Physics Letters B 548 (2002) 88–91

www.elsevier.com/locate/npe

On the symplectic two-form of gravity in terms of Diraceigenvalues

M.C.B. Abdallaa, M.A. De Andradeb, M.A. Santosc, I.V. Vanceaa

a Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, 015405-900 São Paulo, SP, Brazilb Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil

c Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23851-180 Seropédica, RJ, Brazil

Received 26 September 2002; accepted 8 October 2002

Editor: L. Alvarez-Gaumé

Abstract

The Dirac eigenvalues form a subset of observables of the Euclidean gravity. The symplectic two-form in the covariant phasespace could be expressed, in principle, in terms of the Dirac eigenvalues. We discuss the existence of the formal solution of theequations defining the components of the symplectic form in this framework. 2002 Published by Elsevier Science B.V.

One of the major obstacles in quantizing the grav-ity is finding a complete set of observables of it. Re-cently, a certain progress has been made in definingand manipulating covariant observables in various the-ories of gravity [1–4]. Previous works showed that theDirac eigenvalues can be considered as observables ofgravity, too, on manifolds endowed with an Euclid-ean structure [5–7]. The result was generalized to in-clude the localN = 1 supersymmetry in [8–11] andit was shown to be connected to spectral geometryin [12]. However, in order to completely understandthe covariant phase space of the Euclidean gravity interms of the Dirac eigenvalues, one has to know whatis the form of the symplectic two-form in terms of the

E-mail addresses: mabdalla@ift.unesp.br (M.C.B. Abdalla),marco@cbpf.br, marco@gft.ucp.br (M.A. De Andrade),masantos@gft.ucp.br (M.A. Santos),ivancea@ift.unesp.br, vancea@cbpf.br (I.V. Vancea).

observables. The aim of this Letter is to discuss theexistence of a formal solution of this problem.

Let us begin by considering a four-dimensionalcompact manifoldM without boundary endowed withan Euclidean metric fieldgµν . One introduces a tetradfield which maps the metric at each pointx ∈ M

to the local Euclidean metric in the tangent space:gµν(x)=EI

µ(x)EJν (x)δIJ . The covariant phase space

of the theory is given by non-equivalent solutionsof the Einstein equations onM modulo the “gaugetransformations”, i.e., transformations generated bylocal SO(4) times diffeomorphisms. The functions ofthe phase space are observables of the theory. Considernow the Dirac equation

(1)D|ψn〉 = λn|ψn〉,where |ψn〉 is a spinor field (in the Dirac’s bracketnotation) andn is a positive integer (for simplicity,we assume that the Dirac operatorD has no zero

0370-2693/02/$ – see front matter 2002 Published by Elsevier Science B.V.PII: S0370-2693(02)02817-4

M.C.B. Abdalla et al. / Physics Letters B 548 (2002) 88–91 89

eigenvalue). The eigenvaluesλn define a discretefamily of real valued functions on the space of smoothtetradsE and a function fromE into the space ofinfinite sequencesR∞

(2)λn :E −→ R, E → λn[E],(3)λn :E −→ R∞, E →

λn[E].For everyn, λn[E] is invariant under the gauge groupaction on the tetrads [5]. In general,λn do not form aset of coordinates neither on the space of gauge orbitsnor on the phase space [5].

In order to analyze the phase space further, one hasto define the Poisson structure on the set of the eigen-values. This can be achieved by constructing firstly thesymplectic two-form of general relativity [13]

(4)Ω(X,Y )= 1

4

∫Σ

d3σ nρ[Xaµ,

↔∇τ Ybν

]ετabυε

υρµν,

whereXaµ[E] define a vector field on the phase space

and the brackets are given by

(5)[Xaµ,

↔∇τ Ybν

] =Xaµ∇τ Y

bν − Y a

µ∇τXbν .

