Non-commutative geometry and matrix models

0.24 MB·English

by  Harold Steinacker

Non-commutative geometry and matrix models
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5 Harold Steinacker
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FakultätfürPhysik,UniversitätWien
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h Boltzmanngasse5,A-1090Wien,Austria
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- E-mail: [email protected]
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Thesenotesprovideanintroductiontothenoncommutativematrixgeometrywhichariseswithin
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matrix models of Yang-Mills type. Starting from basic examples of compact fuzzy spaces, a
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v general notion of embedded noncommutativespaces (branes) is formulated, and their effective
1 Riemanniangeometryiselaborated. Thisclassofconfigurationsispreservedundersmalldefor-
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5 mations,andisthereforeappropriateformatrixmodels. A realizationofgeneric4-dimensional
5 geometriesissketched,andtherelationwithspectralgeometryandwithnoncommutativegauge
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9 theoryis explained. Ina secondpart, dynamicalaspectsof thesematrixgeometriesareconsid-
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1 ered.Theone-loopeffectiveactionforthemaximallysupersymmetricIKKTorIIBmatrixmodel
1 isdiscussed,whichiswell-behavedon4-dimensionalbranes.
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3rdQuantumGravityandQuantumGeometrySchool
February28-March13,2011
Zakopane,Poland
Speaker.
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(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Non-commutativegeometryandmatrixmodels HaroldSteinacker
1. Introduction
Our basic notion of space and time go back to Einstein. Space-time is described in terms of
a pseudo-Riemannian manifold, whose dynamical metric describes gravity through the Einstein
equations. Thisconceptalsoprovides abasisforquantum fieldtheory, wherethemetricisusually
assumedtobeflat,focusing ontheshort-distance aspectsofthefieldslivingonspace-time.
Despite the great success of both general relativity and quantum field theory, there are good
reasons why we should question these classical notions of space and time. The basic reason is
that nature is governed by quantum mechanics. Quantum mechanics is fundamentally different
from classical physics, and the superposition principle rules out a description in terms of sharply
defined classical objects and states. Since general relativity (GR)couples matter withgeometry, a
superposition ofmatterentailsalsoasuperposition ofgeometries. Wearethusforcedtolookfora
consistent quantum theoryoffieldsandgeometry, henceofgravity.
Constructing a quantum theory of gravity is clearly a difficult task. Not only is general rela-
tivity not renormalizable, there are arguments which suggest that theclassical geometric concepts
are inappropriate at very short distances. A simple folklore argument goes as follows: localizing
anobjectatascaleD xinquantummechanicsrequirestoinvokewave-numbersk 1 ,andthusan
∼ D x
energyoforderE =h¯k h¯ . Nowingeneralrelativity, alocalizedenergyE definesalengthscale
∼ D x
given by the corresponding Schwarzschild radius R GE h¯G. Since observations
Schwarzschild ∼ ≥ D x
inside trapped surfaces do not make sense, one should require (D x) R h¯G, hence
≥ Schwarzschild ≥ D x
(D x)2 h¯G=L2 . Ofcoursetheargument isover-simplistic, howeverarefinedargument [1]sug-
≥ Pl
gests that quantum mechanics combined with GR implies uncertainly relations for the space-time
coordinates at the Planck scale. Even if one does not want to take such “derivations” too serious,
there iscommonconsensus thatspace-time shouldbecome fuzzyorfoam-like atthePlanckscale.
Canonical orloopquantum gravity indeed leadstoanareaquantization atthePlanckscale, andin
stringtheorysomething similarisexpected tohappen [2].
