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Twisted second quantization
Vol. 27 (1989)
REPORTS
TWISTED
ON
SECOND
W. Pusz Department
of Mathematical
MATHEMATICAL
Methods
No. 2
PHYSICS
QUANTIZATION
and S. L. WORONOWICZ in Physics, Faculty Warszawa, Poland
(Received
June
of Physics
University
of Warsaw.
20, 1988)
A formalism of a second quantization procedure based upon the twisted SU(N) group is constructed and the related twisted canonical commutation relations (TCCR) are investigated. In a particular case, these relations reduce to classical CCR. Irreducible representations of TCCR are described. The Stone-von Neumann uniqueness theorem does not hold for the general TCCR.
0. Introduction The notion of a quantum group has been the subject of intensive study in the last years (cf. [3], [4], [S]). In particular, in [lo] the general notion of a compact matrix pseudogroup S,U(N) was studied. This group can be treated as a deformation of the classical SU(N) group and thus a natural question arises whether quantum groups can be used for description of symmetries of quantum-mechanical systems in the same way as classical groups are. The aim of the present paper is a description of the second quantization procedure based upon the S,U(N) group as the symmetry group. It leads to an essential modification of the well-known classical formalism of the Fock space. In particular, ,u-depending coefficients appear in new canonical commutation relations. The basic notions and definitions of the theory of compact matrix pseudogroups and twisted SU(N) group can be found in [lo] and [l I]. For the convenience of the reader we recall some facts and concepts. Let G = (A, u) be a compact matrix pseudogroup, CZ?the algebra generated by matrix elements of u, V a vector, space and u: V-+V@d
(0.1)
a linear map. We say that u is a (smooth right) action of G on the vector space if the following diagram OV@coe u:
“Qlid
V@d
-V@d@O ~2311
Iid@@
232
W. PUSZ
and S. L. WORONOWICZ
(where @ is the comultiplication associated with G) is commutative. If dim I/< co then this notion coincides with that of representation. Remark. Throughout the paper, all tensor products are taken in the algebraic sense (no completion is performed). We shall also write shortly “G-action” meaning smooth right action of G (only such actions will be considered in the paper). If u and w are G-actions on vector space V and IV, respectively, then the direct sum ZI@ w is a G-action on I/@ W defined in the obvious way. By Mor(u, w) we denote the linear space of all operators intertwining u and w: for any linear map s: V--+IV, SE Mor(v, w) if and only if the diagram
is commutative.
If x E V, YE W then UX =
CXiObiY
WY =
CyjObj,
where X~EV, yi E IV, a,, bjesxZ and the formula uaw(x@y)
= Cxi@yj@aibj Lj
defines the G-action on V@ W called the tensor product of ZJ and w. If the vector space V has an additional structure then usually we shall demand this structure to be preserved under the action of G. In particular, if I/= Y is an algebra and u: sp+Y@yc4 (0.2) is a linear map then we say that a is a G-action on the algebra Y if CIis a G-action on the vector space Y and the multiplication mapping m: Y4P03q10q2+q1q2E~ is intertwiting: m E Mor(aa u, cx). This is the case if and only if (0.2) is an algebra homomorphism, i.e. a(q,-q,) = cx(q,)a(q,). If r is a bimodule over such an algebra Y and /I is a G-action on r, then we say that (r, /I) is a couariant Y-bimodule if right and left multiplications mR: r@Y3co@q-+-*qEr, mL:
are intertwining:
.m3r3qmwq~d-,
mR E Mor(P@ u, fi), mLE Mor(uO P(44
= Q)PW?
PC+?) = P(4u(d
/I, /I). This means that
TWISTED
SECOND
233
QUANTIZATION
for any w E r, qE9 (r@ .d is in a natural way a bimodule over Y@d). In this paper we deal with the compact matrix pseudogroup S,U(N), where 0 < ,LL< 1, introduced in [i 11. It is (cf. [ll]) known that the theory of representations of S,U(N) is similar to that of SU(N): irreducible representations are labelled by Young diagrams and formulae for dimensions and multiplicity are the same as in the classical case. In particular, 111 corresponds to the (irreducible) fundamental representation u and uau has a decomposition
This shows that dimMor(ua u, uau) = 2. This space is spanned by two operators: I and or. Let K (dim K = N) be the carrier space of the fundamental representation u of S,U(N) and let {Q, . . . . eN} be the canonical orthonormal basis in K; then PEj 0 si cJp{Ei 0
Ej)
Ej@Ei
=
{
pClej@si+(l -$).si@ej
for for for
i
(0.3)
i > j;
a,*=c~,and
Spo,={l, -cl’}. The eigenspaces of oP are the carrier spaces of the -representations 1-l-l and I_) and ‘analogously as in the case p = 1 we shall say that YE K 0 K is symmetric if a,(u) = v and can easily check that the elements
&iOEiT
pLEi
0
form a basis in the subspace Ei@Ej-/LEj@Ei,
if a,(u) = --k12. Using (0.3), one
antisymmetric
Ej + Fj @ Ei,
i
of symmetric i
(i,j=
elements (i,j=l,2
1,2 ,...,
N)
(0.4)
of K @ K. Similarly, ,...,
N)
(0.5)
form a basis for the antisymmetric subspace. The paper is arranged as follows: In Section 1 we remind the classical second quantization procedure. Section 2 contains our main theory. We show how in a natural way one can define the Fock space and the creation and anihilation operators covariant with respect to the action of S,U(N). We derive the twisted canonical commutation relations (TCCR) satisfied by these operators. In the last section we discuss in full generality the representations of TCCR. The paper is restricted to the Bose case. The fermionic case will be considered in the forthcoming paper [6].
234
W. PUSZ and S. L. WORONOWICZ
1. Classical second quantization The main purpose of this section is the presentation of the classical construction in a manner convenient for further generalization. We stress the role of the symmetry group in this construction. For technical reasons we consider only N-level quantum systems (IV < co). Let K be an N-dimensional Hilbert space with fixed orthonormal basis sN} and let Y(K) denote the tensor algebra over K. The symmetric 1% s 2, *..> tensor algebra is Y(K) = Y(K)/J where the ideal J is generated
by the antisymmetric
elements
of degree
i
&i@&j-&j@Ei,
2: (1.1)
The algebra Y(K) is endowed with the natural grading. The multiples of unity are the only elements of degree 0. The space of elements of degree 1 is canonically isomorphic to K. Let xj be the element of degree 1 corresponding to sj(j = 1, 2, . . . , N). Then Y(K) is the free commutative algebra with unity generated by {x1, x2, . . . , xN}, i.e.’ the algebra of all polynomials of N commuting variables. Let T(K) be the space of differential forms of the first order over the and that the variables x1, x2, , . . , xN. This means that T(K) is a Y(K)-bimodule linear map d: Y(K)
--f T(K)
such that (i) d(q.q’) = (dq).q’+q.(dq’) (ii) any element WET(K)
for all q, q’eY(K), can be written in the following o = ;
(1.2) way
(dxi).qi
(1.3)
i=l
where qi E Y(K) (i = 1, 2, . . . , N) are uniquely determined, (iii) q.(dxj) = (dx,).q (j = 1, 2, . . , IV) for all qEY(K)
(1.4)
is given. For any QJ of the form (1.3) we set
j=l
Clearly, 6: T(K) - Y(K) is a well defined
linear map.
(1.5)
TWISTED
Let
SECOND
(.I .) be a sesquilinear
form
235
QUANTIZATION
on
Y(K).
For
any
CO= :
(dxj)qj
and
j=l N a'=
C
(dx,)qi
belonging
to T(K) we set
j=l
It turns out that there exists one and only one sesquilinear (old&
form on Y(K) such that (1.6)
= (dolq)
for any qEY(K) and OJET(K). In fact, (.I.) coincides with the well-known scalar product in Y(K) (see [9]). To obtain the correspondence with the usual second quantization formalism we consider linear operators o’= 1, .2, . ..) N)
aj: Y(K)-,Y(K) defined
by the formula dq
=
for any q E Y(K) (so ajq can be regarded consider linear operators
as the jth partial derivative
by 6( ~
(dXj)qj)
=
~
(1.8)
U~qj.
j=l
j=l
Clearly, aj’q = xjq for any qEY(K). Formula (1.6) shows that aj’ is the Hermitian conjugation that the variables x1, x2, . . . , xN commute, we see that ai+ a,? zzza,? a’
Taking
the Hermitian
conjugation,
1,2 ,...)
(i,j=
of aj. Remembering
N).
(i,j=
1, 2, ...) N).
using (1.2), (1.7) and (1.8), one can easily verify the canonical UiUj+
(1.9)
we obtain
UiUj=UjUi
Moreover, relations
of q). We also
(j = 1, 2, . . . . N)
aj’: Y(K)+Y(K) introduced
(1.7)
E (dxj)(ajq) j=l
=
6,jZ +
Uj’ Ui.
(1.10)
commutation (1.11)
236
W. PUSZ
The number
a'ndS.L. WORONOWICZ
operator N
(1.12) j=l
maps Y(K)
into Y(K)
and satisfies the following [JV, aj’] = af
,
[JV, aj] = -aj,
For any non-negative
integers
j=1,2
relations 7 ..-> N.
.
we set
Inl, . . . . nj, . . . . nN) = (n,! . ..nj! . ..~1~!)-~‘~x.~...x;j...x~~.
The elements
(1.14) form an orthonormal ajln,,
(1.13)
basis in Y(K)
. . . . nj, . . . . nN) = nf’21n,,
aj’ 1n1, . . . . nj, . . . . nN) =
(nj+
Jlrln 1, . . . , nj, . . . , nN) = (F
. . . . (nj-l),
and we have . . . . n,),
1)1’21nI, . . . . (nj+ l), . . . . n,), nj)lnl,
(1.14)
(1.15)
. . . . nj, . . . . nN).
j=l
The last formula shows that the homogeneous elements of Y(K) are eigenvectors of ./lr, the corresponding eigenvalue coincides with the degree. At the end of this section we would like to discuss the covariance properties of the constructions described above. The space K can be considered as the carrier Hilbert space of the fundamental representation of the group SU(N). Let us notice that the space of antisymmetric elements (1.1) is invariant under the natural action of SU(N) on Y(K). The elements of SU(N) are represented by isomorphisms of the graded algebra Y(K). The actions. canonical embeding K + 9’(K) intertwines the corresponding Finally, one can define the action of SU(N) on T(K) in such a way that T(K) becomes a covariant Y(K)-bimodule and that 6 and d are intertwining operators. The scalar products on Y(K) and T(K) are SU(N)-invariant since they are uniquely determined by (1.6). We can summarize the second quantization procedure for the group SU(N) in the STATEMENT.The second quantization procedure for the group SU(N) produces the sextet
{K J, Y(K), M),
d,
a}
TWISTED
SECOND
237
QUANTIZATION
such that
(i) K is the carrier of Hilbert space of the fundamental
(unitary) representation
of
=J(N); (ii) J is the ideal of the tensor algebra F(K) over K generated by the set of antisymmetric elements of second degree; (iii) Y(K) is the ,factor algebra F(K)/J endowed with the canonical action of SU(N) and the canonical (invariant) scalar product; (iv) T(K) is covariant P’(K)-bimodule of differential forms of degree one and d: Y(K) is the corresponding and f(K);
external derivative intertwining the actions of SU(N) on Y(K) 6: c(K)
(v) is a linear mapping
--f T(K)
-*Y(K)
such that d((dx)q)
= xq
for any XE K and qEY’(K), 6 intertwines the actions of SU(N) Y(K), and 6 coincides with the Hermitian conjugation of d. 2. Twisted
second
on T(K)
and
quantization
In this section we develop the formalism of second quantization based on the twisted SU(N) group. We shall use the ideas presented in Section 1 introducing necessary m+odifications. In particular, the classical differential calculus should be replaced by a non-commutative one. Let K be the carrier Hilbert space of the fundamental representation of S,U(N) (dimK = N), F(K) the tensor algebra over K and
where J, is the ideal in F(K) generated by a $N(N- 1)-dimensional S,U(N)-invariant subspace of K 0 K (this subspace corresponds in the twisted case to the subspace of antisymmetric tensors; cf. (0.5)). F(K) is a graded algebra and carries the natural action of S,U(N). Moreover, the ideal J, is S,U(N)-invariant and any element of J, is a sum of homogeneous elements of J, of the order 3 2. Therefore, we have the induced grading and the induced action of S,U(N) on 5$(K). The subspace of all elements of degree 0 coincides with the set of all multiplies of unity; the subspace of all elements of degree 1 is canonically isomorphic to K and generates $(K). Denoting by to ar, E;, . . ., eN ({ej: j = 1, 2, . . ., N) is Xl, x2, . ..> xN the elements corresponding
W. PUSZ and S. L. WORONOWICZ
238 the canonical
basis in K) and taking
(0.5) we have
into account for
xi.xj = pxj.xi
i
(2.1)
Clearly, YP(K) can be identified with the algebra of all polynomials x1, x2, *.., xN satisfying the relations (2.1). Let Ta be a covariant Y@(K)-bimodule and let d: $(K) be a differentiation, and rP and
of N variables
+ rlr
i.e. d is a linear map intertwining
(2.2) the actions of S,U(N)
on YM(K)
d(a.b) = d(a)b +ad@) for any a, b ES$(K). 41, q2,
ee.9
q,,,E YP(K)
We assume such that
that for any element
OE rp there exist unique
(dx,)q,.
w = i
(2.3)
j=l
Then we say that (r,, The linear map
d) is a first order differential calculus over $(K). 6: r,+LqK)
defined
for any OE~,
(2.4)
of the form (2.3) by 60.I = ;
xjqj
(2.5)
j=l
is well defined and intertwines prove
the actions
of S,U(N)
on r,, and Y@(K). We shall
2.1. Assume that N, > 3. Then there exist precisely two ‘non-isoinorphic differential calculuses over YP(K). In the first case,
THEOREM
first order
for
i < j,
P2 (dxj)xi
for
i=j,
P (dxj)xi
f or
i>j.
p(dxj)xi-(1 xidxj = 1 In the second
-p2)(dxi)xj
(2.6)
case,
xidxj =
p- l(dxj)xi
for
i
pL2(dxj)xi
f or for
i=j,
pml(dxj)xi-(1
-p-‘)(dxi)xj
i>j.
