suresh. k. maran- reality conditions for spin foams

8/3/2019 Suresh. K. Maran- Reality Conditions for Spin Foams

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arXiv:gr-qc/0511014v13

Nov2005

Reality Conditions for Spin Foams

Suresh. K. Maran

November 4, 2005

Abstract

An idea of reality conditions in the context of spin foams (Barrett-Crane models) is developed. The square of areas are the most elementaryobservables in the case of spin foams. This observation implies that sim-plest reality conditions in the context of the Barrett-Crane models is thatthe all possible scalar products of the bivectors associated to the trianglesof a four simplex be real. The continuum generalization of this is thearea metric reality constraint: the area metric is real iff a non-degeneratemetric is real or imaginary. Classical real general relativity (all signa-tures) can be extracted from complex general relativity by imposing thearea metric reality constraint. The Plebanski theory can be modified byadding a Lagrange multiplier to impose the area metric reality conditionto derive classical real general relativity. I discuss the SO(4, C) BF modeland SO(4, C) Barrett-Crane model. It appears that the spin foam modelsin 4D for all the signatures are the projections of the SO(4, C) spin foammodel using the reality constraints on the bivectors.

Contents

1 Introduction 2

2 SO(4, C) General Relativity 4

3 Plebanski theory with Reality Constraint 73.1 Reality Constraint for b = 0 . . . . . . . . . . . . . . . . . . . . . 73.2 Plebanski Action with the reality constraint. . . . . . . . . . . . 9

4 Discretization 104.1 BF theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 BarrettCrane Constraints . . . . . . . . . . . . . . . . . . . . . . 10

4.2.1 Tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2.2 Four Simplex . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Spin foam of the SO(4, C) BF model 12

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6 The SO(4, C) Barrett-Crane Model 156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6.2 The SO(4,C) intertwiner . . . . . . . . . . . . . . . . . . . . . . . 176.3 The Spin Foam Model for the SO(4, C) General Relativity. . . . 196.4 The Features of the SO(4, C) Spin Foam . . . . . . . . . . . . . . 19

7 Spin Foams for 4D Real General Relativity and reality constraints 227.1 The Formal Structure of Barrett-Crane Intertwiners . . . . . . . 227.2 The Real Barrett-Crane Models . . . . . . . . . . . . . . . . . . . 237.3 The Area Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 27

8 Acknowledgement. 28

A Unitary Representations of SL(2,C) 28

B Unitary Representations of SO(4, C) 31

C Unitary Representations of SU(1, 1) 32

D Reality Constraint for Arbitrary Metrics 33

E Field Theory over Group and Homogenous Spaces. 34

1 Introduction

The Ashtekar self-dual formalism [21] in its most intuitive form relates to com-plex general relativity. A set of conditions referred to as reality conditions [21]is to be imposed to go to real general relativity. The reality conditions not only

impose the reality of the physics, but also the Lorentzian signature [21]. Thegoal of this article is to introduce analogous idea of reality conditions for thecovariant version of quantum gravity - the Barrett-Crane model [11].

The Barrett-Crane model [11] is defined by a set of constraints defined onthe bivectors associated to the triangles of a four simplex. The constraints needto be realized at the quantum level to define a quantum state of a tetrahedronor to assign an amplitude to a four simplex. Let us make the bivectors to becomplex which corresponds to SO(4, C) general relativity. Then we need a setof conditions to extract Barrett-Crane models for real general relativity. Inthe Barrett-Crane models the elementary physical observables are the square ofarea eigenvalues associated to the triangles. So the simplest reality conditionsin Barrett-Crane models is to be defined in terms of the squares of the areas.This is equivalent to expecting the inner product of a bivector with itself to be

real. For real general relativity we also need the square of areas correspondingto the sum of the bivectors of the triangles to be real. As will be explainedin this article this is equivalent to expecting the inner product of the bivectorscorresponding to any pair of triangles to be real.

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The continuum generalization of the squares of the areas of triangles ofsimplicial manifold is the Area metric. It can be shown that the reality of an area

metric is equivalent to the reality of a geometry. An area metric can be definedas an inner product of a bivector two-form field with itself. Because of thisthe reality of an area metric can be imposed using a Lagrange multiplier in thePlebanski formulation of SO(4, C) general relativity. The Barrett-Crane modelcorresponds to the discrete analog of the Plebanski formalism. Analogously, itcan be shown that the reality of the bivector inner products discussed before isthe discrete equivalent of the area metric reality constraint.

An attempt by me to rigorously develop and unify the various models forthe Lorentzian general relativity was made in Ref:[18]. The attempt was madeto derive the two models by directly solving the Barrett-Crane constraints. TheBarrett-Crane cross-simplicity constraint operator was explicitly written usingthe Gelfand-Naimarck representation theory ofSL(2, C) [19]. But after numer-ous attempts I could not obtain any solution for the constraint. But the effortsin this research lead to the development of the reality conditions for the spinfoam models. Then as will be discussed in this article the Barrett-Crane mod-els for real general relativity theories for all signatures appears to be relatedto that of the SO(4, C) general relativity through the quantum version of thediscretized area metric reality condition1. In this way we have a unified un-derstanding of the Barrett-Crane models for the four dimensional real generalrelativity theories for all signatures (non-degenerate) and the SO(4, C) generalrelativity.

The layout of this article is as follows:

Section two: I review the Plebanski formulation [2] of SO(4, C) generalrelativity starting from vectorial actions.

Section three: I discuss the area metric reality constraint. After solving thePlebanski (simplicity) constraints[2], I show that, the area metric realityconstraint requires the space-time metric to be real or imaginary for thenon-denegerate case. I modify the vectorial Plebanski actions by addinga Lagrange multiplier to impose the reality constraint.

Section four: I discuss the discretization of the area metric reality con-straint on the simplicial manifolds in the context of the Barrett-Cranetheory [11].

Section five: I discuss the spin foam model for the SO(4, C) BF theory.

Section six: I discuss the SO(4, C) Barrett-Crane model.

Section seven: Using the bivector scalar product reality constraint theBarrett-Crane models for the real general relativity for all signatures andSO(4, C) general relativity are discussed in a unified manner.

1The Barrett-Crane model based on the propagators on the null-cone [14] is an exceptionto this. This needs to b e carefully investigated.

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Appendices: I discuss the necessary representation theories. I discuss thearea metric reality constraint for arbitrary metrics. I also discuss the field

theory over group formalism for the SO(4, C) general relativity.

2 SO(4, C) General Relativity

Plebanskis work [2] on complex general relativity presents a way of recastinggeneral relativity in terms of bivector 2-form fields instead of tetrad fields [4] orspace-time metrics. It helped to reformulate general relativity as a topologicalfield theory called the BF theory with a constraint (for example Reisenberger[20]). Originally Plebanskis work was formulated using spinors instead of vec-tors. The vector version of the work can be used to formulate spin foam modelsof general relativity [20], [8]. Understanding the physics behind this theory sim-plifies with the use of spinors. Here I would like to review the Plebanski theory

for a SO(4, C) general relativity on a four dimensional real manifold startingfrom vectorial actions.

Let me define some notations to be used in this article. I would like to usethe letters i,j,k,l,m,n as SO(4, C) vector indices, the letters a,b,c,d,e,f,g,has space-time coordinate indices, the letters A , B, C, D, E, F as spinorial indicesto do spinorial expansion on the coordinate indices.

In the cases of Riemannian and SO(4, C) general relativity the Lie algebraelements are the same as the bivectors. On arbitrary bivectors aij and bij, Idefine 2

a b =1

2ijkla

ij bkl and

a b =1

2

ikjl aij bkl.

Consider a four dimensional manifold M. Let A be a SO(4, C) connection1-form and Bij a complex bivector valued 2-form on M. I would like to re-strict myself to non-denegerate general relativity in this and the next sectionby assuming b = 14!

abcdBab Bcd = 0. Let F be the curvature 2-form of theconnection A. I define real and complex continuum SO(4, C) BF theory actionsas follows,

ScBF(A, Bij) =

M

abcdBab Fcd and (1)

SrBF(A, Bij , A, Bij ) = Re

M

abcdBab Fcd. (2)

The ScBF is considered as a holomorphic functional of its variables. In SrBFthe variables A, Bij and their complex conjugates are considered as indepen-dent variables. The wedge is defined in the Lie algebra coordinates. The field

2The wedge product in the bivector coordinates plays a critical role in the spin foammodels. This is the reason why the is used to denote a bivector product instead of anexterior product.

