SPLIT CLIFFORD MODULES OVER A HILBERT SPACE
Esther Galina, Aroldo Kaplan and Linda Saal
CIEM-FaMAF, Universidad Nacional de Córdoba
Abstract. We describe the real, complete modules of the Clifford Algebra C(Z),
where Z is a real Hilbert space acting by skew-adjoint operators.
1. Introduction.
Let Z be a real separable prehilbert space and C(Z) the corresponding Clifford
Algebra, i.e., the quotient of the Tensor Algebra of Z by the ideal generated by the
elements of the form z1z2 + z2z1 + 2(z1, z2)1, with z1, z2 " Z. We determine here all
the representations of C(Z) on real separable Hilbert spaces where the elements of
Z ‚" C(Z) act skew-adjointly. Equivalently, all sequences J1, J2, ... of orthogonal,
mutually anticommuting complex structures on a real Hilbert space.
Without loss of generality we may assume that Z is complete as well. Every
such representation gives rise, upon complexification, to a representation of the
algebra CC(Z) = C " C(Z) on a complex Hilbert space, whith Z acting by skew-
hermitian operators. A complete set of representatives of such representations up
to equivalence was given by Gårding and Wightman [GW].
In this article we determine which Gårding-Wightman modules split over R and
describe all their C(Z)-invariant real forms up to equivalence. These effectively
solves the problem, since any C(Z)-module must arise as one of these invariant
forms.
To put the results in perspective, recall that for m = dim Z < ", the question
of when a given CC(Rm)-module, or space of spinors, splits over R, goes back to E.
Cartan and Killing [C], who also provided the answer and geometric interpretation.
The representations of CC(Rm) are direct sums of either one or two finite dimen-
sional irreducible representations, depending on whether m is even or odd. These
remain irreducible over R if m a" 1, 2, mod. (4) and split into a direct sum two dual
irreducibles otherwise. In particular, a CC(Rm)-module splits over R if and only if
m a" 0, 3 mod.(4). We recover this result, at least when m is even.
If dim Z = " the situation is much more complicated, as can be expected.
Rather, the remarkable feature of the infinite Clifford Algebra -as well as its bosonic
cousin, the Heisenberg Lie Algebra, is that its representations can be parametrized
at all and the possible real forms described. In [GW], the resulting set is described
as a true maze . Since both there and here the matters of equivalence and irre-
ducibility are not completely settled, an alternative argument is required to analize
Key words and phrases. representations of Clifford algebras, representations of anticonmuta-
tion relations.
This research was supported in part by CONICET, FONCYT, CONI and UNC.
COR
Typeset by AMS-TEX
arXiv:math.RT/0204117 v3 31 May 2002
2 ESTHER GALINA, AROLDO KAPLAN AND LINDA SAAL
the splitting problem in infinite dimensions. In any case, although Clifford and
Heisenberg Algebras are exceptions to Kirillov s dictum that in infinite dimen-
sions there are no theorems, only examples , it is the examples what may matter
most.
The basic Fermi-Fock representations, familiar to physicists, do not split and in
fact are irreducible over R. But we find others that are irreducible over C, split
over R and the corresponding factors are essentially new, i.e., they do not arise by
restriction of any CC(Z)-module.
Although the examples of the latter kind may fail as models for quantum fields
because they lack a vacuum (or only apply to systems with infinite energy), they
do have natural realizations in L2(T) which are closely related to wavelets and
multilinear operators. The Clifford Algebra acts by singular integral transforms on
the circle and the corresponding splittings can be interpreted in terms of wavelets.
Conversely, this picture describes wavelets as infinite-dimensional spinors.
We are grateful to H. Araki, A. Grunbaum, A. Jaffee, V. Kac and J. Vargas for
their useful advise.
2. The Gårding-Wightman modules.
We describe here the results of [GW] and translate them in the language of
Clifford modules (see also [G], [BSZ]). According to [GW], a representation of the
Canonical Anticommutation Relations (CAR) consists of a countable set of bounded
linear operators ak acting on a separable complex Hilbert space H, satisfying
aj ak + akaj = 0 aj a" + a"aj = ´jk.
k k
The ak and a" are sometimes called the (fermionic) creation and anihilation oper-
k
ators, respectively.
