A general procedure to find ground state solutions for finite $N$ M(atrix) theory. Reduced models and SUSY quantum cosmology

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by  J. L. López

N
A general procedure to find ground state solutions for finite M(atrix) theory.
Reduced models and SUSY quantum cosmology
J. L. Lo´pez and O. Obreg´on
∗ †
Departamento de F´ısica de la Universidad de Guanajuato,
A.P. E-143, C.P. 37150, Le´on, Guanajuato, M´exico.
WeproposeageneralmethodtofindexactgroundstatesolutionstotheSU(N)invariantmatrix
model arising from the quantization of the 11-dimensional supermembrane action in the light-cone
gauge. WeillustratethemethodbyapplyingittolowerdimensionalmodelsandfortheSU(2)group.
Thisapproach can beusedto findground statesolutions tothecomplete9-dimensional model and
for any SU(N) group. The supercharges and the constraints related to the SU(2) symmetry are
5 the relevant operators and they generate a multicomponent wave function. In the procedure, the
1 fermionic degreesof freedom arerepresented bymeansof Dirac-likegammamatrices. Weexhibita
0 relationbetweenthesefiniteN matrixtheorygroundstatesolutionsandSUSYquantumcosmology
2 wavefunctions giving a possible physicalsignificance to thetheory evenfor finiteN

r
a PACSnumbers: 11.25.-w,04.60.Ds,04.65.+e,04.60.Kz,12.60.Jv
M

5 I. INTRODUCTION dimensional non-commutative Super Yang-Mills theory.
In the IKKT framework, the parameter θ that measures
]
h As is known, M-theory should be an eleven dimen- thenon-commutativitybetweenthecoordinatesofspace-
t sional supersymmetric quantum theory from which all time becomes dynamical and in the semiclassicallimit it
-
p superstring theories can be deduced and it can be con- definesaPoissonstructuresuchthatthemetricofspace-
e nectedalsoto11-dimensionalsupergravity. Thisideawas time itself emerges naturally[12–14]. This matrix model
h born since it was realized that the five consistent theo- can be seen as a = 4 noncommutative Super Yang-
[ ries of strings and 11-d supergravity are related among Mills theory on RN4θ. The path integral quantization of

2 them through several types of dualities [1–5]. There are thismatrixmodelisnaturallydefinedbyintegratingover
v interesting approaches and calculations that provide the thespaceofmatricesandinasemiclassicalway,itrepro-
1 possibilitytounderstandsomeimportantfeaturesofthis duces some interesting features of curvedspacetime [15].
2 fundamental theory. There is a concrete formulation of So,ithasbeenclaimedisagoodcandidatetobeaquan-
6
M-theory which has been claimed has the appropriate tum theory of gravity together with other fundamental
1
0 expected properties and it is based on the matrix model interactions [16]. The presence of branes in this theory
. conjecturedin(BFSS[1]). Thismatrixmodelisasuper- [17] becomes also important as it is in the matrix theory
1
symmetricquantummechanicswherethebosonicdegrees of BFSS. The quantization approaches are different for
0
5 of freedom are a finite set of N N Hermitian matrices the two matrix models briefly described above, but they
1 [6]. We can associate a time dep×endent matrix to every have the same common Dp-brane world origin and both
: point in space and these matricial coordinates are obvi- involve the most important characteristic features of a
v
ouslynoncommuting. Theexistenceofdynamicalobjects unified theory of physical interactions including gravity.
i
X of dimensionalities higher than one, like Dp-branes, has
Another, and somehow independent, matrix model
r left aside the fundamental status of particles and strings
a was encountered in the quantization of the classical 11-
andtheoriginofthismatrixmodelanditsconnectionto
dimensional supermembrane action in the light cone
M-theory arises from the dynamics of D0 branes [7–9].
gauge (LCG) [18–21]. These matrix models were stud-
It was conjectured in [1] that M-theory is the N
→ ∞ iedinthecontextofsupersymmetricquantummechanics
limit of the maximally supersymmetric matrix Hamilto-
yearsbefore the quantizationof the supermembrane was
nian emerging from the D0-brane matrix model. The
finally achieved [22]. The action of the supermembrane
conjecturewasextendedin[10]showingthatmatrixthe-
is the generalizationof the Green-Schwarzaction for the
oryisalsomeaningfulforfiniteN,sothesolutionstothe
superstring for a membrane moving in a d-dimensional
matrix model for N finite are relevant.
target superspace, and when d= 11 this can be directly
Inthe recentyears,researchhasbeendoneonanother
relatedto11-dimensionalsupergravity[23]. Itispossible
matrix model arising from noncommutative gauge the-
toregularizethetheorybecauseoftheexistenceofafinite
ories [11], it is called the IKKT matrix model and it
dimensional Hilbert space where the area preserving dif-
can also be related to a large N reduced model of 10-
feomorphisms (APD) invariant states can be expressed.
This APD invariance is translated into a SU(N) sym-
metry in the regularized theory which is equivalent to a
∗ jllopez@fisica.ugto.mx SuperYang-Millsquantummechanicsmatrixmodel[18–
† octavio@fisica.ugto.mx 20]. The Hamiltonian and the associated ground state
2
of these matrix models of supersymmetric quantum me- representation for the fermionic variables has been used
chanics with SU(N) invariance were studied in [22] and inafirstandseveralmodelsofSUSYquantumcosmology
thematrixmodelarisingfromthesupermembraneaction [27,28,30–33]leadingtoprettymuchinterestingfeatures
is exactly one of those. In this type of supersymmetric regarding every physical supersymmetric system treated
models it is of greatimportance to find the ground state in this way. The fermionic degrees of freedom can also
solution[21,22,24,25],andforthematrixmodelwehave be represented by differential operators [34–37]. A rel-
been briefly mentioned here the existence of such state, evant point is that it is possible to solve the complete
even for N finite, has been found only perturbatively. operators system containing both, the bosonic and the
Thesetwomatrixmodels,theonerelatedtoD0-branes fermionic sectors. The fermionic information will then
and the other related to the supermembrane, are not in- be encodedin eachone ofthe entries ofthe multicompo-
dependent. There is a connection between membranes nent wave function and the physical information regard-
and D0-branes first speculated in [26] where it was con- ing the variablesof the system will be givenby means of
sidered that membranes could be regarded as collective the components of this wave function.
