The Hamiltonian formulation of tetrad gravity: Three-dimensional case

ISSN 0202-2893, Gravitation and Cosmology, 2010, Vol. 16, No. 3, pp. 181–194. c© Pleiades Publishing, Ltd., 2010.

The Hamiltonian Formulation of Tetrad Gravity:Three-Dimensional Case¶

A. M. Frolov1*, N. Kiriushcheva2**, and S. V. Kuzmin3***

1Department of Chemistry, University of Western Ontario, London, Canada2Department of Applied Mathematics and Department of Mathematics,

University of Western Ontario, London, Canada3Faculty of Arts and Social Science, Huron University College and Department of Applied Mathematics,

University of Western Ontario, London, CanadaReceived August 1, 2009; in final form, April 26, 2010

Abstract—The Hamiltonian formulation of tetrad gravity in any dimension higher than two, using itsfirst-order form where tetrads and spin connections are treated as independent variables, is discussed,and the complete solution of the three-dimensional case is given. For the first time, applying themethods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra ofPoisson brackets among all constraints is calculated. The algebra of Poisson brackets, among first-classsecondary constraints, locally coincides with Lie algebra of the ISO(2,1) Poincare group. All the first-classconstraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allowus to unambiguously derive the generator of gauge transformations and find the gauge transformationsof the tetrads and spin connections which turn out to be the same as found by Witten without recourseto the Hamiltonian methods [Nucl. Phys. B 311, 46 (1988)]. The gauge symmetry of the tetrad gravitygenerated by the Lie algebra of constraints is compared with another invariance, diffeomorphism. Someconclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular,that diffeomorphism invariance is not derivable as a gauge symmetry from the Hamiltonian formulationof tetrad gravity in any dimension where tetrads and spin connections are used as independent variables.

DOI: 10.1134/S0202289310030011

1. INTRODUCTION

The Hamiltonian formulation of General Relativity(GR) has a history going back to a half-century. Onthe one hand, the Hamiltonian formulation of such ahighly nontrivial theory as GR is a good laboratorywhere general methods of constrained dynamics [1–3] can be studied and some subtle points that cannotbe even seen in simple examples, can be found, inves-tigated and lead to further development of the methoditself. On the other hand, the correct Hamiltonian for-mulation of a theory is a prerequisite to its successfulcanonical quantization. In this paper we consider theHamiltonian formulation of tetrad gravity. Nowadaysit is a more popular formulation of GR, in particular,because it is used in Loop Quantum Gravity [4–6]. The accepted Hamiltonian formulation of tetradgravity leads to the so-called “diffeomorphism con-straint”, or more precisely, the “spatial diffeomor-

*E-mail: [email protected]**E-mail: [email protected]

***E-mail: [email protected]¶The article was submitted by the authors in English.

phism constraint” [4–6] (though the word “spatial”is often omitted in the literature).

Recently it was demonstrated [7–9] that the long-standing problem of having only spatial diffeo-morphism in the Hamiltonian formulation of metricGR [10] is just a consequence of a non-canonicalchange of variables. Without making such changes,the full diffeomorphism invariance of the metric tensoris derivable from the Hamiltonian formulation in alldimensions higher than two (D > 2) [7, 8]. Thisresult suggests the necessity of reconsidering also theHamiltonian formulation of tetrad gravity, especiallybecause the accepted Hamiltonian formulation wasperformed using a change of variables for tetrads [11]similar to metric gravity and this has led to the same“diffeomorphism constraint” which is only spatial.

Moreover, the three-dimensional case of tetradgravity poses additional questions. For instance,what is the gauge symmetry of tetrad gravity in threedimensions? In some papers it is written that thegauge symmetry is the Poincare symmetry [12], inothers that it is the Lorentz symmetry plus diffeo-

181

182 FROLOV et al.

morphism [13],1 or that there exist various waysof defining the constraints of tetrad gravity leadingto different gauge transformations [16]. Accordingto [17], two symmetries are present, and we have todecide “what is a gauge symmetry and what is not”.We think that it is the right question but the answershould not depend on our decision or desire. TheHamiltonian method is a perfect instrument to findthe unique answer to the question what the gaugesymmetry is.

To the best of our knowledge, despite the exis-tence of numerous review articles, living reviews, andbooks, e.g., [18–20], there is no complete Hamilto-nian formulation of tetrad gravity in three dimensions.The only papers that are somehow related to tetradgravity in three dimensions are: the work due toBlagojevic and Cvetkovic [21] where all steps of theDirac procedure were performed, but for the three-dimensional Mielke-Baekler model; and Blagojevicin [22] performed Hamiltonian analysis but for theChern-Simons action. A complete Hamiltonian for-mulation means that all steps of the Dirac procedureshould be performed [1–3]: (i) momenta are intro-duced to all variables leading to the primary con-straints, (ii) the Hamiltonian is found, (iii) the timedevelopment of constraints is considered until (iv) theclosure of the Dirac procedure is reached, (v) allconstraints are classified as being first and secondclass, (vi) second-class constraints are eliminated(Hamiltonian reduction) [23], (vii) a gauge generator,according to the Dirac conjecture [1], is constructedfrom all the first class constraints using one of theavailable methods [24–26]2 , and (viii) this gaugegenerator is used to derive unambiguously the gaugetransformations of all fields. If some of these steps aremissing or implemented incorrectly, then we cannotbe sure that the correct gauge symmetry has beenfound.

The first attempts to interpret GR as a gaugetheory started from the work of Utiyama [28]3 . In hisapproach, in the same way as was done in Yang-Millstheory [30], by postulating the invariance of a systemunder a certain group of transformations it is possibleto introduce a new compensating field, determine theform of interaction, and construct the modified La-grangian which makes the action invariant under theabove transformations. Utiyama applied his method

1 In some papers the Lorentz symmetry plus diffeomorphismare even called the Poincare gauge symmetry (see, e.g., [14,15]).

2 We would like to note that the methods of [25, 26] should beapplied with great caution (see [27]).

3 Actually, before Utiyama’s attempt, Weyl introduced whatwe know now as the gauge invariance principle in his attemptto unify electricity and gravitation [29].

to the gravitational field using local Lorentz transfor-mations for vierbein fields. Later Kibble [31] extendedUtiyama’s prescription by considering the group ofinhomogeneous Lorentz transformations, Poincaregroup (though he switched from the translational pa-rameters of the Poincare symmetry to the parameterswhich “specify a general coordinate transformation”,e.g., diffeomorphism transformation). Localization ofthe Poincare symmetry leads to the Poincare gaugetheory of gravity (PGTG) (see, e.g., [32] and, for thePoincare-Weyl theory, [33]). All these approacheshave the same feature: the aim to construct a theoryfrom a given gauge symmetry rather than to derive agauge symmetry for a given Lagrangian. This is themain disadvantage of such methods since they cannotbe used for systems with unknown a priori gaugeinvariance.

In this paper we explore another approach. We donot relate our analysis to the Chern-Simons action(as was done by Witten in [12]), we do not performany change of variables (even canonical), and do notuse any formulation which is specific to a particulardimension (like Plebanski’s [34] for the dimensionD = 4). Our goal is to start from the first-orderaction of tetrad gravity, in which the tetrads andspin connections are treated as independent variables,follow all steps of the Dirac procedure, without anyassumption of what the gauge symmetry should be,and see what gauge transformations will be derived(or what “decision” the procedure will make). Thestructure of our paper follows steps (i)–(viii) of theprocedure outlined above.