The short-distance aspects of space-time are problematic also within quantum field theory
(QFT),leadingtothewell-knownUVdivergences. Theycanbehandledinrenormalizable QFT’s,
but imply that some low-energy properties of the models are very sensitive to the short-distance
physics. This leads to serious fine-tuning problems e.g. for the mass of scalar (Higgs!) fields,
whichstronglysuggestsnewphysicsatshortdistancesunlessoneiswillingtoacceptananthropic
point of view. Taking into account also gravity leads to even more serious fine-tuning problems,
notably the notorious cosmological constant problem. The point is that vacuum fluctuations in
quantum field theory couple to the background metric, which leads to an induced gravitational
action, inparticular toaninduced cosmological constant. Lacking anynatural subtraction scheme
for these terms, these contributions are strongly UV divergent, or very sensitive to the UV details
of the model. No convincing solution to this problem has been found, which should arise in all
approaches basedongeneral relativityincluding loopquantum gravityandstringtheory.
Given all these difficulties, wewill discuss a radically different approach here. Since the no-
tionsofspace-time andgeometrywereargued tomakesenseonlyatmacroscopic scales, thebasic
degrees offreedom inafundamental quantum theory maybeverydifferent from the macroscopic
ones,whilespace-timeandgeometry“emerge”insomesemi-classicalsense. Thisideaisofcourse
not new, and there are many models where some effective metric emerges in composite systems.
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
However,weneedamodelwhichleadstoauniversal dynamical metriccoupling toa(near-) real-
isticquantum fieldtheory, simpleenoughtoadmitananalytic understanding.
Inthesenotes,wewilldiscusscertainspecificmatrixmodelsofYang-Millstype,whichseem
to realize this idea of emergent geometry and gravity in a remarkably simple way. These models
have been put forward in string theory [3, 4], and may provide a description for the quantum
structure of space-time and geometry. The beauty lies in the simplicity of these models, whose
structure is
SYM =Tr[Xa,Xb][Xa′,Xb′]gaagbb +fermions. (1.1)
′ ′
HereXa,a=1,...,D areasetofhermitian matrices, andwefocusonthecaseofEuclidean signa-
tureg =d inthisarticle. Nonotionofdifferentialgeometryorspace-timeisusedinthisaction.
ab ab
Thesegeometrical structures ariseintermsofsolutions andfluctuations ofthemodels. Theaimof
thisarticleistoclarifythescopeandthemathematicaldescription ofthismatrixgeometry.
Simple examples of such matrix geometries, notably the fuzzy sphere S2 or more general
N
quantized homogeneous spaces including the Moyal-Weyl quantum plane R2n, have been studied
q
in great detail. However to describe the general geometries required for gravity, one cannot rely
on their special group-theoretical structures. The key is to consider generic quantizations of sub-
manifolds or embedded noncommutative (NC) branes M RD in Yang-Mills matrix models [5].
⊂
Thisprovidesasufficientlylargeclassofmatrixgeometriestodescriberealisticspace-times. Their
effective geometry is easy to understand in the ”semi-classical limit“, where commutators are re-
placed by Poisson brackets. M then inherits the pull-back metric gmn of RD, which combines
with the Poisson (or symplectic) structure q mn (x) to form an effective metric Gmn (x). Our task is
thentoelaborate theresulting physics ofthesemodels, andtoidentify thenecessary mathematical
structures tounderstand them.
The aim of these notes is to provide a basic understanding of matrix geometry and its math-
ematical description, and to explain the physical relevance of matrix models. We first recall in
detail some examples of matrix geometries with special symmetries. This includes well-known
examples such as the fuzzy sphere, fuzzy tori, cylinders, and the quantum plane. Wethen explain
how to extract the geometry without relying on particular symmetries. The spectral geometry of
thecanonicalLaplaceoperatorisdiscussed,andcomparedwithasemi-classicalanalysis. Aneffort
ismadetoillustratethescopeandgeneralityofmatrixgeometry. Theremarkablerelationbetween
matrixgeometryandnoncommutative gaugetheory[6,7,8]isalsodiscussed briefly.