Here,Σ is an arbitrary Arnowitt–Deser–Misner sur-face annρ is its normal one-form. The two-formΩ isinvertible only on the space of gauge fixed fields sinceit is degenerate on the space of the solutions of the Ein-stein equations. The coefficients ofΩ are given by thefollowing relation

ΩµνIJ (x, y)=

∫Σ

d3σ nρ[δ(x, x(σ )

)∇ τ δ

(y, x(σ )

)]

(6)× ετ IJυευρµν.

As was already noted in [6], the symplectic two-form(4) can be written in terms of the Dirac eigenvaluesif the map (3) is locally invertible on the phasespace. Then, the coefficientsΩmn of Ω defined by thefollowing relation

(7)Ω =Ωmn dλn ∧ dλm,

can be expressed in terms of (6) as follows

(8)ΩmnTnµI (x)Tm

νJ (y)=Ω

µνIJ (x, y),

where

(9)TnµI (x)= δλn[E]

δEIµ(x)

.

In order to have a complete description of the phasespace of the theory in terms of the Dirac eigenvalues,one has to express the coefficientsΩmn in termsof Ω

µνIJ (x, y), that is to invert (9). To this end, we

introduce the following objects which are well-definedsince the map (3) is invertible (a necessary conditionfor the existence ofΩmn)

(10)UnIµ(x)= δEI

µ(x)

δλn.

A simple algebra shows that the following two rela-tions hold

(11)UnIµ(x)Tn

µI (y)= δµν δ

IJ δ

(4)(x − y),

(12)UnIµ(x)Tn

µI (x)= δmn.

Note that the coefficientsΩmn do not depend explic-itly on the pointx of M. Moreover, since the eigenval-uesλn of D are defined globally onM, Ωmn has thesame property. Therefore, in order to eliminate the de-pendence on the pointsx andy, one has to integratetwice overM when inverting (8). Then, using (10)and (12) one can obtain from (8) the following rela-tion

(13)

Ωmn = 1

V 2M

∫M

Dx

∫M

DyΩµνIJ (x, y)Un

Iµ(x)Um

Iν(y),

whereDx = d4x√g andVM is the four-volume ofM.

Note that in order to obtain the relation (8) from (13)one has either to rescale the relation (12) by a factor ofVM in the r.h.s. or to rescale the delta-function integralonM. In what follows, we are going to use the relation

(14)∫M

Dx f (x)δ(4)(x − y)= VMf (y).

A formal solution of (13) can be given onceUn

Iµ(x)’s are known. To calculate them, we use the

Dirac equation (1). Assume that the Dirac eigen-spinors satisfy the global orthogonality and closure re-lations

(15)〈ψn|ψm〉 = δnm,

(16)∑n

|ψn〉〈ψn| = I,

where the scalar product in the Hilbert space of thevector fields onM is defined as

(17)〈ψ|φ〉 =∫M

Dx⟨ψ(x)

∣∣φ(x)⟩.

90 M.C.B. Abdalla et al. / Physics Letters B 548 (2002) 88–91

Here, 〈ψ(x)|φ(x)〉 is the scalar product in the localspinor fiberSx(M) over x. The local spinor sections|ψn(x)〉 are induced by the fields|ψn〉. We assumefurther that the global fields are defined by integral oflocal spinors

(18)|ψm〉 =∫M

Dx∣∣ψm(x)

⟩.

Then, the bilocal scalar product and the local closurerelations are given by the following relations

(19)⟨ψn(x)

∣∣ψm(y)⟩ = V −1

M δnmδ(4)(x − y),

(20)∑n

∣∣ψn(x)⟩⟨ψn(x)

∣∣ = V−1M I.

The local orthogonality and closure relations must bedefined in order to deal with the local terms in therelation (13).