To view these as representations of CC(Z), we choose a complex structure Ã
on Z compatible with its metric, making (Z, Ã) into a complex-hermitian Hilbert
space and a unitary basis z1, z2, ... of it, so that {zk, Ã(zk)}, is a real orthonormal
basis of Z. Then, the following formulas define a 1-1 correspondence between
representations of the CAR and continuous representations J of the algebra CC(Z)
on H such that the operators Jz, z " Z, are skew-hermitian:
1 1
(2.1) ak = JÃ(z ) + iJz a" = -JÃ(z ) + iJz .
k k k k k
2 2
Note that z Jz is not assumed to be C-linear and that for z " Z, |z| = 1, the
operator Jz is unitary and its square is -I.
Let X be the set of sequences x = (x1, x2, . . . ) where each xi is 0 or 1 (space
of fermionic occupation numbers ). X is a group under componentwise addition
modulo 2. Let " be the subgroup of X consisting of sequences with only finitely
many 1 s. It is generated by the sequences ´k where the k-th component is 1 and
the all others are 0. The Gårding-Wightman (or GW) modules are parametrized
by triples
(2.2) (µ, ½, {ck})
with the following objects as entries:
" µ is a positive measure over the Ã-algebra generated by the sets Xk = {x : xk =
1}, which is quasi-invariant under translations by ". This meansthat its translates
SPLIT CLIFFORD MODULES OVER A HILBERT SPACE 3
µ´(E) = µ(E + ´), ´ " ", all have the same sets of measure zero and implies the
existence of the Radon-Nikodym derivatives dµ´/dµ.
" ½ : X Z+ *" { "} is a measurable function such that ½(x + ´) = ½(x) for all
´ " " and almost all x " X.
" ck(x) : Hx Hx+´ H" Hx is a sequence of unitary operators depending
k
measurably on x and satisfying
c"(x) = ck(x + ´k)
k
(2.3)
ck(x)cl(x + ´k) = cl(x)ck(x + ´l)
The Hilbert space isomorphisms Hx+´ H" Hx are part of the data, although chang-
k
ing them results in unitarely equivalent representations. One may therefore assume
that the family Hx is invariant under translations by ".
The representation associated to a triple (µ, ½, {ck}) is realized in the Hilbert
space
•"
(2.4) H = Hx dµ(x)
X
and defined by the following formulae, where an f " H is regarded as an assigne-
ment x f(x) " Hx:
dµ(x + ´k)
1
(Jz f)(x) = (-1)x +···+xk-1+1ick(x) f(x + ´k)
k
(2.5) dµ(x)
k
(JÃ(z )f)(x) = (-1)x i(Jz f)(x)
k k
The main observation here is that in any representation of the CAR, the oper-
ators Nk = a"ak and Nk = aka" form a conmuting set of projections over the sets
k k
Xk and their complements Xk, respectively, so that
(2.6) (Nkf)(x) = xkf(x) (Nkf)(x) = (1 - xk)f(x)
for all f " H and for almost all x. The direct integral (2.4) is just the corresponding
spectral decomposition.
Setting
Ak = iJz Bk = -iJÃ(z ),
k k
our notation becomes consistent with that of [GW].
3. Real Clifford modules.
If V is a (real) module over C(Z), then C " V is a (complex) module over
C " C(Z) with the C(Z)-invariant decomposition over R
C " V = V •" iV ;
V is an invariant real form of C " V . Hence, to determine all real modules over
C(Z) it is enough to determine all the invariant real forms of the Gårding-Wightman
4 ESTHER GALINA, AROLDO KAPLAN AND LINDA SAAL
modules. This is equivalent to determining all the antilinear operators S : H H
(the corresponding complex-conjugations) which commute with C(Z) and such that
(3.1) S2 = 1, ||Sf|| = ||f||.
As S is a C(Z)-morphism and Ak = iJz and Bk = -iJÃ(z ) by (2.1) and (2.5), we
k k
know that for all k
(3.2) SAk = -AkS and SBk = -BkS.
Hence,
(3.3) SNk = NkS.