excitations of D0-branes. This connection was treated It has been argued that M-theory is a promising can-
ingreatdetailin[1]. The noncommutativespacedefined didate to be a quantum theory of gravity, it is then ex-
bythephasespacecoordinateswherethemembranesex- pected that the matrix models related with it, contain
ist has a basic minimum area, this quantum unit cells of quantum and classical gravitational information. The
space are precisely the D0-branes, each one with a lon- same authors of the conjecture have shown that a spe-
gitudinal momentum of 1/N. The longitudinal direction cificcompactificationofmatrixtheorycorrectlydescribes
is the one compactified in the LCG description. This the properties of a Schwarzschild black hole [38–41] and
relation of duality connects the two matrix models. other works related to matrix theory black holes and its
The first purpose of this work is to exemplify the gen- thermodynamic properties are [42–45].
eral method by finding solutions to the ground state of It has also been claimed that classical cosmological
matrixtheoryreducedmodelsarisingfromthe quantiza- modelscanbededuceddirectlyfromthematrixtheoryof
tion of the classical supermembrane action. Our proce- BFSS[46,47]andspecificallyin[47],usingahomothetic
dure is based on; i) We consider finite N matrix mod- ansatz, the authors were able to derive the Friedmann
els, ii) The relevant operators are the supercharges and equations from the bosonic sector of the classicalHamil-
those related with the SU(N) symmetry. It is no more tonianofmatrixtheory. Followingthiskindofideas,the
necessary to consider the Hamiltonian operator because interestingpointtoushereisthatweshowthatsomesu-
throughthecanonicalalgebraitisaconsequenceofthese persymmetriccosmologicalmodelsarethenhiddeninthe
constraints,iii)Thefermionicdegreesoffreedomarerep- SU(N) invariant supersymmetric matrix model as well.
resentedutilizing a Dirac-likegamma matrixrepresenta- Itisthenexpectedtofindsupersymmetricquantumsolu-
tion [27, 28]. By means of these, the associated super- tions thatcanbe interpretedandrelatedto SUSY quan-
charges and the operators related to the SU(N) symme- tum cosmology models. This is a particularly interest-
trybecomematrixdifferentialoperatorsactingonamul- ing byproduct of our reduced matrix models and shows
ticomponent wave function. One generates a Dirac-like thatthesereducedmodelsarealsosupersymmetricgiven
quantum matrix model formalism. that their corresponding SUSY cosmological models are
With our proposal one can solve any finite N matrix supersymmetric by construction [31, 32]. Also recently
model and for all the bosonic and fermionic degrees of [48],todescribethequantumdynamicsofthesupersym-
freedom. Even though finite N matrix models are phys- metric Bianchi IX cosmological model [28] it has been
ically relevant [10], it is not clear which particular N proposedto fix from the start the six degrees of freedom
should be considered. For large N, our method will gen- describing local Lorentz rotations of the tetrad. The op-
erate a large number of first order coupled differential erational content of the supercharges and Hamiltonian
equations. In order to illustrate how to apply our pro- revealedahiddenhyperbolicKac-Moodystructurewhich
posal, we restrict ourselves to reduced models, for the seems to support the conjecture about a correspondence
SU(N = 2) invariance group. Besides, for the sake of betweensupergravityandthedynamicsofaspinningpar-
simplicity and as a way to demonstrate how this pro- ticle moving on an infinite dimensional coset space. At
cedure works, we will make further assumptions in the thesametimeweshowthatsomereducedmatrixmodels
space of bosonic variables. We will then freeze out some correspond to exact SUSY quantum cosmology models
bosonic degrees of freedom. By doing this one cannot that arise from supergravity. All these relations point
assure that this reduced models will capture all features out to a web of connections between these different top-
of the quantization of finite N matrix theory (“for a re- ics and makeit of interestto searchfor a method to find
lated discussion in the context of quantum cosmology ground state solutions in finite N matrix theory mod-
see; [29]”). Some of the models studied can however be els. Here, we present a general procedure to construct
exactly solved illustrating us about the kind of ground them. We will also be interested in relations between
state solutionstobe expectedfora largerN model. The the solutionsofthese groundstate matrixtheorymodels
methodweproposecanbedirectlyextendedtohigherdi- andwavefunctionscorrespondingtoSUSYquantumcos-
mensionalmodelsandforanySU(N)group. Thiskindof mology. One of the wave function solutions obtained in
3
our reduced models resembles a wave function solution total Hamiltonian and the supercharges themselves are
obtained in the superfield approach to supersymmetric given by
quantum cosmology and it is then argued that the ma-
trix model we consider provides the relevant physicalin-
formation,asclassicalcosmologyseemsalsotoarisefrom { Qα,Qβ} =2δαβH +2g(Γn)αβφnaGa, (2)
bosonic matrix models [46, 47]. Q =(ΓmΛ ) πm+igf (ΣmnΛ ) φmφn,
α a α a abc a α b c
Theworkisorganizedasfollows. FirstinsectionIIwe
showthematrixmodelanditsdescriptionasasupersym- where Σmn = i[Γm,Γn]. Inthe anticommutatorof the
metric SU(N) invariant quantum mechanical model. In superchargesit−a4ppearsthe operator(constraint)related
sectionIIIweillustratethewayinwhichwelookforsolu- totheSU(N)invarianceG whichintermsofthebosonic
a
tions with the matricial representation for the fermionic and fermionic variables is given by
degreesoffreedomina2-dimensionalreducedmodeland
forthe SU(2)group. InsectionIVwe showthe solutions
i~
weobtainforsomeparticularassumptionsonthebosonic G =f (φmπm Λ Λ ). (3)
variables. As we pointed out, this method could be di- a abc b c − 2 bα cα
rectlyextendedtothecomplete9-dimensionaltheoryand
Now, we want to realize the quantization. And as we
foranySU(N)group,althoughforlargeN powerfulcom-
are working with a supersymmetric quantum mechanics
putational tools would be needed. Then in section V we
where the Hamiltonian is the square of the supercharges
make an extension to a 4 dimensional model. It is inter-
anditispositivesemi-definite,thestate Ψ whichobeys
estingtoremarkthatsomeofthe2-dimensionalsolutions | i
H Ψ = 0 is automatically the ground state wave func-
are also solutions to the 4-dimensionalmodel. In section | i
tion. Also, the ground state should be a gauge invariant
VI we show first that in this extension we encounter,
state, it should satisfy G Ψ =0. The commutator and
exactly, a model guessed in [20] and used to analyze a| i
anticommutator relations for the bosonic and fermionic
the spectrum of the matrix Hamiltonian of the quantum
degrees of freedom will be
supermembrane action and then we exhibit some solu-
tions to this extended model. In section VII we discuss
a connection of one of the solutions to the extended 4- [φm,πn]=i~δ δ , Λ ,Λ =δ δ . (4)
dimensionalmodelwithSUSYquantumcosmology. Sec- a b ab mn { aα bβ} ab αβ
tion VIII is devoted to conclusions. ThesuperchargesariseasthesquarerootoftheHamil-
tonian,andasitcanbeseeninthealgebraeq. (2),wecan
solve the equations Q Ψ =0 together with G Ψ =0.