(2.6*)
TWISTED
Remark:
over yfi(K) and (2.6*).
SECOND
239
QUANTIZATION
If N = 2 then there exist two families of first order differential calculi labelled by a complex parameter y. Setting 11= 0, we obtain (2.6)
Proofi Assume that (r/,, d) is a first order differential consider the linear mappings:
calculus over yP(K).
We
and $1 K@Yp(K)3ej@q4dxj)q~l-p actions of (i, j = 1) 2 3 ...> N). Clearly, these mappings intertwine the corresponding S,U(N). Remembering that qj in (2.3) are determined uniquely, we see that $ is S = $-‘.cp: bijective. Therefore $ - I exists and t.he composition S: K@K+K@cYti(K) intertwines
the action
of S,,U(N)
(2.7)
on K @ K and K @Yfl(K).
We know that
where K, is the subspace of homogeneous elements of order n. For p = 1, the representation of S,U(N) acting on K, is irreducible and the corresponding Young diagram (e.g. [l]) consists of n boxes placed in one raw. The same is true for p < 1 (cf. [l 11). The fundamental representation (the action of S,U(N) on K) corresponds to the one-box Young diagram. It is known ([ll], [l] Chapter 8) that 1110 111= 1-10 l_l_l. IL
l~lo’l-J~l...l~l
=~p.l~loI
ntimes
n limes
(2.8) -_ -_I l...lIl. n+ltimes
(2.9) .
Therefore for II # 1 the representations of S,U(N) acting on K 0 K and K 0 K, are disjoint. (This argument does not work if N = 2. Then the representations -_-corresponding to I---I I / and I-- ) I are equivalent.) This means that S(K @ K) c K @ K.
1-l
According
to (2.8), the space of all intertwining
operators
S: K@K+K@K is 2-dimensional:
s = al+@, where (Y, /Jo C and gp is given by (0.3).
(2.10)
240
W. PUSZ
For any i,j=l,2
,...,
N we have XidXj =
and, using (2.10), we obtain plu(dxj)Xi XidXj
=
(‘+
x,x,dx,
formula,
= {c~+P(f
=
+
Xl
for
i
for for
i =j,
a(dXi)Xj
-~“)](dxi)xj+lj~(dxj)xi
(2.11)
i > j
one can check that +Ca+P(l
-P’)1PPvx,bI~
1X2X1+~(a+p)Ca+B(1-~2)1(~X2)X:..
the above expressions
(cf. (2.1)), we get
c+t++(l-p2)] Moreover,
~(s(&iOEj))
-~2)1~+(~+P)lj~2}(dX1)X2X1
PX2XldXI = (~+B)PP”@
Comparing
cP(EiOEj)
P)(dxj)xi
[~+/?(l for some a, DEC. Using the above
and S. L. WORONOWICZ
= 0.
(2.12)
using again (2.11) we obtain: d(x,x,-px2x1) .
= (dx,)x,+x,dx,-~(dX,)X,-~X,dX, = (1 +a-p~2)C(dxI)X2-~L(~X2)X11.
Therefore, 1 +cc-/!I/_? = 0.
(2.13)
Solving the equations (2.12) and (2.13), we see that either tl = ,u2- 1 and b = 1 (then (2.11) coincides with (2.6)) or CI= 0 and fl = p- ’ (then (2.11) coincides with (2.6*). To complete the proof we have to construct the first order differential calculus over YP(K) such that (2.11) holds. Let p be the free right module over the tensor algebra F(K) with N generators d”El, dE,, . . . . JEW. Any element GE r” is of the form c3 = t j=l
where
ql,
introduce products
(2Ej)qj
(2.14)
determined elements of Y(K). We have to q,, . . . . qN are uniquely the Y(K)-bimodule structure on r”. To this end it is sufficient to write the eidd~j (i, j = 1, 2, _. ., N) in the form (2.14). Motivated by (2.11) we set B~(d”Ej)Ei+ Eid”Ej
=
~(~Ei)Ej
for
i
(a + P)(d”Ej)&i
for
i=j,
[cr+/?(l
for
i > j.
-~‘)](JE~)~~+~~(~“E~)E~
(2.15)
TWISTED
SECOND
241
QUANTIZATION
We shall also use the differentiation
introduced
2: Y(K)-+r”
(2.16)
J(l) = 0,
(2.17)
by the formula
.
.
i,=
11,123 *-*,
1,2,...N;
n=
1,2,...
Let i,j = 1, 2, . . . . N and Eij (i < j) denote the elements (0.5). Then in virtue of (2.15), (2.13) and (2.12), we get d”Eij
=
(2.18)
(1 +0Z-~,/A2)[(d”EJ&j-/A(d&j)&J = 0
and one can check that for k = 1, 2,
.
.
.
,
N:
-ap(dEi)&jk+~aB(~&j)&ik+~282(d&k)Eij
for ap~(~&i)&kj+102~2(~&k)&ij+[CI+P(l
Eij~&k
=
i
1
for
i
(2.19)
-~2)1C(~&i)~kj-~(~&j)~ki]
P2~2(~&k)Eij+P~[a+P(1
for
k
for
k=i
or k=j.
Let as before J, be the ideal of F(K) generated by elements (0.5) and let N, be the subset of r” consisting of all elements 6 of the form (2.14) where of r”, l’JB c N, and, in 41, q2, .**, qNE J,. Clearly, N, is a sub-F(K)-bimodule virtue of (2.19), J,r” c N,. Therefore, for any ~1)E F/N, and qE J, we have oq = qw = 0. This means that fa = r/N, can be considered as a bimodule over S$(K) = F(K)/J,. In virtue of (2.18) Jq E J, for any q E J,. Therefore there exists a linear mapping
such that the diagram F(K)YJW
=
~WJ,
1
-
I
_ r
d
Ta = F/N,,
’
242
W. PUSZ
and S. L. WORONOWICZ
(where the vertical arrows denote the canonical projections) is commutative. Clearly, d is a differentiation (the equality d(ab) = (da)b + adb follows immediately from the same formula for d). To finish the proof, let us notice that 9(K) and r” are the carrier spaces of the natural actions of S,U(N). Moreover, the whole construction is S,U(N)-covariant (the form of equations (2.15) is invariant under the action of S,U(N) so r” is a covariant bimodule over 9(K), J, is the invariant ideal in Y(K), N, is the invariant sub-module of r” and d is the intertwining mapping) so YP(K) carries a natural action of S,U(N), rG is a covariant $(K)-bimodule and d is an intertwining operator. n To make further considerations more concrete we deal with the first order differential calculus (I-,, d) over Y,(K) defined by (2.6). Let u; and aj (j=1,2,..., N) be linear mappings acting on YP(K) and introduced by the formulae (cf. (1.7) (1.8)): (2.20)
6( 2 (dxj)qj) = 2 afqj, j=l
j=l
dq = -f (dxj)ajq.
(2.21)
j=l
Like in the untwisted
case (p = 1) we call uf(uj
resp.) .creation (anihilation
resp.)
operators.
Clearly, a,? q = xjq,
j=l,2
(2.22)
3 . ..> IV.
Formula (2.21) means that uj is a partial derivative in the jth direction. Now we look for the canonical commutation relations in the twisted In virtue of (2.22) and (2.1), we get U’
Uj’
=
~Uj+
i
for
Ui+
case. (2.23)
For any i = 1, 2, . . . . N and qEYP(K) d(Xiq) = (dXi)q + Xid~ = (dx,)q + Xi i:
(dxj)Ujq
j=l =
(dXi)q + C ~(dxj)xiajq
j
+~2(dxi)xiaiq
+
+ 1 C~L(dXj)Xi~ja-(‘-~2)(dxi)xj~jql = ~;i(dxj)/m:ujq+(dxi)[I+p2a:a;-l(1-p2)
2 k>i
U:ak]q.
TWISTED
On the other
SECOND
243
QUANTIZATION
hand,
c
d(Xiq) = d(U+q) =
(dXj)ajai+q'
j=l
Comparing
the two expressions,
‘jaT =
we obtain
r
pa+ aj ~+~2a+a~-(l
-p2)
for i #j, 1 &+a, for i = j.
(2.24)
k>i
For any i, j = 1, 2, . . . , N, i > j we consider
the operators
Aij = aiaj-pajai. Using (2.24) one can check that
,a2a: Aij Aija:
p3a+ Aij-~(1-~2)
=
C a: A,j
for
k # i, j,
for
k = i,
for
k =j.
s>i
i
I ~~afA~~-pL(l-_~)
C
a:Ais+(l-p’)
C alAPi
(2.25)
p>i
j
In virtue of (2.22), any monomial in YP(K) can be obtained by multiple action of the creation operators on 1. Since dl = 0, we have a,1 = 0 and Aijl = 0. NOW (2.25) means that Aijq = 0 for any monomial 4 E Y,(K), i.e. ajai = ,uaiaj
for
i
Relations (2.23), (2.24) and (2.26) coincide with (1.9k(1.11) for p = 1. integers. Writing for any integer Now let n,, n2, . . ., nN be non-negative 1 -/P A,(n) = ~ l-p2 and remembering j=l,2 , ..*, N
that d is a derivation
aj(@.
. .x7”. . .x;l)
(2.26)
n (2.27)
and using (2.1), (2.21), (2.6), we get for
L = A,(nj)pk”~“,N . . .xjnj-1. . .x;l.
(2.28)
Let lnlY n 2, *.., nN) = [ fi FactP( j=l
- iii xi,+. . x2xX;’
(2.29)
W. PUSZ and S. L. WORONOWICZ
244 where
Fact,,(n) = fl ~ (Fact,(n) = n!, (2.29) is the twisted Clearly,
=
version
of (1.14)).
(In,, n,, . . .) Q): n,, n,, . .*, nN = 0, 1, 2, . . .}
(2.30)
is a basis in the vector space YP(K). Using (2.22) and (2.28), we obtain X.nk ajlnl, . . a7 nj, . . *> nN) = Ap(nj)“‘P”‘ln,,
afIn,, .
..)
. . . , nj- 1, . . . , TIN),
n,, . . . . no) = A,(nj+ l)1i2$‘jln1,
(2.3 1) . . ., nj+ 1 > . . . . n,L
Let (.I.) be the scalar product on Y*(K) such that the basis (2.30) is orthonormal. Then (cf. (2.31)) aj and UT are mutually Hermitian conjugate a? = uj’, (a;)* = aj. Taking into account (2.20) and (2.21) for any OE~, and 4 E$(K), we get
k4&)rp = (~014 where (.I.),-, is the scalar product
on Tr introduced
(2.32) by (2.33)
Conversely, assume that (. I .) is a scalar product UT = UT, (a;)* = uj and
on Yfl(K) such that (2.32) holds. Then
fxj414’)= (“jfqld) = (4l’jq’)T (41xjd) = (41”f 4’) = (“jq14’) for anyj = 1, 2, . . . . N, q, q’ E Y”,(K). We know that uj (j = 1, 2, . . . , N) decreases the grade of homogeneous elements. Therefore the above formulae show that (. I .) is determined uniquely by (111). This way we have showed that (2.32) defines the scalar product on YP(K) uniquely up to a positive factor. This fact implies that the scalar product is S,U(N) invariant. Let %=ii,d=
;
afaj
(2.34)
j=l
be the (twisted) number operator in the twisted case (cf. (1.12)). It is the linear operator on YP(K) commuting with the action of S,U(N). Using (2.31) we get
Wn1, . . . . nN)= AP(i nj)lnl, . . . . nd. j=l
(2.35)
TWISTED
SECOND
245
QUANTIZATION
Therefore the eigenspaces of % coincide with homogeneous components of Y@(K). Since A,(n) < l/(1 -p’) (n = 0, 1, 2, . . . ), the operator % is bounded. Clearly, this fact implies that d, 6, ai+, aj are also bounded. Let us notice that for ,u = 1, (2.31) and (2.35) coincide with (1.15). It turns out that the first order differential calculus constructed above gives rise to a higher order differential calculus. We would like to make a short remark on this subject. Let r,“” = Yfl(K), Ts@I = r and rFn (for n > 2) be the tensor product (over YP(K)) of n-copies of Tc. We Consider the graded tensor algebra
Let 6 be the ideal in rf
generated
by the elements
pdxi @ dxj + dxj @ dx, dx, 0
k,i,j=
dx, ,
Clearly, 6 is a direct sum of its homogeneous the quotient algebra
for
of the form i
(2.36) 1,2 ,...,
N.
components
of grade > 2. Therefore,
r; = r?p has a natural grading and the set of all elements of grade 0 and 1 coincide with YP(K) and Trc, respectively. The multiplication in ri will be denoted by “. The algebra r; is a generalization of the external algebra. One can easily check that any element QE r; of degree n is of the form e =
dxi’ A dxiZ A . . . A dxinqi,iz,,,i,
C il
where
qiliz...i,
E Yp(K)
are uniquely
determined.