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equations corresponding to the extrema of these actions are

D[aBbc] = 0 andFcd = 0.

BF theories are topological field theories. It is easy to show that the localvariations of solutions of the field equations are gauged out under the symmetriesof the actions [1].

The Plebanski actions for SO(4, C) general relativity is got by adding aconstraint term to the BF actions. First let me define a complex action [20],

ScGR(A, Bij , ) =

M

abcdBab Fcd +

1

2babcdBab Bcd

d4x, (3)

and a real action

SrGR(A, Bij , , A, Bij , ) = Re SC(A, Bij , ). (4)

The complex action is a holomorphic functional of its variables. Here is acomplex tensor with the symmetries of the Riemann curvature tensor such thatabcdabcd = 0. The b is inserted to ensure the invariance of the actions undercoordinate change.

The field equations corresponding to the extrema of the actions SC and Sare

D[aBij

bc] = 0, (5a)

1

2abcdFijcd = b

abcdBijcd and, (5b)

Bab Bcd babcd = 0, (5c)

where D is the covariant derivative defined by the connection A. The fieldequations for both the actions are the same.

Let me first discuss the content of equation (5c) called the simplicity con-straint. The Bab can be expressed in spinorial form as

Bijab = BijABAB + B

ij

ABAB,

where the spinor BAB and BAB are considered as independent variables. Thetensor

Pabcd = Bab Bcd babcd

has the symmetries of the Riemann curvature tensor and its pseudoscalar com-ponent is zero. In appendix A the general ideas related to the spinorial decom-

position of a tensor with the symmetries of the Riemann Curvature tensor havebeen summarized. The spinorial decomposition of Pabcd is given by

Pabcd = B(AB BCD)ABCD + B(AB BCD)AB CD +

b

6

c[ab]d

2+ BAB BAB (AB CD + AB CD),

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where b = BAB BAB + BAB BAB. Therefore the spinorial equivalents of the

equations (5c) are

B(AB BCD) = 0, (6a)

B(AB BCD) = 0, (6b)

BAB BAB + BAB B

AB = 0 and (6c)

BAB BAB = 0. (6d)

These equations have been analyzed by Plebanski [2]. The only difference be-tween my work (also Reisenberger [20]) and Plebanskis work is that I havespinorially decomposed on the coordinate indices of B instead of the vector in-dices. But this does not prevent me from adapting Plebanskis analysis of theseequations as the algebra is the same. From Plebanskis work, we can conclude

that the above equations imply Bij

ab

= [ia

j]

b

where ia are a complex tetrad.Equations (6) are not modified by changing the signs of BAB or/and BAB .These are equivalent to replacing Bab by Bab or

12

cdabBcd which produce

three more solution of the equations [5], [20].The four solutions and their physical nature were discussed in the context

of Riemannian general relativity by Reisenberger [20]. It can be shown thatequation (5a) is equivalent to the zero torsion condition3. Then A must bethe complex Levi-Civita connection of the complex metric gab = ijia

jb on

M. Because of this the curvature tensor Fcdab = Fij

abci

cj satisfies the Bianchi

identities. This makes F to be the SO(4, C) Riemann Curvature tensor. Usingthe metric gab and its inverse gab we can lower and raise coordinate indices.

Let me assume I have solved the simplicity constraint, and dB = 0. Substi-

tute in the action Sthe solutions Bijab = [ia

j]b and A the Levi-Civita connection

for a complex metric gab = a b. This results in a reduced action which is afunction of the metric only,

S(gab) =

d4xbF,

where F is the scalar curvature Fabab ,and b2 = det(gab). This is simply the

Einstein-Hilbert action for SO(4, C) general relativity.The solutions 12

cdabBcd do not correspond to general relativity [5], [20]. If

Bijab = 12

cdabBcd, we obtain a new reduced action,

S() = Re

d4xabcdFabcd,

which is zero because of the Bianchi identity abcdFabcd = 0. So there is no otherfield equation other than the Bianchi identities.

3For a proof please see footnote-7 in Ref.[20].

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3 Plebanski theory with Reality Constraint

3.1 Reality Constraint for b = 0

Let the bivector 2-form field Bijab = [ia

j]b and the space-time metric gab =

ij ia

jb . Then, the area metric [20] is defined by

Aabcd = Bab Bcd (7a)

=1

2ikjl B

ijabB

klcd (7b)

= ga[cgd]b. (7c)

Consider an infinitesimal triangle with two sides as real coordinate vectors Xa

and Yb. Its area A can be calculated in terms of the coordinate bivector Qab =12

X[aYb] as follows

A2 = AabcdQabQcd.

In general Aabcd defines a metric on coordinate bivector fields:< , >= Aabcdabcd

where ab and cd are arbitrary bivector fields.

Consider a bivector 2-form field Bijab = [ia

j]b on the real manifold M defined

in the last section. Let ia be non-degenerate complex tetrads. Let gab =gRab +ig

Iab, where g

Rab and g

Iab are the real and the imaginary parts ofgab = a b.

Theorem 1 The area metric being real

Im(Aabcd) = 0, (8)

is the necessary and the sufficient condition for the non-degenerate metric to be

real or imaginary.

Proof. Equation (8) is equivalent to the following:

gRacgIdb = g

Radg

Icb. (9)

From equation (9) the necessary part of our theorem is trivially satisfied. Let g,gR and gI be the determinants ofgab, g

Rab and g

Iab respectively. The consequence

of equation (9) is that g = gR + gI. Since g = 0, one of gR and gI is non-zero.Let me assume gR = 0 and gacR is the inverse of g

Rab. Let me multiply both the

sides of equation (9) by gacR and sum on the repeated indices. We get 4gIdb = g

Idb,

which implies gIdb = 0. Similarly we can show that gI = 0 implies gRdb = 0. So

we have shown that the metric is either real or imaginary iff the area metric isreal.

Since an imaginary metric essentially defines a real geometry, we have shownthat the area metric being real is the necessary and the sufficient condition forreal geometry (non-degenerate) on the real manifold M. In one of the appendixI discuss this for any dimensions and rank of the space-time metric.

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To understand the nature of the four volume after imposing the area metricreality constraint, consider the determinant of both the sides of the equation

gab = a b,g = b2,

where b = 14! abcdBab Bcd = 0. From this equation we can deduce that b is

not sensitive to the fact that the metric is real or imaginary. But b is imaginaryif the metric is Lorentzian (signature + + + or +) and it is real if themetric is Riemannian or Kleinien (+ + ++, , ++).

The signature of the metric is directly related to the signature of the areametric Aabcd = ga[cgd]b. It can be easily shown that for Riemannian, Kleinienand Lorentzian geometries the signatures type of Aabcd are (6, 0), (4, 2) and(3, 3) respectively.

Consider the Levi-Civita connection

abc =

1

2 gad

[bgcd + cgdb dgbc]

defined in terms of the metric. From the expression for the connection we canclearly see that it is real even if the metric is imaginary. Similarly the Riemanncurvature tensor

Fabcd = [cad]b +

eb[c

ad]e

is real since it is a function of abc only. But Fad

bc = gdeFabce and the scalar

curvature are real or imaginary depending on the metric.In background independent quantum general relativity models, areas are

fundamental physical quantities. In fact the area metric contains the full in-formation about the metric up to a sign 4. If BRab and B

Lab (vectorial indices

suppressed) are the self-dual and the anti-self dual parts of an arbitrary Bijab,

one can calculate the left and right area metrics as

ALabcd = BLab B

Lcd

1

4!efghBLef B

Lghabcd

and

ARabcd = BRab B

Rcd +

1

4!efghBRef B

Rghabcd

respectively [20]. These metrics are pseudo-scalar component free. Reisenbergerhas derived Riemannian general relativity by imposing the constraint that theleft and right area metrics be equal to each other [20]. This constraint is equiv-alent to the Plebanski constraint Bab Bcd babcd = 0. I would like to takethis one step further by utilizing the area metric to impose reality constraintson SO(4, C) general relativity.

4For example, please see the proof of theorem 1 of Ref:[3].

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3.2 Plebanski Action with the reality constraint.