α a
II. MATRIX MODEL Thenthe groundstate|satiisfyingthis,willrespec|taillthe
symmetries; gauge and exact supersymmetry. The rep-
Aswementionedintheintroduction,themodelarising resentation of the canonical momenta corresponding to
fromthequantizationoftheclassicalsupermembraneac- φm is, as usual, πm = i~ ∂ , and given the algebra
a a − ∂φma
tion,oncetheregularizationisachieved,becomesamodel for the fermionic variables eq. (4), we have the freedom
of supersymmetric quantum mechanics [1] that can also to represent the Λ as matrices. This kind of represen-
aα
be obtained from a dimensional reduction of a super tation was used first in SUSY quantum cosmology [27].
Yang-Mills theory with gauge group SU(N) [19, 22], the This is an essential tool [27, 28, 30, 31] in this work and
Hamiltonian of this matrix model [21] is given by as we will show, we will be able to find exact solutions
to some reduced lower dimensional models. The method
we will present can be applied to search for solutions to
1 g2
H = πmπm+ f φmφnf φmφn (1) the 9-dimensionalsupersymmetric matrix model and for
2 a a 4 abc b c ade d e
any SU(N) group. However, as it will be seen, for these
ig~
f Λ (Γm) φmΛ , dimensions and a large N this is a difficult task because
− 2 abc aα αβ b cβ one would need to solve an extremely large number of
m,n= 1,2,...,9 , a,b,c= 1,2,...,N2 1 , coupled partial differential equations.
{ } { − }
α,β = 1,2..., ,
{ N}
where φm and πm are the bosonic degrees of freedom III. THE METHOD OF SOLUTION
a a
and its related momenta respectively, and φm are the
a
components of the SU(N) valued matrices Φm given by To begin with, we will try to find solutions to the
Φm =φmτa whereτa arethebasisgroupelements. Then ground state by solving the corresponding equations
a
a is a group index and m is a dimension index. Λ Q Ψ = 0 and G Ψ = 0 (the Hamiltonian constraint
aα α a
representthe fermonic degrees of freedomand Γm arem is t|hein satisfied) fo|r aiSU(2) symmetry; a = 1,2,3 and
DiracmatricesobeyingtheCliffordalgebra Γm,Γn = for a 2-dimensional reduced model, m = 1,2. Hence,
2δ . The SU(N) group structure consta{nts are f} . we will have six φm bosonic degrees of freedom and the
mn abc a
The algebra between the superchargesas function of the same number for the fermionic ones. In the variables
4
Λ , a is a group index and the α index represents the
aα
number of supercharges which is in accordance with the
numberofcomponentsofaspinorind-dimensions. Then Q1 =Λ11π12+Λ21π22+Λ31π32 (8)
with d = 2, we have two supercharges and consequently
six fermionic variables. The dimension M of the square +Λ12π11+Λ22π21+Λ32π31
M ×M matrix representationΛaα depends on the num- −gΛ12(φ12φ23−φ13φ22)
bmeordoefl fMerm=ion8i.c Tdehgerereesproefsferneteadtoiomn. wSeo,uisnetfhoirs tfihrestΛtoaαy −gΛ22(φ1¯3φ2¯1−φ1¯1φ2¯3)
matrices that satisfies eq. (4) is the following −gΛ32(φ1¯1φ2¯2−φ1¯2φ2¯1) ,
Λ11 = ±1∆0, Λ12 = ±i∆1, Λ21 = ±i∆2, (5) Q2 =Λ11π11+Λ21π21+Λ31π31
√2 √2 √2 Λ12π12 Λ22π22 Λ32π32
− − −
Λ22 = ±i∆3, Λ31 = ±i∆4, Λ32 = ±i∆5. +gΛ11(φ12φ23 φ13φ22)
√2 √2 √2 −
∆µ =γµ σ1, ∆4 =I iσ2, ∆5 =I iσ3, +gΛ21(φ13φ21−φ11π32)
µ=0,1,2⊗,3, ⊗ ⊗ +gΛ31(φ11φ22−φ12φ21) ,
where σi are the Pauli Matrices, γµ are the Dirac ma- and the three operators G (a=1,2,3) related with the
a
trices in a Majorana representation and I is the 4 4 SU(2) gauge symmetry are given by
×
identity matrix. The algebra in eq. (4) is not affected
if we choose the plus, or minus, sign on every matrix in
eq. (5). Noticethatwehavethreekindsofindices,those
correspondingto the superchargesα=1,2,to the group G1 =φ12π31+φ22π32−φ13π21−φ23π22 (9)
a = 1,2,3, and the ones associated to the dimensions i~(Λ21Λ31+Λ22Λ32),
−
m = 1,2. Even when we have three kinds of indices we
iwosfilenlxudpmelinbcoiettrle.yaAllsoafntheexmam, apslei,ntheqe.fi(r5s)t,owfitthhethΛeaαsammaetrkiicneds G2 =φ13πi~11(+Λ3φ123Λπ1121−+φΛ113π231Λ−12)φ,21π32
−
0 0 0 0 0 0 0 i G3 =φ11π21+φ21π22 φ12π11 φ22π12
0 0 0 0 0 0 i −0 i~(Λ11Λ21−+Λ12Λ−22).