Our focus on matrix geometry is justified by the good behavior of certain Yang-Mills matrix
models–moreprecisely,ofonepreferredincarnationgivenbytheIKKTmodel[3]–underquanti-
zation. Thiswillbeexplainedinsection10. TheIKKTmodelissingledoutbysupersymmetryand
its(conjectured)UVfinitenesson4-dimensionalbackgrounds, anditmayprovidejusttherightde-
greesoffreedomforaquantumtheoryoffundamentalinteractions. Alltheingredientsrequiredfor
physicsmayemergefromthemodel,andthereisnoneedtoaddadditional structure. Ourstrategy
is hence to study the resulting physics of these models and to identify the appropriate structures,
while minimizing any mathematical assumptions or prejudices. It appears that Poisson or sym-
plectic structures do play a central role. This is the reason why the approach presented here does
not follow Connes axioms [9] for noncommutative geometry, but we will indicate some relations
whereappropriate.
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
2. Poissonmanifoldsand quantization
Westart by recalling the concept ofthe quantization of aPoisson manifold (M, .,. ), refer-
{ }
ring e.g. to[10]andreferences therein formoremathematical background. APoisson structure is
an anti-symmetric bracket .,. : C(M) C(M) C(M) which is a derivation in each argu-
{ } × →
mentandsatisfiestheJacobiidentity,
fg,h = f g,h +g f,h , f, g,h +cycl.=0. (2.1)
{ } { } { } { { }}
Wewillusually assumethatq mn = xm ,xn isnon-degenerate, thusdefiningasymplecticform
{ }
1
w = q mn−1dxm dxn (2.2)
2
in local coordinates. In particular, the dimension dimM =2n must then be even, and dw =0 is
equivalent totheJacobiidentity.
Itissometimesusefultointroduce anexpansion parameterq ofdimension length2 andwrite
xm ,xn =q mn (x)=qq mn (x) (2.3)
{ } 0
where q mn is some fixed Poisson structure. Given a Poisson manifold, we denote as quantization
0
mapanisomorphism ofvectorspaces
I : C(M) A Mat(¥ ,C)
→ ⊂ (2.4)
f(x) F
7→
whichdepends onthePoissonstructure I Iq ,andsatisfies1
≡
1
I(fg) I(f)I(g) 0 and I(i f,g ) [I(f),I(g)] 0 as q 0. (2.5)
q
− → { } − → →
(cid:16) (cid:17)
HereC(M)denotesasuitablespaceoffunctionsonM,andA isinterpretedasquantizedalgebra
offunctions2 onM. Suchaquantization mapI isnotunique, i.e. thehigher-order termsin(2.5)
arenotunique. SometimeswewillonlyrequirethatI isinjectiveafteraUV-truncationtoCL (M)
defined in termsof aLaplace operator, where L is some UVcutoff. Thisis sufficient for physical
purposes. Inany case, itis clear that q mn –if itexists in nature –mustbe part ofthe dynamics of
space-time. Thiswillbediscussed below.
ThemapI allowstodefinea“star”productonC(M)asthepull-back ofthealgebra A,
f ⋆g:=I−1(I(f)I(g)). (2.6)
It allows to work with classical functions, hiding q mn in the star product. Kontsevich has shown
[11] that such a quantization always exists in the sense of formal power series in q . This is a bit
too weak for the present context since we deal with operator or matrix quantizations. In the case
of compact symplectic spaces, existence proofs for quantization maps in the sense ofoperators as
required here are available [10], and we will not worry about this any more. Finally, the integral
over the classical space is related in the semi-classical limit to the trace over its quantization as
follows
w n
f (2p )nTrI(f). (2.7)
n! ∼
Z
1Theprecisedefinitionofthislimitingprocessisnon-trivial,andtherearevariousdefinitionsandapproaches. Here
wesimplyassumethatthelimitandtheexpansioninq existinsomeappropriatesense.
2A isthealgebrageneratedbyXm =I(xm ),orsomesubalgebracorrespondingtowell-behavedfunctions.