We set de =
C
(-
l)“dx” A dxi2
A ... A
dx’”
A
dqi,iz,.,i,.
il
Then d:
r;-,r;
(2.37)
is a graded derivative, cY(de) = 8~ + 1 for any homogeneous QE r;, on elements of degree 0 the derivative d coincides with (2.2), (2.21) and due to (2.36) and (2.26): d(dQ) = 0
for any QE r,‘J . Clearly, the above construction is fully covariant: the ideal G is invariant with respect to S,U(N) (so there is a natural action of S,U(N) defined on r,^) and (2.37) is an intertwiner.
246
W. PUSZ
and S. L. WORONOWICZ
To introduce creation and anihilation operators we have used the first order differential calculus defined by (2.6). We shall discuss briefly the creation and anihilation operators related to the calculus based on (2.6*). Let us notice that the operation consisting in replacing p and < by ‘p- ’ and >, resp., transforms (2.6) into (2.6*). Moreover, this operation leaves equations (2.1) unchanged. Therefore, applying this operation to equations (2.20)(2.35) we obtain the corresponding formulae of the second quantization formalism based on (2.6*). In particular, equations (2.23), (2.24). (2.26) (2.29), (2.31) and (2.35) take the following form: ,+a,? P-
ajai+ =
= p-1+$
for
‘U+ aj
I+pd2au+ai-(l
_p-‘)
(2.23*)
i >.i,
C a$ak
for i fj, for i=j;
(2.24”)
k
/.l
=
-‘aiUj
. . . . nN) = [ fi
In,, n2,
(2.26*) (2.29*)
Fact,~1(nj)]-1’2x;1x;Z...x”,‘,
j=l
ajln,,
i>j,
for
xnk
-
. .., nj, . . ., nN) = A,m~(nj)112p
k’JInl,
. . . . nj-
1, . . . . n,),
- Znr afln,,
. . . . nj, .... nN) = A,-l(nj+
1)“2p
Wn 1, . . . . nN) = &I(;
""ln,,
nj)(n,,
.... nj+
1, . . . . nN),
. . . . nN).
(2.3 1*)
(2.35*)
j=l
In this case the operators 91, d, 6, aj, a’ are no longer bounded. (2.23*) and (2.26*) coincide with (2.23) and (2.26) resp. 3. Representations
of twisted canonical commutation
Let us notice that
relations
This section is devoted to irreducible representations commutation relations (TCCR). A system {a,, a2, . . . . +, a:, ai, . . . . UN+}
of twisted
canonical
(3.1)
of closed linear operators acting on a Hilbert space X such that aj* 3 aj’ for of K‘CR if: j = 1, 2, . . . . N is called a representation (i) There exists a dense linear subset 9 c .8 which is contained in the domain of any product of two operators from the set (3.1). (ii) The system (3.1) satisfies TCCR on 2, i.e. for any XE 3 a+a,t
X = pa;
Ujai_X
=
Ui+X
/lUiLljX
for
i
(3.2)
i
for
(3.3) for i fj,
aja’x
=
x+~2a+ai.~-(I-~z)
C a:a,x k>i
for
i = j.
(3.4)
TWISTED
(iii) A positive
linear
SECOND
operator
OLJANTIZATION
247
of the form F afaj
(3.5)
j=l
is essentially
self-adjoint
on 9.
The representation (3.1) will be called irreducible (cf. [2] Definition Definition 3.1) if the only operator CEB(X’) such that
2.5, [7]
(XICajY) = (aj’XICY),
(3.6)
11)= (LZjXlCY)
(XICUf
for any x, y~6B and j = 1, 2, . . . . N is a multiple of unity; i.e. C = JU for some 1 E C. Let p, q be non-negative integers such that p +q < N and EE]~‘, 11. We consider the set SE,,,, c RN consisting of the elements of the form 2n1 E,
p2nz E
IJ
_Z+ , ..., ,tt2QE, 0, . . . , 0, ~1-P
where nl, . . ., n,; m,, . . ., m4 are integers n, < n2 < . . . < n,
=
...3
(3.7)
v
m,3m23...>mm,30
and
(P2@1>
7 ..”
such that
If p = 0, then the value of E is irrelevant. N, let ‘j be a linear mapping For j= 1,2,..., Tj@
_n% 2
P2&?j,
Qj+l,
introduced ...2
by the formula
@NJ
(3.8)
for any Q = (Q,, . . . . eN)~RN. THEOREM 3.1. Let E E]P’, 11, p, q be non-negative integers such that p + q < N, S = S&p@ H = 12(S) and {IQ): QES} be the canonical basis in 2. We consider linear operators cj (j = 1, 2, . . ., N) introduced by the formula
cjl@)=
1)1’2 ITJ"@)
(@j-@j+
(3.9)
f or any QES (notice that Q~-Q~+~ = 0 tf T;‘Q$S).
These operators are closable and the closures will be denoted by the same symbols.
[f p+q = N or p = 0, then we set aj = cj, a; = CT (j = 1, 2, . . . , N). Zf p+q < N and ‘p > 0, then we set aj = cj, a; = CT for all j # p and a = e@cp, P
+
ap=e
-i”cp*
(q
E
[O,
27c[).
Then the system {a,, a2, . . . . aN, a:, a,f, . .., a$} is an irreducible representation of TCCR. Conversely, any irreducible representation (3.1) of TCCR is unitarily equivalent to one of the representations described above.
W. PUSZ
248 Proofi Let 9
and S. L. WORONOWICZ
be the linear span of {IQ): QES): One can easily check that cj*l@)=
(3.10)
(P2@j-@j+l)1'21zjQ)
0’=1,2 )....) N) and ~2Qj-ej+ 1 = 0 if rje$S. The proof that the operators introduced by (3.9) and (3.10) (and consequently, of TCCR is purely com{a,, a2, . . ., aN, a:, a:, . . . , a,$}) define a representation putational. The details are left to the reader. The irreducibility follows easily from Proposition 3.5 which is shown later. The rest of this section is devoted to the proof of the last statement of the theorem. Assume now that (3.1) is a representation of TCCR acting on 2. Let YI be the closure of (3.5). Since aj’ aj < %, ajaf
d I+p2afaj
< I+p’%
on 9 for j = 1, 2, . . . . N and 9 is a core for %, we see that D(aj) 3 D(m), D(aj+) 3 I)($%). We would like to show that there exists an essential dense invariant domain for operators (3.1). We start with the following LEMMA 3.2. Let k be a non-negative ajx E D(YI’) and !JIkafx %kajX
integer and XE D(Sk+l).
= af(Z+p291)k~, =
Then aj+xE D(‘Sk), (3.11)
(3.12)
aj[~u2(~ZZ)]kX,
for all j = 1, 2, . . . , N.
Proof: We shall prove (3.12). (3.11) can be proved in the same way. If y, ZEN, then using (i) and (ii), we get (‘JZyIajz) = (afyI~-2(%-z)z)* By the closedness Then we get
argument
the same equality
(‘JZyIajz) = (a~yIp-2(%-Z)z)
for any YE D(g).
This means that ajz ED(%) ~Zajz =
(3.13)
holds for y, z~D(‘%n). Let ZED(‘?R~). =
(_YIaj[~L-2(%-Z)]Z)
and
aj[~-‘(~-Z)]Z.
(3.14)
Suppose now that for some integer yt 2 2 and any x E D (‘W) we have ajx E D(‘W- ‘) and !Rplajx
= aj[p-m2(%-Z)]n-1x.
(3.15)
TWISTED SECOND
249
QUANTIZATION
In particular, if XE D(S”+ 1), then ajx~D(!RnnP1) and (3.15) holds. x E D(S2) and (cf. the first part of the case z = [K2(%-Z)]n-1 a.z = ‘fi”-‘ajx~D($ll). Therefore, u~xED(%“). Using (3.14) and (i.15) with 12 increased by one. The proof of (3.12) is complete
Moreover, in this proof and (3.15)) (3.15), we obtain (by induction). n
Let D, denote the space of analytic vectors for 8. We shall show that ajx E D,,, for any .YED,,. Indeed, using Lemma 3.2 and the inequality aj’ aj d % d (%+ Z)2, we get for k = I, 2, . . . . N: II’JlkUjXl12= ~~uj[~-2(9&-_)]kX/~2 = (Xl[~-2(%-z)]kujfuj[~-2(%-z)]kX) d (~-2kII(%+z)k+1X11)2 and the statement follows. Analogously, using the inequality ujaj+ d Z +p2% ,< (I +p2 ‘%)2, one can show that U~XED, for any XED,. Moreover, relations (3.2)-(3.4) hold for any XE D, (compute the scalar product of both sides of (3.2)-(3.4) with a vector belonging to D, and use the relations uj’ c aj*, uj c (UT)*). Let Sj = uj’uj for j = 1, 2, . . . , N. Clearly, ‘91j(D,) c D,. Let us notice that %x=
i
‘JljxforanyxED
W. Using (3.2)-(3.4), one can prove that the operators
91j
j=l
mutually
on D,.
commute
%jz = ufajufuj so
; j=l
‘%3 d
;
by (3.12) we have on D, that
Moreover,
d uf(Z+p2!R)aj
< afuj%
= 91j%
tRjcJI = ‘$I’. Let (Jr;” d ‘%2k for some integer
k > 1. Then
j=l q(k+l)
and by induction
=
SjyySj
<
Sjgp$qj
=
gpgq@
<
yp+
1)
ST” d S2” for any positive integer n. This means that for XE D,: JplzJxIJ2 = (xpq”x)
< (x)YV”x)
= pVXJ12,
so x is also analytic for all !Rj. Using the FS3 theory ([S], [1] Chapter & 6 Theorem 5) we conclude that ‘Sj 0’ = 1, 2, . . . , N) are essentially selfadjoint D, and mutually strongly commute. PROPOSITION 3.3.For anyj = 1, 2, . . . . N: uj’ = UT. Moreover, a core for uj and u,?. Proof: Let D be a core for ‘% and
We shall prove
x=
{(x,
Y=
I(uj’Y
UjX): 3
x~D}
-y):
11 on
any core for ‘$I is
c bob,
YED}
c A?@%.
that
(ajlD)*= t”j’ ID)*
(3.16)
W. PUSZ
250
To this end it is sufficient E (X 0 Y)‘. Then
and S. L. WORONOWICZ
to show that
X@ Y is dense in S@
(ZlX)+(ulUjX)
Let (z, u)
0,
(3.17)
= 0
(3.18)
=
(ZIqV-MY)
Z.
for any x, ye D. Remembering that D is a core for ‘S and using the estimates u,kuj < %, ajuT < I+$‘%, we obtain the same formula for x, LED. Let XED,. Inserting in (3.18) y = ajx and using (3.17), we get (Z)[Sj+r]X)
= 0.
Since D, is a core for ‘Sj, we conclude that z = 0. Now (3.18) shows that u = 0 and (3.16) follows. Clearly, formula (3.16) implies that (uj’ ID) 3 Combining
this fact with the inclusion
uf .
UT I UT, we see that
(uf lo) = aj*= uj’. Passing
to the adjoint
operators (UjlD)
(3.19)
in (3.16) and using (3.19), we have =
(Uj’
IO)*
=
Uj**
=
Uj.
n
Now we shall analyse the notion of irreducibility. Let C, Q be closed operators acting on 2. Assume that C is bounded. We say that C commutes with Q if CXE D(Q) and QCx = CQx for any x E D(Q). LEMMA 3.4. Let V be rhe set of all bounded a von Neumunn ulgbru.
operators
C satisfying
(3.6). Then %?is
Proof: Clearly, V is a linear *-invariant weakly closed subset of B(Z). We have to show that %? is an algebra. Let CE B(H). Using Proposition 3.3, one can easily check that C satisfies the relations (3.6) if and only if C commutes with the operators (3.1). Using this remark, one shows that C, C, E% for any C,, C, ES’. n PROPOSITION3.5. The following two conditions are equivalent: (i) The system (3.1) is irreducible. (ii) P = 0 and P = I are the only orthogonal projections acting in 2 is a subset of D, invariant under the action of the operators (3.1). Px,
such that PD,
Proof- (i)*(ii). Let P be an orthogonal projection such that for any x, y E D,: PYE D, and ujPx, uj’ PyePD, (j = 1, 2, . . . , N). Then (YIPUjX) = (ufPyIx)
= (Puf Pylx) = (y)PujPx)
= (YIUjPX),
TWISTED
SECOND
251
QUANTIZATION
and finally (3.20)
(YIPUjX) = (aj+ylPx).
This relation is proved for x, y E D,. In virtue of Proposition 3.3, (3.20) holds for any x, YE 9. Using selfadjointness of P and interchanging x and y, we obtain (y(PUj+ X) = (ajy1P.X). Therefore C = P satisfies (3.6) and the irreducibility implies that P = il (i E C) and (ii) follows (P is a projection). (ii)*(i). In virtue of Lemma 3.4, it is sufficient to show that %? contains no nontrivial projections. Let P be an orthogonal projection in %?.Then (cf. the proof of Lemma 3.4) P strongly commutes with aj and uj’ (j = 1, 2, . . , N) and P%x = 9lPx for XE~. Remembering that 9 is a core for 91, we see that P strongly commutes with %. This implies that PD, c D,. Moreover, we have ajPD, = PajD, c PD, and analogously, u,? PD, c PD,. Condition (ii) implies now that P equals 0 or I and the result follows. H Let i Lj =
Q-(1
-(l
Relations
j=
-&iz,
1,2 ) . ..’ N,
k=j
(3.21) and (3.2))(3.4)
(3.2 1)
j=N+l.