0 0 0 0 0 i −0 0  −
1 0 0 0 0 i 0 0 0
Λ11 = √±200 00 0i −0i 00 00 00 00 , (6) areUlsiinnegarthmeamtraixtrdixiffreerpernetsieanltoapteiornatoofrseqa.ct(i5n)g,oanllaonf tehigehmt
 −  componentwavefunction Ψ . Wecanrestrictthismodel
0 i 0 0 0 0 0 0  | i
  even more, setting some of the bosonic variables equal
 i 0 0 0 0 0 0 0 
  to zero, or making some identifications between them.
 
and for this model we choose the following two Γm ma- These will give specific configurations. Using the anti-
commutatorofthe superchargesin eq.(2) andeq. (7) we
trices
can see that it is not always necessary to impose all of
the three G operators, and that we can solve the prob-
a
0 1 1 0 lem for one single operator Q . For every G Ψ = 0,
Γ1 = ,Γ2 = . (7) α a| i
1 0 0 1 and Q Ψ = 0, we have eight coupled partial differen-
α
(cid:18) (cid:19) (cid:18) − (cid:19) | i
tialequationsfortheeightcomponentsofthe wavefunc-
The two supercharges Qα of eq. (2) are given by the tion Ψ = (ψ1,ψ2,ψ3,ψ4,ψ5,ψ6,ψ7,ψ8). For example
| i
following operators the equation Q1 Ψ =0 leads to the eight equations
| i
5
[3f1(φ~)−π31]ψ1+[iπ32−π11+3f2(φ~)]ψ2+[π21−3f3(φ~)]ψ4−[iπ12+π¯22]ψ8 =0, (10)
[3f2(φ~) iπ32 π11]ψ1+[π31 3f1(φ~)]ψ2+[π21 3f3(φ~)]ψ3 [iπ12+π22]ψ7 =0,
− − − − −
[π21 3f3(φ~)]ψ2+[3f1(φ~) π31]ψ3+[iπ32+π11 3f2(φ~)]ψ4+[iπ12+π22]ψ6 =0,
− − −
[π21 3f3(φ~)]ψ1+[π11 iπ32 3f2(φ~)]ψ3+[π31 3f1(φ~)]ψ4+[iπ12+π22]ψ5 =0,
− − − −
[π22 iπ12]ψ4+[3f1(φ~) π31]ψ5+[iπ32 π11+3f2(φ~)]ψ6+[π21 3f3(φ~)]ψ8 =0,
− − − −
[π22 iπ12]ψ3 [iπ32+π11 3f2(φ~)]ψ5+[π31 3f1(φ~)]ψ6+[π21 3f3(φ~)]ψ7 =0,
− − − − −
[iπ12 π22]ψ2+[π21 3f3(φ~)]ψ6+[3f1(φ~) π31]ψ7+[iπ32+π11 3f2(φ~)]ψ8 =0,
− − − −
[iπ12 π22]ψ1+[π21 3f3(φ~)]ψ5 [iπ32 π11+3f2(φ~)]ψ7+[π31 3f1(φ~)]ψ8 =0,
− − − − −
where the functions f (φ~), a = 1,2,3, depend on the Hence, in this configurationthere will be no bosonic po-
a
components of the matrices Φm and give rise to the po- tential and the G operators take the simpler form
a
tentials appearing in the Hamiltonian. These functions
are given by
G1 = i(Λ21Λ31+Λ22Λ32), (14)
−
f1(φ~)=(φ11φ22 φ12φ21), (11) G2 = i(Λ31Λ11+Λ32Λ12),
− −
f2(φ~)=(φ12φ23−φ13φ22), G3 =−i(Λ11Λ21+Λ12Λ22).
f3(φ~)=(φ13φ21 φ11φ23).
− One should apply all the three G operatorsto the wave
a
We can then make the mentioned simplifications to find function, however given eq. (2) and eq. (7) we have
a model with a particular potential, or even easier, for a to consider only the operator G1 acting on the wave
null potential. This last is the first one for which we will function Ψ , and consistently it is sufficient to apply
find a solution. also one o|fithe supercharges, namely Q1 Ψ = 0. Re-
| i
lations between the eight components of Ψ arise and
| i
the wave function as a row vector has the form Ψ =
IV. EXACT SOLUTIONS (ψ1,ψ2,iψ1,iψ2, iψ2, iψ1, ψ2, ψ1), where we|miean
− − − −
by this that, ψ3 = iψ1,ψ4 = iψ2, and so on. Because
Herewewillshowtheexplicitformofthegroundstate of the form of the differential equations for this two di-
wave function for a few models using some specific con- mensional model, it is easier to solve them using polar
figurations of the components of the φm variables. coordinates x = rcosφ,y = rsinφ. The exact solution
a
forthetwoindependentcomponentsofthewavefunction
results in
A. Solution 1
The model is the following. We choose the bosonic
variables to be (in units where ~=1) ψ1 =±arκeiκφ±br−κe−iκφ, (15)
ψ2 = arκeiκφ br−κe−iκφ.
∓ ∓
φ1 =x, φ1 =φ1 =0, φ2 =y, φ2 =φ2 =0, (12)
1 2 3 1 2 3
Here κ is a separation constant and a,b are arbitrary
∂
π1 = i , π1 =0 j =2,3, constantsandwecanchoosethemtohaveanormalizable
1 − ∂x j solution.
∂
π2 = i , π2 =0 j =2,3,
1 − ∂y j
and as a consequence of this choice we have
B. Solution 2
f1(φ~)=0, (13) The following model is chosen in such a way that the
f2(φ~)=0, bosonic partofthe potentialin its correspondingHamil-
tonian takes the form of a one dimensional harmonic os-
f3(φ~)=0. cillator, this is
6
where c1,c2 are constants. We see that the constant
φ1 =x, φ2 =c=constant, (16) C that relates ψ1 and ψ5 depends on the particular
1 2 choice of the constants c1,c2 in the linear combination
φ12 =φ13 =φ21 =φ23 =0, of exponentials. The most general solution of ψ1 can
f1(φ~)=cx, have eight terms, each one with a proportional constant
c ,j =1,2,...,8 and the constantC woulddepend onall
j
f2(φ~)=0, these cj.
f3(φ~)=0.