In the next section we apply the Dirac procedureto the first-order formulation of tetrad GR in anydimensions (D > 2). The first steps are independentof the dimension until we reach the point where apeculiarity of three dimensions appears. From thisstage onwards, where we must consider eliminationof second-class constraints, we restrict our analysisto the three-dimensional case. In Section 3 we per-form Hamiltonian reduction by eliminating second-class primary constraints and the corresponding pairsof canonical variables, and then derive explicit ex-pressions for the secondary first-class constraints andthe Hamiltonian. The closure of the Dirac procedureand Poisson brackets (PB) among all constraints aregiven in Section 4 where it is demonstrated that thePB algebra of secondary first-class constraints coin-cides with the Lie algebra of the ISO(2,1) Poincaregroup. In Section 5, using the Dirac conjectureand the Castellani procedure, we derive the gaugegenerator from all first-class constraints and their PBalgebra which was found in the previous section. Bothgauge parameters presented in the generator turn outto have only internal (“Lorentz”) indices and describerotations and translations in the tangent space, not

GRAVITATION AND COSMOLOGY Vol. 16 No. 3 2010

THE HAMILTONIAN FORMULATION 183

a diffeomorphism. When this gauge generator actson fields, it gives gauge transformations which areequivalent to Witten’s result [12] obtained for D =3 without using the Dirac procedure. This uniquegauge symmetry which has been derived for tetradgravity in its first-order form is compared with an-other non-gauge symmetry, diffeomorphism, in Sec-tion 6. Our consideration is based on the originalvariables, tetrads and spin connections, without mak-ing even a canonical change of variables and withoutspecialization of either variables or the form of theoriginal action to any particular dimension. Thisallows us to use the three-dimensional case as a guidefor higher dimensions and to draw some conclusionsabout the Hamiltonian formulation of tetrad gravityin higher dimensions. This discussion is presented inSection 7. The preliminary results on the analysis ofthe tetrad gravity in higher dimensions are reported in[35–37].

2. THE HAMILTONIANAND THE CONSTRAINTS

To compare our results with previous incompleteattempts of the Hamiltonian formulation, we start ouranalysis from the Einstein-Cartan (EC) Lagrangianof tetrad gravity written in its first-order form (found,e.g., in [38,39])

L = −e(eμ(α)eν(β) − eν(α)eμ(β)

)

×(ων(αβ),μ + ωμ(αγ)ων

(γβ)

), (1)

where the covariant tetrads eγ(ρ) and the spin con-nections ων(αβ) are treated as independent fields, and

e = det(eγ(ρ)). We assume that the inverse eμ(α)

of the tetrad field eγ(ρ) exists and eμ(α)eμ(β) = δ(α)(β) ,

eμ(α)eν(α) = δμν . Indices in brackets (. . . ) denote the

internal (“Lorentz”) indices, whereas indices withoutbrackets are external or “world” indices. Internaland external indices are raised and lowered by theMinkowski tensor η(α)(β) = (−,+,+, . . . ) and the

metric tensor gμν = eμ(α)e(α)ν , respectively. For tetrad

gravity, the first-order form of (1) and second-orderformulations are equivalent in all dimensions, exceptD = 2. (On the Hamiltonian formulation of tetradgravity for D = 2 see [40].) Since we are interestedin obtaining a formulation valid in all dimensions, wewill not specialize our notation to a particular dimen-sion (as, e.g., [12, 34]), imposing only one restriction:D > 2.

To make the analysis more transparent, we rewritethe Lagrangian using integration by parts and intro-ducing a few short notations:

L = eBγ(ρ)μ(α)ν(β)eγ(ρ),μων(αβ)

− eAμ(α)ν(β)ωμ(αγ)ων(γ

β), (2)

where the coefficients Aμ(α)ν(β) and Bγ(ρ)μ(α)ν(β are

Aμ(α)ν(β) ≡ eμ(α)eν(β) − eμ(β)eν(α) (3)

and

Bγ(ρ)μ(α)ν(β) ≡ eγ(ρ)Aμ(α)ν(β)

+ eγ(α)Aμ(β)ν(ρ) + eγ(β)Aμ(ρ)ν(α). (4)

The symmetries of Aμ(α)ν(β) and Bγ(ρ)μ(α)ν(β) followfrom their definitions: e.g., Aμ(α)ν(β) = Aν(β)μ(α),Aμ(α)ν(β) = −Aν(α)μ(β), Aμ(α)ν(β) = −Aμ(β)ν(α). Si-

milar antisymmetry properties hold for Bγ(ρ)μ(α)ν(β) .In (4) the second and third terms can be obtained bya cyclic permutation of the internal indices ραβ →αβρ → βρα (keeping external indices in the samepositions). Bγ(ρ)μ(α)ν(β) can also be presented indifferent forms with cyclic permutations of externalindices (keeping internal indices in the same posi-tion):

Bγ(ρ)μ(α)ν(β) = eγ(ρ)Aμ(α)ν(β)

+ eμ(ρ)Aν(α)γ(β) + eν(ρ)Aγ(α)μ(β) . (5)

These forms, (4) and (5), are useful in calculations.As follows from their antisymmetry, Aμ(α)ν(β) andBγ(ρ)μ(α)ν(β) are equal zero when two external or twointernal indices have the same value. The propertiesof A, B and similar functions are collected in theAppendix.

As in any first-order formulation, the Hamiltoniananalysis of first-order tetrad gravity leads to primaryconstraints equal in number to the number of inde-pendent fields. Introducing momenta for all fields

πμ(α) =δL

δeμ(α),0, Πμ(αβ) =

δL

δωμ(αβ),0,

we obtain the following set of primary constraints:

πγ(ρ) − δ

δeγ(ρ),0

(eBγ(ρ)μ(α)ν(β)eγ(ρ),μων(αβ)

)

= πγ(ρ) − eBγ(ρ)0(α)ν(β)ων(αβ) ≈ 0, (6)

Πμ(αβ) ≈ 0. (7)

The fundamental Poisson brackets are{eμ(α)(x), πγ(ρ)(y)

}= δγ

μδ(ρ)(α)δ(x − y),

{ωλ(αβ)(x), Πρ(μν)(y)

}= Δ(μν)

(αβ)δρλδ(x − y), (8)

where

Δ(μν)(αβ) ≡

12

(δ(μ)(α)δ

(ν)(β) − δ

(ν)(α)δ

(μ)(β)

).


184 FROLOV et al.

(Note that in the text we often write a PB without thefactor δ(x − y)).

From the antisymmetry of Bγ(ρ)μ(α)ν(β) we imme-diately obtain for π0(ρ)

π0(ρ) ≈ 0, (9)

and for πk(ρ)

πk(ρ) − eBk(ρ)0(α)ν(β)ων(αβ) ≈ 0. (10)

Here and everywhere below in our paper we shallapply the usual convention: Greek letters for “space-time” (both internal and external) indices, e.g., α =0, 1, 2, . . . ,D − 1, β = 0, 1, 2, . . . ,D − 1, and Latinletters for “space” indices k = 1, 2, . . . ,D − 1, m =1, 2, . . . ,D − 1, etc.

All primary constraints are now identified, and theHamiltonian density takes the form

H = π0(ρ)e0(ρ)

+(πk(ρ) − eBk(ρ)0(α)ν(β)ων(αβ)

)ek(ρ)

+ Πμ(αβ)ωμ(αβ) − eBγ(ρ)k(α)ν(β)eγ(ρ),kων(αβ)

+ eAμ(α)ν(β)ωμ(αγ)ων(γ

β). (11)

There should be second-class ones among theseprimary constraints because the constraint (10) con-tains the connections ων(αβ), and the PBs of at least

some of them with Πμ(αβ) are nonzero (in particular,{πk(ρ),Πm(αβ)} = −eBk(ρ)0(α)m(β)). To clarify thisand to see what connections are present in (10), wefurther separate πk(ρ) into components (using theantisymmetry of Bγ(ρ)μ(α)ν(β)):

πk(m) − 2eBk(m)0(q)p(0)ωp(q0)

− eBk(m)0(p)n(q)ωn(pq) ≈ 0, (12)

πk(0) − eBk(0)0(p)m(q)ωm(pq) ≈ 0. (13)

This form shows the explicit appearance of particularconnections (ωm(pq) or ωp(q0)) in the primary con-straints. There are no connections with the “tempo-ral” external index in (12) and (13) and, accordingly,the primary constraints Π0(αβ) commute with therest of primary constraints, therefore, the constraintsΠ0(αβ) are first-class at this stage.

One group of constraints, which has the sameform in all dimensions D > 2,

πk(m) − 2eBk(m)0(q)p(0)ωp(q0)

− eBk(m)0(p)n(q)ωn(pq) ≈ 0, Πp(q0) ≈ 0, (14)

form a second-class subset, and using them, onepair of canonical variables (ωp(q0),Πp(q0)) can be now

eliminated. These constraints, (14), are not of aspecial form [3], but they are linear in ωp(q0) and

Πp(q0), and the coefficient in front of ωp(q0) in (14) does

not depend either on ωp(q0) or Πp(q0), so after theirelimination, the PBs among the remaining canonicalvariables will not change (i.e., they are the same asthe Dirac brackets).