Every operator L that conmutes with Nk and Nk for all k is an operator of multi-
plication by an essentially bounded measurable operator-valued function Ć:
(Lf)(x) = (LĆf)(x) = Ć(x)f(x)
(cf. [G] ). In particular, Nk and Nk correspond to multiplication by the character-
istic functions of the sets Xk and Xk respectively (see (2.6)).
From now on, we let H be the complex Clifford module associated to (µ, ½, ck)
as in the previous section.
Lemma 3.4. Suppose that H admits an invariant real structure S. Let E ‚" X be
measurable and ÇE its characteristic function. Then, if
1 - E = {1 - x = (1 - x1, 1 - x2, ...): x " E},
(i) SLÇ = LÇ S.
E 1-E
(ii) Supp(Sf) = 1 - Supp(f).
(iii) (Sf, Sg) = (f, g).
Proof. (i) follows from (3.3) and the fact that Xk and Xk generate the Ã-algebra
where µ is defined. To prove (ii), set F = Supp(f). Then
Supp(Sf) = Supp(SÇF f) = Supp(Ç1-F Sf) ‚" 1 - F.
As S is an involution,
F = Supp(S2f) ‚" 1 - Supp(Sf) ‚" 1 - (1 - F ) = F.
Consequently, all these inclusions are equalities and the assertion follows. For (iii),
as S preserves norm, Re(Sf, Sg) = Re(f, g). Also,
Im(Sf, Sg) = -Re(iSf, Sg) = Re(Sif, Sg) = Re(if, g) = -Im(f, g),
finishing the proof.
If µ is a measure on X, then
µ(E) = µ(1 - E)
Ü
is another.
SPLIT CLIFFORD MODULES OVER A HILBERT SPACE 5
Theorem 3.5. H admits an invariant real form if and only if µ and µ are equiv-
Ü
alent, ½(x) = ½(1 - x) for almost all x " X and there exist a measurable family of
operators
r(x): Hx H1-x H" Hx
which are antilinear, preserve the norm and satisfy
(3.6) r(x)r(1 - x) = 1, r(x)ck(1 - x) = (-1)kck(x)r(x + ´k)
for all k " N and almost all x " X.
Proof. Assume that S is an invariant real structure on H. To see that µ and µ are
Ü
equivalent, it is enough to prove that µ(E) > 0 whenever E ‚" X is measurable and
Ü
µ(E) > 0. To this end, write X = *"n>1En where En = {x : ½(x) = n}. We can
assume that E ‚" En for some n. Let h be a unit vector of Hx for some x " E.
Define the functions f(x) = ÇE(x)h and g(x) = (Sf)(x). As S preserves norms
µ(E) = f = Sf = g
By the Lemma 3.4, the support of g is 1 - E, so
1
2
µ(E) = (g(x), g(x)) dµ(x)
1-E
1
2
= (g(1 - x), g(1 - x)) dµ(x)
Ü
E
Then, µ(E) = 0 implies µ(E) = 0, which is a contradiction.
Ü
To see that ½(x) = ½(1 - x) for almost all x " X, assume the contrary. Then
for some n there exists a set E ‚" En such that µ(E) > 0 and 1 - E ‚" En-1. Let
{hi}n be an orthonormal basis of the n-dimensional Hilbert space Hn (Hx H" Hn
i=1
for all x " E). Set fi(x) = ÇE(x)hi and gi = Sfi for i = 1, . . . , n. We will prove
that for almost all x, {g1(x), . . . , gn(x)} are n non-zero orthogonal vectors in H1-x,
is a contradiction, since we have assumed 1 - E ‚" En-1.
For an arbitrary F ‚" E with µ(F ) > 0, we have that
2 2
0 = LÇ fi = SLÇ fi = LÇ Sfi 2
F F 1-F
2
= Ç1-F (x)gi(x) dµ(x)
X
2
= gi(x) dµ(x)
1-F
Since F is arbitrary and µ and µ are equivalent we have that gi(x) = 0 a.e., and
Ü
hence gi(x) = 0 a.e.
We now prove that ½(x) = ½(1 - x) almost everywhere. If not, there would be
a set of positive measure E such that E ‚" En and 1 - E ‚" En-1, for some n.