Followingthesameprocedureasintheprevioussolution, D. Solution 4
wewillhavetwoindependentcomponents,ψ1andψ2,for
the wave function Ψ = (ψ1,ψ2,ψ2,ψ1,ψ1,ψ2,ψ2,ψ1). This model has three bosonic components, one is con-
| i
The explicit form of these two components is
stant, with a potential different from zero. These are
3cx2 3cx2
ψ1 =(a+b)e 2 +(a−b)e− 2 , (17) φ1 =x, φ1 =0, φ1 =z, (20)
3cx2 3cx2 1 2 3
ψ2 =i(a+b)e 2 i(a b)e− 2 , φ2 =0, φ2 =c, φ2 =0,
− − 1 2 3
where a,b are arbitrary constants, if we choose them to f1(φ~)=cx, c is a constant,
be a= b, then we get a normalizable wave function.
− f2(φ~)= cz,
−
f3(φ~)=0.
C. Solution 3
After imposing the only non-trivial condition G2 Ψ =0
The next model has also a null potential, but it has and solving the equations Q1 Ψ = 0 we have|a iwave
three independent bosonic variables function solution with just one|inidependent component,
ψ1. The form of ψ1 and its relation with the rest of the
φ1 =x, φ1 =y, φ1 =z, (18) components is
1 2 3
φ2 =0, φ2 =0, φ2 =0,
1 2 3
z
ψ1 =exp itan−1 , ψ1 =iψ2 (21)
x
f1(φ~)=0, ψ1 =ψ4 =hψ5 =ψ(cid:16)8. (cid:17)iψ2 =ψ3 =ψ6 =ψ7,
f2(φ~)=0, and a constriction for this model arises, x2+z2 = 1.
f3(φ~)=0. We have seen so far that these reduced lower di3mcen-
sional models have exact solutions. We also found that
We can see from eq. (2) and eq. (7) that for this
forsomeoftheconfigurationswehaveused,thearbitrary
particular configuration it is not necessary to impose
any G ψ = 0 because φ2 = 0, i = 1,2,3. We will constants in the components of the wave function solu-
a| i i tions can be chosen so that they are normalizable. We
try to find a solution to Q2 ψ = 0. This is possi-
| i could make an extension of the method to a higher di-
ble and the eight independent components of the wave
mensionalmodelbut,asthedimensionofthematrixrep-
function reduce to only two, these are ψ1 and ψ5, and
resentation for Λ grows exponentially with the space
the wave function would take the following form Ψ = aα
| i dimensions, it would be hard to handle the complete 9-
(ψ1,ψ1,ψ1,ψ1,ψ5, ψ5, ψ5,ψ5). Besides,thereisacou-
− − dimensional problem. It will be needed a powerful com-
pling between these two components, so they are also
putational method to solve the complete matrix model,
related. The formofthe solutionofψ1 is a linearcombi-
even for the SU(2) group and even harder for a larger
nation of the exponential functions that come out from
group. In the next section, we solve a particular 4-
every particular choice of signs ( ) in the exponential
± dimensional model. Our intention is to show that the
factor exp[ √a x √b y i√a+b z], where a,b are
± ± ± methodcanbedirectlyextendedtomoredimensionsand
constants and because of the coupling, ψ5 results pro-
to any SU(N) group even though, as we mention, these
portional to ψ1, so ψ5 =Cψ1. The form of the constant
extended models will be difficult to handle.
C depends onthe particularform ofψ1. For instance ψ1
and ψ5 could be the following
V. SU(2) 4-DIMENSIONAL EXTENSION
ψ1 =c1e√ax+√by+i√a+bz +c2e√ax+√by−i√a+bz, (19)
(c1+c2)(√b i√a) In the 4-dimensional extension for the SU(2) group
ψ5 = (c1−c2)√a−+b !ψ1, t1h,e2r,e3,aarend12wbitohsodnic=v4aritahbelensuφmmab,ermof=su1p,2e,rc3h,a4r.geas =is
7
4. The number of fermionic variables is also 12 and the search for the ground state solution is much more com-
dimensionM for the matrixrepresentationofthese vari- plicated because we have to solve systems of 64 coupled
ables is M =64. The (64 64)representationwe choose partial differential equations for the 64 components of
×
for the twelve matrices Λ that satisfies the algebra eq. the wave function. In this extension we will also restrict
aα
(4) is the following ourselves to solve some reduced models with particular
assumptions on the bosonic degrees of freedom.
Λ11 =( 1/√2)σ1 1 1 1 1 1, (22)
± ⊗ ⊗ ⊗ ⊗ ⊗
Λ12 =( 1/√2)σ2 1 1 1 1 1, VI. SOLUTIONS TO THE EXTENDED MODEL
± ⊗ ⊗ ⊗ ⊗ ⊗
Λ13 =( 1/√2)σ3 σ1 1 1 1 1,
± ⊗ ⊗ ⊗ ⊗ ⊗ Using the matricialrepresentationin eq. (22) we need
Λ14 =( 1/√2)σ1 σ2 1 1 1 1, to solve, again, systems of matricial partial differential
± ⊗ ⊗ ⊗ ⊗ ⊗
Λ21 =( 1/√2)σ3 σ3 σ1 1 1 1, equations. In this case, every supercharge and gauge
± ⊗ ⊗ ⊗ ⊗ ⊗ symmetry operatoris a (64 64)operatormatrix acting
Λ22 =( 1/√2)σ3 σ3 σ2 1 1 1, on a 64 component wave fun×ction Ψ .