To eliminate this pair, (ωp(q0),Πp(q0)), we must

solve (12) for ωp(q0) in terms of ωn(pq) and πk(m), and

substitute this solution, as well as Πp(q0) = 0, into thetotal Hamiltonian. The solution to Eq. (12) for ωp(q0)

exists in all dimensions D > 2. In fact, it becomesespecially simple if one notices that

Bk(m)0(q)p(0) = −e0(0)Ek(m)p(q), (15)

where

Ek(m)p(q) ≡ γk(m)γp(q) − γk(q)γp(m) (16)

and

γk(m) ≡ ek(m) − ek(0)e0(m)

e0(0), (17)

with the properties

γm(p)em(q) = δ(p)(q) , γn(q)em(q) = δn

m. (18)

The quantity Ek(m)p(q) is also antisymmetric (i.e.,Ek(m)p(q) = −Ep(m)k(q) = −Ek(q)p(m)) and equalszero if k = p or (m) = (q).

For any dimension (D > 2) we can define the in-verse of Ek(m)p(q),

Im(q)a(b) ≡1

D − 2em(q)ea(b) − em(b)ea(q). (19)

It is easy to check that

Im(q)a(b)Ea(b)n(p) = En(p)a(b)Ia(b)m(q) = δn

mδ(p)(q) .

(20)

Using the above notation, the solution of (12) canbe written in the form

ωk(q0) = − 12ee0(0)

Ik(q)m(p)πm(p)

+1

2e0(0)Ik(q)m(p)B

m(p)0(a)n(b)ωn(ab). (21)

Hence we see that the constraint (12) can besolved for ωp(q0) in any dimension D > 2 because ofthe existence of the inverse Ik(q)m(p) and because ofthe same number of equations and unknowns in (12),i.e. [πm(p)] = [ωk(q0)] = (D − 1)2 (where [X] indi-cates the number of components of a field X).



The second constraint, (13), cannot be solved un-ambiguously for ωk(pq) because the number of equa-

tions, [πk(0)] = D − 1, and the number of unknowns,[ωm(pq)] = 1

2(D − 1)2(D − 2), are, in general, differ-ent. This difference depends on the space-time di-mension:[ωm(pq)

]−

[πk(0)

]=

12(D − 1)2(D − 2) − (D − 1)

=12(D − 1)D(D − 3). (22)

In dimensions D > 3 we can choose only a subsetof these variables for elimination, which is not uniqueand, more importantly, this procedure will destroy thecovariant form of the constraints. It also creates diffi-culties in consistent elimination of these fields. Thecomponents of momenta (primary constraints) thatwould be left after this elimination would presumablylead to secondary constraints that could be elimi-nated at the next stage of the Hamiltonian reduction(solving this problem in different order or mixing aprimary second-class pair with pairs of primary andsecondary constraints is a difficult task). The detail ofthis analysis for D > 3 can be found in [35, 37].

If D = 3, the difference in (22) is zero. We have[πk(0)] = [ωm(pq)] = 2, or two equations in two un-knowns in (13). This drastically simplifies the calcu-lations. What is important, we have unambiguouslyone more pair of second-class primary constraints,and all connections with “spatial” external indicesand their conjugate momenta are eliminated at thisstage, leading immediately to the Hamiltonian andprimary constraints which have vanishing PBs. In thenext section we analyze this case (D = 3).

3. HAMILTONIAN ANALYSIS OF TETRADGRAVITY IN D = 3

From this point onwards, we specialize to thecase of D = 3. The same number of equations andunknowns in (12) and (13) allows us to eliminate allconnections with “spatial” external indices by solv-ing the primary second-class constraints. More-over, there are additional simplifications that occuronly for D = 3. First of all, in this dimension theconstraint (12) becomes simpler because the secondterm is zero (there are three “spatial” internal in-dices in Bγ(m)μ(p)ν(q), and if D = 3, at least two ofthem must be equal which, based on the propertiesof Bγ(ρ)μ(α)ν(β), gives Bk(m)0(p)n(q) = 0). This alsoleads to separation of the components of the spinconnections among the primary constrains. Eqs. (12)and (13) become

πk(m) − 2eBk(m)0(q)p(0)ωp(q0) ≈ 0, (23)

πk(0) − eBk(0)0(p)m(q)ωm(pq) ≈ 0. (24)

The Hamiltonian in this case is

H = π0(ρ)e0(ρ)

+(πk(0) − eBk(0)0(p)m(q)ωm(pq)

)ek(0)

+(πk(m) − 2eBk(m)0(q)p(0)ωp(q0)

)ek(m)

+ Πm(αβ)ωm(αβ) + Π0(αβ)ω0(αβ)

− eBγ(ρ)k(α)ν(β)eγ(ρ),kων(αβ)

+ eAμ(α)ν(β)ωμ(αγ)ων(γ

β). (25)

One group of constraints allows us to perform theHamiltonian reduction:

Πm(p0) = 0, (26)

ωk(q0) = − 12ee0(0)

Ik(q)m(p)πm(p). (27)

Similarly, for the second group of constraints we have

Πm(pq) = 0, (28)

and we need to solve (24) for ωm(pq). Using the sym-

metries of Bγ(ρ)μ(α)ν(β) and (15), we can rewrite (24)in the form

πk(0) + ee0(0)Ek(q)m(p)ωm(qp) = 0. (29)

For D = 3, there are only two independent com-ponents of ωm(pq): ω1(12) and ω2(12). Writing explic-itly (29) in components and using the antisymmetry ofEk(q)m(p), the solution of this equation can be foundand presented in a short, manifestly “covariant” form

ωm(pq) = − 12ee0(0)

Im(p)k(q)πk(0). (30)

Note that (30) is the result of peculiarities of thethree-dimensional case, contrary to (12) which isvalid in any dimension D > 2.

Substitution of (26), (27) and (28), (30) into (11)gives us the reduced Hamiltonian with a fewer num-ber of canonical variables and primary constraints:

H = π0(ρ)e0(ρ) + Π0(αβ)ω0(αβ)

− L(ωm(pq)=ωm(pq)(πk(0)), ωk(q0)=ωk(q0)(πm(p))), (31)

where we have explicitly separated the terms propor-tional to ω0(αβ) in the canonical Hamiltonian:

Hc = −L(“without velocities”)

= −2eBγ(ρ)k(p)m(0)eγ(ρ),kωm(p0)

− eBγ(ρ)k(p)m(q)eγ(ρ),kωm(pq)

+ eAn(p)m(q)ωn(p0)ωm(0

q)


186 FROLOV et al.

+ 2eAn(0)m(q)ωn(0r)ωm(r

q)

+(2eA0(α)m(γ)ωm

(βγ)

− eBγ(ρ)k(α)0(β)eγ(ρ),k

)ω0(αβ). (32)

Note that in (32) there are no terms quadraticin the connections with “spatial” external indices ifD = 3. It is a result of the antisymmetry of Aμ(α)ν(β)

and the spin connections ωγ(αβ) in α and β, and sincefor D = 3 a “spatial” index can take only two values,1 and 2.

The reduced total Hamiltonian is

HT = π0(ρ)e0(ρ) + Π0(αβ)ω0(αβ)

+ Hc

(eμ(α), π

m(α), ω0(αβ)

), (33)

where, after a few simple rearrangements, the canon-ical Hamiltonian for D = 3 becomes

Hc = e0(ρ),kπk(ρ) − e0(ρ)1

4ee0(0)Im(q)n(r)π

m(q)

×(η(ρ)(0)πn(r) − 2η(ρ)(r)πn(0)

)− 1

2

(πk(α)e

(β)k

− πk(β)e(α)k + 2eBγ(ρ)k(α)0(β)eγ(ρ),k

)ω0(αβ). (34)

To summarize, after the reduction, we have aHamiltonian with simple primary constraints π0(ρ)

and Π0(αβ), and all PBs among them are zero (asthey are just fundamental, canonical, variables ofthis formulation). With such a simple Hamiltonianand a trivial PB algebra of primary constraints,the secondary constraints follow immediately fromconservation of the primary constraints:

π0(ρ) ={

π0(ρ),HT

}= − δHc

δe0(ρ)≡ χ0(ρ), (35)

Π0(αβ) ={

Π0(αβ),HT

}

= − δHc

δω0(αβ)≡ χ0(αβ). (36)

An explicit expressions for χ0(ρ) in (35) is

χ0(ρ) = πk(ρ),k +

14ee0(0)

Im(q)n(r)πm(q)

×(η(ρ)(0)πn(r) − 2η(ρ)(r) pin(0)

)(37)

(note that, because of (30), the form of χ0(ρ) is alsospecific for D = 3 only).