-g-9,
imply that for j = 1, 2, ajfaj 3 0,
=
Lj-Lj+l
/12Lj-Lj+1
. . . , N:
=ajajf
(3.22)
3 0,
(3.23)
and Lja:
=
Ljak = Let m?,?
a: Lj
for j > k,
1 p2a: Lj
for j d k,
L, =
for j > k,
akLj
(3.25)
for j d k.
i pp2akLj
e2, . . . . Q,) be the common
(3.24)
spectral measure for the family (L,, L,,
j @&(Ql,
k&
. ..r@k. . ..> @N).
RN
For any XEH
we set d~cc,(e,>e2,
Let XED,.
We shall prove
. .
> ed
=
(xlWel>
e2,
. . . > edx).
that
+,,A dPa,+x(~k@)
‘@) = =
(@k
-@k+
(~2@k-@k+
(3.26)
M&A
(3.27)
dh&),
for all k = 1, 2, . . ., N (where zk is the transformation
introduced
by (3.8)).
252
W. PUSZ and S. L. WORONOWICZ
Indeed,
for any sl, s2, . . . , SUER we have
i E hen j ,““I
dp&-*@I,
.a., P-*Q~, e,c+t, .‘.f e,v)
RN
Se
=
&a,&1,
...,
@k?
t?k+l,
-..Y
@iv)
RN
i(
=
(ukx)Ie
x
p*s,L,+
.I1
&lLd E PI=&+* ukx)>
and using (3.25) and (3.22) we see that above expression N N i TXs,L i 1 s& (akxJukenzL x) = (x1(&-&+ l)e n=l x) i en2l X
=
j
equals
hen
(@k-@k+&&&‘l,
*..T
@k, @k+l,
...)
@IV).
R”
In a similar way, using (3.24) and (3.23), we prove formula (3.27). We shall consider the subgroup .Y c GL(N, R) generated by the transformations Ti, r2, .‘., rN. Let x = [I-l,
-$)u
(0) u($,
1-J
and X” be the Cartesian product of N copies of X. One can easily check that any orbit of Y intersects XN of precisely one point. In other words, XN parametrizes the space of orbits, XN = RN/F.
(3.28)
(Warning: The above identification concerns only the Bore1 structure of considered spaces; RN/F as a topological space is not Hausdorff.) Let ZE 2, dfi, be the projection of dpz onto XN (cf. (3.28)), QE supp dp,, 6 be a neighbourhood of 4 in XN, 0 be the set-theoretical sum of all orbits belonging to 6, Z(0) be the subspace of all vectors x E %’ such that the measure dpcL,is concentrated on 0 and P, be an orthogonal projection on Z(0). Then (zlPoz) > 0 and PC # 0. Clearly, P,D, c D, (P, commutes with %) %‘(O) n D, 2 POD,. On the other hand, Therefore, if x E S(0) n D,, then x = P,xEP~D,. Y?(O) n D, = POD,. Formulae (3.26), (3.27) show now that POD, is invariant under the action of aj and aj’ ij= 1,2, . . . . N). Assume that the representation (3.1) is irreducible. Then using Proposition 3.5, we see that P, = I and Z(0) = %. This means that the spectral measure d-E(.) is concentrated on 9. By considering a decreasing sequence of neighbourhoods 6
TWISTED SECOND
253
QUANTIZATION
shrinking to 4, we conclude that the spectral measure dE(.) is concentrated on the one orbit VEX N. Since any orbit of Y is denumerable, the family {L, , L,, . . . , LN} has a pure-point common spectrum. Let S denote the pure-point common spectrum of {L,, L,, . . . , LN}, and K = ((@I, @2, ..*y @N)ERN: @j>, @j+l, P2@j 2 @j+l, j=
1, 27 ..*y N}.
In virtue of (3.22) and (3.23) S c K. We remind that &?N+ 1 = -( 1 -p2)-l. For any QE RN, Xe will denote the common eigenspace of {L,, L,, . . . , &} corresponding to the system of eigenvalues Q and 2 = @ xe. Let PES
We shall prove PROPOSITION 3.6. Assume that the representation (3.1) is irreducible. 7hen S = SE,,., for some E E 1~2, 11and some non-negative integers p, q such that p + q < N.
Proof: Uncomplicated combinatorial analysis shows that SE,P,Qare the only sets S satisfying the following three conditions: (i) For any QES and tET if e$aK, then t@ES. (ii) For any QES and tET if t@EK, then @ES. (iii) For any eo, QES there exist sequences t,, t,, . . . , t, E T and er, ez, . . . . , Q, = QES such that ek = t,Qk_l Therefore we shall prove Let QES and $E&@,
(k = 1, 2, . . ., n).
that S satisfies conditions /1$1/= 1: Lk$ = @,ti?
(i)(iii).
k = 1, 2, ...,N.
Using (3.24) and (3.25), we see that
In virtue
L,afII/
= (zje),aT$,
j, k = 1, 2, . . . . N,
(3.29)
L,aj$
= (73 ‘&aj$,
j, k = 1, 2, ...,N.
(3.30)
of (3.22) and (3.23) we obtain I/afIC/112=~2Qi-gj+1, Ilaj~ll’ =Qj-Qj+l,
j= j=
1, 2, ..-, N, 1, 2, ...y N.
(3.3 1) (3.32)
Let &aK. Then the above expressions are different from zero and this and (3.29), (3.30) shows that rje, z,:‘QES for j = 1, 2, . . . . N. Let rje~ K for some jE { 1, 2, ...,N}. We consider three cases:
W. PUSZ
254
and S. L. WORONOW!CZ
ej > 0. Then ~‘ej-ej+1 > ~4ej-ej+l = p2(rje)j-(rje)j+ r 3 0 since Z~QEK. Formula (3.31) shows that UT $ # 0 and (cf. (3.29)) zjeeS. ej = 0. If ek = 0 for all k 0. Then rje = 7i~ and (see the previous case) rj@ = r,@Es. ej < 0. Then ,‘ej-,j+1 > ej-Qj+1 >, 0, since QEK. Therefore (see the first case) zj~ ES. Let ~~‘QEK for somejE{l, 2, . . . . Nj. This case can be treated in the similar manner. The condition (iii) follows immediately from irreducibility (cf. Proposition 3.5). m
Proof’cf the last part cf Theorem 3.1. Let (3.1) be an irreducible representation of TCCR, S = SE,,,, (cf. (3.7)) be the pure-point common spectrum of {L,, L,, . . . , LN} (cf. Proposition 3.6). Let Q, @‘ES. We say that the pair (Q, Q’) is a close pair if there exists t E T such that Q’ = CQ. Since the transformations zj (j = p, . . . , N-q - 1) coincide with zNey on S, we can always assume that t E TP,4 = {zj, zi I: jEJp,q} where J,,, = (1, . ..) p-l, N-q, N-q+l, . ..) N}. For any close pair (e, Q’) we define an operator
z e’,e
=
ai
if
Q’ = zj 1Q and j E J,,,,
uj’
if
Q’ =
ZjQ
and j E J,%, .
(3.33)
It follows from the proof of Proposition 3.6 that if (Q, Q’) is a close pair and $ E ze, $ # 0, then Ze,,e$~A?,, and ;5eJ.e$ # 0. If eI, ez, .., Q,, Q,,+~ = er is a sequence of elements of S such that (Q~,Q,,,+~) is a close pair for all m = 1, 2, . . ., n and $EX~~, $ #O, then (3.34) where A>O. Indeed,.let t,, t,, . . . . t, be the unique sequence of elements of TP,4 such that ek+ 1 = tkek (k = 1, 2, . . . , n). Clearly, for any j E J,,,, any zj appears in this sequence as many times as z_,,’ does. Therefore (cf. (3.2) and (3.3)) the operator on the left-hand side of (3.34) equals (up to a positive factor) to a product of operators of the form uj’ aj and ajnj+ (j E J,,,). Using then (3.22) and (3.23), we see that $ is an eigenvector of u,? aj and ujuj’ (with strictly positive eigenvalue) and (3.34) follows. Now we take a sequence er, e2, . . of elements of S such that: (i) (Q,, en+ r) is a close pair for any y1= 1, 2, . . (ii) If (Q, Q’) is any close pair in S, then g = Q,, Q’ = Q,+ 1 for some ~1. Let us choose a non-zero element I+!I E Xe, . We introduce a sequence of non-zero vectors +r, ti2, . . . defined by the recurrence formula ti nfl
--z
en+ 1.e” tin.
TWISTED
SECOND
255
QUANTIZATlON
For any QE S there exists n such that e = q,. We set I@)= 1111/,11-‘4k. In virtue of (3.34), the right-hand side of this definition is independent of n. It follows immediately from the above construction that I&+ 1) = GYG+
of the choice
(3.35)
,,e,le,)
where CI, is a normalization factor. Let e E S and JE J,,,. Assume that 7,: ’ QE S! Then (Q, ~~7’ e) is a close pair and there exists a positive integer IZ such that q = q,, ~~7‘q = en+ 1. In this case Z en+1,etl-- aj. Using (3.35), we obtain
ail@) = %l~I”@). The normalization
factor
CI, can easily be computed.
&? =
Using (3.23), we have
ll~jl@)l12 = k?k?j+aj@) = @j-@j+l
(3.36)
ajk?) =
(3.37)
and finally for j E J,,,: (@j-@j+l)1'21Zj1@).
We proved this formula assuming that ~~7‘@ES. However, this formula is also valid = 0 and (cf. (3.36)) both sides of (3.37) ifeES and zJ’q$S, since in this case qj-qj+r vanish. Now we consider three cases. (i) iV=p+q.ThenJ,,,={1,2 ,..., NJ and formula (3.37) describes the action of all aj (j = 1, 2, . . . , N) . (ii) p = 0. Let us notice that (cf. (3.37)) aj’ aj = Lj- Lj+ 1 = 0 for j = 1, 2, . . . , N-q - 1. Then aj = 0 for suchj and since formula (3.37) is also valid in this case, we have that (3.37) is true for all j = 1, 2, . . . , N. (iii) p+qO.Asbeforewehaveaj=Oforj=p+l,...,N-q-l.Thus to complete the proof we have to describe the action of ap. Let c,le) = e;‘21r;1e).
(3.38)
We shall show that ap = eiqccp for some cpE [0, 27r[. At first we notice that the relations L,, 1 = 0 = LNpq, (3.22) and (3.23) imply that %&+ =p2apfap=p2Lp and aNfpqaNpq = aN_qai_q
= -LN_q+l.
(3.39)
Let Q = a,ai_q.
(3.40)
256
W. PUSZ and S. L. WORONOWICZ
Then
QaN_, = -a,L,_,+l
and
a, = -QaN_4LN!q+l
= -QLNlq+laN_q.
(3.41)
Using (3.25), we see that Q*Q = -p2LpLN_q+1 The operator --P’L~L~_~+ 1 is invertible spectrum of L,, L,, . . . , LN). Therefore,
= QQ*.
(cf. formula
(3.7) describing
Q = W-P~L&N-~+IP~,
the common
(3.42)
where W is a unitary operator acting on YE’. Remembering that aj = 0 for j = p + 1, . . . , N-q + 1 and using (3.2)-(3.4) one can check that W commutes with aj and a; ij = 1, 2, . . . , N). The irreducibility implies now that W= eiVI for some ~E[O, 2n[. Now using (3.42) and (3.41), we have: up = e’“( -p2 L,L,1,+
l)1i2a,_,
and a,lQ) = ei’P(-~2LpLNlq+1)1’2(-eN_q+1)1’21zN’,e) = ei’PQi/21TNlyQ)= ei’Pek’21r,‘e)
It follows easily from the presented proof that the irreducible representations of TCCR determined by SE,p,q (and eventually by cp) are pairwise non-equivalent. It is interesting that contrary to the classical case (Stone-von Neumann uniqueness theorem) we obtained a large family of representations. The Fock representation considered in Section 2 corresponds to p = 0 and 4 = N (the value of E is irrelevant). REFERENCES [I] [2] [3] [4] [S] [6] [7]
Barut, A.O., Rgczka, R.: Theory of Group Representations and Applications, PWN-Polish Scientific Publishers 1977. Borchers, H. J., YngvaFon, J.: Commun. Math. Phys. 42 (1975), 231-252. Drinfeld, V. S.: Quantum Groups, will appear in Proceedings ICM-1986. Enock, M., Schwartz, J. M.: Bull. Sot. Math. Frunce, Supllment mhoire 44 (1975), l-144. Flato, M., Simon, J., Shellmann, H., Sternheimer, D.: Ann&s Scient. Eco/e Normale Sup. 4’ serie 5 (1972), 423434. Pusz, W.: Twisted canonical anticommutation relations (to appear in Reports on Mathematical Physics). Powers, R.: Commun. Math. Phys. 21 (1971), 85-124.
TWISTED
SECOND
[S] Schwartz, J. M.: J. Fun&. Anal. 34 (1979), 370-406. [9] Slowikowski, W.: “Commutative Wick algebras I. The Bargman, Proceedings
of the Conference
257
QUANTIZATION
on Vector Space, Measures
Wiener and Fock algebras”, Dublin 1977. Lecture
and Applications.
