Topological parameters in gravity

Topological parameters in gravity Romesh K. Kaul1, ∗ and Sandipan Sengupta1, † 1

The Institute of Mathematical Sciences

CIT Campus, Chennai-600 113, INDIA.

arXiv:1106.3027v2 [gr-qc] 20 Jun 2011

Abstract We present the Hamiltonian analysis of the theory of gravity based on a Lagrangian density containing Hilbert-Palatini term along with three topological densities, Nieh-Yan, Pontryagin and Euler. The addition of these topological terms modifies the symplectic structure non-trivially. The resulting canonical theory develops a dependence on three parameters which are coefficients of these terms. In the time gauge, we obtain a real SU (2) gauge theoretic description with a set of seven first class constraints corresponding to three SU (2) rotations, three spatial diffeomorphism and one to evolution in a timelike direction. Inverse of the coefficient of Nieh-Yan term, identified as Barbero-Immirzi parameter, acts as the coupling constant of the gauge theory. PACS numbers: 04.20.Fy, 04.60.-m, 04.60.Ds, 04.60.Pp

∗

Electronic address: [email protected]

†

Electronic address: [email protected]

1

I.

INTRODUCTION

Addition of total divergence terms to the Lagrangian density does not change the classical dynamics described by it; the Euler-Lagrange equations of motion are unaltered. In the Hamiltonian formulation, these total divergences reflect themselves as canonical transformations, resulting in the change of the phase space. This changes the symplectic structure and Hamiltonian of the system, yet the Hamilton’s equations of motion remain equivalent to the Euler-Lagrange equations of the Lagrangian formulation. While the classical dynamics is not sensitive to the total divergence terms in the Lagrangian density, the quantum theory may depend on these. The canonical transformation of classical Hamiltonian formulation are implemented in the quantum theory through unitary operators on the phase space and the states. However, there are special situations where we find topological obstructions in such a unitary implementation. In such cases these total divergences do affect the quantum dynamics. Therefore, to have non-trivial implications in the quantum theory, the total divergence terms have to be topological densities. This is a necessary requirement, but not sufficient. There are several known examples of topological terms which have serious import in the quantum theory. A well-known case is the Sine-Gordon quantum mechanical model [1] where an appropriate effective topological term can be added to the Lagrangian density to reflect the non-perturbative properties of the quantum theory. In this model, we have a periodic potential with infinitely many degenerate classical ground states. With each of these, we associate a perturbative vacuum state labeled by an integer n related to the winding number of homotopy maps S 1 → S 1 characterized by the homotopy group Π1 (S 1 ) which is the set of integers Z. The physical quantum vacuum state, so called θ-vacuum, is non-perturbative in nature and is given by a linear superposition of these perturbative vacua with weights given by phases exp(inθ) where angular variable θ, properly normalized, is the coefficient of the effective topological density term in the Lagrangian density. The physical quantities in the quantum theory depend on this parameter. For example, the quantum vacuum energy, besides the usual zero-point energy, has a contribution due to quantum tunneling processes between various perturbative vacua, which depends on θ. In field theory, we have an example of such a topological parameter θ in the theory of strong interactions, namely QCD. Here also, we have infinitely many degenerate classical 2

ground states labeled by integers n associated with the winding numbers of homotopy maps S 3 → S 3 characterized by the homotopy group Π3 (SU(3)) ≡ Z. The quantum vacuum (θvacuum) is a linear superposition of the perturbative vacua associated with these classical ground states. Associated effective topological term in the Lagrangian density is Pontryagin density of SU(3) gauge theory with coefficient θ. This leads to θ dependent CP violating contributions to various physical quantities. However, there are stringent phenomenological constraints on the value of θ. For example, from possible CP-violating contribution to the electric-dipole moment of the neutron, this parameter is constrained by experimental results to be less than 10−10 radians. In gravity theory in 3+1 dimensions, there are three possible topological terms that can be added to the Lagrangian density. Two of these, the Nieh-Yan and Pontryagin densities, are P and T odd, and the third, Euler density, is P and T even. Associated with these are three topological parameters. In order to understand their possible import in the quantum theory, it is important to set up a classical Hamiltonian formulation of the theory containing all these terms in the action. In ref.[2], such an analysis has been presented for a theory based on Lagrangian density containing the standard Hilbert-Palatini term and the NiehYan density [3]. The resulting theory, in time gauge, has been shown to correspond to the well-known canonical gauge theoretic formulation of gravity based on Sen-AshtekarBarbero-Immirzi real SU(2) gauge fields [4]. Here inverse of the coefficient of Nieh-Yan term is identified with the Barbero-Immirzi parameter γ. Thus the analysis of ref.[2] has provided a clear topological interpretation for γ, realizing a suggestion made earlier in [5] that this parameter should have a topological origin. The framework of [2] involving Nieh-Yan density supersedes the earlier formulation of Holst [6]. Detailed Hamiltonian analysis of the theory with Holst term for pure gravity is provided in ref.[7] and that including spin 1/2 fermions in ref.[8]. This discussion has also been extended to supergravity theories [9]. Since Holst term is not topological, inclusion of matter necessitates matter dependent modification of the Holst term so that original equations of motion stay unaltered. On the other hand, the analysis containing Nieh-Yan density [2], besides explaining the topological origin of the Barbero-Immirzi parameter, provides a universal prescription for inclusion of arbitrary matter without any need for further modifications of the topological Nieh-Yan term which is given in terms of the geometric quantities only. As elucidations of these facts, this analysis has been extended to the theory including 3