The last constraint, (36), is obviously

χ0(αβ) =12πk(α)e

(β)k − 1

2πk(β)e

(α)k

+ eBn(ρ)k(α)0(β)en(ρ),k. (38)

We will call (37) and (38) the “translational” and“rotational” constraints, respectively, for reasons thatwill become clear at the end of the analysis.

4. CLOSURE OF THE DIRAC PROCEDURE

To prove that the Dirac procedure closes, we mustfind the time development of secondary constraintsand check whether they produce new constraints ornot. If tertiary constraints arise, we have to continuethe procedure until no new constraints appear. Ifthe PBs of the secondary constraints with the totalHamiltonian are zero or proportional to constraintsalready present, then the procedure stops [1]. Thetime development of the first-class constraints andthe PBs amongst them and with HT are sufficientto find the gauge transformations of all canonicalvariables [24].

We first compute the PBs amongst the con-straints. The primary constraints π0(ρ) and Π0(αβ)

have vanishing PBs amongst themselves:{π0(ρ),Π0(αβ)

}= 0. (39)

The rotational constraint has obviously a zero PBwith the primary constraint that generates it:{

Π0(μν), χ0(αβ)}

= 0. (40)

The PB of this constraint, χ0(αβ), with the secondprimary constraint is also zero:{

π0(ρ), χ0(αβ)}

= 0. (41)

With this result it is obvious that the only contributionto the secondary translational constraint comes fromvariation of that part of the Hamiltonian (34) whichis not proportional to the spin connections with a“temporal” external index. Since there are no con-tributions proportional to the connection ω0(μν) in thesecondary translational constraints (37), its PB withthe primary rotational constraint is zero:{

χ0(α),Π0(μν)}

= 0. (42)

The PB among the secondary and primary transla-tional constraints must be calculated. The result is{

π0(ρ), χ0(α)}

= 0. (43)

These vanishing PBs among all primary and sec-ondary constraints simplify the analysis. We can al-most immediately express the canonical Hamiltonianas a linear combination of secondary constraints plusa total spatial derivative:

Hc = −e0(ρ)χ0(ρ) − ω0(αβ)χ

0(αβ)



+(e0(ρ)π

k(ρ))

,k. (44)

Taking into account the PBs among primary andsecondary constraints, the constraints (35) and (36)follow from variation of Hc.

Calculation of PBs among secondary constraintsis straightforward though tedious, and the presenceof derivatives of en(ρ) requires the use of test functions(see, e.g., [41]). We obtain{

χ0(ρ), χ0(γ)}

= 0, (45)

{χ0(αβ), χ0(ρ)

}=

12η(β)(ρ)χ0(α)

− 12η(α)(ρ)χ0(β), (46)

{χ0(αβ), χ0(μν)

}=

12η(β)(μ)χ0(αν) − 1

2η(α)(μ)χ0(βν)

+12η(β)(ν)χ0(μα) − 1

2η(α)(ν)χ0(μβ). (47)

Note that in the calculations of (45) and (46)we also used the fact that the form of χ0(ρ), (30),is peculiar to D = 3. However, the PBs of χ0(αβ)

among themselves, (47), and with the primary con-straints, (40) and (41), are found without referenceto D = 3, and, as was shown in [35], these PBsremain valid in all dimensions (D > 2). In paperson group theory the brackets (46) and (47) usuallyappear without 1

2 . It is easy to remove this factor ifwe replace χ0(αβ) with 2χ0(αβ). However, we do notmake this replacement here because when derivinggauge transformations using the method of [24] itis important to find out what secondary constraintis produced exactly by the time development of thecorresponding primary constraint, (38).

It is very simple to calculate the time develop-ment of the secondary constraints χ0(ρ) and χ0(αβ),because Hc is proportional to these constraints (44),and we have only simple local PB (there are no deriva-tives of δ functions among them, and this allows us touse the associative properties of the PB). The result is

χ0(γ) ={

χ0(γ),Hc

}

=12ω0(αβ)

(η(β)(γ)χ0(α) − η(α)(γ)χ0(β)

), (48)

χ0(μν) ={

χ0(μν),Hc

}

= −12e0(ρ)

(η(ν)(ρ)χ0(μ) − η(μ)(ρ)χ0(ν)

)

− 12ω0(αβ)

(η(α)(μ)χ0(βν) − η(β)(μ)χ0(αν)

+ η(α)(ν)χ0(μβ) − η(β)(ν)χ0(μα))

. (49)

The above relations, (48) and (49), show that nonew constraints appear. This completes the proofthat the Dirac procedure is closed. All constraints(π0(ρ),Π0(αβ), χ0(ρ), χ0(αβ)) are first-class after elim-ination of ωk(pq) and ωp(q0), and, moreover, the PBsof the secondary constraints, (45)–(47), form anISO(2,1) Poincare algebra. It is a well defined Liealgebra: there are only structure constants, no non-local PBs, and it is closed “off-shell”. (In this respect,it is similar to the gauge invariance of the Yang-Millstheory.) The same result was obtained by Witten [12]though he did not use the Dirac procedure, insteadhe constructed the theory starting from the Poincarealgebra.

Now let us evaluate the degree of freedom (DOF)in the case of 3D tetrad gravity after eliminating ωk(pq)

and ωp(q0). Using the relation DOF = #(fields) −#(FC constraints), we obtain

DOF =[eμ(ρ)

]+

[ω0(αβ)

]−

([π0(ρ)

]

+[Π0(αβ)

]+

[χ0(ρ)

]+

[χ0(αβ)

])= 0, (50)

as expected for the D = 3 case.In the literature, there are some discussions

about the Hamiltonian formulation of pure three-dimensional tetrad gravity, but, in fact, a completeHamiltonian analysis has never been performed be-fore. In particular, in [17] the Poincare algebrais given but there is no explicit form of either theconstraints or the Hamiltonian. In [12], the analysiswas done by comparing three-dimensional tetradgravity with the Chern-Simons theory, the gaugetransformations were given but without derivation.A Hamiltonian analysis of the Chern-Simons actionwas done in [22]. We have not found any work(including reviews and books which are dedicated tothe D = 3 case, e.g. [20]) where the gauge trans-formations of three-dimensional tetrad gravity werederived from the first-class constraints according toone of the known procedures [24–26] (see footnote 2).Such a derivation is the subject of the next section.

5. GAUGE TRANSFORMATIONSFROM THE CASTELLANI PROCEDURE

We will derive the gauge transformations arisingfrom the first-class constraints by using the Castel-lani procedure. This procedure [24] is based on aderivation of gauge generators which are defined bychains of first-class constraints. One starts with pri-mary first-class constraint(s), i = 1, 2, . . . , and con-

structs the chain(s) ε(n)i Gi

(n), where ε(n)i are time


188 FROLOV et al.

derivatives of order (n) of the i-th gauge parameter(the maximum value of (n) is fixed by the length ofthe chain). In the case under consideration (three-dimensional tetrad gravity) with only primary andsecondary constraints n = 0, 1. The number of in-dependent gauge parameters is equal to the num-ber of first-class primary constraints. Note that thechains are unambiguously constructed once the pri-mary first-class constraints are determined.

For tetrad gravity, we have two chains of con-straints starting from the translational (π0(ρ)) and ro-tational (Π0(αβ)) primary first-class constraints. Ac-cording to the Castellani procedure, the generator isgiven by

G = G(ρ)(1) t(ρ) + G

(ρ)(0)t(ρ) + G

(αβ)(1) r(αβ)

+ G(αβ)(0) r(αβ). (51)

Here t(ρ) and r(αβ) are gauge parameters which, aswe will show later, parametrize the translational androtational gauge symmetries, respectively. Note thatthese gauge parameters have internal indices only. Itis clear even now that from the first-class constraintsof tetrad gravity it is impossible to derive a generatorof the diffeomorphism invariance for tetrads and spinconnections4 . The diffeomorphism gauge parameterξμ is of a very different nature. It is a “world” vectorbecause it has an external index, whereas t(ρ) andr(αβ) are “world” scalars. We will discuss the relationbetween the gauge symmetry of tetrad gravity anddiffeomorphism invariance in the next section.