Notes in Mathematics, Vol. 644. Springer 1978. [lo] Woronowicz, S. L.: Commun. Math. Phys. 111 (1987), 613-665. Cl l] Woronowicz, S. L.: Tannaka-Krein duality for compact matrix pseudogroups. Invent. Math. 93 (1988), 35.-76.
Twisted SU(N) groups,
REPORTS
TWISTED
ON
SECOND
W. Pusz Department
of Mathematical
MATHEMATICAL
Methods
No. 2
PHYSICS
QUANTIZATION
and S. L. WORONOWICZ in Physics, Faculty Warszawa, Poland
(Received
June
of Physics
University
of Warsaw.
20, 1988)
A formalism of a second quantization procedure based upon the twisted SU(N) group is constructed and the related twisted canonical commutation relations (TCCR) are investigated. In a particular case, these relations reduce to classical CCR. Irreducible representations of TCCR are described. The Stone-von Neumann uniqueness theorem does not hold for the general TCCR.
0. Introduction The notion of a quantum group has been the subject of intensive study in the last years (cf. [3], [4], [S]). In particular, in [lo] the general notion of a compact matrix pseudogroup S,U(N) was studied. This group can be treated as a deformation of the classical SU(N) group and thus a natural question arises whether quantum groups can be used for description of symmetries of quantum-mechanical systems in the same way as classical groups are. The aim of the present paper is a description of the second quantization procedure based upon the S,U(N) group as the symmetry group. It leads to an essential modification of the well-known classical formalism of the Fock space. In particular, ,u-depending coefficients appear in new canonical commutation relations. The basic notions and definitions of the theory of compact matrix pseudogroups and twisted SU(N) group can be found in [lo] and [l I]. For the convenience of the reader we recall some facts and concepts. Let G = (A, u) be a compact matrix pseudogroup, CZ?the algebra generated by matrix elements of u, V a vector, space and u: V-+V@d
(0.1)
a linear map. We say that u is a (smooth right) action of G on the vector space if the following diagram OV@coe u:
“Qlid
V@d
-V@d@O ~2311
Iid@@
232
W. PUSZ
and S. L. WORONOWICZ
(where @ is the comultiplication associated with G) is commutative. If dim I/< co then this notion coincides with that of representation. Remark. Throughout the paper, all tensor products are taken in the algebraic sense (no completion is performed). We shall also write shortly “G-action” meaning smooth right action of G (only such actions will be considered in the paper). If u and w are G-actions on vector space V and IV, respectively, then the direct sum ZI@ w is a G-action on I/@ W defined in the obvious way. By Mor(u, w) we denote the linear space of all operators intertwining u and w: for any linear map s: V--+IV, SE Mor(v, w) if and only if the diagram
is commutative.
If x E V, YE W then UX =
CXiObiY
WY =
CyjObj,
where X~EV, yi E IV, a,, bjesxZ and the formula uaw(x@y)
= Cxi@yj@aibj Lj
defines the G-action on V@ W called the tensor product of ZJ and w. If the vector space V has an additional structure then usually we shall demand this structure to be preserved under the action of G. In particular, if I/= Y is an algebra and u: sp+Y@yc4 (0.2) is a linear map then we say that a is a G-action on the algebra Y if CIis a G-action on the vector space Y and the multiplication mapping m: Y4P03q10q2+q1q2E~ is intertwiting: m E Mor(aa u, cx). This is the case if and only if (0.2) is an algebra homomorphism, i.e. a(q,-q,) = cx(q,)a(q,). If r is a bimodule over such an algebra Y and /I is a G-action on r, then we say that (r, /I) is a couariant Y-bimodule if right and left multiplications mR: r@Y3co@q-+-*qEr, mL:
are intertwining:
.m3r3qmwq~d-,
mR E Mor(P@ u, fi), mLE Mor(uO P(44
= Q)PW?
PC+?) = P(4u(d
/I, /I). This means that
TWISTED
SECOND
233
QUANTIZATION
for any w E r, qE9 (r@ .d is in a natural way a bimodule over Y@d). In this paper we deal with the compact matrix pseudogroup S,U(N), where 0 < ,LL< 1, introduced in [i 11. It is (cf. [ll]) known that the theory of representations of S,U(N) is similar to that of SU(N): irreducible representations are labelled by Young diagrams and formulae for dimensions and multiplicity are the same as in the classical case. In particular, 111 corresponds to the (irreducible) fundamental representation u and uau has a decomposition
This shows that dimMor(ua u, uau) = 2. This space is spanned by two operators: I and or. Let K (dim K = N) be the carrier space of the fundamental representation u of S,U(N) and let {Q, . . . . eN} be the canonical orthonormal basis in K; then PEj 0 si cJp{Ei 0
Ej)
Ej@Ei
=
{
pClej@si+(l -$).si@ej
for for for
i
(0.3)
i > j;
a,*=c~,and
Spo,={l, -cl’}. The eigenspaces of oP are the carrier spaces of the -representations 1-l-l and I_) and ‘analogously as in the case p = 1 we shall say that YE K 0 K is symmetric if a,(u) = v and can easily check that the elements
&iOEiT
pLEi
0
form a basis in the subspace Ei@Ej-/LEj@Ei,
if a,(u) = --k12. Using (0.3), one
antisymmetric
Ej + Fj @ Ei,
i
of symmetric i
(i,j=
elements (i,j=l,2
1,2 ,...,
N)
(0.4)
of K @ K. Similarly, ,...,
N)
(0.5)
form a basis for the antisymmetric subspace. The paper is arranged as follows: In Section 1 we remind the classical second quantization procedure. Section 2 contains our main theory. We show how in a natural way one can define the Fock space and the creation and anihilation operators covariant with respect to the action of S,U(N). We derive the twisted canonical commutation relations (TCCR) satisfied by these operators. In the last section we discuss in full generality the representations of TCCR. The paper is restricted to the Bose case. The fermionic case will be considered in the forthcoming paper [6].
234
W. PUSZ and S. L. WORONOWICZ
1. Classical second quantization The main purpose of this section is the presentation of the classical construction in a manner convenient for further generalization. We stress the role of the symmetry group in this construction. For technical reasons we consider only N-level quantum systems (IV < co). Let K be an N-dimensional Hilbert space with fixed orthonormal basis sN} and let Y(K) denote the tensor algebra over K. The symmetric 1% s 2, *..> tensor algebra is Y(K) = Y(K)/J where the ideal J is generated
by the antisymmetric
elements
of degree
i
&i@&j-&j@Ei,
2: (1.1)
The algebra Y(K) is endowed with the natural grading. The multiples of unity are the only elements of degree 0. The space of elements of degree 1 is canonically isomorphic to K. Let xj be the element of degree 1 corresponding to sj(j = 1, 2, . . . , N). Then Y(K) is the free commutative algebra with unity generated by {x1, x2, . . . , xN}, i.e.’ the algebra of all polynomials of N commuting variables. Let T(K) be the space of differential forms of the first order over the and that the variables x1, x2, , . . , xN. This means that T(K) is a Y(K)-bimodule linear map d: Y(K)
--f T(K)
such that (i) d(q.q’) = (dq).q’+q.(dq’) (ii) any element WET(K)
for all q, q’eY(K), can be written in the following o = ;
(1.2) way
(dxi).qi
(1.3)
i=l
where qi E Y(K) (i = 1, 2, . . . , N) are uniquely determined, (iii) q.(dxj) = (dx,).q (j = 1, 2, . . , IV) for all qEY(K)
(1.4)
is given. For any QJ of the form (1.3) we set
j=l
Clearly, 6: T(K) - Y(K) is a well defined
linear map.
(1.5)
TWISTED
Let
SECOND
(.I .) be a sesquilinear
form
235
QUANTIZATION
on
Y(K).
For
any
CO= :
(dxj)qj
and
j=l N a'=
C
(dx,)qi
belonging
to T(K) we set
j=l
It turns out that there exists one and only one sesquilinear (old&
form on Y(K) such that (1.6)
= (dolq)
for any qEY(K) and OJET(K). In fact, (.I.) coincides with the well-known scalar product in Y(K) (see [9]). To obtain the correspondence with the usual second quantization formalism we consider linear operators o’= 1, .2, . ..) N)
aj: Y(K)-,Y(K) defined
by the formula dq
=
for any q E Y(K) (so ajq can be regarded consider linear operators
as the jth partial derivative
by 6( ~
(dXj)qj)
=
~
(1.8)
U~qj.
j=l
j=l
Clearly, aj’q = xjq for any qEY(K). Formula (1.6) shows that aj’ is the Hermitian conjugation that the variables x1, x2, . . . , xN commute, we see that ai+ a,? zzza,? a’
Taking
the Hermitian
conjugation,
1,2 ,...)
(i,j=
of aj. Remembering
N).
(i,j=
1, 2, ...) N).
using (1.2), (1.7) and (1.8), one can easily verify the canonical UiUj+
(1.9)
we obtain
UiUj=UjUi
Moreover, relations
of q). We also
(j = 1, 2, . . . . N)
aj’: Y(K)+Y(K) introduced
(1.7)
E (dxj)(ajq) j=l
=
6,jZ +
Uj’ Ui.
(1.10)
commutation (1.11)
236
W. PUSZ
The number
a'ndS.L. WORONOWICZ
operator N
(1.12) j=l
maps Y(K)
into Y(K)
and satisfies the following [JV, aj’] = af
,
[JV, aj] = -aj,
For any non-negative
integers
j=1,2
relations 7 ..-> N.
.
we set
Inl, . . . . nj, . . . . nN) = (n,! . ..nj! . ..~1~!)-~‘~x.~...x;j...x~~.
The elements
(1.14) form an orthonormal ajln,,
(1.13)
basis in Y(K)
. . . . nj, . . . . nN) = nf’21n,,
aj’ 1n1, . . . . nj, . . . . nN) =
(nj+
Jlrln 1, . . . , nj, . . . , nN) = (F
. . . . (nj-l),
and we have . . . . n,),
1)1’21nI, . . . . (nj+ l), . . . . n,), nj)lnl,
(1.14)
(1.15)
. . . . nj, . . . . nN).
j=l
The last formula shows that the homogeneous elements of Y(K) are eigenvectors of ./lr, the corresponding eigenvalue coincides with the degree. At the end of this section we would like to discuss the covariance properties of the constructions described above. The space K can be considered as the carrier Hilbert space of the fundamental representation of the group SU(N). Let us notice that the space of antisymmetric elements (1.1) is invariant under the natural action of SU(N) on Y(K). The elements of SU(N) are represented by isomorphisms of the graded algebra Y(K). The actions. canonical embeding K + 9’(K) intertwines the corresponding Finally, one can define the action of SU(N) on T(K) in such a way that T(K) becomes a covariant Y(K)-bimodule and that 6 and d are intertwining operators. The scalar products on Y(K) and T(K) are SU(N)-invariant since they are uniquely determined by (1.6). We can summarize the second quantization procedure for the group SU(N) in the STATEMENT.The second quantization procedure for the group SU(N) produces the sextet
{K J, Y(K), M),
d,
a}
TWISTED
SECOND
237
QUANTIZATION
such that
(i) K is the carrier of Hilbert space of the fundamental
(unitary) representation
of
=J(N); (ii) J is the ideal of the tensor algebra F(K) over K generated by the set of antisymmetric elements of second degree; (iii) Y(K) is the ,factor algebra F(K)/J endowed with the canonical action of SU(N) and the canonical (invariant) scalar product; (iv) T(K) is covariant P’(K)-bimodule of differential forms of degree one and d: Y(K) is the corresponding and f(K);
external derivative intertwining the actions of SU(N) on Y(K) 6: c(K)
(v) is a linear mapping
--f T(K)
-*Y(K)
such that d((dx)q)
= xq
for any XE K and qEY’(K), 6 intertwines the actions of SU(N) Y(K), and 6 coincides with the Hermitian conjugation of d. 2. Twisted
second
on T(K)
and
quantization
In this section we develop the formalism of second quantization based on the twisted SU(N) group. We shall use the ideas presented in Section 1 introducing necessary m+odifications. In particular, the classical differential calculus should be replaced by a non-commutative one. Let K be the carrier Hilbert space of the fundamental representation of S,U(N) (dimK = N), F(K) the tensor algebra over K and
where J, is the ideal in F(K) generated by a $N(N- 1)-dimensional S,U(N)-invariant subspace of K 0 K (this subspace corresponds in the twisted case to the subspace of antisymmetric tensors; cf. (0.5)). F(K) is a graded algebra and carries the natural action of S,U(N). Moreover, the ideal J, is S,U(N)-invariant and any element of J, is a sum of homogeneous elements of J, of the order 3 2. Therefore, we have the induced grading and the induced action of S,U(N) on 5$(K). The subspace of all elements of degree 0 coincides with the set of all multiplies of unity; the subspace of all elements of degree 1 is canonically isomorphic to K and generates $(K). Denoting by to ar, E;, . . ., eN ({ej: j = 1, 2, . . ., N) is Xl, x2, . ..> xN the elements corresponding
W. PUSZ and S. L. WORONOWICZ
238 the canonical
basis in K) and taking
(0.5) we have
into account for
xi.xj = pxj.xi
i
(2.1)
Clearly, YP(K) can be identified with the algebra of all polynomials x1, x2, *.., xN satisfying the relations (2.1). Let Ta be a covariant Y@(K)-bimodule and let d: $(K) be a differentiation, and rP and
of N variables
+ rlr
i.e. d is a linear map intertwining
(2.2) the actions of S,U(N)
on YM(K)
d(a.b) = d(a)b +ad@) for any a, b ES$(K). 41, q2,
ee.9
q,,,E YP(K)
We assume such that
that for any element
OE rp there exist unique
(dx,)q,.
w = i
(2.3)
j=l
Then we say that (r,, The linear map
d) is a first order differential calculus over $(K). 6: r,+LqK)
defined
for any OE~,
(2.4)
of the form (2.3) by 60.I = ;
xjqj
(2.5)
j=l
is well defined and intertwines prove
the actions
of S,U(N)
on r,, and Y@(K). We shall
2.1. Assume that N, > 3. Then there exist precisely two ‘non-isoinorphic differential calculuses over YP(K). In the first case,
THEOREM
first order
for
i < j,
P2 (dxj)xi
for
i=j,
P (dxj)xi
f or
i>j.
p(dxj)xi-(1 xidxj = 1 In the second
-p2)(dxi)xj
(2.6)
case,
xidxj =
p- l(dxj)xi
for
i
pL2(dxj)xi
f or for
i=j,
pml(dxj)xi-(1
-p-‘)(dxi)xj
i>j.