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
Embedded noncommutative spaces. Now consider a Poisson manifold embedded in RD. De-
notingtheCartesiancoordinate functionsonRD withxa,a=1,...,D,theembeddingisencodedin
themaps
xa: M ֒ RD, (2.8)
→
so that xa C(M). Given a quantization (2.4) of the Poisson manifold (M, .,. ), we obtain
∈ { }
quantized embeddingfunctions
Xa:=I(xa) A Mat(¥ ,C) (2.9)
∈ ⊂
given by specific (possibly infinite-dimensional) matrices. This defines an embedded noncommu-
tative space, or a NC brane. These provide a natural class of configurations or backgrounds for
the matrix model (1.1), which sets the stage for the following considerations. The map (2.4) then
allows to identify elements of the matrix algebra with functions on the classical space, and con-
versely the commutative space arise as a useful approximation of some matrix background. Its
Riemannian structurewillbeidentifiedlater.
Note that given some arbitrary matrices Xa, there is in general no classical space for which
this interpretation makes sense. Nevertheless, we will argue below that this class of backgrounds
is in asense stable and preferred bythe matrix model action, and this concepts seems appropriate
tounderstand thephysical contentofthematrixmodelsunderconsideration here.
Let us discuss the semi-classical limit of a noncommutative space. In practical terms, this
means that every matrix F will be replaced by its classical pre-image I 1(F)=: f, and commu-
−
tators willbereplaced byPoissonbrackets. Thesemi-classical limitprovides theleading classical
approximation of the noncommutative geometry, and will be denoted as F f . However one
∼
can go beyond this semi-classical limitusing e.g. the star product, which allows tosystematically
interpret theNCstructureinthelanguage ofclassicalfunctions andgeometry, ashigher-order cor-
rections in q to the semi-classical limit. The matrix model action (1.1) can then be considered as
adeformed action on someunderlying classical space. Thisapproximation isuseful ifthehigher-
ordercorrections inq aresmall.
3. Examples ofmatrix geometries
In this section we discuss some basic examples of embedded noncommutative spaces de-
scribed by finite or infinite matrix algebras. The salient feature is that the geometry is defined
byaspecificsetofmatricesXa,interpretedasquantizedembeddingmapsofasub-manifoldinRD.
3.1 Prototype: thefuzzysphere
ThefuzzysphereS2 [12,13]isaquantization ormatrixapproximationoftheusualsphereS2,
N
withacutoffintheangular momentum. Wefirstnotethatthealgebra offunctions ontheordinary
sphere can be generated by the coordinate functions xa of R3 modulo the relation (cid:229) 3 xaxa =1.
a=1
ThefuzzysphereS2 isanon-commutativespacedefinedintermsofthreeN Nhermitianmatrices
N ×
Xa,a=1,2,3subject totherelations
[Xa,Xb]= i e abcXc , (cid:229) 3 XaXa=1l (3.1)
√C
N a=1
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
where C = 1(N2 1) is the value of the quadratic Casimir of su(2) on CN. They are realized
N 4 −
by the generators of the N-dimensional representation (N) of su(2). The matrices Xa should be
interpreted asquantized embeddingfunctions intheEuclideanspaceR3,
Xa xa: S2֒ R3. (3.2)
∼ →
They generate an algebra A =Mat(N,C), which should be viewed as quantized algebra of func-
∼
tionsonthesymplecticspace(S2,w )wherew isthecanonicalSU(2)-invariant symplecticform
N N
on S2 with w =2p N. The best way to see this is to decompose A into irreducible representa-
N
tionsundertheadjointactionofSU(2),whichisobtainedfrom
R
S2 =(N) (N¯) = (1) (3) ... (2N 1)
N ∼
⊗ ⊕ ⊕ ⊕ −
= Yˆ0 ... YˆN 1 . (3.3)
0 m−
{ }⊕ ⊕{ }
Thisprovidesthedefinitionofthefuzzyspherical harmonicsYˆl,anddefinesthequantization map
m
I : C(S2) A = Mat(N,C)
→
Yˆl, l<N (3.4)
Yl m
m 7→ 0, l N
(
≥
It follows easily that I(i xa,xb )=[Xa,Xb]where , denotes the Poisson brackets correspond-
{ } { }
ing to the symplectic form w = Ne xadxbdxc on S2. Together with the fact that I(fg)
N 2 abc →
I(f)I(g) for N ¥ (which is not hard to prove), I(i f,g )N→¥ [I(f),I(g)] follows. This
→ { } →
meansthatS2 isthequantization of(S2,w ). Itisalsoeasytoseethefollowingintegralrelation
N N
2p Tr(I(f))= w f, (3.5)
N
Z
S2
consistent with (2.7). Therefore S2 is the quantization of (S2,w ). Moreover, there is a natural
N N
Laplaceoperator3 onS2 definedas
N
(cid:3)=[Xa,[Xb,.]]d (3.6)
ab
which is invariant under SU(2); infact itis nothing but the quadratic Casimirof the SU(2)action
on S2. It is then easy to see that up to normalization, its spectrum coincides with the spectrum of
the classical Laplace operator on S2 up to the cutoff, and the eigenvectors are given by the fuzzy
spherical harmonicsYˆl.