The functions G(1) in (51) are the primary con-straints

G(ρ)(1) = π0(ρ), G

(αβ)(1) = Π0(αβ), (52)

and G(0) are defined using the following relations [24]:

G(ρ)(0)(x) = −

{π0(ρ)(x), HT

}

+∫ [

α(ρ)(γ)(x, y)π0(γ)(y)

+ α(ρ)(αβ)(x, y)Π0(αβ)(y)

]d2y, (53)

G(αβ)(0) (x) = −

{Π0(αβ)(x),HT

}

+∫ [

α(αβ)(γ) (x, y)π0(γ)(y)

+ α(αβ)(νμ) (x, y)Π0(νμ)(y)

]d2y, (54)

4 Diffeomorphism was also not derivable from the constraintstructure of the Chern-Simons action [22].

where the functions α(..)(..)(x, y) should be chosen in

such a way that the chains end at the primary con-straints {

Gσ(0),HT

}= primary. (55)

To construct the generator (51), we must find

α(..)(..)(x, y) using the condition (55). This calculation,

because of the simple PBs among the constraintswhen D = 3, is straightforward:

{G

(ρ)(0)(x),HT

}=

{− χ0(ρ)(x)

+∫ [

α(ρ)(γ)(x, y)π0(γ)(y)

+ α(ρ)(αβ)(x, y)Π0(αβ)(y)

]d2y, HT

}= 0, (56)

{G

(αβ)(0) (x), HT

}=

{− χ0(αβ)(x)

+∫ [

α(αβ)(γ) (x, y)π0(γ)(y)

+ α(αβ)(νμ) (x, y)Π0(νμ)(y)

]d2y, HT

}= 0, (57)

where HT can be replaced by Hc = −e0(σ)χ0(σ) −

ω0(σλ)χ0(σλ) because the PBs among the primary

constraints themselves and among primary and sec-ondary constraints are zero.

From (56) and (57) and the PBs among first-class

constraints we find all functions α(..)(..)(x, y) in (53),

(54),

α(ρ)(αβ)(x, y) = 0, (58)

α(ρ)(γ)(x, y) = ω

ρ)0(γδ(x − y), (59)

α(αβ)(γ) (x, y)

=12

(e(α)0 δ

(β)(γ) − e

(β)0 δ

(α)(γ)

)δ(x − y), (60)

α(αβ)(νμ) (x, y)

=(ω0

(αμ)δ(β)

(ν) − ω0(α

ν)δ(β)(μ)

)δ(x − y). (61)

This completes the derivation of the generator (51) asnow

G(ρ)(0) = −χ0(ρ) + ω

ρ)0(γπ0(γ) (62)

and

G(αβ)(0)

= −χ0(αβ) +12

(e(α)0 δ

(β)(γ)

− e(β)0 δ

(α)(γ)

)π0(γ)



+ ω0(α

μ)Π0(βμ) − ω0

(βμ)Π

0(αμ). (63)

Substitution of (52), (62), and (51) into (51) gives

G = π0(ρ)t(ρ) +(−χ0(ρ) + ω0

(ργ)π

0(γ))

t(ρ)

+ Π0(αβ)r(αβ) +[− χ0(αβ)

+12

(e(α)0 δ

(β)(γ) − e

(β)0 δ

(α)(γ)

)π0(γ)

+ ω0(α

μ)Π0(βμ) − ω0(β

μ)Π0(αμ)

]r(αβ). (64)

Now using

δ(field) = {G, field}, (65)

we can find the gauge transformations of the fields.

For example, for δω0(σλ) one finds

δω0(σλ) = −r(σλ) −(ω0

(αλ)δ(β)

(σ) − ω0(α

σ)δ(β)(λ)

)r(αβ).

(66)

This result is the same as Witten’s [12] for spin con-nections with the “ temporal” external index, ω0(σλ).Witten used a different notation which is specific to3D, while we will present the transformations of thefields in a covariant form. Note that δω0(σλ) dependsonly on the rotational parameter. In the gauge trans-formation of the tetrads, both parameters are present,

δe0(λ) = −t(λ) − ωρ)0(λt(ρ)

− 12

(e(α)0 δ

(β)(λ) − e

(β)0 δ

(α)(λ)

)r(αβ). (67)

Eq. (65) gives for δek(λ):

δek(λ) =δχ0(ρ)

δπk(λ)t(ρ) +

δχ0(αβ)

δπk(λ)r(αβ)

= −t(λ),k − ωρ)k(λt(ρ)

− 12

(e(α)k δ

(β)(λ) − e

(β)k δ

(α)(λ)

)r(αβ). (68)

(Here we have substituted πk(λ) in terms of ωk(αβ)

from (23) and (24)).

We can write together (67) and (68) as a singlecovariant equation:

δeγ(λ) = −t(λ),γ − ωρ)γ(λt(ρ)

− 12

(e(α)γ δ

(β)(λ) − e(β)

γ δ(α)(λ)

)r(αβ). (69)

This gauge transformation, (69), also confirms Wit-ten’s result [12], but in [12] it was not derived.

To obtain δωk(σλ), we first need to find δπγ(λ).

Eq. (65) gives for δπγ(λ):

δπ0(λ) =12

(η(α)(λ)π0(β)

− η(β)(λ)π0(α))

r(αβ), (70)

δπm(λ) =12

(η(α)(λ)πm(β) − η(β)(λ)πm(α)

)r(αβ)

+ eBm(λ)n(α)0(β)r(αβ),n. (71)

Note that both equations, (70) and (71), can be writ-ten in a single covariant form:

δπγ(λ) =12

(η(α)(λ)πγ(β) − η(β)(λ)πγ(α)

)r(αβ)

+ eBγ(λ)n(α)0(β)r(αβ),n. (72)

Using (27) and (30), together with the transfor-mation properties of eγ(λ) (69) and πγ(λ) (72), we canobtain the gauge transformation of δωk(σλ):

δωk(σλ) = −r(σλ),k

−(ωk

(αλ)δ(β)

(σ) − ωk(β

σ)δ(α)(λ)

)r(αβ). (73)

Now combining (66) and (73), we get the covariantequation for δωγ(σλ)

δωγ(σλ) = −r(σλ),γ

−(ωγ

(αλ)δ(β)

(σ) − ωγ(β

σ)δ(α)(λ)

)r(αβ). (74)

The gauge transformations of the field variables eγ(λ)

and ωγ(σλ) have been expressed in a covariant form.To summarize, our analysis has confirmed Wit-

ten’s result: for D = 3, we have obtained the samegauge transformations for eγ(λ) and ωγ(σλ) as in [12].From our analysis, which is based on the Diracprocedure, we have derived the gauge transforma-tions (69) and (74) generated by the first-classconstraints for tetrad gravity for D = 3. The PBsof the secondary first-class constraints form thePoincare algebra ISO(2,1). It is not surprising sinceequivalent formulations of the same theory shouldproduce the same result, as, e.g., Lagrangian andHamiltonian formulations. For D = 3, the canonicalHamiltonian (44) is a linear combination of the sec-ondary first-class constraints which we have calledtranslational and rotational, and this is consistentwith there being zero degrees of freedom (50). Wesee that the notorious “diffeomorphism constraint”(neither the full nor the “spatial” one) does not arisein the course of the Hamiltonian analysis of tetradgravity in D = 3. We would also like to mention thatBlagojevic [22], performing a Hamiltonian analysis ofthe Chern-Simons action and using the Castellani


190 FROLOV et al.

procedure to find the gauge invariance, stated that“the diffeomorphisms are not found” and concluded:“Thus, the diffeomorphisms are not an independentsymmetry [Italics by M.B.]”. In the next sectionwe will compare the gauge invariance found hereusing the Hamiltonian analysis with diffeomorphisminvariance.