Since µ and µ are equivalent, µ(1 - E) > 0. Since {fi}n is an orthogonal set in
Ü
i=1
H, so is {LÇ fi}n for any given subset F ‚" E. By Lemma (3.1), the elements
F i=1
SLÇ fi = LÇ gi are also orthogonal. Therefore, for all i = j
F 1-F
(gi(x), gj (x))dµ(x) = 0.
6 ESTHER GALINA, AROLDO KAPLAN AND LINDA SAAL
Since µ H" µ and F is arbitrary, we conclude that for almost all x " 1 - E,
Ü
2
(gi(x), gj (x)) = 0. On the other hand, µ(F ) = gj(x) dµ(x) for almost all
Ü
1-F
x " 1-F . Therefore gj = 0, so that the gj cannot vanish. Since E has dimension
n - 1, this is a contradiction. We conclude that ½(x) = ½(1 - x) for almost all x.
Up to identifications, we may now assume that Hx = H1-x. Define the operator
T : H H by
dµ(x)
Ü
(T f)(x) = f(1 - x)
dµ(x)
T is C-linear, unitary and satisfies the relations
2
T = 1 T Nk = NkT
The product ST is then an antilinear operator on H which conmutes with all the
Nk and Nk. Therefore it can be represented as multiplication by a measurable
operator-valued function r(x),
(ST f)(x) = r(x)f(x)
where r(x) : Hx Hx = H1-x is antilinear and preserves norm. Since T and S
are involutions, we obtain, respectively,
dµ(x)
Ü
(Sf)(x) = r(x)(T f)(x) = r(x) f(1 - x).
dµ(x)
and
r(x)r(1 - x) = 1.
S anticommutes with the operators Ak and Bk, because it commutes with J. There-
fore
1 1
r(x)(-1)x +···+xk-1+k-1ck(1 - x) = (-1)x +···+xk-1+1ck(x)r(x + ´k)
and r(x)ck(1 - x) = (-1)kck(x)r(x + ´k) follows.
Conversely, if µ and µ are equivalent, ½(x) = ½(1 - x) a.e. and there exist
Ü
antilinear operators r(x): Hx Hx that preserve norm and satisfy (3.6) a.e., we
can define T and S as above. Is straightforward to see that S is in fact an invariant
real form of H.
If dim Z = 2m < ", µ and µ are equivalent because they are quasi-invariant
Ü
discrete measures concentrated in the same coset of " (namely, X = " itself)
[GW]. From (3.5) we deduce
m(m+1)
2
r(1) = (-1) r(0).
Assuming, as we may, that r(0) is the standard conjugation on C, we see that H
splits /R if and only if m(m + 1)/2 is an even integer, that is, for m a" 0, 3 modulo
4, as is well known.
SPLIT CLIFFORD MODULES OVER A HILBERT SPACE 7
In the infinite-dimensional case there are always plenty of solutions r(x) to the
equations (3.6). They can all be described by fixing a full set A ‚" X of represen-
tatives of X modulo " *" (" + 1), defining r(a) on A arbitrarely and extending it
to all of X according to the rules
r(1 - a) = r(a)-1, r(x + ´k) = (-1)kck(x)-1r(x)ck(1 - x).
In this sense, any GW module with ½ invariant and µ quasi-invariant under trans-
lation by 1, splits over R . But since the set A is not measurable, the resulting
operators r(x) may not be either. The difficulty of deciding whether a given GW
module actually splits, i.e., whether there is a measurable r(x), depends on the
module, as we see next. Assume now that Z is infinite dimensional and separable.
Corollary 3.7. If µ is discrete and H is irreducible over C, then it is irreducible
over R. In particular, this is the case for the Fermi-Fock representation.
Proof. A proper invariant real subspace U ‚" H must be a real form of H. Indeed,
U )" iU is complex, invariant and proper, hence equal to {0} by irreducibility. By
the same reason, U •" iU is complex, invariant and not {0}, hence equal to H. Since
the measure µ is discrete and H is irreducible, µ is concentrated in some translate
xo + ", xo " X [GW]. As X is infinite, 1 - (xo + ") and xo + " are disjoint. This
implies that µ and µ are not equivalent and therefore H cannot have an invariant
Ü
real form.
