally implements the same gauge transformation as the 1.h.s. (cf. our remarks above (6.9)). Also, we have proved that the l.h.s., multiplied by the wave function renormalization constant (4~r~) -I/2 and a factor F( -Q)=e i~°, converges to the free Dirac field ~ ( 0 , r), in the precise sense of Lemma 4.7. Now from (3.46) we have
lim "0~'(x) = 2~'O(x - r), (6.42) ,--.o
so that
• s(0, r )= Nexp[-2i~r I dy O(sy-sr)Js(O, y)+ iTr I dy J°(0, y)]. (6.43)
Hence, using (6.2) and then (6.28),
• +(0, r) -- N exp - i~r '/2 dy ~b(0, y ) - i~1/2~0(0, r )+ T ] ' (6.44)
~_(0, r) = N exp i~r a/2 dy ~b(0, y)+ i~rl/2q~(O, r ) + - ~ ] . (6.45)
Here, we have interpreted the boundary terms i~r~/2~0(-~) and -i7r~/2~o(~) as ilrO/2, which is natural in view of (6.29). Comparing with (6.39), we con- clude that ~ equals ~_. Also, ~+~ equals ~+, if one is willing to put exp[i~r[dyJl(O, y) - i~ rO]= 1; again, this is formally the natural thing to do.
72 A.L. CAREY AND S. N. M. RUIJSENAARS
Thus, we have shown that our results confirm the interpretation of the expression at the r.h.s, of (6.39) as the free Dirac field.
At this point it should be emphasized that the formal expressions and cal- culations we have just indulged in can easily lead to contradictions. For instance, in the formal approach one concludes from (6.8) that the implementers of gauge transformations of the form
(0 'v' ~ ) a n d (~ en, 0 )
can be written as exponentials of J+ and Jr_, resp. The expression at the r.h.s, of (6.41) is a case in point. Since J+(0, x) and J_(0, y) commute, one is led to the conclusion that such implementers always commute. However, this conclusion is in error when the winding numbers of e i'~ and e iw are both odd, cf. (5.17) and its proof.
Formally, (6.43), (6.44) also entail Coleman's relation (6.40). Thus, one might regard this formula as 'proved' too, in the sense of having been shown to symbolize a relation between well-defined mathematical objects. However, we do not take this point of view where (6.40) is concerned. The point is, that one can and should make sense of the r.h.s, of (6.40) in terms of an implementer like
exp(it dF(H))qls,l,~(sx)ql_~,l,~(-sx)* exp(-i t dF(H)), (6.46)
and for m > 0 our convergence result by no means implies that this (rescaled) implementer converges to the quadratic form obtained by normal ordering the l,h.s.; this will probably involve making precise short-distance expansions.
After this discussion, it may be in order to add that we do regard the field q~ for m > 0 as the sine-Gordon field for/3 = -2~r l/2. It was first introduced and studied by Wightman [16], who proved that it is a local operator-valued tempered distribution. (Our q~ is a scalar multiple of his tr.) It is not hard to conclude from our results that it is also cyclic in the vacuum sector. Indeed, already the suitably smeared time-zero fields q~(0, x) and ~b(0, x) are cyclic, which can be seen on combining Theorem 4.8 with (6.34), (6.35).
As regards the (suitably interpreted) object exp (2i~r~/2A~), this has recently (-~, ~) [38, Ch. been proved to be a field satisfying all Wightman axioms for A ~
4]. (Here, cyclicity in the vacuum sector is still open, however.) The open problem of making nonperturbative sense of (6.40) is tied up with open problems concerning the existence and properties of the exponential for I)t 1/> ½, cf. also [36, Section 4B]:
7. Applications to Kac-Moody Algebras
7.1• PRELIMINARIES
As pointed out first by Frenkel [10], the above current algebras are related to the representation theory for a class of Kac-Moody Lie algebras, viz., the so-called
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 73
affine algebras or loop algebras. A brief discussion of the latter, which also serves to fix our notation, now follows. (For more information, see [57,63] and references given there.)