± ⊗ ⊗ ⊗ ⊗ ⊗ | i
Λ23 =(±1/√2)σ3⊗σ3⊗σ3⊗σ1⊗1⊗1, selAecstatnweoxavmarpialeblwesebtoegbinewdiiffthertehnetfforlolomwiznegrom,oφd1el.=Wxe,
1
Λ24 =(±1/√2)σ3⊗σ3⊗σ3⊗σ2⊗1⊗1, φ22 = y. In this particular model, we see from eq. (2)
Λ31 =( 1/√2)σ3 σ3 σ3 σ3 σ1 1, and eq. (23) that it is not necessary to impose the re-
± ⊗ ⊗ ⊗ ⊗ ⊗ strictions G Ψ =0 for any G . We need then to solve,
a a
Λ32 =(±1/√2)σ3⊗σ3⊗σ3⊗σ3⊗σ2⊗1, for example,|Q1i|Ψi = 0. As in the previous models the
Λ33 =( 1/√2)σ3 σ3 σ3 σ3 σ3 σ1, components of the wave function can be identified and
± ⊗ ⊗ ⊗ ⊗ ⊗ one gets a solution with just two independent compo-
Λ34 =(±1/√2)σ3⊗σ3⊗σ3⊗σ3⊗σ3⊗σ2, nents ψ1, andψ2. The independent components (ψ1,ψ2)
appear finally coupled in only two differential equations
where σi,i = 1,2,3, are the Pauli matrices. The (4 4)
Dirac representation for the Γm matrices we use is × ∂ψ2 ∂ψ2
i + 3xyψ1 =0, (25)
∂x ∂y −
Γ1 = 0I −0I , Γ2 = iσ02 −i0σ2 , (23) i∂∂ψx1 − ∂∂ψy1 +3xyψ2 =0.
(cid:18)− (cid:19) (cid:18) (cid:19)
These two coupled components (ψ1,ψ2) are related to
I 0 0 iσ3 other six components of the wave function in the follow-
Γ3 = 0 I , Γ4 = iσ3 −0 . ing manner
(cid:18) − (cid:19) (cid:18) (cid:19)
In this case I is the 2 2 identity matrix. Then, for
example, the first superc×harge takes the following form ψ8 =iψ1,ψ7 =iψ2,ψ19 = iψ2, (26)
−
ψ20 =iψ1,ψ21 = ψ1,ψ22 =ψ2.
−
Q1 = Λ13π11 Λ23π21 Λ33π31 Λ14π12 (24) There are also other pairs of components coupled in the
− − − −
Λ24π22 Λ34π32+Λ11π13+Λ21π23 same way, here we write all these pairs of coupled com-
− − ponents
+Λ31π33 iΛ13π14 iΛ23π24 iΛ33π34
− − −
igΛ11(φ12φ43 φ13φ42) igΛ21(φ13φ41 φ11φ43)
−igΛ31(φ11φ42−φ12φ41)−gΛ12(φ12φ23 −φ13φ22) (ψ1,ψ2),(ψ3,ψ4),(ψ9,ψ10),(ψ11,ψ12), (27)
− − − − (ψ33,ψ34),(ψ35,ψ36),(ψ41,ψ42),(ψ43,ψ44).
gΛ22(φ13φ21 φ11φ23) gΛ32(φ11φ22 φ12φ21)
− − − −
+igΛ12(φ22φ43 φ23φ42)+igΛ22(φ23φ41 φ21φ43) If we identify all these pairs as one, then all the rest of
− − the components are related to the only independent pair
+igΛ32(φ21φ42−φ22φ41)+gΛ13(φ12φ33−φ13φ32) (ψ1,ψ2) in a similar way as in eq. (26) and the wave
+gΛ23(φ13φ31 φ11φ33)+gΛ33(φ11φ32 φ12φ31) function has only two independent components. This,
− − now exact, reduced model obtained following our gen-
−igΛ13(φ32φ43−φ33φ42)−igΛ23(φ33φ41−φ31φ43) eralprocedurewouldcorrespondto the onlytwo compo-
igΛ33(φ31φ42 φ32φ41)+gΛ14(φ22φ33 φ23φ32) nent one guessed in [20] to analyze the spectrum of the
− − −
+gΛ24(φ23φ31 φ21φ33)+gΛ34(φ21φ33 φ22φ31). Hamiltonian eq. (1). Our quantization approach leads
− − to similar features. It allows however an exact solution
In this extension we have a richer structure and we can which satisfies also the Hamiltonian operator including
searchformoreinterestingpotentials. Wenoticethatthe the fermionic sector at once. A continuum spectrum of
8
the Hamiltonian is an expected feature in the connec- The rest of the components are also coupled in the same
tion between the supermembrane and D0-branes [1, 20]. way and give the same solution, these pairs of coupled
For several possible choices of the bosonic variables one components are
gets trivial solutions. Here we show solutions to some
non-trivial models.
(ψ1,ψ37), (ψ2,ψ38), (ψ3,ψ39), (ψ4,ψ40),
(ψ5,ψ33), (ψ6,ψ34), (ψ7,ψ35), (ψ8,ψ36),
A. Solution 1
(ψ9,ψ45), (ψ10,ψ46), (ψ11,ψ47), (ψ12,ψ48),
(ψ13,ψ41), (ψ14,ψ42), (ψ15,ψ43), (ψ16,ψ44),
Another choice of a model with three bosonic compo-
nents is the following (ψ17,ψ53), (ψ18,ψ54), (ψ19,ψ55), (ψ20,ψ56),
(ψ21,ψ49), (ψ22,ψ50), (ψ23,ψ51), (ψ24,ψ52),
φ1 =x, φ2 =y, φ3 =c=constant (28) (ψ25,ψ61), (ψ26,ψ62), (ψ27,ψ63), (ψ28,ψ64),
2 2 3
(ψ29,ψ57), (ψ30,ψ58), (ψ31,ψ59), (ψ32,ψ60).
with all other bosonic variables equal to zero. In this
case,weshouldimposetheconditionG3 Ψ =0,withthe
| i
consequencethatmanycomponentsofthewavefunction Itcanbeseenthatthereareonlytwoindependenteigen-
vanish. Theonlycomponentsthataredifferentfromzero vector solutions, one of them is normalizable and the
are other one is not. In the next section we will see that
thislastsolutioncanbeinterpretedasthesupersymmet-
ric quantum solution of a cosmologicalmodel.
ψ1,ψ2,ψ3,ψ4,ψ21,ψ22,ψ23,ψ24, (29)
ψ41,ψ42,ψ43,ψ44,ψ61,ψ62,ψ63,ψ64,
VII. CONNECTION WITH SUSY QUANTUM
and solving Q1 Ψ = 0, leads to the solution which is in
| i COSMOLOGY
this case
As it was discussed in the introduction, M(atrix) the-
3c 3c
ψ1 =Aexp (x2 y2) , ψ21 = iAexp (x2 y2) , ory is related with the theory underlying the five consis-
2 − − 2 − tent theories of strings and 11-dimensional supergravity,
(cid:20) (cid:21) (cid:20) (cid:21)
(30) namely M-theory. It is then expected that the matrix
ψ1 =ψ2 =ψ3 =ψ4 =ψ41 =ψ42 =ψ43 =ψ44 models would be related to gravity as well. A relation
between gravity and matrix models is that these models
ψ21 =ψ22 =ψ23 =ψ24 = ψ61 = ψ62 = ψ63 = ψ64.