m
Inthisspecial example,(3.3)allowstoconstruct aseriesofembeddings ofvectorspaces
A A ... (3.7)
N N+1
⊂ ⊂
with norm-preserving embedding maps. This allows to recover the classical sphere by taking the
inductive limit. While this is a very nice structure, we do not want to rely on the existence of
such explicit series of embeddings. We emphasize that even finite-dimensional matrices allow to
approximate aclassicalgeometry toahighprecision, asdiscussed furtherinsection 3.7.
3Thesymbol(cid:3)isusedheretodistinguishthematrixLaplaceoperatorfromtheLaplacianD onsomeRiemannian
manifold.Itdoesnotindicateanyparticularsignature.
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
3.2 Thefuzzytorus
The fuzzy torus T2 can be defined in terms of clock and shift operators U,V acting on CN
N
with relations UV = qVU for qN = 1, with UN =VN = 1. They have the following standard
representation
0 1 0 ... 0 1
0 0 1 ... 0 e2p iN1
   
U = ... , V = e2p iN2 . (3.8)
   
0 ... 0 1  ... 
1 0 ... 0  e2p iNN−1
   
   
These matrices generate the algebra A =Mat(N,C), which can be viewed as quantization of the
∼
function algebraC(T2)onthesymplectic space(T2,w ). Onewaytorecognize thestructure ofa
N
torusisbyidentifying aZ Z symmetry,definedas
N N
×
Z A A (3.9)
N
× →
(w k,f ) Ukf U k (3.10)
−
7→
and similarly for the other Z defined by conjugation with V. Under this action, the algebra of
N
functions A =Mat(N,C)decomposes as
A =⊕Nn,−m=10UnVm (3.11)
into harmonics i.e. irreducible representations. This suggests to define the following quantization
map:
I : C(T2) A = Mat(N,C) (3.12)
→
einj eimy q−nm/2UnVm, |n|,|m|<N/2
7→ 0, otherwise
(
which is compatible with the Z Z symmetry and satisfies I(f )=I(f)†. The underlying
N N ∗
×
Poisson structure on T2 is given by eij ,eiy = 2p eij eiy (or equivalently j ,y = 2p ), and it
{ } N { } −N
iseasytoverifythefollowingintegralrelation
N
2p Tr(I(f))= w f, w = dj dy (3.13)
N N 2p
Z
T2
consistent with(2.7). ThereforeT2 isthequantization of(T2,w ).
N N
The metric is an additional structure which goes beyond the mere concept of quantization.
Here we obtain it by considering T2 as embedded noncommutative space in R4, by defining 4
hermitian matrices
X1+iX2:=U, X3+iX4:=V (3.14)
whichsatisfytherelations
(X1)2+(X2)2=1=(X3)2+(X4)2,
(X1+iX2)(X3+iX4)=q(X3+iX4)(X1+iX2). (3.15)
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
Theycanagainbeviewedasembeddingmaps
Xa xa : T2֒ R4 (3.16)
∼ →
and wecan writex1+ix2 =eij , x3+ix4 =eiy inthesemi-classical limit. Thisallowstoconsider
thematrixLaplaceoperator(3.6),andtocomputeitsspectrum:
(cid:3)f =[Xa,[Xb,f ]]d (3.17)
ab
=[U,[U†,f ]]+[V,[V†,f ]]=4f Uf U† U†f U Vf V† V†f V (3.18)
− − − −
(cid:3)(UnVm)= c([n]2+[m]2)UnVm c(n2+m2)UnVm,
q q
− ∼ −
1
c=(q1/2 q 1/2)2 (3.19)
− − ∼ N2
where
qn/2 q n/2 sin(np /N)
−
[n] = − = n (“q-number”) (3.20)
q q1/2 q 1/2 sin(p /N) ∼
−
−
Thus the spectrum of the matrix Laplacian (3.6) approximately coincides4 with the classical case
below the cutoff. Therefore T2 with the embedding defined via the above embedding (3.14) has
N
indeed thegeometryofatorus.