Let g be a complex simple Lie subalgebra of gl(N, C). Then the affine Lie algebra associated to g may be described as the vector space
A(g) = C(t, t - l )@ g @ C z (7.1)
with the bracket
[ t k @ L (~ az, i t@ M ( ~ bz] = t k+t Q [ L , M]@ kSk._t(Tr LM)z,
k , l ~ Z , L , M ~ g , a , b ~ C . (7.2)
(More precisely, this defines a derived affine Lie algebra, cf. also Subsection 8.3.) Here, C(t, t -1) denotes the Laurent polynomials in an indeterminate t, and z is
a central element. Hence, A(g) may be viewed as a central extension of the Lie algebra
L(g) =-- C(t, t - l )@ g, (7.3)
whose bracket is given by
[t k @ L, fl Q M] = t k+' Q [ L , M]. (7.4)
Of course, the above definitions of L(g) and A(g) also make sense for nonsimple g. Below we shall consider in particular the case g = g/(N, C). In the physics literature not only A(g) with g simple, but also L(g) and A(g) with g arbitrary are referred to as Kac-Moody algebras.
Of particular importance in the representation theory of A(g) is the so-called basic representation. This is a representation of A(g) with a special property that renders it unique up to isomorphism. Let us denote the basic representation by B and the vector space on which it is defined by ,~. Then B may be characterized by the existence of a cyclic vector 1) E ~ with the property
B ( t k @ L @ a z ) l l = aCgl-l, Vk>~0, VLE g, V a ~ C , (7.5)
where Cg is a positive constant depending only on g. Specifically, we mention the cases
cg = 1, g = sl(N, C), V N > 1, (7.6)
cg=½, g = o ( 2 N , C ) , o ( 2 N - 3 , C), V N > 3 . (7.7)
7.2. THE MASSLESS CASE
Let us now turn to the connection with the gauge groups and current algebras studied above. First, we note that we may define two representations of L(g) on the Hilbert space ~ = LZ(R, dx )EQcN of the one-particle Dirac theory by
74
setting
{)(o~+(tk@L)=--(o'-k~ ")L ~),
A. L. CAREY AND S. N. M. RUIJSENAARS
k 6 Z , L e g (7.8)
~(o)_( tk @ L ) = ( ~ 0 ~rk(" )L)
cf. Subsection 3.3. We use the subscript (0) here, since these representations of L(g) do not lead to Fock space representations of A(g) when m > 0.
However, assuming m -- 0 henceforth, the operators
[~(o),(t k @ L @ az) - dI'(p(o)s(t k @ L)) + a U (7.9)
are not only well defined, but also represent A(g). To prove this, we must first show that the off-diagonal parts of p(o)s(tk@L) are Hilbert-Schmidt. Now the matrix multiplication operators
(L ~ ) a n d (~ 0L) o n ~
have this property (for m = 0, but not for m > 0, cf. Subsection 3.2), and if we subtract these from the r.h.s, of (5.78), the resulting operators have HS off- diagonal parts, since the functions
B . ( x ) - o'"(x)- 1
= ( x - i ) " Z* \ ~ 1 - 1 , n e (7.10)
belong to H~(R). To prove that /5(o)5 represents A(g), we compare (5.21) with (7.2), and recall
(3.25) and Proposition 2.2. This readily leads to the conclusion that it suffices to show that
dx B'k(x)B~(x) = 2 7rik 3k,_ 1 . (7.11)
But the 1.h.s. may be written, using (7.10) and (3.50),
Idx ( x ' I _ , , k 1
\ x + i] (x + i) 2 ~ i = 27rik dxF-k(x)Ft(x), (7.12)
so that (7.11) follows from the orthonormality of the functions F, in LZ(R). Thus, our assertion is proved.
If g consists of skew-symmetric matrices, we can represent A(g) on the neutral Fock space o~a(~(+) by setting
P(o)s (t k @ L @ az) =- d['(p(o)s (t k @ L)) + a a (7.13) 2 "
Indeed, this follows as in the charged case, using (2.131) instead of (2.86).
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 75
We proceed by studying/5 and 15 on ~a(Y(s) and ,,~a(~-), resp. We begin by
observing that
Pm)s(tkQL(~)az)t)=afl, Vk>~O, VL~gI(N,C), N ~ l , V a e C (7.14)
whereas
/5(O)s(tk @ LOaz) l ~ a =-~[l, Vk>-O, VL~o(N,C), N>~2, Va~C. (7.15)
Indeed, these relations are obvious for k = 0; they follow for k > 0 by recalling (5.8) and noting
supp/~-k = (-0% 0], Vk > 0. (7.16)
Now we introduce the (nonclosed) subspace ~ ( g ) of the vacuum sector, obtained by acting with polynomials in operators from /5 and P, resp., on f~. Then we are prepared for the following theorem.