Next, I would like to proceed to modify SO(4, C) general relativity actionsdefined before to incorporate the area metric reality constraint. The new actionsare defined as follows:

Sc(A,B, B, , q) =

M

abcdBab Fcdd4x + CS + CR, (10)

andSr(A,B, A, B,, , q) = Re S(A,B, B, , q),

where

CS =

Mr

b

2abcdBab Bcdd

4x (11)

and

CR =

M

|b|

2 q

abcd

Im (Bab Bcd) d4

x. (12)

The field abcd is the same as in the last section. The field qabcd is real with thesymmetries of the Riemann curvature tensor. The CR is the Lagrange multiplierterm introduced to impose the area metric reality constraint.

The field equations corresponding to the extrema of the actions under theA and variations are the same as given in section two. They impose the

condition Bijab = [ia

j]b or

[ia

j]b and A be the Levi-Civita connection for

the complex metric. The field equations corresponding to the extrema of theactions under the qabcd variations are Im(Bab Bcd) = 0. This, as we discussedbefore, imposes the condition that the metric gab = a b be real or imaginary5.

Let me assume I have solved the simplicity constraint, the reality constraint

and dB = 0. Substitute the solutions Bijab = [ia

j]b and A the Levi-Civita

connection for a real or imaginary metric gab = a b in the action S . Thisresults in a reduced action which is a function of the tetrad ia only,

S() = Re

d4xbF.

where F is the scalar curvature Fabab . Recall that F is real or imaginary depend-ing on the metric. This action reduces to Einstein-Hilbert action if both themetric and space-time density are simultaneously real or imaginary. If not, itis zero and there is no field equation involving the curvature Fabcd tensor otherthan the Bianchi identities.

If Bijab = [ia

j]b , we get a new reduced action,

S() = Re

d

4

x

abcd

Fabcd, (13)

which is zero because of the Bianchi identity abcdFabcd = 0. So there is no otherfield equation other than the Bianchi identities.

5Also for Bijab

= [ia

j]b

, it can be verified that the reality constraint implies that themetric gab = a b be real or imaginary.

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4 Discretization

4.1 BF theoryConsider that a continuum manifold is triangulated with four simplices. Thediscrete equivalent of a bivector two-form field is the assignment of a bivectorBijb to each triangle b of the triangulation. Also the equivalent of a connectionone-form is the assignment of a parallel propagator geij to each tetrahedrone. Using the bivectors and parallel propagators assigned to the simplices, theactions for general relativity and BF theory can be rewritten in a discrete form[7]. The real SO(4, C) BF action can be discretized as follows [6]:

S(Bb, ge) = Re

b

Bijb lnHbij. (14)

The Hb is the holonomy associated to the triangle b. It will be quantized to getan spin foam model later as done by Ooguri in section five.

4.2 BarrettCrane Constraints

The bivectors Bi associated with the ten triangles of a four simplex in a flatRiemannian space satisfy the following properties called the Barrett-Crane con-straints [11]:

1. The bivector changes sign if the orientation of the triangle is changed.

2. Each bivector is simple.

3. If two triangles share a common edge, then the sum of the bivectors isalso simple.

4. The sum of the bivectors corresponding to the edges of any tetrahedronis zero. This sum is calculated taking into account the orientations of thebivectors with respect to the tetrahedron.

5. The six bivectors of a four simplex sharing the same vertex are linearlyindependent.

6. The volume of a tetrahedron calculated from the bivectors is real andnon-zero.

The items two and three can be summarized as follows:

Bi Bj = 0 i,j,

where A B = IJKLAIJBKL and the i, j represents the triangles of a tetrahe-dron. Ifi = j, it is referred to as the simplicity constraint. If i = j it is referredas the cross-simplicity constraints.

Barrett and Crane have shown that these constraints are sufficient to restricta general set of ten bivectors Eb so that they correspond to the triangles of a

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geometric four simplex up to translations and rotations in a four dimensionalflat Riemannian space [11].

The Barrett-Crane constraints theory can be easily extended to the SO(4, C)general relativity. In this case the bivectors are complex and so the volumecalculated for the sixth constraint is complex. So we need to relax the conditionof the reality of the volume.

We would like to combine the area metric reality constraint with the Barrett-Crane Constraints. For this we must find the discrete equivalent of the areametric reality condition. For this let me next discuss the area metric realitycondition in the context of three simplices and four simplices. I would liketo show that the discretized area metric reality constraint combined with theBarrett-Constraint constraint requires the complex bivectors associated to athree or four simplex to describe real flat geometries.

4.2.1 TetrahedronConsider a tetrahedron t. Let the numbers 0 to 3 denote the vertices of thetetrahedron. Let me choose the 0 as the origin of the tetrahedron. Let Bij bethe complex bivector associated with the triangle 0ij where i and j denote oneof the vertices other than the origin and i < j. Let B0 be the complex bivectorassociated with the triangle 123. Then similar to Riemannian general relativity[11], the Barrett-Crane constraints6 for SO(4, C) general relativity imply that

Bij = ai aj , (15a)

B0 = B12 B23 B34, (15b)

where ai, i = 1 to 3 are linearly independent complex four vectors associated tothe links 0i of the three simplex. Let me choose the vectors ai, i = 1 to 3 to be

the complex vector basis inside the tetrahedron. Then the complex 3D metricinside the tetrahedron is

gij = ai aj , (16)

where the dot is the scalar product on the vectors. This describes a flat complexthree dimensional geometry inside the tetrahedron. The area metric is given by

Aijkl = gi[kgl]j .

The coordinates of the vectors ai are simply

a1 = (1, 0, 0),

a2 = (0, 1, 0),

a3 = (0, 0, 1).

Because of this all of the six possible scalar products made out of the bivectorsBij are simply the elements of the area metric. From the discussion of the last

6We do not require to use the fifth Barrett-Crane constraint since we are only consideringone tetrahedron of a four simplex.

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section the reality of the area metric simply requires that the metric gij be realor imaginary. Since B0 is also defined by equation (15b) its inner product with

itself and other bivectors are real. Thus in the context of a three simplex, thediscrete equivalent of the area metric reality constraint is that the all possiblescalar products of bivectors associated with the triangles of a three simplex bereal.

4.2.2 Four Simplex

In the case of a four simplex s there are six bivectors Bij . There are four B0type bivectors. Let Bi denote the bivector associated to the triangle made byconnecting the vertices other than the origin and vertex i. The Barrett-Craneconstraints imply equation (15a) with i, j = 1 to 4. There is one equation foreach Bi similar to equation (15b). Now the metric gij = ai aj describes a com-plex four dimensional flat geometry inside the four simplex s. Now assuming we

are dealing with non-degenerate geometry, the reality of the geometry requiresthe reality of the area metric. Similar to the three dimensional case, the com-ponents of the area metric are all of the possible scalar products made out ofthe bivectors Bij . The scalar products of the bivectors Bi among themselves orwith Bij s are simple real linear combinations of the scalar products made fromBij s. So one can propose that the discrete equivalent of the area metric realityconstraint is simply the condition that the scalar product of these bivectors bereal. Let me refer to the later condition as the bivector scalar product realityconstraint.

Theorem 2 The necessary and sufficient conditions for a four simplex with realnon-degenerate flat geometry are 1) The SO(4, C) Barrett-Crane constraints7

and 2) The reality of all possible bivector scalar products.

Proof. The necessary condition can be shown to be true by straight forwardgeneralization of the arguments given by Barrett and Crane [11] and applicationof the discussions in the last paragraph. The sufficiency of the conditions followfrom the discussion in the last paragraph.

5 Spin foam of the SO(4, C) BF model

Consider a four dimensional submanifold M. Let A be a SO(4, C) connection1-form and Bij a complex bivector valued 2-form on M. Let F be the curvature2-form of the connection A. Then the real continuum BF theory action defined

7The SO(4, C) Barrett-Crane constraints differ from the real Barrett-Crane constraints by

the following:

1. The bivectors are complex, and

2. The condition for the reality of the volume of tetrahedron is not required.

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in section two is,

SBF(A, Bij ,

A,

Bij) = Re

M B F, (17)

where A, Bij and their complex conjugates are considered as independent freevariables. This classical theory is a topological field theory. This property alsoholds on spin foam quantization as will be discussed below.