(2.6*)
TWISTED
Remark:
over yfi(K) and (2.6*).
SECOND
239
QUANTIZATION
If N = 2 then there exist two families of first order differential calculi labelled by a complex parameter y. Setting 11= 0, we obtain (2.6)
Proofi Assume that (r/,, d) is a first order differential consider the linear mappings:
calculus over yP(K).
We
and $1 K@Yp(K)3ej@q4dxj)q~l-p actions of (i, j = 1) 2 3 ...> N). Clearly, these mappings intertwine the corresponding S,U(N). Remembering that qj in (2.3) are determined uniquely, we see that $ is S = $-‘.cp: bijective. Therefore $ - I exists and t.he composition S: K@K+K@cYti(K) intertwines
the action
of S,,U(N)
(2.7)
on K @ K and K @Yfl(K).
We know that
where K, is the subspace of homogeneous elements of order n. For p = 1, the representation of S,U(N) acting on K, is irreducible and the corresponding Young diagram (e.g. [l]) consists of n boxes placed in one raw. The same is true for p < 1 (cf. [l 11). The fundamental representation (the action of S,U(N) on K) corresponds to the one-box Young diagram. It is known ([ll], [l] Chapter 8) that 1110 111= 1-10 l_l_l. IL
l~lo’l-J~l...l~l
=~p.l~loI
ntimes
n limes
(2.8) -_ -_I l...lIl. n+ltimes
(2.9) .
Therefore for II # 1 the representations of S,U(N) acting on K 0 K and K 0 K, are disjoint. (This argument does not work if N = 2. Then the representations -_-corresponding to I---I I / and I-- ) I are equivalent.) This means that S(K @ K) c K @ K.
1-l
According
to (2.8), the space of all intertwining
operators
S: K@K+K@K is 2-dimensional:
s = al+@, where (Y, /Jo C and gp is given by (0.3).
(2.10)
240
W. PUSZ
For any i,j=l,2
,...,
N we have XidXj =
and, using (2.10), we obtain plu(dxj)Xi XidXj
=
(‘+
x,x,dx,
formula,
= {c~+P(f
=
+
Xl
for
i
for for
i =j,
a(dXi)Xj
-~“)](dxi)xj+lj~(dxj)xi
(2.11)
i > j
one can check that +Ca+P(l
-P’)1PPvx,bI~
1X2X1+~(a+p)Ca+B(1-~2)1(~X2)X:..
the above expressions
(cf. (2.1)), we get
c+t++(l-p2)] Moreover,
~(s(&iOEj))
-~2)1~+(~+P)lj~2}(dX1)X2X1
PX2XldXI = (~+B)PP”@
Comparing
cP(EiOEj)
P)(dxj)xi
[~+/?(l for some a, DEC. Using the above
and S. L. WORONOWICZ
= 0.
(2.12)
using again (2.11) we obtain: d(x,x,-px2x1) .
= (dx,)x,+x,dx,-~(dX,)X,-~X,dX, = (1 +a-p~2)C(dxI)X2-~L(~X2)X11.
Therefore, 1 +cc-/!I/_? = 0.
(2.13)
Solving the equations (2.12) and (2.13), we see that either tl = ,u2- 1 and b = 1 (then (2.11) coincides with (2.6)) or CI= 0 and fl = p- ’ (then (2.11) coincides with (2.6*). To complete the proof we have to construct the first order differential calculus over YP(K) such that (2.11) holds. Let p be the free right module over the tensor algebra F(K) with N generators d”El, dE,, . . . . JEW. Any element GE r” is of the form c3 = t j=l
where
ql,
introduce products
(2Ej)qj
(2.14)
determined elements of Y(K). We have to q,, . . . . qN are uniquely the Y(K)-bimodule structure on r”. To this end it is sufficient to write the eidd~j (i, j = 1, 2, _. ., N) in the form (2.14). Motivated by (2.11) we set B~(d”Ej)Ei+ Eid”Ej
=
~(~Ei)Ej
for
i
(a + P)(d”Ej)&i
for
i=j,
[cr+/?(l
for
i > j.
-~‘)](JE~)~~+~~(~“E~)E~
(2.15)
TWISTED
SECOND
241
QUANTIZATION
We shall also use the differentiation
introduced
2: Y(K)-+r”
(2.16)
J(l) = 0,
(2.17)
by the formula
.
.
i,=
11,123 *-*,
1,2,...N;
n=
1,2,...
Let i,j = 1, 2, . . . . N and Eij (i < j) denote the elements (0.5). Then in virtue of (2.15), (2.13) and (2.12), we get d”Eij
=
(2.18)
(1 +0Z-~,/A2)[(d”EJ&j-/A(d&j)&J = 0
and one can check that for k = 1, 2,
.
.
.
,
N:
-ap(dEi)&jk+~aB(~&j)&ik+~282(d&k)Eij
for ap~(~&i)&kj+102~2(~&k)&ij+[CI+P(l
Eij~&k
=
i
1
for
i
(2.19)
-~2)1C(~&i)~kj-~(~&j)~ki]
P2~2(~&k)Eij+P~[a+P(1
for
k
for
k=i
or k=j.
Let as before J, be the ideal of F(K) generated by elements (0.5) and let N, be the subset of r” consisting of all elements 6 of the form (2.14) where of r”, l’JB c N, and, in 41, q2, .**, qNE J,. Clearly, N, is a sub-F(K)-bimodule virtue of (2.19), J,r” c N,. Therefore, for any ~1)E F/N, and qE J, we have oq = qw = 0. This means that fa = r/N, can be considered as a bimodule over S$(K) = F(K)/J,. In virtue of (2.18) Jq E J, for any q E J,. Therefore there exists a linear mapping
such that the diagram F(K)YJW
=
~WJ,
1
-
I
_ r
d
Ta = F/N,,
’
242
W. PUSZ
and S. L. WORONOWICZ
(where the vertical arrows denote the canonical projections) is commutative. Clearly, d is a differentiation (the equality d(ab) = (da)b + adb follows immediately from the same formula for d). To finish the proof, let us notice that 9(K) and r” are the carrier spaces of the natural actions of S,U(N). Moreover, the whole construction is S,U(N)-covariant (the form of equations (2.15) is invariant under the action of S,U(N) so r” is a covariant bimodule over 9(K), J, is the invariant ideal in Y(K), N, is the invariant sub-module of r” and d is the intertwining mapping) so YP(K) carries a natural action of S,U(N), rG is a covariant $(K)-bimodule and d is an intertwining operator. n To make further considerations more concrete we deal with the first order differential calculus (I-,, d) over Y,(K) defined by (2.6). Let u; and aj (j=1,2,..., N) be linear mappings acting on YP(K) and introduced by the formulae (cf. (1.7) (1.8)): (2.20)
6( 2 (dxj)qj) = 2 afqj, j=l
j=l
dq = -f (dxj)ajq.
(2.21)
j=l
Like in the untwisted
case (p = 1) we call uf(uj
resp.) .creation (anihilation
resp.)
operators.
Clearly, a,? q = xjq,
j=l,2
(2.22)
3 . ..> IV.
Formula (2.21) means that uj is a partial derivative in the jth direction. Now we look for the canonical commutation relations in the twisted In virtue of (2.22) and (2.1), we get U’
Uj’
=
~Uj+
i
for
Ui+
case. (2.23)
For any i = 1, 2, . . . . N and qEYP(K) d(Xiq) = (dXi)q + Xid~ = (dx,)q + Xi i:
(dxj)Ujq
j=l =
(dXi)q + C ~(dxj)xiajq
j
+~2(dxi)xiaiq
+
+ 1 C~L(dXj)Xi~ja-(‘-~2)(dxi)xj~jql = ~;i(dxj)/m:ujq+(dxi)[I+p2a:a;-l(1-p2)
2 k>i
U:ak]q.
TWISTED
On the other
SECOND
243
QUANTIZATION
hand,
c
d(Xiq) = d(U+q) =
(dXj)ajai+q'
j=l
Comparing
the two expressions,
‘jaT =
we obtain
r
pa+ aj ~+~2a+a~-(l
-p2)
for i #j, 1 &+a, for i = j.
(2.24)
k>i
For any i, j = 1, 2, . . . , N, i > j we consider
the operators
Aij = aiaj-pajai. Using (2.24) one can check that
,a2a: Aij Aija:
p3a+ Aij-~(1-~2)
=
C a: A,j
for
k # i, j,
for
k = i,
for
k =j.
s>i
i
I ~~afA~~-pL(l-_~)
C
a:Ais+(l-p’)
C alAPi
(2.25)
p>i
j
In virtue of (2.22), any monomial in YP(K) can be obtained by multiple action of the creation operators on 1. Since dl = 0, we have a,1 = 0 and Aijl = 0. NOW (2.25) means that Aijq = 0 for any monomial 4 E Y,(K), i.e. ajai = ,uaiaj
for
i
Relations (2.23), (2.24) and (2.26) coincide with (1.9k(1.11) for p = 1. integers. Writing for any integer Now let n,, n2, . . ., nN be non-negative 1 -/P A,(n) = ~ l-p2 and remembering j=l,2 , ..*, N
that d is a derivation
aj(@.
. .x7”. . .x;l)
(2.26)
n (2.27)
and using (2.1), (2.21), (2.6), we get for
L = A,(nj)pk”~“,N . . .xjnj-1. . .x;l.
(2.28)
Let lnlY n 2, *.., nN) = [ fi FactP( j=l
- iii xi,+. . x2xX;’
(2.29)
W. PUSZ and S. L. WORONOWICZ
244 where
Fact,,(n) = fl ~ (Fact,(n) = n!, (2.29) is the twisted Clearly,
=
version
of (1.14)).
(In,, n,, . . .) Q): n,, n,, . .*, nN = 0, 1, 2, . . .}
(2.30)
is a basis in the vector space YP(K). Using (2.22) and (2.28), we obtain X.nk ajlnl, . . a7 nj, . . *> nN) = Ap(nj)“‘P”‘ln,,
afIn,, .
..)
. . . , nj- 1, . . . , TIN),
n,, . . . . no) = A,(nj+ l)1i2$‘jln1,
(2.3 1) . . ., nj+ 1 > . . . . n,L
Let (.I.) be the scalar product on Y*(K) such that the basis (2.30) is orthonormal. Then (cf. (2.31)) aj and UT are mutually Hermitian conjugate a? = uj’, (a;)* = aj. Taking into account (2.20) and (2.21) for any OE~, and 4 E$(K), we get
k4&)rp = (~014 where (.I.),-, is the scalar product
on Tr introduced
(2.32) by (2.33)
Conversely, assume that (. I .) is a scalar product UT = UT, (a;)* = uj and
on Yfl(K) such that (2.32) holds. Then
fxj414’)= (“jfqld) = (4l’jq’)T (41xjd) = (41”f 4’) = (“jq14’) for anyj = 1, 2, . . . . N, q, q’ E Y”,(K). We know that uj (j = 1, 2, . . . , N) decreases the grade of homogeneous elements. Therefore the above formulae show that (. I .) is determined uniquely by (111). This way we have showed that (2.32) defines the scalar product on YP(K) uniquely up to a positive factor. This fact implies that the scalar product is S,U(N) invariant. Let %=ii,d=
;
afaj
(2.34)
j=l
be the (twisted) number operator in the twisted case (cf. (1.12)). It is the linear operator on YP(K) commuting with the action of S,U(N). Using (2.31) we get
Wn1, . . . . nN)= AP(i nj)lnl, . . . . nd. j=l
(2.35)
TWISTED
SECOND
245
QUANTIZATION
Therefore the eigenspaces of % coincide with homogeneous components of Y@(K). Since A,(n) < l/(1 -p’) (n = 0, 1, 2, . . . ), the operator % is bounded. Clearly, this fact implies that d, 6, ai+, aj are also bounded. Let us notice that for ,u = 1, (2.31) and (2.35) coincide with (1.15). It turns out that the first order differential calculus constructed above gives rise to a higher order differential calculus. We would like to make a short remark on this subject. Let r,“” = Yfl(K), Ts@I = r and rFn (for n > 2) be the tensor product (over YP(K)) of n-copies of Tc. We Consider the graded tensor algebra
Let 6 be the ideal in rf
generated
by the elements
pdxi @ dxj + dxj @ dx, dx, 0
k,i,j=
dx, ,
Clearly, 6 is a direct sum of its homogeneous the quotient algebra
for
of the form i
(2.36) 1,2 ,...,
N.
components
of grade > 2. Therefore,
r; = r?p has a natural grading and the set of all elements of grade 0 and 1 coincide with YP(K) and Trc, respectively. The multiplication in ri will be denoted by “. The algebra r; is a generalization of the external algebra. One can easily check that any element QE r; of degree n is of the form e =
dxi’ A dxiZ A . . . A dxinqi,iz,,,i,
C il
where
qiliz...i,
E Yp(K)
are uniquely
determined.