The last step is to check the invariance of theLagrangian under the gauge transformations. Ac-tually, it is not necessary since the derivation of thegauge transformations has been performed in sucha way that the Lagrangian should be automaticallyinvariant, however, we will check this for consistency.It is not difficult to show that the transformationsunder rotation, the part in (69) proportional to r(αβ)

and (74), give δrL = 0 in all dimensions (D > 2).Note that in derivation of the rotational constraints wedid not use any peculiarity of the D = 3 case. We alsoconfirm in [36] that, using the Lagrangian methods,the same transformations under rotation arise in alldimensions (D > 2), and they leave the Lagrangianinvariant. As we have shown in [36], the transfor-mations under translation are different in dimensionsD > 3. It is not evident that the Lagrangian (1),which has the same form in all dimensions (D > 2),is invariant under translational transformations whichare specific to D = 3. From (69) and (74) we can seethat in the D = 3 case they are

δteγ(λ) = −t(λ),γ − ωρ)γ(λt(ρ), δtωγ(σλ) = 0. (75)

A proof that the EC Lagrangian (1) is invariantunder (75) is given in the Appendix.

6. GAUGE INVARIANCE VERSUSDIFFEOMORPHISM INVARIANCE

FOR TETRAD GRAVITY IN D = 3

We would like to mention again that we will call“gauge symmetry” the invariance that follows fromthe structure of the first-class constraints of theHamiltonian formulation of a theory. In particular, inthe Hamiltonian analysis of the EC action in D = 3we found that the gauge symmetry is translation androtation in the tangent space. But we know thatthe Lagrangian (1) is also invariant under diffeomor-phisms. Let us compare these invariances.

We will use the particular form of a diffeomorphismtransformation given by [42, 43]5

δgμν = −ξμ;ν − ξν;μ, (76)

5 In the mathematical literature, the term diffeomorphismrefers to a mapping from one manifold to another whichis differentiable, one-to-one, onto, and with a differentiableinverse.

or, in another equivalent form,

δgμν = −ξμ,ν − ξν,μ

+ gαβ (gμβ,ν + gνβ,μ − gμν,β) ξα, (77)

where ξμ is the diffeomorphism parameter (which isa “world” vector) and the semicolon denotes a co-variant derivative. In the literature on the Hamilto-nian formulation of metric General Relativity the word“diffeomorphism” is often used as an equivalent to thetransformation (76), which is similar to gauge trans-formations in ordinary field theories. It is precisely inthis sense that diffeomorphism invariance was derivedin the Hamiltonian analysis of the Einstein-Hilbertaction (metric gravity) for D > 2 for the second-order [7–9] and first-order [27] forms, without anyneed for a noncovariant and/or field-dependent redef-inition of the parameter ξμ.

A transformation similar to (77) can also be de-rived for the tetrad field eγ(λ). One way is to use therelation between the metric tensor gμν and the tetradseγ(λ)

gμν = eμ(λ)e(λ)ν . (78)

From (78) it follows that

δeν(λ) =12eμ(λ)δgμν . (79)

If we substitute (77) into (79) and use ξρ = gραξα, weobtain

δeν(λ) = −eρ(λ)ξρ,ν − eν(λ),ρξ

ρ. (80)

Another way of deriving the transformation (80)is to use the fact that eν(λ) is a “world” vector andtransforms under general coordinate transformationsas

e′ν(λ)(x′) =

∂x′ν

∂xγeγ(λ)

(x). (81)

For infinitesimal transformationsxμ → x′μ = xμ + ξμ(x) (82)

Eq. (81) can be written as

e′μ(λ)(x′) = e′μ(λ)(x) + ξμ

,γeγ(λ)(x) + O(ξ2). (83)

Combining the Taylor expansion of e′μ(λ)(x′)

e′μ(λ)(x′) = e′μ(λ)[x

γ + ξγ(x)]

= e′μ(λ)(x) + e′μ(λ),γξγ + O(ξ2) (84)

with (83) and replacing e′μ(λ),γ with eμ(λ),γ , the trans-

formation δeμ(λ)(x) reads

δeμ(λ)(x) = e′μ(λ)(x) − eμ

(λ)(x)

= eγ(λ)ξ

μ,γ − eμ

(λ),γξγ . (85)



Using δ(η(λ)(γ)eν(λ)e

μ(γ)

)= 0, it is easy to obtain the

transformation δeν(λ) given by (80).This perpetrated “gauge” transformation of eν(λ),

(80) can be found in many papers on tetrad gravity,e.g. [11, 20, 38] as well as in Witten’s paper [12].However, the only gauge transformation of eν(λ)

following from the Hamiltonian formulation is givenby (69) and is not a diffeomorphism.

As stated in [12], the gauge transformation (69)and the diffeomorphism (80) “are equivalent”. How-ever, this equivalence needs an imposition of severeadditional conditions: (i) a field-dependent redefini-tion of the gauge parameter ξβ = eβ(ρ)t(ρ); (ii) keep-ing only the translational invariance and disregardingthe rotational invariance; (iii) using the equations ofmotion (“on-shell” invariance). It is difficult to acceptsuch an “equivalence” and voluntarily replace thederived ISO(2,1) gauge symmetry of tetrad gravitywith diffeomorphism plus Lorentz invariance or, evenworse, with only a “spatial” diffeomorphism, as isoften presented in the literature. As we have alreadyshown, a gauge invariance of a theory can be foundexactly if one follows the Dirac procedure in whichone casts the theory into a Hamiltonian form, findsall constraints, the PBs among them, and classifiesthem as first-class or second-class, derives the gaugegenerator from the first-class constraints, and finallyuses this gauge generator to find gauge transforma-tions of variables in the theory. Using this procedure,we have derived the gauge transformations of tetradseγ(λ) (69) and spin connections ωρ(αβ) (74). More-over, the algebra of secondary first-class constraintsgives unambiguously the Poincare algebra ISO(2,1),(45)–(47), not an algebra of diffeomorphisms andLorentz rotations. We have to conclude that thegauge invariance of the tetrad gravity in three di-mensions is a Poincare symmetry. The diffeomor-phism (80) is a symmetry of the Einstein-Cartan ac-tion, but it is not a gauge symmetry derived from thefirst-class constraints in the Hamiltonian formulationof tetrad gravity [36].

It is not surprising that metric and tetrad grav-ity theories have different gauge symmetries sincethey are not equivalent. Einstein in his article ontetrad (n-bein) gravity wrote [44]: “The n-bein fieldis determined by n2 functions hμ

a [tetrads eμ(α), in our

notation], whereas the Riemannian metric is deter-mined by n(n + 1)/2 quantities. According to (3)[gμν = hμah

aν ], the metric is determined by the n-bein

field but not vice versa”. So, the attempt to deducea gauge transformation of eγ(λ) from the diffeomor-phism invariance of gμν is a wrong way. But it shouldbe possible to deduce a transformation of gμν fromthat of eγ(λ) (the “vice versa” of Einstein).

To compare the results of (69) and (80), with-out forcing an equivalence by imposing the restric-tions (i)–(iii) mentioned after (85), we rewrite (69) ina slightly different form. First of all, from the equationof motion δL/δωμ(αβ) = 0 it follows

Bε(α)μ(λ)σ(ρ)eε(α),μ − Aσ(λ)ν(β)ω(ρνβ)

+ Aσ(ρ)ν(β)ω(λνβ) = 0. (86)

Solving (86) for ω(ρνβ), we can express it in terms of

eν(λ) and its derivatives:

ω(αβ)σ =

12eσ(λ)eε(ρ),μ

(η(ρ)(β)Aε(α)μ(λ)

+ η(ρ)(α)Aε(λ)μ(β) − η(ρ)(λ)Aε(β)μ(α))

, (87)

or in a more familiar form,

ω(αβ)σ =

12

[eε(α)

(e(β)ε,σ − e(β)

σ,ε

)

− eε(β)(e(α)ε,σ − e(α)

σ,ε

)

−e(ρ)σ eε(β)eμ(α)

(eε(ρ),μ − eμ(ρ),ε

)]. (88)

Substitution of (87) into (69) gives

δeγ(λ) = −t(λ),γ − 12

(eγ(λ),ε − eε(λ),γ

)eε(ρ)t(ρ)

− 12e(α)γ eε

(λ)

(eε(α),μ − eμ(α),ε

)eμ(ρ)t(ρ)

− 12

(eε(μ),γ − eγ(μ),ε

)eε(λ)t

(μ)

− 12

(e(α)γ δ

(β)(λ)

− e(β)γ δ

(α)(λ)

)r(αβ). (89)

From (78) we can find that

δgμν = e(λ)μ δeν(λ) + e(λ)

ν δeμ(λ). (90)

Now from δeμ(λ), (89), and from (90) we can obtainδgμν :

δgμν = −(e(ρ)μ t(ρ)

),ν−

(e(ρ)ν t(ρ)

),μ

+ (gμβ,ν + gνβ,μ − gμν,β) eβ(ρ)t(ρ), (91)

which after the redefinition ξβ = eβ(ρ)t(ρ) leadsto (77). Note that the contributions with the rota-tional parameter r(αβ) from (89) completely cancelout in (91) as well as some terms proportional to t(ρ),without imposing any conditions. We do not need theadditional restrictions (ii)–(iii); only a field-dependentredefinition of the parameters (i) is needed. Thus it ispossible to obtain the diffeomorphism invariance ofgμν from the Poincare symmetry of eγ(λ), but not viceversa. A field-dependent redefinition of the gauge


192 FROLOV et al.

parameter is necessary, but it is a consequence ofthe non-equivalence of metric and tetrad gravities.A similar redefinition had to be introduced when weconsidered two-dimensional metric and tetrad gravi-ties in [40], despite the fact that the gauge symmetryof two-dimensional gravity (both metric and tetrad)is very different from that in higher dimensions.