The Spin foam model for the SO(4, C) BF theory action can be derived fromthe discretized BF action by using the path integral quantization as illustratedin Ref:[6] for compact groups. Let be a simplicial manifold obtained by atriangulation of M. Let ge SO(4, C) be the parallel propagators associatedwith the edges (three-simplices) representing the discretized connection. LetHb =

ebge be the holonomies around the bones (two-simplices) in the four

dimensional matrix representation of SO(4, C) representing the curvature. LetBb be the 4 4 antisymmetric complex matrices corresponding to the dual Lie

algebra of SO(4, C) corresponding to the discrete analog of the B field. Thenthe discrete BF action is

Sd = Re

bM

tr(Bb ln Hb),

which is considered as a function of the Bbs and ges. Here Bb the discreteanalog of the B field are 4 4 antisymmetric complex matrices correspondingto dual Lie algebra of SO(4, C). The ln maps from the group space to the Liealgebra space. The trace is taken over the Lie algebra indices. Then the quantumpartition function can be calculated using the path integral formulation as,

ZBF() = b

dBbdBb exp(iSd)e dge=

b

(Hb)

e

dge, (18)

where dge is the invariant measure on the group SO(4, C). The invariant mea-sure can be defined as the product of the bi-invariant measures on the left andthe right SL(2, C) matrix components. Please see appendix A and B for moredetails. Similar to the integral measure on the Bs an explicit expression for thedge involves product of conjugate measures of complex coordinates.

Now consider the identity

(g) =1

648

d tr(T(g))d, (19)

where the T(g) is a unitary representation of SO(4, C), where = (L, R)

such that nL + nR is even, d = |LR|2

. The details of the representationtheory is discussed in appendix B. The integration with respect to d in theabove equation is interpreted as the summation over the discrete ns and theintegration over the continuous s.

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By substituting the harmonic expansion for (g) into the equation (18) wecan derive the spin foam partition of the SO(4, C) BF theory as explained in

Ref:[1] or Ref:[6]. The partition function is defined using the SO(4, C) inter-twiners and the {15} symbols.

The relevant intertwiner for the BF spin foam is

ie =

1

2

3

4

.

The nodes where the three links meet are the Clebsch-Gordan coefficients ofSO(4, C). The Clebsch-Gordan coefficients of SO(4, C) are just the productof the Clebsch-Gordan coefficients of the left and the right handed SL(2, C)

components. The Clebsch-Gordan coefficients of SL(2, C) are discussed in thereferences [19] and [31].The quantum amplitude associated with each simplex s is given below and

can be referred to as the {15} symbol,

{15} =23

24

3

34

4

45

5

15

1 12

2

14

25

13

35

i j

k

l

m

.

The final partition function is

ZBF() =

{b,e}

b

db648

s

ZBF(s)

b

db

e

de, (20)

where the ZBF(s) = {15} is the amplitude for a four-simplex s. The db =

|LR|2

term is the quantum amplitude associated with the bone b. Here eis the internal representation used to define the intertwiners. Usually e isreplaced by ie to indicate the intertwiner. The set {b,e} of all bs and esis usually called a coloring of the bones and the edges. This partition functionmay not be finite in general.

It is well known that the BF theories are topological field theories. A priorione cannot expect this to be true for the case of the BF spin foam modelsbecause of the discretization of the BF action. For the spin foam models of theBF theories for compact groups, it has been shown that the partition functionsare triangulation independent up to a factor [13]. This analysis is purely basedon spin foam diagrammatics and is independent of the group used as long the

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BF spin foam is defined formally by equation (18) and the harmonic expansionin equation (19) is formally valid. So one can apply the spin foam diagrammatics

analysis directly to the SO(4, C) BF spin foam and write down the triangulationindependent partition function as

Z

BF() = n4n3ZBF()

using the result from [13]. In the above equation n4, n3 is number of four bubblesand three bubbles in the triangulation and

= SO(4,C)(I)

=1

648

d2d.

The above integral is divergent and so the partition functions need not be finite.

The normalized partition function is to be considered as the proper partitionfunction because the BF theory is supposed to be topological and so triangula-tion independent.

6 The SO(4, C) Barrett-Crane Model

6.1 Introduction

My goal here is to systematically construct the Barrett-Crane model of theSO(4, C) general relativity. In the previous section I discussed the SO(4, C) BFspin foam model. The basic elements of the BF spin foams are spin networksbuilt on graphs dual to the triangulations of the four simplices with arbitraryintertwiners and the principal unitary representations of SO(4, C) discussed in

appendix B. These closed spin networks can be considered as quantum states offour simplices in the BF theory and the essence of these spin networks is mainlygauge invariance. To construct a spin foam model of general relativity thesespin networks need to be modified to include the Plebanski Constraints in thediscrete form.

A quantization of a four-simplex for the Riemannian general relativity wasproposed by Barrett and Crane [11]. The bivectors Bi associated with the tentriangles of a four-simplex in a flat Riemannian space satisfy the propertiescalled the Barrett-Crane constraints8. They have been listed in section 4.2which are repeated below for convenience:

1. The bivector changes sign if the orientation of the triangle is changed.

2. Each bivector is simple.

3. If two triangles share a common edge, then the sum of the bivectors isalso simple.

8I would like to refer the readers to the original paper [11] for more details.

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4. The sum of the bivectors corresponding to the edges of any tetrahedronis zero. This sum is calculated taking into account the orientations of the

bivectors with respect to the tetrahedron.

5. The six bivectors of a four-simplex sharing the same vertex are linearlyindependent.

6. The volume of a tetrahedron calculated from the bivectors is real andnon-zero.

The items two and three can be summarized as follows:

Bi Bj = 0 i,j,

where A B = IJKLAIJBKL and the i, j represents the triangles of a tetrahe-

dron. Ifi = j, it is referred to as the simplicity constraint. If i = j it is referred

as the cross-simplicity constraints.Barrett and Crane have shown that these constraints are sufficient to restrict

a general set of ten bivectors Eb so that they correspond to the triangles of ageometric four-simplex up to translations and rotations in a four dimensionalflat Riemannian space.

The Barrett-Crane constraints theory can be trivially extended to the SO(4, C)general relativity. In this case the bivectors are complex and so the volume cal-culated for the sixth constraint is complex. So we need to relax the conditionof the reality of the volume.

A quantum four-simplex for Riemannian general relativity is defined byquantizing the Barrett-Crane constraints [11]. The bivectors Bi are promotedto the Lie operators Bi on the representation space of the relevant group andthe Barrett-Crane constraints are imposed at the quantum level. A four-simplexhas been quantized and studied in the case of the Riemannian general relativ-ity before [11]. All the first four constraints have been rigorously implementedin this case. The last two constraints are inequalities and they are difficult toimpose. This could be related to the fact that the Riemannian Barrett-Cranemodel reveal the presence of degenerate sectors [29], [26] in the asymptotic limit[25] of the model. For these reasons here after I would like to refer to a spinfoam model that satisfies only the first four constraints as an essential Barrett-Crane model, While a spin foam model that satisfies all the six constraints as arigorous Barrett-Crane model.

Here I would like to derive the essential SO(4, C) Barrett-Crane model.For this one must deal with complex bivectors instead of real bivectors. Theprocedure that I would like to use to solve the constraints can be carried over

directly to the Riemannian Barrett-Crane model. This derivation essentiallymakes the derivation of the Barrett-Crane intertwiners for the real and thecomplex Riemannian general relativity more rigorous.

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6.2 The SO(4,C) intertwiner

The group SO(4, C) is locally isomorphic toSL(2,C)SL(2,C)

Z2 . An element B ofthe Lie algebra space ofSO(4, C) can be split into the left and the right handedSL(2, C) components,

B = BL + BR. (21)

There are two Casimir operators for SO(4, C) which are IJKLBIJBKL and

IK JL BIJBKL , where IK is the flat Euclidean metric. In terms of the left

and right handed split I can expand the Casimir operators as

IJKLBIJBKL = BL BL BR BR and

IK JL BIJBKL = BL BL + BR BR,

where the dot products are the trace in the SL(2, C) Lie algebra coordinates.

The bivectors are to be quantized by promoting the Lie algebra vectors to Lieoperators on the unitary representation space of SO(4, C) SL(2,C)SL(2,C)

Z2.

The relevant unitary representations ofSO(4, C) SL(2, C) SL(2, C)/Z2 arelabeled by a pair (L, R) such that nL +nR is even (appendix B). The elementsof the representation space DL DR are the eigen states of the Casimirs andon them the operators reduce to the following:

IJKLBIJBKL =

2L 2R

2I and (22)

IK JL BIJBKL =

2L + 2R 2

2I. (23)

The equation (22) implies that on DL DR the simplicity constraint BB = 0is equivalent to the condition L = R. I would like to find a representationspace on which the representations of SO(4, C) are restricted precisely by L =R. Since a representation is equivalent to representations [19], L =+R case is equivalent to L = R [19].