We set de =
C
(-
l)“dx” A dxi2
A ... A
dx’”
A
dqi,iz,.,i,.
il
Then d:
r;-,r;
(2.37)
is a graded derivative, cY(de) = 8~ + 1 for any homogeneous QE r;, on elements of degree 0 the derivative d coincides with (2.2), (2.21) and due to (2.36) and (2.26): d(dQ) = 0
for any QE r,‘J . Clearly, the above construction is fully covariant: the ideal G is invariant with respect to S,U(N) (so there is a natural action of S,U(N) defined on r,^) and (2.37) is an intertwiner.
246
W. PUSZ
and S. L. WORONOWICZ
To introduce creation and anihilation operators we have used the first order differential calculus defined by (2.6). We shall discuss briefly the creation and anihilation operators related to the calculus based on (2.6*). Let us notice that the operation consisting in replacing p and < by ‘p- ’ and >, resp., transforms (2.6) into (2.6*). Moreover, this operation leaves equations (2.1) unchanged. Therefore, applying this operation to equations (2.20)(2.35) we obtain the corresponding formulae of the second quantization formalism based on (2.6*). In particular, equations (2.23), (2.24). (2.26) (2.29), (2.31) and (2.35) take the following form: ,+a,? P-
ajai+ =
= p-1+$
for
‘U+ aj
I+pd2au+ai-(l
_p-‘)
(2.23*)
i >.i,
C a$ak
for i fj, for i=j;
(2.24”)
k
/.l
=
-‘aiUj
. . . . nN) = [ fi
In,, n2,
(2.26*) (2.29*)
Fact,~1(nj)]-1’2x;1x;Z...x”,‘,
j=l
ajln,,
i>j,
for
xnk
-
. .., nj, . . ., nN) = A,m~(nj)112p
k’JInl,
. . . . nj-
1, . . . . n,),
- Znr afln,,
. . . . nj, .... nN) = A,-l(nj+
1)“2p
Wn 1, . . . . nN) = &I(;
""ln,,
nj)(n,,
.... nj+
1, . . . . nN),
. . . . nN).
(2.3 1*)
(2.35*)
j=l
In this case the operators 91, d, 6, aj, a’ are no longer bounded. (2.23*) and (2.26*) coincide with (2.23) and (2.26) resp. 3. Representations
of twisted canonical commutation
Let us notice that
relations
This section is devoted to irreducible representations commutation relations (TCCR). A system {a,, a2, . . . . +, a:, ai, . . . . UN+}
of twisted
canonical
(3.1)
of closed linear operators acting on a Hilbert space X such that aj* 3 aj’ for of K‘CR if: j = 1, 2, . . . . N is called a representation (i) There exists a dense linear subset 9 c .8 which is contained in the domain of any product of two operators from the set (3.1). (ii) The system (3.1) satisfies TCCR on 2, i.e. for any XE 3 a+a,t
X = pa;
Ujai_X
=
Ui+X
/lUiLljX
for
i
(3.2)
i
for
(3.3) for i fj,
aja’x
=
x+~2a+ai.~-(I-~z)
C a:a,x k>i
for
i = j.
(3.4)
TWISTED
(iii) A positive
linear
SECOND
operator
OLJANTIZATION
247
of the form F afaj
(3.5)
j=l
is essentially
self-adjoint
on 9.
The representation (3.1) will be called irreducible (cf. [2] Definition Definition 3.1) if the only operator CEB(X’) such that
2.5, [7]
(XICajY) = (aj’XICY),
(3.6)
11)= (LZjXlCY)
(XICUf
for any x, y~6B and j = 1, 2, . . . . N is a multiple of unity; i.e. C = JU for some 1 E C. Let p, q be non-negative integers such that p +q < N and EE]~‘, 11. We consider the set SE,,,, c RN consisting of the elements of the form 2n1 E,
p2nz E
IJ
_Z+ , ..., ,tt2QE, 0, . . . , 0, ~1-P
where nl, . . ., n,; m,, . . ., m4 are integers n, < n2 < . . . < n,
=
...3
(3.7)
v
m,3m23...>mm,30
and
(P2@1>
7 ..”
such that
If p = 0, then the value of E is irrelevant. N, let ‘j be a linear mapping For j= 1,2,..., Tj@
_n% 2
P2&?j,
Qj+l,
introduced ...2
by the formula
@NJ
(3.8)
for any Q = (Q,, . . . . eN)~RN. THEOREM 3.1. Let E E]P’, 11, p, q be non-negative integers such that p + q < N, S = S&p@ H = 12(S) and {IQ): QES} be the canonical basis in 2. We consider linear operators cj (j = 1, 2, . . ., N) introduced by the formula
cjl@)=
1)1’2 ITJ"@)
(@j-@j+
(3.9)
f or any QES (notice that Q~-Q~+~ = 0 tf T;‘Q$S).
These operators are closable and the closures will be denoted by the same symbols.
[f p+q = N or p = 0, then we set aj = cj, a; = CT (j = 1, 2, . . . , N). Zf p+q < N and ‘p > 0, then we set aj = cj, a; = CT for all j # p and a = e@cp, P
+
ap=e
-i”cp*
(q
E
[O,
27c[).
Then the system {a,, a2, . . . . aN, a:, a,f, . .., a$} is an irreducible representation of TCCR. Conversely, any irreducible representation (3.1) of TCCR is unitarily equivalent to one of the representations described above.
W. PUSZ
248 Proofi Let 9
and S. L. WORONOWICZ
be the linear span of {IQ): QES): One can easily check that cj*l@)=
(3.10)
(P2@j-@j+l)1'21zjQ)
0’=1,2 )....) N) and ~2Qj-ej+ 1 = 0 if rje$S. The proof that the operators introduced by (3.9) and (3.10) (and consequently, of TCCR is purely com{a,, a2, . . ., aN, a:, a:, . . . , a,$}) define a representation putational. The details are left to the reader. The irreducibility follows easily from Proposition 3.5 which is shown later. The rest of this section is devoted to the proof of the last statement of the theorem. Assume now that (3.1) is a representation of TCCR acting on 2. Let YI be the closure of (3.5). Since aj’ aj < %, ajaf
d I+p2afaj
< I+p’%
on 9 for j = 1, 2, . . . . N and 9 is a core for %, we see that D(aj) 3 D(m), D(aj+) 3 I)($%). We would like to show that there exists an essential dense invariant domain for operators (3.1). We start with the following LEMMA 3.2. Let k be a non-negative ajx E D(YI’) and !JIkafx %kajX
integer and XE D(Sk+l).
= af(Z+p291)k~, =
Then aj+xE D(‘Sk), (3.11)
(3.12)
aj[~u2(~ZZ)]kX,
for all j = 1, 2, . . . , N.
Proof: We shall prove (3.12). (3.11) can be proved in the same way. If y, ZEN, then using (i) and (ii), we get (‘JZyIajz) = (afyI~-2(%-z)z)* By the closedness Then we get
argument
the same equality
(‘JZyIajz) = (a~yIp-2(%-Z)z)
for any YE D(g).
This means that ajz ED(%) ~Zajz =
(3.13)
holds for y, z~D(‘%n). Let ZED(‘?R~). =
(_YIaj[~L-2(%-Z)]Z)
and
aj[~-‘(~-Z)]Z.
(3.14)
Suppose now that for some integer yt 2 2 and any x E D (‘W) we have ajx E D(‘W- ‘) and !Rplajx
= aj[p-m2(%-Z)]n-1x.
(3.15)
TWISTED SECOND
249
QUANTIZATION
In particular, if XE D(S”+ 1), then ajx~D(!RnnP1) and (3.15) holds. x E D(S2) and (cf. the first part of the case z = [K2(%-Z)]n-1 a.z = ‘fi”-‘ajx~D($ll). Therefore, u~xED(%“). Using (3.14) and (i.15) with 12 increased by one. The proof of (3.12) is complete
Moreover, in this proof and (3.15)) (3.15), we obtain (by induction). n
Let D, denote the space of analytic vectors for 8. We shall show that ajx E D,,, for any .YED,,. Indeed, using Lemma 3.2 and the inequality aj’ aj d % d (%+ Z)2, we get for k = I, 2, . . . . N: II’JlkUjXl12= ~~uj[~-2(9&-_)]kX/~2 = (Xl[~-2(%-z)]kujfuj[~-2(%-z)]kX) d (~-2kII(%+z)k+1X11)2 and the statement follows. Analogously, using the inequality ujaj+ d Z +p2% ,< (I +p2 ‘%)2, one can show that U~XED, for any XED,. Moreover, relations (3.2)-(3.4) hold for any XE D, (compute the scalar product of both sides of (3.2)-(3.4) with a vector belonging to D, and use the relations uj’ c aj*, uj c (UT)*). Let Sj = uj’uj for j = 1, 2, . . . , N. Clearly, ‘91j(D,) c D,. Let us notice that %x=
i
‘JljxforanyxED
W. Using (3.2)-(3.4), one can prove that the operators
91j
j=l
mutually
on D,.
commute
%jz = ufajufuj so
; j=l
‘%3 d
;
by (3.12) we have on D, that
Moreover,
d uf(Z+p2!R)aj
< afuj%
= 91j%
tRjcJI = ‘$I’. Let (Jr;” d ‘%2k for some integer
k > 1. Then
j=l q(k+l)
and by induction
=
SjyySj
<
Sjgp$qj
=
gpgq@
<
yp+
1)
ST” d S2” for any positive integer n. This means that for XE D,: JplzJxIJ2 = (xpq”x)
< (x)YV”x)
= pVXJ12,
so x is also analytic for all !Rj. Using the FS3 theory ([S], [1] Chapter & 6 Theorem 5) we conclude that ‘Sj 0’ = 1, 2, . . . , N) are essentially selfadjoint D, and mutually strongly commute. PROPOSITION 3.3.For anyj = 1, 2, . . . . N: uj’ = UT. Moreover, a core for uj and u,?. Proof: Let D be a core for ‘% and
We shall prove
x=
{(x,
Y=
I(uj’Y
UjX): 3
x~D}
-y):
11 on
any core for ‘$I is
c bob,
YED}
c A?@%.
that
(ajlD)*= t”j’ ID)*
(3.16)
W. PUSZ
250
To this end it is sufficient E (X 0 Y)‘. Then
and S. L. WORONOWICZ
to show that
X@ Y is dense in S@
(ZlX)+(ulUjX)
Let (z, u)
0,
(3.17)
= 0
(3.18)
=
(ZIqV-MY)
Z.
for any x, ye D. Remembering that D is a core for ‘S and using the estimates u,kuj < %, ajuT < I+$‘%, we obtain the same formula for x, LED. Let XED,. Inserting in (3.18) y = ajx and using (3.17), we get (Z)[Sj+r]X)
= 0.
Since D, is a core for ‘Sj, we conclude that z = 0. Now (3.18) shows that u = 0 and (3.16) follows. Clearly, formula (3.16) implies that (uj’ ID) 3 Combining
this fact with the inclusion
uf .
UT I UT, we see that
(uf lo) = aj*= uj’. Passing
to the adjoint
operators (UjlD)
(3.19)
in (3.16) and using (3.19), we have =
(Uj’
IO)*
=
Uj**
=
Uj.
n
Now we shall analyse the notion of irreducibility. Let C, Q be closed operators acting on 2. Assume that C is bounded. We say that C commutes with Q if CXE D(Q) and QCx = CQx for any x E D(Q). LEMMA 3.4. Let V be rhe set of all bounded a von Neumunn ulgbru.
operators
C satisfying
(3.6). Then %?is
Proof: Clearly, V is a linear *-invariant weakly closed subset of B(Z). We have to show that %? is an algebra. Let CE B(H). Using Proposition 3.3, one can easily check that C satisfies the relations (3.6) if and only if C commutes with the operators (3.1). Using this remark, one shows that C, C, E% for any C,, C, ES’. n PROPOSITION3.5. The following two conditions are equivalent: (i) The system (3.1) is irreducible. (ii) P = 0 and P = I are the only orthogonal projections acting in 2 is a subset of D, invariant under the action of the operators (3.1). Px,
such that PD,
Proof- (i)*(ii). Let P be an orthogonal projection such that for any x, y E D,: PYE D, and ujPx, uj’ PyePD, (j = 1, 2, . . . , N). Then (YIPUjX) = (ufPyIx)
= (Puf Pylx) = (y)PujPx)
= (YIUjPX),
TWISTED
SECOND
251
QUANTIZATION
and finally (3.20)
(YIPUjX) = (aj+ylPx).