THEOREM 7.1. The representation/5(o)~(A(g)) on ~ ( g ) C a~).~, given by (7.9), is the basic representation of A(g), when g = sl(N, C), ' qN> 1. The representation /~m)~(A(g)) on ~ ( g ) C ~ . ~ , given by (7.13), is the basic representation of A(g), when g = o(2N, C), o ( 2 N - 3, C), V N > 3.
Proof. This follows by inspection from (7.5)-(7.7) and (7.14), (7.15). [ ]
For other g one obtains in general tensor products of the basic reprsentation. For instance, one may consider /sco)s for g = o(2N, C), o ( 2 N - 3 , C) with N > 3. Then one gets a twofold tensor product. (Recall the Schwinger term is twice as large as for /b~o)~.) Similarly, /5(o)~(A(o(3, C))), viewed as corresponding to sl(2, C ) = 0(3, C), gives rise to a twofold tensor product of the basic represen- tation Pm),(A(sl(2, C))), since the traces of the squared 0(3, C) matrices are four times as large as those of the corresponding sl(2, C) matrices, etc.
Next, we discuss the 'size' of the closure of ,~(g). To reduce this to previous results, we first note that one already obtains ff,(g), if only polynomials in P~o)s((t k - I)(~L) are used, with k ~ 7 * and L~ g. (To see this, recall that the operators dF(Q O L) and dF(I t~) L), resp., annihilate f).) To fix the thoughts, we now consider a special case: We claim that one has
ff~(gl(N, C))- ~ = 5%.~,, V N ~ > 1. (7.17)
It might appear that this is obvious from our discussion of the smeared currents in Subsection 6.2 (specifically, from Theorem 6.1), since one has
cf. (6.11). However, there is a snag here, since the linear hull of the smearing functions B, is a proper subspace of H~(R).
76 A.L. CAREY AND S. N. M. RUIJSENAARS
To obviate this snag, we appeal to the d['-bound (2.53). This bound entails that we need only show that this linear hull is dense in Hi(R) in the H1(R) topology. Indeed, it is readily verified that a,---~a in H~(R) implies Ill a , , , - As [[I ---, 0.
However, this density property is proved in the following proposition, so that (7.17) indeed follows from Theorem 6.1.
PROPOSITION 7.2. The closure of the vector space
K = LH{Bo}.~z. (7.19)
in the H1(R) topology equals Hi(R). Proof. The Hx(R) topology may be viewed as the graph norm topology for the
self-adjoint operator i(d/dx) on L2(R). Thus it is enough to show that K is a core for i(d/dx). To prove this, we note first that we may write
{(xS/)T K = LH . (7.20) l ~ l
Now it is easy to verify that
I I ( d ) " ( 1 ) / 1 [ ( / + n - l ) , dxx ) - ~ ~< C, (1 - 1)! (7.21)
Hence, K consists of analytic vectors for i(d/dx). But one also has
K = LH{F,}n~z, (7.22)
cf. (3.50), so that K is dense in L2(R). Thus, K is a core for i(d/dx) by virtue of Nelson's analytic vector theorem. [ ]
cf. Theorem 5.1. Similarly, one gets in the neutral case
o%s(o(N, C))- = LHII~(LeSO(N)0)~, VN/> 2,
= ,~n VN/> 3. (7.24)
More generally, it will now be clear that the size question discussed here is in essence the same as the size question discussed in Subsection 5.3. Possibly, the questions raised in the latter context can be answered by exploiting the extensive lore on affine algebras.