− − − − describe the properties of Schwarzschild black holes af-
with A a real constant. tercompactificationsontorusofvariousdimensions [38–
45]. It has also been proposed that one can construct
a cosmologicalmodel corresponding to matrix theory, in
B. Solution 2 [46] the authors study the relation between this matrix
model and Newtonian cosmology. In [47], using a ho-
Thenextonedimensionalmodelhasamoreinteresting mothetic ansatz, the authors were able to get the Fried-
form, it has four bosonic components and three of them mann equations from the classical equations of matrix
are constants. The choice of variables is theory. Here, it is important to remark that in all these
cosmological models, the fermionic sector is absent and
that they have been worked out only with classical ma-
φ42 =x, φ12 =φ43 =c, φ13 =d, (31) trix equations. We are dealing with the quantization of
the matrix model of the supermembrane directly related
to the matrix model of D0-branes. Then we can expect
where c and d are constants and the rest of the bosonic
that, given that we are working with the quantization
degrees of freedom are set to zero. The conditions
of a supersymmetric model, our solutions would be re-
Ga Ψ =0 are not requiredand the solution to Q1 Ψ = lated to the solutions that have been obtained in differ-
| i | i
0 leads to pairs of coupled components, for instance
entapproachestoSUSYquantumcosmology[27,28,30–
(ψ1,ψ37)thatcanbeplacedinatwodimensionalbaseof 32]. As it was pointed in [28], there are several ways
two eigenvectors in the form
to carry out the quantization of a supersymmetric cos-
mological model. One of this options is to use the su-
ψψ317 =c1 1i exp −3 c2x− dx22 (32) pFeRrWfielcdoasmpporlooagcicha,lgmenoedrealliz(ainngdtohtehmerinciossumpeorlospgaiccaelomfothde-
(cid:18) (cid:19) (cid:18)− (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) els of interest) introducing the corresponding superpart-
1 dx2
+c2 i exp 3 c2x− 2 . ntheer scuopoerrdsiynmatmesetorfictigmeneerta→liza(tti,oηn,oη¯f)tahnedFwRWorkainctgiownitinh
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
9
termsofthe superfields correspondingtothe scalefactor In the superfield formulation one introduces the su-
andthe lapse function respectivelyR(t) R(t,η,η¯) and perpartner coordinates of time in the superspace t
N(t) N(t,η,η¯)[31,32]. We canactual→lynotethatone (t,η,η¯). The superspace valued Taylor series expansio→n
→
ofoursolutionsresemblesthe supersymmetriccosmolog- of the superfields is
ical solution found in [32] with this last approach.
Fromthe superchargesandthe SU(2) symmetry oper-
ators we can calculate the Hamiltonian of every model N(t,η,η¯)=N(t)+iηψ¯(t)+iη¯ψ(t)+V(t)ηη¯, (38)
we have solved using eq. (2). For the solutions in R(t,η,η¯)=R(t)+iηλ¯(t)+iη¯λ(t)+B(t)ηη¯,
the extended 4-dimensional model the Hamiltonians are
(64 64) non-diagonalmatrix operators whose diagonal where ψ(t),ψ¯(t) are the complex gravitino, and V(t)
×
terms are related with its bosonic part. Let us first take is a U(1) gauge field. B(t) in the superfield R(t,η,η¯)
a look at the second solution of our extended model eq. is an auxiliary degree of freedom and λ(t),λ¯(t) are the
(32) but identify now the variables and constants in this
fermionic partners of the scale factor R(t). The super-
case as
symmetric generalization of eq. (36) is
φ42 =R(t), φ12 =φ43 =rM3c, φ13 = √3κGc3, (33) S = −2cG2 N−1RDη¯RDηR+ c32√GκR2−McR dηdη¯dt,
Z (cid:20) (cid:21)
(39)
where κ is a constant, c is the speed of light and G is
the gravitational constant. We will also recover for this where Dη¯ = −∂∂η¯ −iη∂∂t and Dη = ∂∂η +iη¯∂∂t are the
solution the constant ~. The bosonic (diagonal) part of supercovariant derivatives of the global “small” super-
symmetryofthe generalizedparametercorrespondingto
the Hamiltonian is
the time variable t. When the canonical formalism is
applied to the supersymmetric Lagrangian it was shown
κc6 2Mc4√κ [31, 32] that the classical total Hamiltonian is given by
H =p2 + R2 R+M2c2, i=1,2,...,64
ii R G2 − G
(34)
i i 1
and the solution in eq. (32) for the coupled (ψ1,ψ37) Hc =NH + ψ¯S ψS¯+ VF, (40)
2 − 2 2
components of the wave function become
where H is the Hamiltonian of the system, S and S¯ are
ψψ317 =c1 1i exp − M~cR− √2Gκc~3R2 tψh,eψ¯suapnedrcVhaarrgeensoawndLFagrisanagUe(m1u)lrtoiptaliteirosn. gHe,nSe,raS¯toarn.dNF,
(cid:18) (cid:19) (cid:18)− (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) are first class constraints, hence the physical state wave
(35)
functions Ψ(R) are obtained from the conditions
1 Mc √κc3
+c2 i exp ~ R− 2G~ R2 .