The Barrett-Crane intertwiner for Riemannian general relativity has beensystematically quantized in Ref:[18]. Since the representation theory for SO(4, C)is similar to that of SO(4, R), the systematic derivation can be generalized toSO(4, C) general relativity9.

The components of the Barrett-Crane intertwiner |

i

Di Di

can

9Readers can refer to the preprint Ref:[42] for details of derivations of SO(4, C) Barrett-Crane intertwiner.

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be written using the Gelfand-Naimarck representation theory [19] as:

| =

CS3

4

2

3

1 n n

n n

dn.

Since SL(2, C) CS3, using the following graphical identity:

SL(2,C)

g

g

g

g1

2

3

4

dg =

1

2

3

4

2

3

4

1

84

d,

the Barrett-Crane solution can be rewritten as

| =

2

2

1

1 4

4

3

3

84

d,

which emerges as an intertwiner in the familiar form in which Barrett and Craneproposed it for the Riemannian general relativity. It can be clearly seen thatthe simple representations for SO(4, R) (JL = JR) has been replaced by thesimple representation of SO(4, C) (L = R).

All the analysis done until for the SO(4, C) Barrett-Crane theory can bedirectly applied to the Riemannian Barrett-Crane theory. The correspondencesbetween the two models are listed in the following table10:

Property SO(4, R) BC model SO(4, C) BC model

Gauge group SO(4, R) SL(2,C)SL(2,C)Z2

SO(4, C) SU(2)SU(2)Z2

Representations JL

, JR

L

, RSimple representations JL = JR L = R

Homogenous space S3 SU(2) CS3 SL(2, C)

10BC stands for Barrett-Crane. For L and R we have nL + nR = even.

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6.3 The Spin Foam Model for the SO(4, C) General Rela-tivity.

The SO(4, C) Barrett-Crane intertwiner derived in the previous section canbe used to define a SO(4, C) Barrett-Crane spin foam model. The amplitudeZBC(s) of a four-simplex s is given by the {10}SO(4,C) symbol given below:

{10}SO(4,C) =

12

25

34

35

45

14

BC

BC

BC

BCBC

21

5

4

3

2324

13

15

, (24)

where the circles are the Barrett-Crane intertwiners. The integers representthe tetrahedra and the pairs of integers represent triangles. The intertwinersuse the four s associated with the links that emerge from it for its definitionin equation (24). In the next subsection, the propagators of this theory aredefined and the {10} symbol is expressed in terms of the propagators in thesubsubsection that follows it.

The SO(4, C) Barrett-Crane partition function of the spin foam associatedwith the four dimensional simplicial manifold with a triangulation is

Z() ={b}

b

d2b648

s

Z(s), (25)

where Z(s) is the quantum amplitude associated with the 4-simplex s and thedb adopted from the spin foam model of the BF theory can be interpreted asthe quantum amplitude associated with the bone b.

6.4 The Features of the SO(4, C) Spin Foam

Areas: The squares of the areas of the triangles (bones) of the triangulationare given by IK JL B

IJBKL . The eigen values of the squares of the areasin the SO(4, C) Barrett-Crane model from equation (41) are given by

IK JL BIJb B

KLb =

2 1

I

= n2

2

2 1 + in I.One can clearly see that the area eigen values are complex. The SO(4, C)Barrett-Crane model relates to the SO(4, C) general relativity. Since inthe SO(4, C) general relativity the bivectors associated with any two di-mensional flat object are complex, it is natural to expect that the areas

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defined in such a theory are complex too. This is a generalization of theconcept of the space-like and the time-like areas for the real general rela-

tivity models: Area is imaginary if it is time-like and real if it is space-like.

Propagators: Laurent and Freidel have investigated the idea of expressingsimple spin networks as Feynman diagrams [32]. Here we will apply thisidea to the SO(4, C) simple spin networks. Let be a triangulated threesurface. Let ni CS

3 be a vector associated with the ith tetrahedron ofthe . The propagator of the SO(4, C) Barrett-Crane model associatedwith the triangle ij is given by

Gij (ni, nj ) = T r(Tij(g(ni))T

ij(g(nj )))

= T r(Tij(g(ni)g1(nj ))),

where ij is a representation associated with the triangle common to the

ith and the jth tetrahedron of . If X and Y belong to CS3 then

trg(X)g(Y)1

= 2X.Y,

where X.Y is the Euclidean dot product and tr is the matrix trace. If = et and 1

are the eigen values of g(X)g(Y)1 then,

+ 1 = 2X.Y

X.Y = cosh(t).

From the expression for the trace of the SL(2, C) unitary representations,(appendix A, [19]) I have the propagator for the SO(4, C) Barrett-Cranemodel calculated as

Gij (ni, nj) =cos(ijij + nij ij)

|sinh(ij + iij )|2 ,

where ij + iij is defined by ni.nj = cosh(ij + iij). Two importantproperties of the propagators are listed below.

1. Using the expansion for the delta on SL(2, C) I have

CS3(X, Y) = SL(2,C)(g(X)g1(Y))

=1

84

Tr(T(g(X)g

1(Y))d,

where the suffix on the deltas indicate the space in which it is defined.Therefore G(X, Y)) = 8

4CS3(X, Y).

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2. Consider the orthonormality property of the principal unitary repre-sentations of SL(2, C) given by

CS3Tz1z11(g(X))T

z2z22

(g(X))dX

=84

11(1 2)(z1 z1)(z2 z2),

where the delta on the s is defined up to a sign of them. From thisI have

CS3G1(X, Y)G2(Y, Z)dY =

84

11(1 2)G1(X, Z).

The {10} symbol can be defined using the propagators on the complexthree sphere as follows:

Z(s) =

xkCS3

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Imposition of the Barrett-Crane constraints on the BF theory spinfoam, suppresses the terms corresponding to the non-simple repre-

sentations. If only the simple representations are allowed in the righthand side, it is no longer peaked at the identity. This means thatthe partition function for the SO(4, C) Barrett-Crane model involvescontributions only from the non-flat connections which has local in-formation.

In the asymptotic limit study of the SO(4, C) spin foams in sectionfour of Ref:[42] the discrete version of the SO(4, C) general relativity(Regge calculus) is obtained. The Regge calculus action is clearlydiscretization dependent and non-topological.

The real Barrett-Crane models that are discussed in the next section arethe restricted form of the SO(4, C) Barrett-Crane model. The above rea-soning can be applied to argue that they are also discretization dependent.

7 Spin Foams for 4D Real General Relativityand reality constraints

7.1 The Formal Structure of Barrett-Crane Intertwiners

Let me briefly discuss the formal structure of the Barrett-Crane intertwinerof the SO(4, C) general relativity for the purpose of the developing spin foammodels for real general relativity theories. It has the following elements:

A gauge group G,

A homogenous space X of G,

A G invariant measure on X and,

A complete orthonormal set of functions which call as Tfunctions whichare maps from X to the Hilbert spaces of a subset of unitary representa-tions of G:

T : X D,

where is a representation of G. The Tfunctions correspond to thevarious unitary representations under the transformation of X under G.The Tfunctions are complete in the sense that on the L2 functions on Xthey define invertible Fourier transforms. The T functions are writtenusing its components in a linear vector basis of representation D.

Formally Barrett-Crane intertwiners are quantum states associated toclosed simplicial two surfaces defined as an integral of a outer product ofTfunctionson the space X:

=

X

T(x)dX x

D.

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It can seen that is gauge invariant under G because of the invariance of themeasure dX x.

7.2 The Real Barrett-Crane Models

Consider a four-simplex with complex bivectors Bi, i = 1 to 10 associated withits triangles. The discrete equivalent of the area metric reality constraint isthe bivector scalar product reality constraint. Then the bivector scalar productreality constraint requires

Im(Bi Bj ) = 0 i,j.

I would like to formally reduce the Barrett-Crane models for real generalrelativity from that of the SO(4, C) Barrett-Crane model by using the bivectorscalar products reality constraint. Precisely I plan to use the following three

ideas to reduce the Barrett-Crane models:

1. The formal structure of the reduced intertwiners should be the same asthat of the SO(4, C) Barrett-Crane model,

2. The eigen value of the Casimir corresponding to the square of the area ofany triangle must be real. I would like to refer to this as the self-realityconstraint11,

3. The eigen values of the square of area Casimir corresponding to the rep-resentations associated with the internal links of the intertwiner must bereal. I would like to refer to this as the cross-reality constraint.

