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C°°{M) C{Td) which yields the identification with C°°(M)9. Second, one uses the previous identification to deform the C°°(M)-module of sections T(M, S) of the spinor bundle S —> M. Since the action T is isometric, up to a double covering of T d , the spinor bundle is T d -equivariant. The space of invariants T(M, S)e := {T(M, S) ® C ( T d ) e ) r x L _ 1 is therefore a C°°(M)e-module stable by the operator D ® I. The restriction of the latter on T(M, S)g then defines the deformed Dirac operator D$.
We end this subsection by stressing the fact that the way Connes and Dubois-Violette define a deformation of the Dirac operator via an action of Td essentially relies on two crucial properties. First, the invariance of Rieffel's deformed product on C(Td)e under the (left or right) regular representation (this allows to define the algebra (or module) structure on the space of invariant elements in the tensor product). Second, the fact that the action T of T d on M is isometric.
5.2. Symmetric
spaces and universal
deformation
formulae
In this section we recall a consruction of a universal (strict) deformation formula for the actions of a non-Abelian Lie group. This formula was obtained via geometric methods based on symplectic symmetric spaces. We begin by recalling the notion of symplectic symmetric space, we then pass to the specific example which will be relevant here. A symplectic symmetric space 7 ' 8 is triple (M, u, s) where (M, w) a symplectic manifold and s : M x M — > M is a smooth map such that for each point x G M the partial map sx : M —> M : y i-> sx(y) := s(x, y) is an involutive symplectic diffeomorphism which admits x as isolated fixed point. Under these hypotheses, the following formula defines a symplectic torsionfree affine connection V on M: w x ( V x F , Z) := \xx.w{Y
+ sx*Y, Z);
(16)
where X, Y and Z are smooth vector fields on M. This connection is the only one being invariant under the symmetries {SX}X^M which turns out to be the geodesic symmetries. The group G(M) generated by the products
28
P.BIELIAVSKY,
S.DETOURNAY, P.SPINDEL and M.ROOMAN
sx o sy is a Lie group of transformations of M which acts transitively on M. It is called the transvection group of M. The particular example we will be concerned with is the one where the transvection group is the Poincare group P^i := 5 0 ( 1 , 1 ) xR 2 . It is a three dimensional solvable Lie group whose generic (two dimensional) coadjoint orbits are symplectic symmetric spaces admitting Px,i a s tranvection group. They are all equivalent to oneanother (Pi,i/M as homogeneous spaces) and topologically E 2 . Let (M, ui) be such an orbit and let V be tis canonical connection. Such a space turns out to be stricly geodesically convex in the sense that between two points in M there is a unique geodesic. It has moreover the property that given any three points x, y and z in M, there's a unique fourth point t in M satisfying sxsyszt
= t.
(17)
This allows to define the three point phase of M as the three point function S € C°°(M x M x M, R) 23 given by S(x,y,z):=
[ J
u
(18)
A(t,szt,syszt)
where A(x, y, z) denotes the geodesic triangle with vertices x, y and z. The three point amplitude A G C°°(M x M x M,R) 20 ' 6 is defined as the ratio
'XA(t,s t,s 2
i ; 5 z t)
UJ
V J±{x,y,z)w J Both S and A are invariant under the diagonal action of the symmetries. For compactly supported u,v e C£°(M), one sets A
u *e v(x) := H / y
fa
Xf which satisfies the Leibniz rule, X(fg) — (Xf)g + fXg. Derivations will be denoted by X, Y, ea and so forth and the set of all derivations by Der(.4). A simple example is the algebra of 2 x 2 complex matrices M2 with (redundant) generators the Pauli matrices. The algebra is of dimension four, the center is of dimension one and Der(M2) is of dimension three with basis consisting of three derivations ea = &&aa: eaf = [aa,f}.