7.3. THE MASSIVE CASE
For m > 0 one can also obtain representations of A(g), provided one starts from different single-particle representations of L(g). Indeed, as already mentioned in passing, the L(g) representations at the r.h.s, of (7.8) violate the HS condition for m > 0. Now, one must incorporate the global gauge transformations in the form
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 77
(0 L O ) t o ensure that the oil-diagonal parts vanish. There are twoways t o d o
this. The first possibility is to set
o.k(.) L , k ~ Z , L e g . (7.25)
Then the operators
Plm)( t k @ L @ az) =- dF(p¢.,,(t k @ L)) + 2aU (7.26)
represent A(g), which follows in by now familiar fashion. If the matrices in g are skew-symmetric, one can also represent A(g) on the neutral Fock space by replacing d[" by d[" and omitting the 2 at the r.h.s, of (7.26). It would be of interest to study these representations on the cyclic component of the vacuum. We shall forego this here, however, since the above results are not readily applicable to this: If, e.g., g = gl(N, C), the gauge group corresponding to (7.26) is, so to speak, a mixture of II+,0 and II_,0.
The second possibility consists in introducing the L(g) representations
,, o c / | L o'k(') L O ) t k ~ Z ' L c g (7.27)
O("-( tk ® L ) =- ( \0
Then one gets A(g) representations on o~a(~i0 by setting
/5,~,)~(tk @ L @ az) ~ d['(p(.o,(t k @ L)) + a R, (7.28)
as is readily verified. Again, for skew-symmetric g one also gets representations on ffa(~÷) through
P(m)s( t k @ L @ az ) - d['(p(,,)~ (/k @ L)) + 2 L (7.29)
In this case, our previous results do lead to interesting information, in particular as concerns the size of the cyclic subspaces ff~(g) (which are defined just as for m -~ 0) and as regards the Fock space gauge groups generated by exponentiating the self-adjoint part of the Lie algebras (7.28) and (7.29).
First, we note the equalities
~s(gl(N, C))- = ~g, VN>~ 1 (7.30)
in the charged Fock space, and
~ , (o (N ,C) ) -= ~8, VN~>3 (7.31)
in the neutral one. Indeed, this follows from Theorem 6.3 by exploiting Pro- position 7.2 just as we did for m = 0.
78 A.L . CAREY AND S. N. M. RUIJSENAARS
Secondly, we discuss the unitary groups obtained from /5<r,0, and /5c,,,), (recall Proposition 2.1 and Proposition 2.5 in this connection). We shall again restrict our attention to the cases g=gl (N,C) , N ~ > 1, and g = o ( N , C ) , N~>3, and denote the corresponding groups by (~ and ~ , resp.
Our aim is to show these groups generate hyperfinite type III~ factors on the vacuum sectors 5~) and J:(7, resp. It is expedient to precede this result by some observations concerning @~ and its commutant. The corresponding remarks for @, will be obvious from this, and hence are omitted. (In fact, most of what follows holds true more generally.) Let us introduce the global gauge group
°//(N)---(F(~ Q U) I UE U(N)}. (7.32)
(cf. Subsection 3.3 for the notation used here.) Also, let us set
H~,0,k -- ~ ('1 II~,o. (7,33)
Clearly, one may then write
~ = q/(N)II~,o.k. (7.34)
First, we observe that II~,o belongs to the strong closure of II~.0,k. Indeed, this follows from Proposition 7.2 by recalling Proposition 2.1 and the fact that strong convergence of self-adjoint generators on a common core entails strong con- vergence of the corresponding one-parameter groups. Thus, using Theorem 4.8,
l-I',,o,k = 1-I~.o.
Secondly, we note that the map
~.: ns (v )~ f(n ® u)n~(v)r(n ® u*),
(7.35)
U c U(N), v ~ L~U(N). (7.36)
extends to an automorphism of II~. Indeed, the unitary at the r.h.s, may be written H~(UvU*), and hence belongs to II~. Combining (7.34) and (7.35), it follows that
~'~ = E(rl"~), (7.37)
where
E(-) -= IU(N) a.( ') d/z(u), (7.38)
with d/x the Haar measure on U(N). In words, ~'s consists precisely of the globally gauge invariant elements of [I'-'s. Thus, for N > 1, (~'s is not cyclic on fl in ~o. Note this is also evident from the relation F(0 @ U)f~ = ~1, combined with the fact that the operator F(O @ U) - D ~ ~ is nonzero on fro when U ~ e~0. (Of course, for N = 1 one has ~', = Fl"~,o, so that f~ is cyclic.)
Let us now state and prove the result mentioned above.
THEOREM 7.3. The groups ~ and ~ generate hyper]inite type IIIt factors on J~ C ~ ( ~) and ~g C ~ ( ~(+), resp.