The first idea sets a formal ansatz for the reduction process. The simple and

symmetric nature of the SO(4, C) (or SO(4, R)) Barrett-Crane intertwiner and,the work done in Ref:[32], Ref:[17] and Ref:[33] can be considered as evidencesfor the formal structure to be the general form of structure of intertwiners forall signatures. Here we assume this idea as a hypothesis.

The square of the area of a triangle is simply the scalar product of thebivector of a triangle with itself. Second condition is the quantum equivalentof the reality of the scalar product of a bivector associated with a triangle withitself. Once the second condition is imposed the third condition is the quantumequivalent of the reality of the scalar product of the two bivectors of any twotriangle of a tetrahedron12.

11I would like to mention that the areas being real necessarily does not mean that thebivectors must also be real.

12We have ignored to impose reality of the scalar products of the bivectors associated to

any two triangles of the same four simplex which intersect at only at one vertex. This isbecause these constraints appears not to be needed for a formal extraction of the Barrett-Crane models of real general relativity from that of SO(4, C) general relativity describedin this section. Imposing these constraints may not be required because of the enormousredundancy in the bivector scalar product reality constraints. This issue need to be carefullyinvestigated

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My goal is to use the above principles to derive reduced Barrett-Crane modelsand later one can convince oneself by identifying and verifying that the Barrett-

Crane constraints are satisfied for a subgroup ofSO(4, C) for each of the reducedmodel.

In general by reducing a certain Hilbert space associated with the repre-sentations of a group G by some constraints, the resultant Hilbert space neednot contain the states gauge invariant under G. In that case one can look forgauge invariance states under subgroups of G. In our case we will find that thesuitable quantum states extracted by adhering to the above principles are gaugesymmetry reduced versions of SO(4, C) Barrett-Crane states. They are gaugeinvariant only under the real subgroups of SO(4, C).

Let P be a formal projector which reduces the Hilbert space DL DRto a reduced Hilbert space such that the reality constraints are satisfied. Letme assume as an ansatz that now the complex three sphere is replaced by itssubspace X due to projection. Now I expect, the projected SO(4, C) Barrett-Crane intertwiner is spanned by the following states for all i satisfying thereality constraints:

X =

xX

i

P Ti(g(x))dg(x),

where dg(n) is the reduced measure of dg(n) on X. The imposition of the self-reality constraints expressed at the quantum level sets i or ni to be zero on eachvertex of the SO(4, C) Barrett-Crane intertwiner. Let me rewrite the projectedintertwiner as follows.

X =

x,yX

1,2

P T1(g(x))X (x, y)3,4

P T1(g(y))dX g(x)dX g(y),

where X (x, y) is the delta function on X. Since X is a subspace of SL(2, C) aharmonic expansion can be derived for (x, y) using the unitary representationsof SL(2, C). Since the intertwiner must obey the cross reality constraint theharmonic expansion must only contain simple representations of SL(2, C) ( orn is zero).

For the Fourier transform defined by P T(g(x)) to be complete and orthonor-mal I must have

Q

tr(P T(g(x))P T(g(y)))d = X (x, y),

where Q is the set of all simple representations13 of SL(2, C) required for theexpansion. Only the simple representations ofSL(2, C) must be used to satisfy

the cross-reality constraints. Thus, the number of reduced intertwiners deriv-able is directly related to the possible solutions for this equation (subjected toBarrett-Crane constraints).

13One could also call the simple representations of SL(2, C) as the real representations sinceit corresponds to the real areas and the real homogenous spaces. But I will avoid this to avoidany possible confusion.

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The equation of a complex three sphere is

x2 + y2 + z2 + t2 = 1.

There are four different topologically different maximally connected real sub-spaces of CS3 such that the harmonic (Fourier) expansions on these spaces usethe simple representations of SL(2, C) only. They are namely, the three sphereS3, the real hyperboloid H+, the imaginary hyperboloid H and the Kleinienhyperboloid14 K3. Each of these subspace X are maximal real subspaces ofCS3.They are all homogenous under the action of a maximal real subgroup15 GX ofSO(4, C). There exists a GX invariant measure dX (x). The reduced bivectorsacting on the functions on X effectively take values in the Lie algebra of GX .Since the measure dX (n) is invariant, the reduced intertwiner is gauge invariant.So the intertwiner X must correspond to the quantum general relativity forthe group GX .

Let the coordinates of n = (x, y, z, t) be restricted to real values here afterin this section. Let me discuss the various reduced intertwiners:

1. = 0 case: This uses only the = (0, n) representations only. Thiscorresponds to X = S3, satisfying

x2 + y2 + z2 + t2 = 1,

which is invariant under SO(4, R). So this case corresponds to the Rie-mannian general relativity. The appropriate pro jected Tfunctions arethe representation matrices of SU(2) S3 and the reduced measure isthe Haar measure of SU(2). The intertwiner I get is the Barrett-Craneintertwiner for the Riemannian general relativity. Here the s has been

replaced by the J

s and the complex three sphere by the real three sphere.The case of going from the SO(4, C) Barrett-Crane model to the Rieman-nian Barrett-Crane model is intuitive. It is a simple process of going fromcomplex three sphere to its subspace the real three sphere.

2. n = 0 case: This uses = (, 0) representations only: This correspondsto X as a space-like hyperboloid (only one sheet) with GX = SO(3, 1, R):

x2 + y2 + z2 t2 = 1.

The intertwiner now corresponds to the Lorentzian general relativity. Thisintertwiner was introduced in [14]. The unitary representations of theLorentz group on the real hyperboloid have been studied by Gelfand andNaimarck [19], from which the Tfunctions are

T(x)[] = [.x]1

2i1,

14By Kleinien hyperboloid I refer to the space described by x2 + y2 z2 t2 = 1 for realx,y,z and t.

15The real group is maximal in the sense that there is no other real topologically connectedsubgroup of SO(4, C) that is bigger.

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where null cone intersecting t = 1 plane in the Minkowski space.Here replaces (z1, z2) in the Tfunction T(g(x))(z1, z2)of the SO(4, C)

Barrett-Crane Model. An element g SO(3, 1) acts as a shift operator asfollows:

gT(x)[] = T(gx)[]

= T(x)[g1].

This intertwiner was first introduced in [14].

3. Combination of (0, n) and(, 0) representations: There are two possiblemodels corresponding to this case. One of them has X as the Kleinienhyperboloid defined by

x2 + y2 z2 t2 = 1,

with GX = SO(2, 2, R). Here the X is isomorphic to SU(1, 1) SL(2, R).The intertwiner now corresponds to Kleinien general relativity ( + + signature). The Tfunctions are of the form T(k(n))(z1, z2) where z1and z2 takes real values only (please refer to appendix C ), = 0 and kis an isomorphism from the Kleinien hyperboloid to SU(1, 1) defined by

k(n) =

x iy z itz + it x + iy

.

The representations corresponding to the n = 0 and = 0 cases arequalitatively different. The representations corresponding to = 0 arecalled the continuous representations and those to n = 0 are called

the discrete representations. The action of g SO(2, 2, R) on theTfunctions isgT(k(x)) = T(k(g(x)),

where g(x) is the result of action of g on x X.

4. The second model using both (0, n) and(, 0) representations: This corre-sponds to the time-like hyperboloid with GX = SO(3, 1),

x2 y2 z2 t2 = 1,

where two vectors that differ just by a sign are identified as a single pointof the space X. The corresponding spin foam model has been introducedby Barrett and Crane [11]. It has been derived using a field theory over

group formalism by Rovelli and Perez [16]. Similar to the previous case, Ihave both continuous and discrete representations, with the Tfunctionsgiven by

T(x)[] = [.x]1

2i1,

Tn(x)[l(a, )] = exp(2in)(a.),

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where the l(a, ) is an isotropic line16 on the imaginary hyperboloid alongdirection going through a point a on the hyperboloid and the is the

distance between l(a, ) and l(x, ) given by cos = a.x, where the dot isthe Lorentzian scalar product. I have for g SO(3, 1, R),

gTn(x)[l(a, )] = Tn(x)[l(a,g)]

= Tn(g1x)[l(a, )],

and the action of g on continuous representations are defined similar toequation (26). The corresponding spin foam model has been introducedand investigated before by Rovelli and Perez [16].