The next step is to projectΩmn onto the basisformed by the eigenspinors ofD. Since the coeffi-cients of the symplectic form in the basis formed byλn are globally defined onM, the projection shouldbe performed onto the basis|ψn〉 rather than onto|ψn(x)〉. By using the relations (15), (16), (18), (19)and (20), one can easily show that the components ofΩmn are given by

(21)

[Ωmn]st = V 2M

∫Dx

∑r,k

[Ω

µνIJ (x, x)

]sr

[Un

Iµ(x)

]rk

× [Um

jν(x)

]kt,

where we are using the following shorthand notations

[Ωmn]st = 〈ψs |[Ωmn]|ψt 〉,(22)

[Ω

µνIJ (x, y)

]sr

= ⟨ψs(x)

∣∣[ΩµνIJ (x, y)

]∣∣ψr(y)⟩,

(23)[Un

Iµ(x)

]rk

= ⟨ψr(x)

∣∣[UnIµ(x)

]∣∣ψk(x)⟩.

Given the manifoldM, one could calculate, in princi-ple, the matrix elements of[Ωµν

IJ (x, x)] after comput-ing the spectrum of the Dirac operator. What is left arethe matrix entries from (23). To obtain them we derivethe local Dirac equation with respect to the eigenvalueλm. The resulting relation has the following form

(24)V 2M

∑k

[Um

Iµ(x)

]rk

[D

µI (x)

]kn

= δmnδrn,

for all m,r andn. Here, the sum is overk only and

(25)[D

µI (x)

]kn

= ⟨ψk(x)

∣∣[DµI (x)

]∣∣ψn(x)⟩.

These terms are determined by the eigenspinors ofD

and by noting that

DµI (x)

(26)

= iγ J

−Eν

I (x)EµJ (x)

(∂ν +ωνKL(x)σ

KL)

− 1

2EνJ (x)

×[E

ρI (x)E

µK(x)

(∂νEρL(x)− ∂ρEνL(x)

)+ δLI ∂σ

(E

µK(x)δ

σν −Eσ

K(x)δµν

)+ (

δMI δµν EτK(x)E

σL(x)

−EτI (x)E

µK(x)E

σL(x)E

Mν (x)

−EτK(x)E

σI (x)E

µL(x)E

Mν (x)

)× ∂σEτM(x)

− ∂ρ(E

µK(x)E

ρL(x)EνI (x)

)

− (K ↔ L)]σKL

,

where ωµIJ (x) are the components of the spin-connection in the spin bundleS(M) overM, γ I are thetangent-space Dirac matrices andσ IJ = 1

4[γ I , γ J ]. Inprinciple, one can compute the matrix elements of (26)if the vielbein is fixed and the eigenspinors ofD areknown. Therefore, one has to solve the system (24) inorder to find[Ωmn]st .

In general, the Dirac operator may have an infi-nite set of eigenspinors onM, which makes the sys-tem (24) infinite, too. A necessary and sufficient con-dition for the determinant det[Dµ

I (x)]kn be absolutelyconvergent [14] is that the product

∏k |[Dµ

I (x)]kk|converges absolutely and there is a non-negative in-teger numberp such that

(27)∑

k

[∑l

∣∣[DµI (x)

]kl

∣∣p] 1p−1

, k = l,

be convergent, too. Let us assume that this is the case.By examining Eq. (24) we observe that there are twopossibilities: (i)m= n and (ii)m = n, respectively. Inthe first case, one can construct an inverse of the matrix[Dµ

I (x)]kn from the set of matricesV 2M [Um

Iµ(x)] only

if the determinant of the latter is different from zero.This fact allows us to write some of the elements ofV 2M [Um

Iµ(x)] as

(28)[Un

Iµ(x)

]kn

= V −2M

[D

µI (x)

]−1kn

,

M.C.B. Abdalla et al. / Physics Letters B 548 (2002) 88–91 91

where[DµI (x)]−1

kn represent the elements of the inverseof [Dµ

I (x)]kn. These elements exist if the comple-ment of each element of[Dµ

I (x)] converges. The rela-tion (28) holds only for certain matrix elements fromthe full set of elements ofall matricesV 2

M [UmIµ(x)].