7. CONCLUSION

In his book [1], Dirac wrote “I feel that therewill always be something missing from them[non-Hamiltonian methods] which we can onlyget by working from a Hamiltonian”. We feelthat the Hamiltonian method not only allows oneto find something that can be missed when usingother methods but also protects us from “finding”something that might be attributed to a theory butreally not there. In particular, the gauge invariance ofa theory should follow from the Dirac procedure, andany guess (even an intelligent one) must be supportedby calculations.

The results reported in this paper confirm Dirac’sold conjecture [1]. The Hamiltonian formulation oftetrad gravity considered here using the Dirac pro-cedure leads to self-consistent and unambiguous re-sults. In particular, this approach gives a unique an-swer to the question of what is the true gauge invari-ance of tetrad gravity and eliminates any possibilityof being able to “choose” a gauge invariance basedon either a belief, desire or common wisdom, becausethe gauge invariance should be derivable from theunique constraint structure of this (or any) theory.This constraint structure can be modified only by anon-canonical change of variables that immediatelydestroys any connection with an original theory and,at best, can be considered as some model not relatedto the tetrad gravity (see the discussion on a non-canonical change of variables for tetrad gravity inSection 5 of [27]).

After Hamiltonian reduction, solving the second-class constraints and eliminating nonphysical (re-dundant) variables, the remaining first-class con-straints form a Lie algebra. This is exactly what onecan expect for a local field theory, and it is preciselywhat one needs to quantize it. The results presentedin this paper were mainly obtained for the D = 3 casewhich has been treated completely. The Hamiltoniananalysis of the EC action in higher dimensions is inprogress [35, 37], and further developments will bereported elsewhere.

However, even the results of this paper provideenough information to form some conclusions about

the Hamiltonian formulation of tetrad gravity inhigher dimensions, based on the following reasoning:

—the Lagrangian of the first-order formulation oftetrad gravity (2) with tetrads and spin connectionstreated as independent fields gives equivalent equa-tions of motion to those in its second-order counter-part, and this can be demonstrated in any dimensionsD > 2 in a covariant form and without any recourseto a particular dimension, and does not show anypeculiarities in D = 3;

—the primary first-class constraints are a part ofthe generator and unambiguously define the gaugeparameters in (51); in any dimension, because of theantisymmetry of Bγ(ρ)μ(α)ν(β), there is a translationalconstraint π0(ρ), and the corresponding gauge pa-rameter cannot have a “world” index that we wouldneed if diffeomorphism were a gauge invariance deriv-able from the first class constraints;

—the explicit form of the rotational constraintχ0(αβ) (38) is not peculiar to D = 3 and is written ina covariant form, that is why it remains unchanged indimensions D > 3 [35, 37];

—the PBs among the rotational constraints (47)were also calculated without using any specific prop-erty of D = 3;

—the final form of the gauge transformations oftetrad gravity for D = 3 can be cast into a covariantdimension-independent form (69), (74);

—the secondary translational constraint χ0(ρ) hasa form specific to D = 3 (the second term in (37))but it does not mean that it will be modified in higherdimensions in such a way that it will destroy thetranslational invariance in the internal space.

Based on the above arguments, we can con-clude that diffeomorphism invariance is not a gaugesymmetry derived from the first-class constraints oftetrad gravity, neither in D = 3 nor in higher dimen-sions, and translational and rotational invariance areexpected in all dimensions (D > 2).

These conclusions for D > 3 can be called con-jectures. The only way to prove or disprove them isto apply the Dirac procedure and explicitly find theHamiltonian, eliminate all second-class constraintsand use all the remaining first-class constraints tobuild the generators that will produce the true gaugeinvariance of tetrad gravity in higher dimensions.6

Some preliminary results are reported in [35–37].

6 Of course, a non-canonical change of variables should be ex-cluded at any step of the calculations, and any manipulationthat destroys the equivalence with the original theory are notpermissible.



APPENDIX

ABC PROPERTIES AND TRANSLATIONALINVARIANCE OF THE EC LAGRANGIAN

Here we collect the properties of the ABC func-tions that are useful in the Hamiltonian analysis [35,37] as well as in the Lagrangian approach [36] to theEinstein-Cartan action.

These functions are generated by consecutivevariation of the n-bein density eeμ(α):

δ

δeν(β)

(eeμ(α)

)= e

(eμ(α)eν(β) − eμ(β)eν(α)

)

= eAμ(α)ν(β) , (A1)

δ

δeλ(γ)

(eAμ(α)ν(β)

)= eBλ(γ)μ(α)ν(β) , (A2)

δ

δeτ(σ)

(eBλ(γ)μ(α)ν(β)

)

= eCτ(σ)λ(γ)μ(α)ν(β) . . . . (A3)

The first important property of these density func-tions is their total antisymmetry: interchange of twoindices of the same nature (internal or external), e.g.,

Aν(β)μ(α) = −Aν(α)μ(β) = −Aμ(β)ν(α), (A4)

the same being valid for B, C, etc. In calculationspresented here, nothing is needed beyond C.

The second important property is their expansionusing an external index:

Bτ(ρ)μ(α)ν(β) = eτ(ρ)Aμ(α)ν(β)

+ eτ(α)Aμ(β)ν(ρ) + eτ(β)Aμ(ρ)ν(α), (A5)

Cτ(ρ)λ(σ)μ(α)ν(β)

= eτ(ρ)Bλ(σ)μ(α)ν(β) − eτ(σ)Bλ(α)μ(β)ν(ρ)

+ eτ(α)Bλ(β)μ(ρ)ν(σ) − eτ(β)Bλ(ρ)μ(σ)ν(α) , (A6)

or with an internal index:

Bτ(ρ)μ(α)ν(β) = eτ(ρ)Aμ(α)ν(β)

+ eμ(ρ)Aν(α)τ(β) + eν(ρ)Aτ(α)μ(β), (A7)

Cτ(ρ)λ(σ)μ(α)ν(β)

= eτ(ρ)Bλ(σ)μ(α)ν(β) − eλ(ρ)Bμ(σ)ν(α)τ(β)

+ eμ(ρ)Bν(σ)τ(α)λ(β) − eν(ρ)Bτ(σ)λ(α)μ(β) . (A8)

The third property involves their derivatives:(eAν(β)μ(α)

),σ =

δ

δeλ(γ)

(eAν(β)μ(α)

)eλ(γ),σ

= eBλ(γ)ν(β)μ(α)eλ(γ),σ , (A9)

(eBτ(ρ)ν(β)μ(α)

),σ

=δ

δeλ(γ)

(eBτ(ρ)ν(β)μ(α)

)eλ(γ),σ

= eCτ(ρ)λ(γ)ν(β)μ(α)eτ(ρ),σ . (A10)

The antisymmetry of B, both in the external andinternal indices, and the antisymmetry of ω in itsinternal indices lead to

Bτ(ρ)μ(α)ν(β)ωμ(αγ)ων(γ

σ)ω(στβ) = 0, (A11)

and

eBτ(ρ)μ(α)ν(β)ων(αβ),μτ = 0. (A12)

The above properties considerably simplify the cal-culations. The list of ABC properties can be ex-tended, but for our purpose the above relations areadequate.