(cid:18) (cid:19) (cid:20)(cid:18) (cid:19)(cid:21) HΨ(R)=0, SΨ(R), S¯Ψ(R)=0, FΨ(R)=0. (41)
Now we review how this solution arised in the con-
text of SUSY quantum cosmology. Following [31, 32] we These constraints have a dependence on R(t) and
rewrite here the supersymmetric action of the FRW cos- λ(t),λ¯(t). The explicit dependence of the Hamiltonian
mological model with a perfect fluid of barotropic equa- and supercharges on this variables is the following
tion of state p=γρ. For the case of dust γ =0 and null
cosmologicalconstant Λ=0 such action is given by
G κc4R M2G
H = π2 +c2M√κ (42)
−2c2R R− 2G − 2R
c2 dR 2 κc4 c√κ MG
S = R +N R NMc2 dt (36) λ¯λ λ¯λ,
"−2NG dt 2G − # − 2R − 2cR2
Z (cid:18) (cid:19)
iG1/2 c2√κR1/2 MG1/2
S = π + λ,
where κ = 1,1,0 is the curvature constant. N(t) and cR1/2 R− G1/2 R1/2
− (cid:18) (cid:19)
R(t) are the lapse function and the scale factor respec-
iG1/2 c2√κR1/2 MG1/2
tively and M is the mass parameterof dust. This action S¯= π + λ¯,
cR1/2 R G1/2 − R1/2
as given is invariant under reparametrizations of time (cid:18) (cid:19)
[t t+a(t)] when N(t),R(t) transformin the following
→
way
where π is the canonical momentum of the variable
R
R. As a consequence of the superalgebra between the
dR d(aN) Hamiltonian and the supercharges, the physical states
δR=a , δN = . (37)
dt dt are found by applying only the supercharge operatorsto
10
thestatewavefunctionΨ(R). Usingthefollowingmatrix after we showed the coincidence of the solutions, that
representation for the fermionic variables this makes evident certain connection between solutions
in matrix theory and solutions in cosmological models
arising from supergravity.
1
λ=√~σ , λ¯ = √~σ+, σ = (σ1 σ2), (43)
− − ± 2 ±
Ψ(R) becomes a two component wave function Ψ(R) = VIII. CONCLUSIONS
(Ψ1,Ψ2). The corresponding superquantum solutions
[32] for these two components are
We have considered finite N matrix models, the rel-
evant operators are the supercharges and those corre-
Mc √κ sponding to the SU(N) symmetry. The Hamiltonian op-
Ψ1 =Cexp ~ R− 2G~R2 , (44) erator is a consequence of them. The fermionic degrees
(cid:20)(cid:18) (cid:19)(cid:21) offreedomarerepresentedasDirac-likegammamatrices
Mc √κ
Ψ2 =C˜exp − ~ R− 2G~R2 . [g2e7n,er2a8l].prBocyedmueraentsoofifndtheisx,acwtegrwoeurnedasbtlaetetosoplurotipoonssetoa
(cid:20) (cid:18) (cid:19)(cid:21)
SU(N) matrix models. We have shownhere some super-
We can see that our solution eq. (35) and this one ob-
symmetric reducedmodels andfoundtheir exactground
tained in the superfield approach eq. (44), are indeed
state solutions being some of them normalizable. The
the same. The diagonal part of the matrix Hamiltonian
methodweproposeallowsthepossibilitytofindexactso-
for every matrix model is associated to the bosonic sec-
lutionsbymeansoftheconstraintsrelatedtotheSU(N)
tor of the system. If we want to find the solution to
invariance and the corresponding supercharges, contain-
HΨ(R) = 0, we can rewrite the Hamiltonian constraint
ing both the bosonic and the fermionic sectors. Even
in eq. (42) and see that the bosonic part of this Hamil-
though we have solved reduced models, the extension to
tonian, which is the one that does not depend on the
fermionicvariables(λ,λ¯)isthesameasthediagonalpart any dimension and to any SU(N) gauge invariant group
is possible following the proposed procedure. One could
of the matrix Hamiltonian in eq. (34) of our particular
then search for exact solutions to the full 9-dimensional
matrix model. We have then found the same solution in
matrix model and for any finite N although powerful
SUSY quantum cosmology and in matrix theory follow-
computationaltoolswouldbeneededbecausethedimen-
ing our general procedure to solve matrix theory.
sionalityofthematricesinvolvedmaketheproblemhard
The behavior of the Ψ1 component is the expected
tohandle. Giventhestructureofthesuperchargesforall
when R R , where R is the maximum radius
→ sup sup the alloweddimensions as functions of the momenta and
of the universe. One of the components converges as
the potentials of the variables involved, we expect that
R while the other one diverges,but the convergent
→∞ some solutions to the lower dimensional models would
solution can be isolated with the right choice of the con-
alsobesolutionstocertainhigherdimensionalones. This
stants. The scalar product of the Ψ1 component in the
is the case for the solutions we have shown here. Some
measure R1/2dR is normalizable
of the solutions found to the 2-dimensional model are
alsosolutions to the 4-dimensionalextension. A reduced
Rsup model guessed and investigated in [20] results as an ex-
1=C2 Ψ2R1/2dR, (45)
1 actsub-model inour formalism. A particularinteresting
0
Z result is a relation between one of our solutions in the
=C2 Rsup e2MR−√G˜κR2 R1/2dR, extended model with supersymmetric quantum cosmol-
0 ogy. This guarantees that the reduced matrix model is
Z (cid:16) (cid:17)
supersymmetric. Theserelationsreinforcethemethodof
and in both cases the norm of the wave function can be
quantizationwe proposein this work for finite N matrix
defined with the inner product
theory [10]. Besides, we encounter possible relations be-
tweenseveraltopicsconnectedwithourwork;ontheone
Rsup hand, according to our results, there is a connection be-
ΨΨ = Ψ†ΨR1/2dR. (46) tweenreducedmatrixtheorymodelsandSUSYquantum
h | i
Z0 cosmology, also, already in [47] the FRW cosmological
This review allowed us to know how to get a supersym- models have been related with classical bosonic matrix
metric quantum solution for a particular cosmological theory. EvenSchwarzschildblackholes[38–41]andtheir
model in the superfield approach, and we were able to thermodynamical properties [42–45] have been obtained
relate it with ground state solutions we have got in the by compactifications in matrix theory. On the other
matrixmodel,inparticularforthesecondsolutionofthe hand, in a recent work [48] the authors show evidence
extended model. It was then important to show in some ofthe conjecturerelatingsupergravityandthe dynamics
detail how these supersymmetric quantum solutions are ofaspinningparticlemovinginaninfinitecosetspaceby
obtainedandhowthey arerelatedwithpossiblynormal- means of the quantum dynamics of the supersymmetric
izable solutions in matrix theory. We want to remark, Bianchi IX cosmological model [28] and the operational