However, this solution is not unique since one can con-struct different matrices proportional to the inverse of[Dµ

I (x)]kn by taking different elements from the set ofall matrix elements of all matricesV 2

M [UmIµ(x)]. The

rest of elements can be grouped into matrices whichare not proportional to the inverse of[Dµ

I (x)]−1kn . The

entriesV 2M [Um

Iµ(x)]kl of these matrices do not satisfy

the relation (24) since the product of the matrices with[Dµ

I (x)]kn is zero. Consequently, the determinant ofthe latter should be simultaneously different from zeroin order to have an inverse and equal to zero in order todetermine the elements of the other matrices. There-fore, the termsV 2

M [UmIµ(x)]kl that can be calculated

depend on the value of the determinant of[DµI (x)]kn.

The best control on them is when the determinant isdifferent from zero. In this case, the second possibilitym = n is also discarded because the existence of so-lutions of (24) under this hypothesis implies that thedeterminant of[Dµ

I (x)]kn converges to zero.To conclude, one can formally solve Eq. (8) as in

(13). If the determinant of[DµI (x)]kn is different zero,

the terms[UmIµ(x)]kl from (13) can be determined up

to some freedom in picking them up from the set ofall coefficients of all matrices[Um

Iµ(x)]. It is not

clear if this undeterminacy is related in some way tothe fact that we are dealing with the set of smoothvielbeins instead of the gauge fixed ones. However,one way of removing the undeterminacy is to fixsome of the terms to zero. The rest of them are givenby Eq. (28) and they are proportional (up to anV 2

M

term) to the elements of the inverse matrix[DµI (x)]−1

kn .These are the only terms that can be determined. Whenthe determinant of[Dµ

I (x)]kn converges to zero, the

existence of the matrix[UmIµ(x)] is guaranteed, but

its elements are undetermined. For a complete controlover the symplectic two-form in the parametrization interms of the Dirac eigenvalues, more relations besidethe Dirac equation should be imposed on the theory.

Acknowledgements

M.A.S. and I.V.V. would like to thank J.A. Helayel-Neto for hospitality at CBPF during the preparationof this work. I.V.V. would like to acknowledge toN. Berkovits and O. Piguet for useful discussion and toFAPESP for support within a postdoctoral fellowship.

References

[1] R. De Pietri, C. Rovelli, Class. Quantum Grav. 12 (1995) 1279,gr-qc/9406014.

[2] A. Perez, C. Rovelli, gr-qc/0104034.[3] C. Rovelli, Phys. Rev. D 65 (2002) 044017, gr-qc/0110003.[4] C. Rovelli, Phys. Rev. D 65 (2002) 124013, gr-qc/0110035.[5] G. Landi, C. Rovelli, Phys. Rev. Lett. 78 (1997) 3051, gr-

qc/9612034.[6] G. Landi, C. Rovelli, Mod. Phys. Lett. A 13 (1998) 479, gr-

qc/9708041.[7] G. Landi, gr-qc/9906044.[8] I.V. Vancea, Phys. Rev. Lett. 79 (1997) 3121;

I.V. Vancea, Phys. Rev. Lett. 80 (1998) 1355, Erratum, gr-qc/9707030.

[9] I.V. Vancea, Phys. Rev. D 58 (1998) 045005, gr-qc/9710132.[10] C. Ciuhu, I.V. Vancea, Int. J. Mod. Phys. A 15 (2000) 2093,

gr-qc/9807011.[11] N. Pauna, I.V. Vancea, Mod. Phys. Lett. A 13 (1998) 3091, gr-

qc/9812009.[12] S.I. Vacaru, math-ph/0205023.[13] A. Ashtekar, L. Bombelli, O. Reula, in: M. Francaviglia (Ed.),

Mechanics, Analysis and Geometry: 200 Years after Lagrange,Elsevier, Amsterdam, 1991.

[14] St. Bóbr, Math. Z. 10 (1921) 1.