(2.5)
We notice that the Leibniz rule is here the Jacobi identity. We see also that the left multiplication aaeb of the derivation e\, by the generator aa no longer satisfies the Jacobi identity: it is not a derivation. The vector space Der(M2) is not a left M2 module. This property is generic. If X is a derivation of an algebra A and h an element of A, then hX is not necessarily a derivation: hX(fg) = h(Xf)g + hfXg ? h(Xf)g + fhXg
(2.6)
if hf ± fh. Although derivations do not form a left module, one can introduce associated elements known as differential forms which form a bimodule; they can be multiplied from the left and from the right. We shall therefore express as much as possible physical quantities using the latter. We define here a 1-form w is a linear map CJ : Der(.4) —• A. The set of 1-forms ^(A) has a bimodule structure, that is, if u> is a 1-form, fw and uif are also 1-forms. The elements of CI1 (A) will be typically denoted by w, 6, £, 77. The important step is the definition of a differential d\ it is a linear map from functions to 1-forms, d : A —> 01(«4) which obeys the Leibniz rule. In general fdg ^ dgf but we shall introduce later special forms 6a which commute with the algebra. The exterior product £and r\ of two 1-forms £ and 77 is a 2-form. There is no reason to assume the exterior product antisymmetric. We mention also that one can deduce the structure of the algebra of all forms from that 'of the module of 1-forms. The map d can be extended to all forms if one require that d2 = 0. We should stress that in general one can associate many differential calculi to a given algebra. There is then a variety of possibilities to define a differential. One problem is how to determine or at least restrict it by imposing some physical requirements. We shall use here a modification of the moving frame formalism and show that so defined differential calculi over an algebra admit
Dynamics
39
of Fuzzy Spaces
essentially a unique metric and linear connection. We shall fix therefore the differential calculus by requiring that the metric have the desired classical limit. The idea is to define an analogue of a parallelizable manifold, which therefore has a globally defined frame. The frame is defined either as a set of vector fields ea or as a set of 1-forms 6a dual to them. The metric components with respect to the frame are then constant. We choose a set of n derivations ea which we assume to be inner generated by 'momenta' pa: eaf = \paj}.
(2.7)
We suppose that the momenta generate also the whole algebra A. Since the center is trivial, this means that an element which commutes with all momenta must be a complex number. An alternative way is to use the 1-forms 6a dual to ea such that relation ^(eb) = 6?
(2-8)
holds. To define the left hand side of this equation we define first the differential, exactly as in the classical case, by the condition df(ea) = eaf.
(2.9)
The left and right multiplication by elements of the algebra A are defined by fdg = fea9ea,
dgf = eagf9a.
(2.10)
Since every 1-form can be written as sum of such terms the definition is complete. In particular, since f6a(eb) = fSi = (eaf)(eb),
(2.11)
we conclude that the frame necessarily commutes with all the elements of the algebra A; this is a characteristic feature. If one does not insist on using differential calculi defined by inner derivations this condition can be generalized to include frames which commute only modulo an algebra morphism. For a recent discussion of this possibility we refer to 10 . In the case of the algebra Mi considered above, the module of 1-forms is generated by three elements daa defined as the maps d<Ja(eb) = ebaa = [ab, aa}.
(2.12)
The maps acdcra and daaac are defined respectively as acdaa(eb)
= <Jc[ab,cra},
daaac(eb)
= [ab, aa]ac.
(2.13)
40
M. BURIC and J.MADORE
Obviously, acdaa ^ daaac. The 1-form 0 defined as 0 = ~Pa0a
(2.14)
can be considered as an analog of the Dirac operator in ordinary geometry. It implements the action of the exterior derivative on elements of the algebra. That is df = ~ [0, f] = IpaO", f] = \Pa, f] 0a-
(2.15)
The differential is real if (df)* = df*. This is assured if the derivations e„ are real: eaf* = (e Q /)*, which is the case if the momenta pa are antihermitean. From the definitions one has 0a* = 0°, 0* = -0. Furthermore, (/£)* = C A (£/)* = / T , and (£77)* = -rj*C- Note that whereas the product of two hermitean elements is hermitean only if they commute, the product of two hermitean 1-forms is hermitean only if they anticommute. The implementation of the differential structure as we have given is just as arbitrary as before since it amounts to a choice of the momenta. In some cases, the construction of the frame is not difficult. In the example (2.2) one can choose the differential such that [a^efcr^] = 0; a frame is 0a = 5fdxl since dxl commute with all elements of the algebra. The most general form is 0a = Afdx1 with A" real numbers. The duality relations give the momenta Si = 0a(eb) = Sfdx\eb)
= Sfebix') = 6?\pb, x%
(2.16)
that is, Pb = -tiUki1*1-
(2-17)
In order to discuss noncommutative limit, (2.2) should in fact be rewritten as [x\x>]=ikJii,
(2.18)
introducing the parameter % to describe the fundamental area scale on which noncommutativity becomes important. The k is presumably of order of the Planck area Gh; the commutative limit is denned by k —> 0. The momenta read then Pa = JfrTrtX'' and they are singular in the limit k —> 0.