3.3 FuzzyCPN
A straightforward generalization of the fuzzy sphere leads to the fuzzy complex projective
space CPn, which is defined in terms of hermitian matrices Xa, a=1,2,...,n2+n subject to the
N
relations
i
[Xa,Xb]= fabXc , dc XaXb=D Xc, X Xa=1l (3.21)
c ab N a
C
N′
(adopting a sum convention).pHere fab are the structure constants of su(n+1), dabc is the totally
c
symmetricinvariant tensor, andC ,D aregroup-theoretical constants whicharenotneeded here.
N′ N
These relations are realized by the generators of su(n+1) acting on irreducible representations
withhighest weight(N,0,...,0) or(0,0,...,N), withdimension d . Again,thematrices Xa should
N
beinterpreted asquantized embedding functions intheEuclidean spacesu(n+1)=Rn2+n,
∼
Xa xa: CPn֒ Rn2+n. (3.22)
∼ →
Theygenerate analgebra A =Mat(d ,C),whichshould beviewedasquantized algebra offunc-
∼ N
tions on the symplectic space (CPn,Nw ) where w is the canonical SU(n)-invariant symplectic
formonCPn. Itiseasytowritedownaquantization mapanalogous to(3.4),
I : C(CPn) A (3.23)
→
usingthedecompositionofA intoirreduciblerepresentationsofsu(n+1). Again,thereisanatural
LaplaceoperatoronCPn definedasin(3.6)whosespectrumcoincides withtheclassical oneupto
N
thecutoff. Asimilarconstruction canbegivenforanycoadjoint orbitofacompactLiegroup.
4Itisinterestingtonotethatmomentumspaceiscompactifiedhere,reflectedintheperiodicityof[n]q.
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
3.4 TheMoyal-Weyl quantumplane
TheMoyal-WeylquantumplaneR2nisdefinedintermsof2n(infinite-dimensional) hermitian
q
matricesXa L(H )subjecttotherelations
∈
[Xm ,Xn ]=iq mn 1l (3.24)
where q mn = q nm R. Here H is a separable Hilbert space. This generates the (n-fold ten-
− ∈
sor product of the) Heisenberg algebra A (or some suitable refinement of it, ignoring operator-
technical subtleties here), which can be viewed as quantization ofthe algebra of functions on R2n
usinge.g. theWeylquantization map
I : C(R2n) R2n Mat(¥ ,C)
q
→ ⊂ (3.25)
eikm xm eikm Xm .
7→
Sinceplanewavesareirreduciblerepresentationsofthetranslationgroup,thismapisagaindefined
m
as intertwiner of the symmetry group as in the previous examples. Of course, the matrices X
should be viewed as quantizations of the classical coordinate functions Xm xm : R2n R2n.