The Fermi-Fock representation corresponds to the triple (µ, 1, 1), where µ(´k) =
1 for all k and µ(X - ") = 0, hence it is supported in the discrete set ".
Corollary 3.8. Let µX be the Haar measure on X. Then the representation asso-
ciated to the triple (µX, 1, 1) is irreducible over R.
Proof. Irreducibility over C follows from the ergodicity of the Haar measure, as
argued in [GW]. Irreducibility over R follows from Theorem 4 in [G], which implies
that any function r(x) satisfying
r(x + ´k) = (-1)kr(x)
must be non-measurable.
While determining if a GW module has an invariant real form may not always
be easy, it is still possible to obtain all the real C(H) modules up to orthogonal
equivalence. The next and final Theorem shows that any pair consisting of a GW
module together with an invariant real form, is unitarely equivalent to one of the
following standard type.
Let
•"
K = Kx dµ(x)
X
be a direct integral of real Hilbert spaces satisfying
Kx+´ = Kx, K1-x = Kx
for all ´ " " and almost all x " X. With H = C " K and Hx = C " Kx,
•"
H = Hx dµ(x).
8 ESTHER GALINA, AROLDO KAPLAN AND LINDA SAAL
Clearly, Hx+´ = Hx = H1-x, Kx is a canonical real form of Hx and K one of H.
Denote by Å» the corresponding conjugations. By definition, for f " H,
Å»
f(x) = f(x).
This determines corresponding conjugations on the endomorphisms A of these
spaces by (f) = A(fÅ»).
Theorem 3.9. If (µ, ½, {ck}) is a GW module modeled on the Hilbert space H with
µ H" µ, ½(1 - x) = ½(x), ck(1 - x) = (-1)kck(x),
Ü
then
dµ(1 - x)
HR = {f " H : f(x) = f(1 - x)}
dµ(x)
is an invariant real form of H. Conversely, any pair consisting of a GW module
together with an invariant real form, is unitarely equivalent to one of that type.
Proof.
For the first assertion we just need to apply Theorem (3.5) with r(x)f(x) = f(x).
For the converse, start with any split module and let S, T, r(x), be as in the
proof of (3.5). In it we have fixed identifications Èx : Hx H1-x in order to
dµ(x)
Ü
define the operator T . Explicitely, letting Åš(x) = , the actual operator is
dµ(x)
(Ä f)(x) = Åš(x)f(1 - x) and
-1 -1
(T f)(x) = Èx (Ä f)(x) = Åš(x)Èx (f(1 - x)).
Ü
If Èx : Hx H1-x is another identification, the corresponding operator will be
-1
Üf)(x) Ü-1((Ä Ü-1
(T = Èx f)(x)) = Èx ć% Èx ć% Èx ((Ä f)(x)) = hx((T f)(x))
Ü-1
with hx = Èx ć% Èx unitary on Hx.
Define u(x) = Hx Hx by
u(x)f(x) = r(x)f(x).
These are C-linear and norm-preserving, hence unitary and measurable in x, be-
cause r(x) is so. Change the chosen identification Hx H" H1-x by u(x), so that the
new operator T will be
Üf(x)
T = u(x)-1(T f)(x).
For this identification and the equivalent real structure
Ü
S = u-1Su,
the function r(x) is
Ü
Ü Üf(x)
r(x)f(x) = ST = u-1Suu(x)-1(T f)(x) = u(x)-1ST f(x) = u(x)-1r(x)f(x) = f(x).
Ü
SPLIT CLIFFORD MODULES OVER A HILBERT SPACE 9
This effectively shows that, up to equivalence, we may take r(x) to be conjugation
with respect to any basic flat field of real forms.
Since the operators ck(x) do not depend on the identification Hx = H1-x, the
relations (3.6) must still hold for the new r(x), which translates into ck(1 - x) =
Ü
(-1)kck(x).
As an application, let us describe the simplest irreduciblerepresentation of CC(Z)
that splits over R., i.e., as a representation of C(Z). The two irreducible summands
will be equivalent and truly real, in the sense that they do not arise from any
complex representation of CC(Z) by restriction of the scalars.