( 2 - 19 )
Dynamics
41
of Fuzzy Spaces
Since the frame is given by 8a = Sfdx1, this space can be naturally thought of as the noncommutative generalization of flat space. The momenta are linear in the coordinates and hence \Pa,Pb) = J ^ ^ J ^
[**. *] = Kab,
Kab = - 1 j £ .
(2.20)
In general only by explicit construction can one show that the frame exists. In the case of the Lie algebra (2.3), for example, one sees that the 1-forms dx1 do not define a frame because they do not commute with the algebra. In the example of Mi with Pauli matrices as momenta, the frame which is the solution to the equation (2.8) is seen to be 0a = \uhaadab.
(2.21)
n
This construction can be repeated for the algebra Mn of n x n complex matrices. As a last example we consider the algebra generated by two hermitean elements x and y related by [x, y] = -2i%ny.
(2.22)
This is related to the Jordanian deformation 12 of R 2 with deformation parameter h = ilcfi2. The /u is the gravitational mass scale; the associated length G\i vanishes with k. To find the frame, we rewrite this as follows (x + ik[i)y = y(x — iUfi).
(2.23)
The differential must satisfy dxy + (x + ikfx) dy = dy(x — ikfi) + y dx.
(2.24)
We shall impose separately the conditions dxy = ydx,
(x + ik[i)dy = dy (x —ikfi).
(2.25)
The first of these relations suggests that dx can be taken as a frame element, in fact f(y)dx as well as dx. We set 61 = f(y)dx. Rewriting (2.25) as (a; + i%n)yy~l dy = yy~xdy y~xy(x - iS/x),
(2.26)
we see that we can take 92 = —(fiy)~1dy. We assume that 8a commute with x and y. The duality relations (2.8) determine f{y). They read
f(y)\pi,x] = l, 1
{w)~ \pi,y] = o,
f(y)\p2,x} = 0, (/iz/)-1[P2,y] = - i ,
42
M. BURIC and J.MADORE
and reduce to (2.25) for
The frame therefore is given by 01 = (fiy^dx,
62 = -(fiy^dy.
(2.29)
The corresponding calculus is the covariant one 12 . The momenta are proportional to the coordinates so their commutation relation is: \pi,P2]=Wi.
(2.30)
A short calculation shows that the frame elements anticommute. The line element ds2 = ±(§1)2 ± (§2)2
(2.31)
of the commutative limit is that of the Lobachevski plane or of (anti) de Sitter space depending on the choice of signature. 3. Rindler space: frame rotations As an example of a frame for which such a noncommutative extension does not exist we consider the 2-dim Rindler frame which is defined in one-half of 2-dim Minkowski space. We shall use this example to illustrate the fact that not all moving frames are suitable for 'quantization'; some are more suitable than others. Let /i be a parameter with dimensions of mass and proportional to the Rindler acceleration. The commutative Rindler frame is given by 0° = \ixdl and 0l — dx; the commutative Minkowski frame is 6'° = di' and 6'1 = dx'. The local Lorentz rotation from the former to the latter is defined by e,a = A - l a 0 &
(3 J)
with ^
=
/cosh fxi sinh fj,i\ \ sinh id cosh fit J
The classical coordinate transformation from the Rindler coordinates to the Minkowski coordinates is given, for x > 0 by x1 = x cosh /it,
i' = xsinh/xf.
It is of course not to be confused with the rotation.
(3.3)
Dynamics
43
of Fuzzy Spaces
We shall first show that the Rindler frame is not a suitable frame; there are no dual momenta. If momenta pa did exist then they would necessarily satisfy the relations \po,t] = (fix)-1,
\p0,x] = 0,
[Pi,*] = 0,
\p1,x] = l.
But one easily sees that the solution is not a quadratic algebra. In fact if one set bo, Pi] = An and [t, x] = ikJ01, one finds from the Jacobi identities that [L 0 i, t] = bo, [Pi, *]] - b i . bo, *]] = M _ 1 z~ 2 ,
(3.5)
[LQ1, a;] = bo, b i , x\\ ~ b i , bo, x\] = °-
(3-6)
The Loi commutes with x and therefore belongs to the algebra generated by x. Prom the commutation with t one finds zfc/i(-r-£oi)J 01 = -x'2. ax
(3.7)
Similarly one has iH]po, J 0 1 ] = [bo, t],x]-
[bo, x],t] = 0,
i*\pi,J01] = [\put],x]-[\pux],t] 01
=0
from which one concludes that J is constant. We shall set J deduce therefore, neglecting integration constants, that 1 m = —x' 1 .