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 79
Proof. First, we note that the modular group exp(it In A) associated to (II", 12) commutes with the global gauge group. (Indeed, A is of the form F ( A @ i ) with A :~ = 0, cf., e.g., [45].) Next, recall ft is cyclic in ~0 for lI'].o, cf. Theorems 4.8 and 4.10. Afortiori, f~ is cyclic for ~3", so that 12 is separating for q3~. Hence, 12 is separating and cyclic for 'g'~ restricted to ff~ --- (~'~12)-. But then it follows that exp(it ln ~) I ~ is the modular group associated to (~g'~ I ~ ,12) . Since A has purely continuous spectrum on 121, ~3'~ I if., is a factor and is of type III~. Hence ~'s and ~3" are type III~ factors on ~0. Since l'I"_~ is hyperfinite, ~3'~ and q3'~ are
hyperfinite, cf. (7.37). [ ]
As in the massless case, the situation regarding subalgebras of gl(N,O) and o(N,O) is much less clear. Again, studying this from the viewpoint of Kac- Moody algebras might well lead to information that is hard to obtain with our
approach.
8. Concluding Remarks
8.1. WIENER-HOPF OPERATORS
As we have seen in Subsection 3.1 and used repeatedly, in the massless case the projections P~ commute with ~5, so that attention may be restricted to 9~ s --- ~ - ~ L2(R, dx) in many respects. On the space ~ , P~ equals the projection on the Hardy space H~ of functions holomorphic i~ the upper (8 = +) and lower (8 = - ) half-plane, as is clear from Subsection 3.l. Thus the operators/~Tr+(u)16~ on 9~ ~+, u • L~U(I), are (Fourier transforms of) Wiener-Hopf operators. (For this and other notions occurring below, cf. the books by Douglas [58, 59] and references given there.)
The formula
(Yl, 1q+(¢'~)12) = exp( -¼(a , a)), a e V (8.1)
from Subsection 5.1 (cf. (5.10), (5.11)) can be exploited to obtain known information on these operators in a novel way, as we shall now detail.
PROPOSITION 8.1. Let w(u) denote the winding number of u e LeU(1). Then one has
dim Ker 7r+(u)~ = ½ (sgn w(u) - 8)w(u), Vu e LeU(1). (8.2)
Proof. If w(u)=O, then u is of the form e i~', ~ e V . In view of (8.1), the vacuum expectation value of the implementer I-l+(e i~) is nonzero. However, this holds if and only if Ker 1r+(ei~)6~ = 0, 6 = +, - (cf., e.g., [12, Section 5]), so that (8.2) follows when w(u)= 0. Also, we know already that (8.2) holds true when u = o-", n E Z (cf. Subsection 3.4, in particular, the shift property (3.55) with e = 1, r = 0 ) . Now assume that u is an arbitrary loop with w(u) = n > 0 . Then w(uo'-") =0 , so that Ker 1r+(uo'-")++ ={0}. Since or" leaves H2+ invariant, it follows that Ker w+(u)++ = {0}, so that Ker 1r÷(u)*_ = {0i by unitarity. In view of
80 A. L. CAREY AND S. N. M. RUIJSENAARS
(2.67) this implies that the charge of 7r+(u) equals dim Ker 7r+(u)__. But one has H+(u) = H+(uo'-")H+(o'") up to a phase, so that dim Ker 7r+(u)__ = n = w(u). Hence, (8.2) follows for w(u)> 0, and, analogously, for w(u)< O. []
In this proof of (8.2) we have substituted the theory of implementers for known properties of Hardy spaces and Fredholm operators. The former theory also leads to another result.
PROPOSITION 8.2. For any a ~ V one has
det[Tr+(e-")--rr+(e~)--] = e x p [ - ~ I ° dkkl&(k)[2]. (8.3)
Proof. To prove this, we employ the explicit formula for ['(U) in the case where Ker U~ = {0}, 6 = +, - . It reads
f~(U) = det( t + Z+_*Z÷_)-~I/2~E¢(Z), is.4)
where Z is defined by (2.75) and Ec(Z) by (2.77) (see [12, Theorem 4.1]). Thus one has
The relation (8.3) may be viewed as a special case of a result obtained by Widom in his study of the Szeg6 limit theorem [60, Theorem 7.1]. (His work deals with Toeplitz operators, but through the Cayley transform it can easily be translated to the Wiener-Hopf context.)