Let µ = µX be the Haar measure on X and ½ = 1, so that Hx = C, Kx = R,
H = L2(X)
(dropping the reference to the Haar measure) and the conjugation Å» is given by
Å»
f(x) = f(x). As ck s, which are now functions from X to the circle T, choose
c2 (x) = 1
4 +3
c4 +1(x) = (-1)x .
4 +1
c4 +3(x) = (-1)x
Theorem 3.11. These functions satisfy (2.3) and, therefore, (2.5) defines a repr e-
sentation of CC(Z) in L2(X). This representation is irreducible. The real subspace
L2(X)R = {f " L2(X) : f(1 - x) = f(x)}
is invariant and irreducible under C(Z). Furthermore, the representation so ob-
tained does not arise from any representation of CC(Z) by restriction of the scalars.
Proof. The irreducibility of the corresponding GW module H follows from the
ergodicity of the Haar measure, as in (3.8).
It is straightforward to check that the functions ck satisfy (2.3) and (3.10), so,
by Theorem (3.9), the corresponding J must leave the real form HR invariant. Of
course, this can be deduced by direct calculation as well, using (2.5). HR must
be irreducible under C(Z), since any closed invariant subspace generates a closed
CC(Z)-invariant subspace in H.
Finally, suppose that the representation of C(Z) in HR could be extended to
one" CC(Z) (in HR itself). Denote by T the operation representing multiplication
of
by -1: T is an orthogonal complex structure in HR commuting with C(Z). Its
unique C-linear extension to all of H = HR •" iHR is unitary and commutes with
all the J. As we have already mentioned, this implies that T is given pointwise, by
an operator-valued measurable function k(x): (T f)(x) = k(x)f(x). In the present
case, k(x) is complex valued. Since k(x)2 = -1, we can write it as
k(x) = (x)i
for some measurable : X {Ä…1}. The condition for T to leave invariant the real
form HR and to commute with the representation J amount to, respectively,
(1 - x) = - (x) (x + ´k) = (x)
10 ESTHER GALINA, AROLDO KAPLAN AND LINDA SAAL
for almost all x and all k. The second equation implies that is actually constant on
each "-equivalence class. [According to [Go] ???], must then be constant almost
everywhere, contradicting the first equation. Hence, no such T can exist.
Representations with these parameters can be realized on the standard L2(T)
of complex-valued functions on the circle since, as a measure space, (X, µX ) is the
same as the interval (0, 1) -hence to the circle, equipped with the Lebesgue measure
and dimHx = 1. In this identification, translations in X do not correspond to rigid
rotations of T. However, the operation x 1 - x in X (changing all the elements
of x) does correspond to x 1 - x on (0, 1) which, on T ‚" C, becomes ordinary
complex conjugation. The operators ck can then be viewed as functions
ck : T T,
Ü
satifying, of course, (2.3). According to Theorem 3.9, those that split over R can
be realized with
Å»
L2(T)R = {f " L2(T) : f(t) = f(t)}.
as the invariant real form. The condition on the functions ck for this real form to
be invariant under the J s becomes
Å»
(3.10) ck(t) = (-1)kck(t).
for t " T.
References
[BD] T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Springer-Verlag,
1985.
[BSZ] J. Baez, I. Segal and Z. Zhaou, Introduction of algebraic and Constructive Quantum
Field Theory, Princeton University Press, 1992.
[C] E. Cartan, Nombres complexes, Encyclopédie des Sciences Mathématiques 1 (1908),
329 468.
[G] V. Ya. Golodets, Classification of representations of the anti-commutation relations,
Russ. Math. Surv. 24 (1969), 1 63.
[GKS] E. Galina, A. Kaplan and L. Saal, Infinite dimensional quadratic forms admitting com-
position (to appear).
[GW] L. Gårding and A. Wightman, Representations of the anticommutation relations, Proc.
Natl. Acad. Sci. USA 40 (1954), 617 621.
CIEM - FAMAF, Universidad Nacional de ordoba, Ciudad Universitaria, (5000)
Ć
Córdoba, Argentina
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