(3.8) (3.9) 01
= 1. We
(3.10)
But the duality relations (3.4) require
and thus one easily sees that L01 = -ie'*"* 0
(3.12)
which is not a quadratic expression in po and p\. The expressions (3.11) for the momenta seem quite different from the corresponding commutative expressions for the derivations e^ dual to the frame: e 0 = (fix^do,
ei = di.
(3.13)
44
M. BURIC and
J.MADORE
However in both cases one obtains the same action on the generators of the algebra. In particular \p0,t] = (jtx)-1.
e0i=(iJ,x)-\
(3.14)
Here one appreciates the importance of the space-time commutation relation. Although the momenta pa dual to the frame which we have used do not satisfy a quadratic relation it is easy to introduce another set pa which do. We define the new momenta by the equations Po = -^re-ik™,
p!=Pl.
(3.15)
They obey the commutation relation \po,Pi} = ^ .
(3.16)
From (3.11) one see also that pa are related to the coordinate generators by the transformations x = — iftpo,
t — ikp\.
(3-17) a
l
The frame defined by the new momenta is given by 6 = 5?dx ; it is a Minkowski-like frame in Rindler coordinates. We put here a bar on the differential to emphasize that the calculus is different. In spite of the apparent nonlocality in the transformation (3.15) the action of both po and po [po, / ] - {»x)-ld0f,
[po, / ] - dof
(3.18)
is local. We now compare the differential calculi defined by two different frames, Rindler and Minkowski, and related by a noncommutative frame rotation of the type (3.1). Both frames can be used to define a differential calculus; each differential calculus has at most one basis as frame. Let 8a be a global moving frame for some 2-dimensional commutative geometry and let {6,a} be the set of all moving frames Q'a such that
for some local Lorentz rotation A. The set of noncommutative versions will be described then each by a frame 6,a which we shall suppose related by g,a
=
A-laeb
(3
20)
for the corresponding 'noncommutative' local Lorentz rotation. In the special cases we have been considering one can restrict the matrices A to the
Dynamics
of Fuzzy Spaces
45
subset the elements of which depend on but one generator so they are welldefined. In more general situations the definition would require elaboration. It is clear that if [/, 9'a] = 0 then [f,6a] = lf,Aab}A-lbc6c.
(3.21)
This can also be written as a rule eaf = Kifhrlbcec
(3.22)
for relating the left- and right-module structures. In general then each local rotation defines a different calculus. The equivalence class of (commutative) moving frames gives rise to a set of inequivalent frames which have the same classical limit. If one wishes to consider one calculus, defined, say, by the condition [/, 9a] = 0 then each of the bases 9'a satisfies the relation 9,af = K-llfKbce'c.
(3.23)
That is, 9'a is not the frame for the same calculus unless A is a global rotation (a constant matrix). Note that the Rindler metric can be considered equivalent to the 2-dim Kasner metric. The Kasner metric in dimension-4, for a special value of the parameters, is flat and a moving frame can be chosen which for these values become the ordinary fiat frame: 9° = di,
91 = dx-
i^xdi,
92 = dy,
93 = dz.
(3.24)
We refer here to {6°, 9 } as the 2-dim Kasner moving frame. By a change of variables
x-*i,
i^i~lx.
(3.25)
the Rindler frame can be brought to this form. 4. Kasner: noncommutative corrections We shall here study the noncommutative corrections of the Kasner metric as denned in (3.24). For convenience instead of x and i we choose as classical variables t and 4> = rxx.
(4.1)
As we have already learned, not all moving frames attached to a metric are suitable for quantization; in this case the appropriate differential calculus is that determined by the flat Minkowski frame. The classical frame rotation
46
M. BURIC and J.MADORE
from the Minkowski moving frame 6a to the Kasner moving frame fja is given by 770 = c o s h ^ ^ ° - s i n h ^ 1 , 7j1 = -sinh^6» 0 + c o s h ^ 1 .
(4 2)
'
The relation between the two coordinate systems is given by x' = ismh(f), 2
i' =icosh(j>.
(4.3)
2
It follows that P = i' - x' ; the origin of the Kasner time coordinate, exactly at the flat-space values of the parameters and because of the singular nature of the transformation, becomes a null surface. In the noncommutative case we choose the symmetric ordering; therefore the change of generators (4.3) becomes x' = tsinhd) — hit, sinh