In the table below the representations and the homogenous spaces associatedwith various Barrett-Crane intertwiners (models) in four dimensions have beensummarized.

Model Representations Homogenous SpaceSO(4, C) model L = R =

n2 + i Complex three-sphere

SO(4, R) model = 0 Discrete irreps Real three-sphereSO(3, 1) model n = 0 Continuos irreps Space-Like HyperboloidSO(3, 1) model n = 0 = 0 Time-Like HyperboloidSO(2, 2) model n = 0 = 0 Kleinien Hyperboloid

From the table, it can be clearly seen that the representations used forthe intertwiners for real general relativity are various possible combinations ofrepresentations of SL(2, C) simply restricted by the reality condition n = 0.Also all the homogenous spaces of the intertwiners of the real general relativitytheories are simply the all possible real cross-sections (maximal) of the complexthree sphere. From this point of view the intertwiners for the real general

relativity theories listed in the table makes a complete set.

7.3 The Area Eigenvalues

Using the Tfunctions described above, the intertwiners for real general rela-tivity can be constructed. Using these intertwiners, spin foam models (Barrett-Crane) for the real general relativity theories of the various different signaturescan be constructed. The square of the area of a triangle of a four-simplex for allsignatures associated with a representation is described by the same formula17,

IK JL BIJBKL =

2 1

I

=

n2

2 2 1

I,

16A line on an imaginary hyperboloid [19] is the intersection of a 2-plane of the Minkowskispace with it. The line is called isotropic if the Lorentzian distance between any two pointson it is zero. An isotropic line l is described by the equation x = s + x0, x is the variablepoint on l, x0 is any fixed point on l, and is a null-vector. For more information please referto [19]

17Please refer to the end of appendix C regarding the differences between the Casimers ofSL(2, C) and SU(1, 1).

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where only of n and is non-zero. The square of the area is negative or pos-itive depending on whether or n is non-zero. The negative (positive) sign

corresponds to a time-like (space-like) area.

8 Acknowledgement.

I thank Allen Janis, George Sparling and John Baez for correspondences.

A Unitary Representations of SL(2,C)

The Representation theory ofSL(2,C) was developed by Gelfand and Naimarck[19]. Representation theory ofSL(2, C) can be developed using functions on C2

which are homogenous in their arguments18. The space of functions D isdefined as functions f(z

1, z

2) on C2 whose homogeneity is described by

f(az1, az2) = a11a21f(z1, z2),

for all a = 0, where is a pair (1, 2). The linear action of SL(2, C) on C2

defines a representation ofSL(2, C) denoted by T. Because of the homogeneityof functions of D, the representations T can be defined by its action on thefunctions (z) of one complex variable related to f(z1, z2) D by

(z) = f(z, 1).

There are two qualitatively different unitary representations of SL(2, C): theprincipal series and the supplementary series, of which only the first one isrelevant to quantum general relativity. The principal unitary irreducible repre-sentations ofSL(2,C) are the infinite dimensional. For these 1 = 2 =

n+i2

,

where n is an integer and is a real number. In this article I would like to labelthe representations by a single complex number = n2 + i

2 , wherever necessary.

The T representations are equivalent to T representations [19].Let g be an element of SL(2,C) given by

g =

,

where ,, and are complex numbers such that = 1. Then the Drepresentations are described by the action of a unitary operator T(g) on thesquare integrable functions (z) of a complex variable z as given below:

T(g)(z) = (z1 + )1(z1 + )

1(z +

z + ). (27)

This action on (z) is unitary under the inner product defined by

((z), (z)) =

(z)(z)d2z,

18These functions need not be holomorphic but infinitely differentiable may be except atthe origin (0, 0).

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where d2z = i2dz dz and I would like to adopt this convention everywhere.Completing D with the norm defined by the inner product makes it into a

Hilbert space H.Equation (27) can also be written in kernel form [15],

T(g)(z1) =

T(g)(z1, z2)(z2)d

2z2,

Here T(g)(z1, z2) is defined as

T(g)(z1, z2) = (z1 + )1(z1 + )

1(z2 g(z1)), (28)

where g(z1) =z1+z1+

. The Kernel T(g)(z1, z2) is the analog of the matrix rep-resentation of the finite dimensional unitary representations of compact groups.An infinitesimal group element, a, ofSL(2,C) can be parameterized by six real

numbers k and k as follows [40]:

a I+i

2

3k=1

(kk + kik),

where the k are the Pauli matrices. The corresponding six generators of the representations are the Hk and the Fk. The Hk correspond to rotations andthe Fk correspond to boosts. The bi-invariant measure on SL(2, C) is given by

dg =

i

2

3d2d2d2

||2 =

i

2

3d2d2d2

||2 .

This measure is also invariant under inversion in SL(2,C). The Casimir oper-

ators for SL(2, C ) are given by

C = det

X3 X1 iX2

X1 + iX2 X3

and its complex conjugate C where Xi = Fi + iHi. The action of C (C) onthe elements of D reduces to multiplication by 21 1 (

22 1).The real and

imaginary parts of C are another way of writing the Casimirs. On D theyreduce to the following

Re(C) =

2 +

n

4

2 1

I,

Im(C) = nI.

The Fourier transform theory on SL(2,C) was developed in Ref:[19]. Iff(g)is a square integrable function on the group, it has a group Fourier transformdefined by

F() =

f(g)T(g)dg, (29)

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where is F() is linear operator defined by the kernel K(z1, z2) as follows:

F()(z) =

K(z, z)(z)d2z.

The associated inverse Fourier transform is

f(g) =1

84

T r(F()T(g

1))d, (30)

where the

d indicates the integration over and the summation over n.From the expressions for the Fourier transforms, I can derive the orthonormalityproperty of the T representations,

SL(2,C)

Tz1z11(g)Tz2

z22(g)dg =

84

11(1 2)(z1 z1)(z2 z2),

where T is the Hermitian conjugate of T.The Fourier analysis on SL(2, C) can be used to study the Fourier analysis

on the complex three sphere CS3. Ifx = (a,b,c,d) CS3 then the isomorphismg : CS3 SL(2, C) can be defined by the following:

g(x) =

a + ib c + id

c + id a ib

.

Then, the Fourier expansion of f(x) L2(CS3) is given by

f(x) =1

84

T r(F()T(g(x)

1)d

and its inverse isF() =

f(g)T(g(x))dx,

where the dx is the measure on CS3. The measure dx is equal to the bi-invariantmeasure on SL(2, C) under the isomorphism g.

The expansion of the delta function on SL(2, C) from equation (30) is

(g) =1

84

tr [T(g)] d. (31)

Let me calculate the trace tr [T(g)]. If = e+i and1

are the eigen values ofg then

tr [T(g)] =1 2 + 1 2

| 1

|2

,

which is to be understood in the sense of distributions [19]. The trace can beexplicitly calculated as

tr [T(g)] =cos( + n)

2 |sinh( + i)|2. (32)

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Therefore, the expression for the delta on SL(2, C) explicitly is

(g) = 184

n

d(n2 + 2) cos( + n)

|sinh( + i)|2 . (33)

Let us consider the integrand in equation (30). Using equation (29) in it wehave

T r(F()T(g1)) =

f(g)T r(T(g)T(g

1))dg

=

f(g)T r(T(gg

1))dg. (34)

But, since the trace is insensitive to an overall sign of , so are the terms of theFourier expansion of the L2 functions on SL(2, C) and CS3.

B Unitary Representations ofSO(4, C)

The group SO(4, C) is related to its universal covering group SL(2, C)SL(2, C)

by the relationship SO(4, C) SL(2,C)SL(2,C)Z2

. The map from SO(4, C) toSL(2, C) SL(2, C) is given by the isomorphism between complex four vectorsand GL(2, C) matrices. If X = (a,b,c,d) then G : C4 GL(2, C) can bedefined by the following:

G(X) =

a + ib c + id

c + id a ib

.

It can be easily inferred that det G(X) = a2+ b2 + c2 + d2 is the Euclidean normof the vector X. Then, in general a SO(4, C) rotation of a vector X to another

vector Y is given in terms of two arbitrary SL(2, C) matrices gA

L B, gA

R B SL(2, C) by

G(Y)AA

= g AL B gA

R B GAB (X),

where GAB(X) is the matrix elements ofG(X). The above transformation does

not differentiate between (LAB, RA

B ) and (LAB, R

A

B ) which is responsible for

the factor Z2 in SO(4, C) SL(2,C)SL(2,C)

Z2.