For m > 0, it is no longer possible to restrict oneself to 9~. In that case, the operators ~r.(u)~ on ~ are generalized Wiener-Hopf operators [61]. We are not aware of any results in the operator-theoretic context that lead to m > 0 analogs of Propositions 8.1 and 8.2. In particular, it appears that the arguments used in the literature to prove (generalizations of) (8.2) and (8.3) cannot be extended to the m > 0 case. However, our methods do lead to a partial analog of Proposition 8.1, which says that the equality of winding number and Fredholm index persists for m > 0 .
PROPOSITION 8.3. For any u ~ L¢U(1) one has
w(u) = dim Ker 7r+(u)__- dim Ker 7r+(u)_*_. (8.7)
Proof. This formula follows again from the theory of implementers (in parti- cular, (2.67)) and the fact that (8.7) holds for u = ~r, as proved in Lemma 3.5. [ ]
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 81
It is an open question whether lhe stronger result (8.2) is valid here, too. This is tied up with the problem of obtaining a useful analog of Proposition 8.2, or, put differently, with the problem of: obtaining more information on the generating functional E~,,o(a), cf. (5.10). (These problems are in essence the same, since (8.6) holds for m > 0, too, as is readily seen.) For instance, Et,,~(a) ~ 0 entails in the same way as for m = 0 that Ker -tr+(ei~)~ = {0}, 6 = +, - .
In the above we have restricted ourselves to loops in U(N) with N = 1. For N > 1 one is dealing with matrix-valued Wiener-Hopf operators, and similar remarks can be made. In particular, (8.7) follows by the same arguments for any u ~ LeU(N). Also, one can probably get results for more general classes of single-particle multiplication operators by using continuity arguments and polar decomposition. We have not pursued this, however.
8.2. UNIQUENESS OF STANDARD KINKS
As we have stressed before, the key to the structure analysis of the gauge groups IIs and IIs.0 is the fact that our approximate Dirac fields converge to the free Dirac fields, in a topology that is strong enough to legitimize invoking well- known properties of the latter fields. We recall that the approximate fields are in essence implementers of special one-particle gauge transformations, which we have dubbed 'standard kinks'. These kinks correspond to winding number one loops, defined through the approximate step functions
"tl~(x) = "rr+ 2 Arctan x - r (v), (8.8)
cf. Subsection 3.4.
The question now arises in how far this choice of step function is critical. For instance, one could take instead of r/~' the functions
O~'(x) - 27rJ dy j,(x - y)O(y - r), (8.9)
where j, is an approximate identity. Is it true that the corresponding im- plementers are approximate Dirac fields in the same sense as before?
We believe that the answer to this question is yes, but we do not see any direct way to prove this. The only avenue that appears promising to us involves a detour, viz., making suitable use of what we have proved already for the choice (8.8). However, this might also be a considerable enterprise, especially in the massive case.
i
We could push through ~he special choice (8.8) by virtue of its 'solubility', as embodied in the explicit formulas of Subsections 3.4 and 3.5, and the resulting detailed information on the implementers. In the massless case this is the unique choice for which the implementers have the simple structure (4.14) and (4.38).
82 A.L. CAREY AND S. N. M. RUIJSENAARS
We proceed by making this statement precise and proving it in the charged case (the neutral case can be handled analogously).
Assume that u 6 LeU(1) is a winding number one loop. Then we know from (8.2) that U - ~r+(u) satisfies (2.68) and (2.69). Thus, the implementer I '(U) has the form (2.76).
PROPOSITION 8.4. The tail term exp(Z+_a*b*) in Ec(Z) equals the identity if and only if u = exp(ir/~).