∼ →
Similarasinquantummechanics, itiseasytoseethatthenoncommutativeplanewavessatisfythe
Weylalgebra
eikm Xm eipm Xm =e2iq mn km pn ei(km +pm )Xm (3.26)
Itisalsoeasytoobtainanexplicitformforthestarproductdefinedbytheabovequantizationmap:
itisgivenbythefamousMoyal-Weylstarproduct,
(f ⋆g)(x)= f(x)e2iq mn ←¶−m −→¶ n g(x) (3.27)
inobviousnotation. Thisgivese.g.
i
xm ⋆xn =xm xn + q mn
2
[xm ,xn ] =iq mn . (3.28)
⋆
Inthisexample,wenotethat
[Xm ,.]=:iq mn ¶ n (3.29)
provides a reasonable definition of partial derivatives in terms of inner derivations on A, pro-
vided q mn is non-degenerate (which we will always assume). This is justified by the observation
[Xm ,eikn Xn ]= q mn kn eikm Xm togetherwiththeidentificationI. Thereforethesepartialderivatives
−
andinparticular thematrixLaplacian
(cid:3)=[Xm ,[Xn ,.]]d mn =−q mm ′q nn ′d mn ¶ m ′¶ n ′ ≡−L −NC4Gmn ¶ m ¶ n ,
Gmn :=L 4NCq mm ′q nn ′d m ′n ′, L 4NC:= detq mn−1 (3.30)
q
coincide via I with the commutative Laplacian for the metric Gmn . Therefore Gmn should be
considered aseffective metricofR4.
q
The Moyal-Weyl quantum plane differs from our previous examples in one essential way:
the underlying classical space is non-compact. This means that the matrices become unbounded
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Non-commutativegeometryandmatrixmodels HaroldSteinacker
operators acting on an infinite-dimensional separable Hilbert space. The basic difference can be
seenfromtheformula
(2p )nTrF q mn−1 f , (3.31)
∼ | |
ZM
which together with q mn−1 =const implies that the trace diverges as a consequence of the infinite
| |
symplecticvolume. Locally(i.e. fortest-functions f withcompactsupport,say),thereisnoessen-
tial difference between the compact fuzzy spaces described before and the Moyal-Weyl quantum
plane. This reflects the Darboux theorem, which states that all symplectic spaces of a given di-
mension are locally equivalent. Thus from the point of view of matrix geometry, R2n is simply a
q
non-compact versionofafuzzyspace.
3.5 Thefuzzycylinder
Finally,thefuzzycylinder S1 x Risdefinedby[14]
×
[X1,X3]=ix X2, [X2,X3]= ix X1,
−
(X1)2+(X2)2=R2, [X1,X2]=0. (3.32)
DefiningU :=X1+iX2 andU†:=X1 iX2,thiscanbestatedmoretransparently as
−
UU† = U†U =R2
[U,X3] = x U, [U†,X3]= x U† (3.33)
−
Thisalgebrahasthefollowingirreducible representation5
U n = R n+1 , U† n =R n 1
| i | i | i | − i
X3 n = x nn , n Z, x R (3.34)
| i | i ∈ ∈
onaHilbertspace H , where n formanorthonormal basis. Wetakex R,since theXi areher-
| i ∈
mitian. Then the matrices X1,X2,X3 can be interpreted geometrically as quantized embedding
{ }
functions
X1+iX2 Reiy3
: S1 R֒ R3. (3.35)
X3 ∼ x3 × →
! !
Thequantization mapisgivenby
I : C(S1 R) S1 x R Mat(¥ ,C) (3.36)
× → × ⊂
eipx3einy3 einx /2eipX3Un, (3.37)
7→
whichpreservestheobviousU(1) Rsymmetry. ThisdefinesthefuzzycylinderS1 x R. Itisthe
× ×
quantization of T S1 with canonical Poisson bracket eiy3,x3 = ix eiy3, or x3,y3 =x locally.
∗
{ } − { }
Itsgeometrycanberecognized eitherusingtheU(1) Rsymmetry,orusingthematrixLaplacian
×
(cid:3)=[Xa,[Xb,.]]d whichhasthefollowingspectrum
ab
(cid:3)eipX3Un= 4R2sin2(px /2)+n2x 2 eipX3Un px ≪1 R2p2+n2 x 2eipX3Un, (3.38)
∼
(cid:16) (cid:17)
consistent withtheclassical spectrum forsmallmomenta. Th(cid:0)erefore the(cid:1)effectivegeometry isthat
ofacylinder.
5Moregeneralirreduciblerepresentationsareobtainedbya(trivial)constantshiftX3 X3+c.
→
10