The unitary representation theory of the group SL(2, C) SL(2, C) is easilyobtained by taking the tensor products of two Gelfand-Naimarck representationsof SL(2, C). The Fourier expansion for any function f(gL, gR) of the universalcover is given by

f(gL, gR) =1

648 LLRRF(L, R)T(g1L )T(g1R )dLdR,

where L =nL+iL

2 and R =nR+iR

2 . The Fourier expansion on SO(4, C)is given by reducing the above expansion such that f(gL, gR) = f(gL, gR).From equation (32) I have

tr [T(g)] = (1)ntr [T(g)] ,

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where = n+i2 . Therefore

f(gL, gR) = 184

LLRRF(L, R)(1)

nL+nRT(g1L )T(g

1R )dLdR.

This implies that for f(gL, gR) = f(gL, gR), I must have (1)nL+nR = 1.From this, I can infer that the representation theory of SO(4, C) is deducedfrom the representation theory of SL(2, C) SL(2, C) by restricting nL + nRto be even integers. This means that nL and nR should be either both oddnumbers or even numbers. I would like to denote the pair (L, R) (nL + nReven) by .

There are two Casimir operators available for SO(4, C), namely IJKLBIJBKL

and IK JL BIJBKL . The elements of the representation space DL DR arethe eigen states of the Casimirs. On them, the operators reduce to the following:

IJKLBIJBKL = 2L

2R

2and (35)

IK JL BIJBKL =

2L + 2R 2

2. (36)

C Unitary Representations ofSU(1, 1)

The unitary representations ofSU(1, 1) SL(2, R), given in Ref:[41], is definedsimilar to that ofSL(2, C). The main difference is that the D are now functions(z) on C1. The representations are indicated by a pair = (, ), is the parityof the functions ( is 0 for even functions and 12 for odd functions) and is acomplex number defining the homogeneity:

(az) = |a|2 sgn(a)2(z),

where a is a real number. Because of homogeneity the D functions can berelated to the infinitely differentiable functions (ei) on S1 where is thecoordinate on S1. The representations are defined by

T(g)(ei) = (ei + )+(ei + )(

z +

z + ). (37)

There are two types of the unitary representations that are relevant for quan-tum general relativity: the continuous series and the discrete series. For thecontinuous series = (i 12 , ), where is a non-zero real number. Let medenote the continuous series representations with suffix or prefix c, for exampleTc.

There are two types of discrete series representations which are indicatedby signs . They have their respective homogeneity as = (l,

l ) where

l = 1 is defined by the condition l l is an integer. Let me denote

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the representations as T+l and T

l . The T+

l (T

l ) representations can be re-expressed as linear operators on the functions +(z)((z)) on C

1 that are

analytical inside (outside) the unit circle. The Tl (g) are defined as

Tl (g)(z) = |z + |2l (

z +

z + ).

The inner products are defined by

(f1, f2)c =1

2

20

f1(ei)f2(ei)d,

(f1, f2)l+ =

1

(2l 1)

|z|1

(1 |z|)

2l2

f1(z)f2(z)

dzdz

2i .

The Fourier transforms are defined for the unitary representations by

Fc() =

f(g)Tc(g)dg,

F+(l) =

f(g)T+l (g)dg, and

F(l) =

f(g)Tl (g)dg,

where dg is the bi-invariant measure on the group.The inverse Fourier transform is defined by

f(g) =1

42

l 1

2N0

(l + 12 )T r[F+(l)(T

+l (g)) + F

(l)(T

l (g))]

+

0

T r[F()T ] tanh ( + i)d

.

The T(,) is equivalent to T(1,). The Casimir operator for the T represen-tations (all) can be defined similar to SU(2) and its eigen values are

C = ( + 1),

where the comes from = (, ). The in this section is related to the inthe representations of SL(2, C) by = + 12 . The expressions for the Casimirsof the two groups differ by a factor of 4.

D Reality Constraint for Arbitrary Metrics

Here we analyze the area metric reality constraint for a metric gac of arbitraryrank in arbitrary dimensions, with the area metric defined as Aabcd = ga[cgd]b.Let the rank of gac be r.

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If the rank r = 1 then gab is of form ab for some complex non zero co-vectora. This implies that the area metric is zero and therefore not an interesting

case.Let me prove the following theorem.

Theorem 3 If the rank r of gac is 2,then the area metric reality constraintimplies the metric is real or imaginary. If the rankr of gac is equal to 1, thenthe area metric reality constraint implies gac = ab for some complex = 0and real non-zero co-vector a.

The area metric reality constraint implies

gRacgIdb = g

Radg

Icb. (38)

Let gAC be a r by r submatrix of gac with a non zero determinant, where thecapitalised indices are restricted to vary over the elements of gAC only. Now wehave

gRACgIDB = g

RADg

ICB . (39)

From the definition of the determinant and the above equation we have

det(gAC) = det(gRAC) + det(g

IAC).

Since det(gAC) = 0 we have either det(gRAC) or det(gIAC) not equal to zero. Let

me assume gRac = 0. Then contracting both the sides of equation (39) with theinverse of gRAC we find g

IDB is zero. Now from equation (38) we have

gRACgIdB = g

RAdg

ICB = 0. (40)

Since the Rank of gRAC 2 we can always find a gRAC = 0 for some fixed A and

C. Using this in equation (40) we find gIdB is zero. Now consider the following:

gRACgIdb = g

RAdg

ICb . (41)

we can always find a gRAC = 0 for some fixed A and C. Using this in equation(41) we find gIdb = 0. So we have shown that if g

Rac = 0 then g

Idb = 0. Similarly

if we can show that if gIac = 0 then gRdb = 0.

E Field Theory over Group and HomogenousSpaces.

One of the problems with the Barrett-Crane model for general relativity is its

dependence on the discretization of the manifold. A discretization indepen-dent model can be defined by summing over all possible discretizations. Witha proper choice of amplitudes for the lower dimensional simplices the BF spinfoams can be reformulated as a field theory over a group (GFT) [ 6]. Similarly,the Barrett-Crane models can be reformulated as a field theory over the ho-mogenous space of the group [24]. Consider a tetrahedron. Let a group element

34


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gi be associated with each triangle i of the tetrahedron. Let a real field (g1, g2,g3, g4) invariant under the exchange of its arguments be associated with the

tetrahedron. Let the field be invariant under the simultaneous (left or right)action of a group element g on its variables. Then the kinetic term is defined as

K.E =

4i=1

dgi2.

To define the potential term, consider a four-simplex. Let gi, where i =1 to 10 be the group elements associated with its ten triangles. With eachtetrahedron e of the four-simplex, associate a field which is a function of thegroup elements associated with its triangles. Denote it as e. Then the potentialterm is defined as

P.E =

=

5!

10i=1

dgi

5e=1

e,

where is an arbitrary constant.Now the action for a GFT can be defined as

S() = K.E+ P.E =

4i=1

dgi2 +

5!

10i=1

dgi

5e=1

e.

The action has two terms, namely the kinetic term and the potential terms.The Partition function of the GF T is

Z =

DeS().

Now, an analysis of this partition function yields the sum over spin foam parti-tions of the four dimensional BF theory for group G for all possible triangula-tions. From the analysis of the GFT we can easily show that this result is validfor G = SO(4, C) with the unitary representations defined in the appendix B.

Let us assume is invariant only under the simultaneous action of an elementof a subgroup H of G. Then, if G = SO(4, R) and H = SU(2) we get GFTsfor the Barrett-Crane model19 [24]. Similarly, if G = SL(2, C) and H = SU(2)or SU(1, 1), we can define GFT for the Lorentzian general relativity [ 16], [15].The representation theories of SO(4, C) and SL(2, C) has similar structure tothose of SO(4, R) and SU(2) respectively. So the GFT with G = SO(4, C)and H = SL(2, C) should yield the sum over triangulation formulation of the

SO(4, C) Barrett-Crane model. The details of this analysis and its variationswill be presented elsewhere.

19Depending on whether we are using the left or right action of G on , we get two differentmodels that differ by amplitudes for the lower dimensional simplices [24].

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suresh. k. maran- reality conditions for spin foams

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