Let us check the invariance of the Einstein-CartanLagrangian (1) under the translational transforma-tion in D = 3 (75):

δtL = δt

[−eAμ(α)ν(β)

(ων(αβ),μ + ωμ(ασ)ων

(σβ)

)]

= −δt

(eAμ(α)ν(β)

)(ων(αβ),μ + ωμ(ασ)ων

(σβ)

)

= −eBγ(λ)μ(α)ν(β)δteγ(λ)


(σβ)

)

= −eBγ(λ)μ(α)ν(β)(−t(λ),γ − ω

ρ)γ(λt(ρ)

)

×(ων(αβ),μ + ωμ(ασ)ων

(σβ)

).

After extracting a total derivative, it leads to the fol-lowing:

δtL

= −[eBγ(λ)μ(α)ν(β)t(λ)


(σβ)

)],γ

+(eBγ(λ)μ(α)ν(β)

),γ

t(λ)


(σβ)

)

+ eBγ(λ)μ(α)ν(β)t(λ)ων(αβ),μγ + eBγ(λ)μ(α)ν(β)t(λ)

×(ωμ(ασ),γων

(σβ) + ωμ(ασ)ω

(σνβ),γ

)

− eBγ(λ)μ(α)ν(β)t(ρ)ωρ)γ(λων(αβ),μ

− eBγ(λ)μ(α)ν(β)t(ρ)ωρ)γ(λωμ(ασ)ων

(σβ). (A13)

The first term in (A13) is a total derivative, thesecond term is zero in D = 3. It is because the deriva-tive

(eBγ(λ)μ(α)ν(β)

),γ

is proportional to C which isantisymmetric in its four internal and four externalindices, but in D = 3 there are only three distinctindices: 0, 1, 2. The third term in (A13) is also zerobecause of (A12). It is not evident that the fourth andfifth terms cancel. From the antisymmetry of B it fol-lows that λ �= α �= β in these terms; in addition, from


194 FROLOV et al.

the antisymmetry of ω in internal indices and from thefact that in D = 3 there are only three distinct indiceswe can conclude that in the fifth term ρ should beequal to either α or β (note that it is a peculiarity ofD = 3 only). If we consider both cases, ρ = α andρ = β, and relabel the dummy indices in the fifth term,we find that the fifth and fourth terms cancel. Thelast, sixth, term is not exactly in the form of (A11).However, if we use again the antisymmetry of B andω and the peculiarity of D = 3 (only three distinctindices), then we have also only two cases: eitherρ = α or ρ = β. In both cases we can apply (A11)for the sixth term. Finally, under translational trans-formations of (75) in D = 3 we have

δtL = −[eBγ(λ)μ(α)ν(β)t(λ)

×(ων(αβ),μ + ωμ(ασ)ων

(σβ)

)],γ

,

or, using the definition of B (A5) and the Ricci ten-sor Rμν(αβ) = ων(αβ),μ − ωμ(αβ),ν + ωμ(ασ)ων

(σβ) −

ων(ασ)ωμ(σ

β), we have

δtL = −[e(eγ(λ)eμ(α)eν(β) + eγ(α)eμ(β)eν(λ)

+eγ(β)eμ(λ)eν(α))

t(λ)Rμν(αβ)

],γ

.

ACKNOWLEDGMENTS

We would like to thank D.G.C. McKeon for helpfuldiscussions during the preparation of the paper andreading the manuscript.

REFERENCES1. P. A. M. Dirac, Lectures on Quantum Mechanics

(Belfer Graduate School of Sciences, Yeshiva Univer-sity, New York, 1964).

2. K. Sundermeyer, Constrained Dynamics, LectureNotes in Physics (Springer, Berlin, 1982), vol. 169.

3. D. M. Gitman and I. V. Tyutin, Quantization ofFields with Constraints (Springer, Berlin, 1990).

4. R. Gambini and J. Pullin, Loops, Knots, Gauge The-ories and Quantum gravity (Cambridge UniversityPress, Cambridge, 1996).

5. C. Rovelli, Quantum Gravity (Cambridge UniversityPress, Cambridge, 2004).

6. T. Thiemann, Modern canonical quantum generalrelativity (Cambridge University Press, Cambridge,2007).

7. N. Kiriushcheva, S. V. Kuzmin, C. Racknor, andS. R. Valluri, Phys. Lett. A 372, 5101 (2008).

8. N. Kiriushcheva and S. V. Kuzmin, arXiv:0809.0097.9. A. M. Frolov, N. Kiriushcheva, and S. V. Kuzmin,

arXiv:0809.1198.10. C. J. Isham and K. V. Kuchar, Ann. Phys. 164, 316

(1985).11. S. Deser and C. Isham, Phys. Rev. D 14, 2505 (1976).

12. E. Witten, Nucl. Phys. B 311, 46 (1988).13. S. Carlip, Phys. Rev. D 42, 2647 (1990).14. D. Grensing and G. Grensing, Phys. Pev. D 28, 286

(1983).15. S. A. Ali, C. Cafaro, S. Capozziello, and Ch. Corda,

Int. J. Theor. Phys. 48, 3426 (2009).16. J. M. Charap, M. Henneaux, and J. E. Nelson, Class.

Quantum Grav. 5, 1405 (1988).17. H.-J. Matschull, Class. Quant. Grav. 16, 2599

(1999).18. H.-J. Matschull, Class. Quant. Grav. 12, 2621

(1995).19. S. Carlip, Living Rev. Relativity 8, 1 (2005);

http://www.livingreviews.org/lrr-2005-1.20. S. Carlip, Quantum Gravity in 2 + 1 Dimensions

(Cambridge University Press, Cambridge, 1998).21. M. Blagojevic and B. Cvetkovic, in “Trends in Gen-

eral Relativity and Quantum Cosmology”, vol. 2 (ed. C. V. Benton, Nova Science Publisher, New York,2006), p. 85; gr-qc/0412134.

22. M. Blagojevic, Gravitation and Gauge Symmetries(Institute of Physics Publishing, Bristol and Philadel-phia, 2002).

23. N. Kiriushcheva and S. V. Kuzmin, Ann. Phys. 321,958 (2006).

24. L. Castellani, Ann. Phys. 143, 357 (1982).25. M. Henneaux, C. Teitelboim, and J. Zanelli, Nucl.

Phys. B 332, 169 (1990).26. R. Banerjee, H. J. Rothe, and K. D. Rothe, Phys. Lett.

B 479, 429 (2000).27. N. Kiriushcheva and S. V. Kuzmin, arXiv:0912.3396.28. R. Utiyama, Phys. Rev. 101, 1597 (1956).29. H. Weyl, Physics 15, 323 (1929).30. C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954).31. T. W. B. Kibble, Journal of Math. Phys. 2, 212 (1961).32. B. N. Frolov, Grav. Cosmol. 10, 116 (2004).33. O. V. Babourova, B. N. Frolov, and V. Ch. Zhukovsky,

Phys. Rev. D 74, 064012 (2006).34. J. F. Plebanski, J. Math. Phys. 18, 2511 (1977).35. N. Kiriushcheva and S. V. Kuzmin, arXiv:0907.1553.36. N. Kiriushcheva and S. V. Kuzmin, arXiv:0907.1999

(to appear in Gen. Rel. Grav.).37. N. Kiriushcheva and S. V. Kuzmin, arXiv:0912.5490.38. J. Schwinger, Phys. Rev. 130, 1253 (1963).39. L. Castellani, P. van Nieuwenhuizen, and M. Pilati,

Phys. Rev. D 26, 352367 (1982).40. R. N. Ghalati, N. Kiriushcheva, and S. V. Kuzmin,

Mod. Phys. Lett. A 22, 17 (2007).41. S. V. Kuzmin and D. G. C. McKeon, Ann. Phys. 138,

495 (2005).42. L. D. Landau and E. M. Lifshitz, The Classical The-

ory of Fields (fourth ed., Pergamon Press, Oxford,1975).

43. M. Carmeli, Classical Fields: General Relativityand Gauge Theory (World Scientific, New Jersey,2001).

44. A. Einstein, Sitzungsber. preuss. Akad. Wiss.,Phys.-Math. K1, 217 (1928), and in TheComplete Collection of Scientific Papers(Nauka, Moscow, 1966), Vol. 2, p. 223; Englishtranslation is available from: http:/www.lrz-muenchen.de/‘aunzicker/ae1930.htm, andA. Unzicker, T. Case, arXiv:physics/0503046.


The Hamiltonian formulation of tetrad gravity: Three-dimensional case

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