Proof. We need only show that Z+_ = 0 implies u = ~r~. Assuming Z+_ = 0, it follows from the definition (2.75) that U_+ = 0. Hence, uH2+ C H2+. Thus, under the Cayley transform (9 = "O(x), or, equivalently, z = ( x - i ) / ( x + i), u(x) turns into a continuous loop fi(19), which is the boundary value of a function f(z) that is holomorphic in Izl < 1. Since fi(@) has winding number one, f(z) has one simple zero z0 in I z l < l by the argument principle. Next, set g(z)=- f((z + z0)/(1 + ~oz))/z. Then g(z) is holomorphic and nonvanishing on Izl < 1 and has a continuous boundary value with modulus one. Hence, by the maximum principle, Ig(z)l~<l and II/g(z)l<~l, so that g ( z ) = e i~ in the unit disc. A straightforward calculation, using f ( 1 ) = l, then shows that u(x) equals try(x) (where r + iE = i(1 + z0)/(1 - Zo)). Thus, our claim is proved. [ ]
There is a second way in which the choice (8.8) can be singled out when m = 0: The n-point functions of the approximate fields corresponding to (8.8) are related to those of the free massless Dirac fields in what appears to be the simplest way possible. Roughly speaking, the 'ie' in the latter distributions, which symbolizes a distributional boundary value as e ~ 0, is replaced by "ie' with • > 0, • being determined by the 'cutoff' parameters •~ in the approximate fields. This is a consequence of [42], where, more generally, approximate fields for the massless Thirring model having this property were found.
For m > 0 the tail term is never trivial. Furthermore, we do not even know how to determine explicitly the 2-point function of our approximate fields. In this case the only criterion known to us which does single out the choice (8.8) is our inability to deal directly with other choices, for instance (8.9).
8.3. VERTEX AND VIRASORO OPERATORS
As we have discussed below (5.19), the gauge group Hs,o belongs to the strong closure of the linear hull of its subgroup Hs(LeTN), which consists of 'diagonal' gauge transformations. Heuristically, this can be understood as follows. The element qls.j.,(sr)°lls,k,,(sr) * of the latter group, multiplied by (47re) -1, converges in the sense of forms to :~*j(0, r)~.k(O, r): for any j-~ k = 1 . . . . . N, as can be seen from our convergence results. (Note that the double dots may as well be omitted, since j~-k . ) But when this form is smeared with a ( r ) c H~(R), one obtains an element of the complex Lie algebra of H~.o corresponding to an
FERMION GAUGE GROUPS, CURRENT ALGEBRAS, KAC-MOODY ALGEBRAS 83
'off-diagonal' gauge transformation, cf. (6.11). Of course, this argument does not prove the extension property, but it does help in understanding why it holds.
In previous literature on the above loop group and loop algebra represen- tations for m = 0, the extension property just discussed plays a much more important role [4, 5, 7-11]. Crudely speaking, a Heisenberg algebra (or CCR algebra, in a mathematical physicist's language) is used as the starting-point for the construction of the representation. This algebra is a subalgebra of the loop algebra, and the remaining part is then obtained through an object that is called the vertex operator in the context of dual resonance models [62]. The vertex operator only depends on operators from the Heisenberg algebra, so that the extension property is manifest.
Another class of operators playing a crucial role in dual resonance and string theory are the Virasoro operators [62]. They form a projective representation of (a basis of) the vector fields on the unit circle S 1. G. Segal proved (using an argument due to Kazhdan) that any ditIeomorphism of S ~ is unitarily imple- mentable and studied the resulting groups and related loop groups in [4]. Subsequently, Frenkel used Segal's Virasoro operators to construct his spinor representations of loop algebras [10].
It would be of interest to establish whether our new representations for m > 0 also admit analogs of the vertex and Virasoro operators. (The fact that our work uses R and not S ~ constitutes a minor problem, which can be remedied by using the Cayley transform.) In this connection, we point out that for m > 0 diffeomorphisms of S 1 again give rise to automorphisms, not only of the CAR algebra, but also of the various loop algebras, since the Schwinger term (cocycle) is mass-independent. However, we do not know whether these automorphisms are unitarily implementable; Segal's method of proof does not appear to extend to the massive case. In particular, it is not clear to us whether the rotations of S 1 are implementable for m > 0, or, equivalently, whether our massive represen- tations of the derived altine Lie algebras extend to the full afline Lie algebras.
Acknowledgements
We have benefited from discussions with C. A. Hurst, L.-E. Lundberg, G. B. Segal, and A. S. Wightman. S. R. would like to thank G. B. Segal for useful correspondence, and D. Robinson for an invitation to the Mathematics Depart~ ment of the Institute of Advanced Studies at the Australian National University, where the present collaboration was begun. A. L. Carey thanks A. S. Wightman for hospitality at Princeton during the early stages of this work.
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