This relation is proved for x, y E D,. In virtue of Proposition 3.3, (3.20) holds for any x, YE 9. Using selfadjointness of P and interchanging x and y, we obtain (y(PUj+ X) = (ajy1P.X). Therefore C = P satisfies (3.6) and the irreducibility implies that P = il (i E C) and (ii) follows (P is a projection). (ii)*(i). In virtue of Lemma 3.4, it is sufficient to show that %? contains no nontrivial projections. Let P be an orthogonal projection in %?.Then (cf. the proof of Lemma 3.4) P strongly commutes with aj and uj’ (j = 1, 2, . . , N) and P%x = 9lPx for XE~. Remembering that 9 is a core for 91, we see that P strongly commutes with %. This implies that PD, c D,. Moreover, we have ajPD, = PajD, c PD, and analogously, u,? PD, c PD,. Condition (ii) implies now that P equals 0 or I and the result follows. H Let i Lj =
Q-(1
-(l
Relations
j=
-&iz,
1,2 ) . ..’ N,
k=j
(3.21) and (3.2))(3.4)
(3.2 1)
j=N+l.
-g-9,
imply that for j = 1, 2, ajfaj 3 0,
=
Lj-Lj+l
/12Lj-Lj+1
. . . , N:
=ajajf
(3.22)
3 0,
(3.23)
and Lja:
=
Ljak = Let m?,?
a: Lj
for j > k,
1 p2a: Lj
for j d k,
L, =
for j > k,
akLj
(3.25)
for j d k.
i pp2akLj
e2, . . . . Q,) be the common
(3.24)
spectral measure for the family (L,, L,,
j @&(Ql,
k&
. ..r@k. . ..> @N).
RN
For any XEH
we set d~cc,(e,>e2,
Let XED,.
We shall prove
. .
> ed
=
(xlWel>
e2,
. . . > edx).
that
+,,A dPa,+x(~k@)
‘@) = =
(@k
-@k+
(~2@k-@k+
(3.26)
M&A
(3.27)
dh&),
for all k = 1, 2, . . ., N (where zk is the transformation
introduced
by (3.8)).
252
W. PUSZ and S. L. WORONOWICZ
Indeed,
for any sl, s2, . . . , SUER we have
i E hen j ,““I
dp&-*@I,
.a., P-*Q~, e,c+t, .‘.f e,v)
RN
Se
=
&a,&1,
...,
@k?
t?k+l,
-..Y
@iv)
RN
i(
=
(ukx)Ie
x
p*s,L,+
.I1
&lLd E PI=&+* ukx)>
and using (3.25) and (3.22) we see that above expression N N i TXs,L i 1 s& (akxJukenzL x) = (x1(&-&+ l)e n=l x) i en2l X
=
j
equals
hen
(@k-@k+&&&‘l,
*..T
@k, @k+l,
...)
@IV).
R”
In a similar way, using (3.24) and (3.23), we prove formula (3.27). We shall consider the subgroup .Y c GL(N, R) generated by the transformations Ti, r2, .‘., rN. Let x = [I-l,
-$)u
(0) u($,
1-J
and X” be the Cartesian product of N copies of X. One can easily check that any orbit of Y intersects XN of precisely one point. In other words, XN parametrizes the space of orbits, XN = RN/F.
(3.28)
(Warning: The above identification concerns only the Bore1 structure of considered spaces; RN/F as a topological space is not Hausdorff.) Let ZE 2, dfi, be the projection of dpz onto XN (cf. (3.28)), QE supp dp,, 6 be a neighbourhood of 4 in XN, 0 be the set-theoretical sum of all orbits belonging to 6, Z(0) be the subspace of all vectors x E %’ such that the measure dpcL,is concentrated on 0 and P, be an orthogonal projection on Z(0). Then (zlPoz) > 0 and PC # 0. Clearly, P,D, c D, (P, commutes with %) %‘(O) n D, 2 POD,. On the other hand, Therefore, if x E S(0) n D,, then x = P,xEP~D,. Y?(O) n D, = POD,. Formulae (3.26), (3.27) show now that POD, is invariant under the action of aj and aj’ ij= 1,2, . . . . N). Assume that the representation (3.1) is irreducible. Then using Proposition 3.5, we see that P, = I and Z(0) = %. This means that the spectral measure d-E(.) is concentrated on 9. By considering a decreasing sequence of neighbourhoods 6
TWISTED SECOND
253
QUANTIZATION
shrinking to 4, we conclude that the spectral measure dE(.) is concentrated on the one orbit VEX N. Since any orbit of Y is denumerable, the family {L, , L,, . . . , LN} has a pure-point common spectrum. Let S denote the pure-point common spectrum of {L,, L,, . . . , LN}, and K = ((@I, @2, ..*y @N)ERN: @j>, @j+l, P2@j 2 @j+l, j=
1, 27 ..*y N}.
In virtue of (3.22) and (3.23) S c K. We remind that &?N+ 1 = -( 1 -p2)-l. For any QE RN, Xe will denote the common eigenspace of {L,, L,, . . . , &} corresponding to the system of eigenvalues Q and 2 = @ xe. Let PES
We shall prove PROPOSITION 3.6. Assume that the representation (3.1) is irreducible. 7hen S = SE,,., for some E E 1~2, 11and some non-negative integers p, q such that p + q < N.
Proof: Uncomplicated combinatorial analysis shows that SE,P,Qare the only sets S satisfying the following three conditions: (i) For any QES and tET if e$aK, then t@ES. (ii) For any QES and tET if t@EK, then @ES. (iii) For any eo, QES there exist sequences t,, t,, . . . , t, E T and er, ez, . . . . , Q, = QES such that ek = t,Qk_l Therefore we shall prove Let QES and $E&@,
(k = 1, 2, . . ., n).
that S satisfies conditions /1$1/= 1: Lk$ = @,ti?
(i)(iii).
k = 1, 2, ...,N.
Using (3.24) and (3.25), we see that
In virtue
L,afII/
= (zje),aT$,
j, k = 1, 2, . . . . N,
(3.29)
L,aj$
= (73 ‘&aj$,
j, k = 1, 2, ...,N.
(3.30)
of (3.22) and (3.23) we obtain I/afIC/112=~2Qi-gj+1, Ilaj~ll’ =Qj-Qj+l,
j= j=
1, 2, ..-, N, 1, 2, ...y N.
(3.3 1) (3.32)
Let &aK. Then the above expressions are different from zero and this and (3.29), (3.30) shows that rje, z,:‘QES for j = 1, 2, . . . . N. Let rje~ K for some jE { 1, 2, ...,N}. We consider three cases:
W. PUSZ
254
and S. L. WORONOW!CZ
ej > 0. Then ~‘ej-ej+1 > ~4ej-ej+l = p2(rje)j-(rje)j+ r 3 0 since Z~QEK. Formula (3.31) shows that UT $ # 0 and (cf. (3.29)) zjeeS. ej = 0. If ek = 0 for all k
Proof’cf the last part cf Theorem 3.1. Let (3.1) be an irreducible representation of TCCR, S = SE,,,, (cf. (3.7)) be the pure-point common spectrum of {L,, L,, . . . , LN} (cf. Proposition 3.6). Let Q, @‘ES. We say that the pair (Q, Q’) is a close pair if there exists t E T such that Q’ = CQ. Since the transformations zj (j = p, . . . , N-q - 1) coincide with zNey on S, we can always assume that t E TP,4 = {zj, zi I: jEJp,q} where J,,, = (1, . ..) p-l, N-q, N-q+l, . ..) N}. For any close pair (e, Q’) we define an operator
z e’,e
=
ai
if
Q’ = zj 1Q and j E J,,,,
uj’
if
Q’ =
ZjQ
and j E J,%, .
(3.33)
It follows from the proof of Proposition 3.6 that if (Q, Q’) is a close pair and $ E ze, $ # 0, then Ze,,e$~A?,, and ;5eJ.e$ # 0. If eI, ez, .., Q,, Q,,+~ = er is a sequence of elements of S such that (Q~,Q,,,+~) is a close pair for all m = 1, 2, . . ., n and $EX~~, $ #O, then (3.34) where A>O. Indeed,.let t,, t,, . . . . t, be the unique sequence of elements of TP,4 such that ek+ 1 = tkek (k = 1, 2, . . . , n). Clearly, for any j E J,,,, any zj appears in this sequence as many times as z_,,’ does. Therefore (cf. (3.2) and (3.3)) the operator on the left-hand side of (3.34) equals (up to a positive factor) to a product of operators of the form uj’ aj and ajnj+ (j E J,,,). Using then (3.22) and (3.23), we see that $ is an eigenvector of u,? aj and ujuj’ (with strictly positive eigenvalue) and (3.34) follows. Now we take a sequence er, e2, . . of elements of S such that: (i) (Q,, en+ r) is a close pair for any y1= 1, 2, . . (ii) If (Q, Q’) is any close pair in S, then g = Q,, Q’ = Q,+ 1 for some ~1. Let us choose a non-zero element I+!I E Xe, . We introduce a sequence of non-zero vectors +r, ti2, . . . defined by the recurrence formula ti nfl
--z
en+ 1.e” tin.
TWISTED
SECOND
255
QUANTIZATlON
For any QE S there exists n such that e = q,. We set I@)= 1111/,11-‘4k. In virtue of (3.34), the right-hand side of this definition is independent of n. It follows immediately from the above construction that I&+ 1) = GYG+
of the choice
(3.35)
,,e,le,)
where CI, is a normalization factor. Let e E S and JE J,,,. Assume that 7,: ’ QE S! Then (Q, ~~7’ e) is a close pair and there exists a positive integer IZ such that q = q,, ~~7‘q = en+ 1. In this case Z en+1,etl-- aj. Using (3.35), we obtain
ail@) = %l~I”@). The normalization
factor
CI, can easily be computed.
&? =
Using (3.23), we have
ll~jl@)l12 = k?k?j+aj@) = @j-@j+l
(3.36)
ajk?) =
(3.37)
and finally for j E J,,,: (@j-@j+l)1'21Zj1@).
We proved this formula assuming that ~~7‘@ES. However, this formula is also valid = 0 and (cf. (3.36)) both sides of (3.37) ifeES and zJ’q$S, since in this case qj-qj+r vanish. Now we consider three cases. (i) iV=p+q.ThenJ,,,={1,2 ,..., NJ and formula (3.37) describes the action of all aj (j = 1, 2, . . . , N) . (ii) p = 0. Let us notice that (cf. (3.37)) aj’ aj = Lj- Lj+ 1 = 0 for j = 1, 2, . . . , N-q - 1. Then aj = 0 for suchj and since formula (3.37) is also valid in this case, we have that (3.37) is true for all j = 1, 2, . . . , N. (iii) p+q
(3.38)
We shall show that ap = eiqccp for some cpE [0, 27r[. At first we notice that the relations L,, 1 = 0 = LNpq, (3.22) and (3.23) imply that %&+ =p2apfap=p2Lp and aNfpqaNpq = aN_qai_q
= -LN_q+l.
(3.39)
Let Q = a,ai_q.
(3.40)
256
W. PUSZ and S. L. WORONOWICZ
Then
QaN_, = -a,L,_,+l
and
a, = -QaN_4LN!q+l
= -QLNlq+laN_q.
(3.41)
Using (3.25), we see that Q*Q = -p2LpLN_q+1 The operator --P’L~L~_~+ 1 is invertible spectrum of L,, L,, . . . , LN). Therefore,
= QQ*.
(cf. formula
(3.7) describing
Q = W-P~L&N-~+IP~,
the common
(3.42)
where W is a unitary operator acting on YE’. Remembering that aj = 0 for j = p + 1, . . . , N-q + 1 and using (3.2)-(3.4) one can check that W commutes with aj and a; ij = 1, 2, . . . , N). The irreducibility implies now that W= eiVI for some ~E[O, 2n[. Now using (3.42) and (3.41), we have: up = e’“( -p2 L,L,1,+
l)1i2a,_,
and a,lQ) = ei’P(-~2LpLNlq+1)1’2(-eN_q+1)1’21zN’,e) = ei’PQi/21TNlyQ)= ei’Pek’21r,‘e)
It follows easily from the presented proof that the irreducible representations of TCCR determined by SE,p,q (and eventually by cp) are pairwise non-equivalent. It is interesting that contrary to the classical case (Stone-von Neumann uniqueness theorem) we obtained a large family of representations. The Fock representation considered in Section 2 corresponds to p = 0 and 4 = N (the value of E is irrelevant). REFERENCES [I] [2] [3] [4] [S] [6] [7]
Barut, A.O., Rgczka, R.: Theory of Group Representations and Applications, PWN-Polish Scientific Publishers 1977. Borchers, H. J., YngvaFon, J.: Commun. Math. Phys. 42 (1975), 231-252. Drinfeld, V. S.: Quantum Groups, will appear in Proceedings ICM-1986. Enock, M., Schwartz, J. M.: Bull. Sot. Math. Frunce, Supllment mhoire 44 (1975), l-144. Flato, M., Simon, J., Shellmann, H., Sternheimer, D.: Ann&s Scient. Eco/e Normale Sup. 4’ serie 5 (1972), 423434. Pusz, W.: Twisted canonical anticommutation relations (to appear in Reports on Mathematical Physics). Powers, R.: Commun. Math. Phys. 21 (1971), 85-124.
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SECOND
[S] Schwartz, J. M.: J. Fun&. Anal. 34 (1979), 370-406. [9] Slowikowski, W.: “Commutative Wick algebras I. The Bargman, Proceedings
of the Conference
257
QUANTIZATION
on Vector Space, Measures
Wiener and Fock algebras”, Dublin 1977. Lecture
and Applications.
Notes in Mathematics, Vol. 644. Springer 1978. [lo] Woronowicz, S. L.: Commun. Math. Phys. 111 (1987), 613-665. Cl l] Woronowicz, S. L.: Tannaka-Krein duality for compact matrix pseudogroups. Invent. Math. 93 (1988), 35.-76.
Twisted SU(N) groups,