Dirac fermions in ref.[2] and to supergravity theories in ref.[10]. In a quantum framework, the implications of a topological term in the Lagrangian can also be understood through a rescaling of the wave functional by a topologically non-trivial phase factor. This procedure has been used for QCD [11] where, as mentioned above, the properties of the non-perturbative θ-vacuum are effectively represented by a SU(3) Pontryagin density term in the Lagrangian. The rescaling of wave functional is provided by the exponential of SU(3) Chern-Simons three-form with iθ as its coefficient. This framework can be extended to the gravity theory where we have a corresponding wave functional scaling associated with the Nieh-Yan density. However, for the pure gravity (without any matter couplings), the standard Dirac quantization, where the second class constraints are implemented before quantization, is not appropriate. This is so because second class constraints of pure gravity imply vanishing of the torsion, which results in making the rescaling trivial. Instead, as discussed in [12], the Gupta-Bleuler and coherent state quantization methods are well suited for the purpose. These methods are quite general and can be used for gravity theory with or without matter. However, for matter-couplings leading to non-vanishing torsion, e.g. Dirac fermions, the Dirac quantization, as has been discussed earlier in ref.[13], can also be adopted for this purpose. Hamiltonian analysis of the first order (anti-) self-dual Lagrangian density for gravity including the Pontryagin density of complex SU(2) (anti-) self-dual gauge fields has first been reported by Montesinos in [14]. In the time gauge, the Sen-Ashtekar complex SU(2) connection stays unchanged, but its conjugate momentum field gets modified by the presence of the Pontryagin term. Recently, in [15], this analysis has also been done for gravity theories containing Holst, Nieh-Yan, Euler and Pontryagin terms. This study concludes that, in the time gauge, real SU(2) gauge theoretical formulation is possible only if the Pontryagin and Euler terms are absent; the Pontryagin density can be added consistently only in the complex SU(2) gauge formulation leading to a canonical analysis in accordance with results of Montesinos [14]. In the following, we present a classical Hamiltonian analysis for theory of gravity based on Hilbert-Palatini Lagrangian supplemented with all the three possible topological terms in (1 + 3) dimensions, namely, Nieh-Yan, Pontryagin and Euler classes. Unlike [15], in view of results of [2] and the remarks already made above, we shall not add the Holst term, which is not a topological density. We demonstrate that, in the time gauge, we do have 4

a real SU(2) gauge theory with its coupling given by inverse of the coefficient of Nieh-Yan term. The canonical theory also depends on two additional arbitrary parameters, the coefficients of Pontryagin and Euler terms in the Lagrangian density. These parameters are not subjected to any restrictions. A formulation of the theory presented involves the standard Sen-Ashtekar-Barbero-Immirzi real SU(2) connections Aia , which depend only on the coefficient of the Nieh-Yan term, as the canonical fields. Associated conjugate momentum fields, instead of being densitized triads of the standard canonical theory, are modified and depend on the coefficients of the Nieh-Yan, Pontryagin and Euler terms. There are second class constraints in the description, essentially reflecting the fact that the extrinsic curvature is not independent. Correspondingly, for this constrained Hamiltonian system, the Dirac brackets analysis is developed. Dirac brackets of the phase variables do not exhibit the same algebraic structure as those of the standard canonical theory of gauge fields Aia and densitized triads Eia ; the new variables are not related to them by a canonical transformation. However, it is possible to construct another set of phase variables which are canonical transforms of the standard variables (Aia , Eia ). In this framework, both new gauge fields and their conjugate momentum fields are modified and develop dependences on all three topological parameters. The canonical formulation described in terms these new phase variables is presented in detail.

II.

TOPOLOGICAL COUPLING CONSTANTS IN GRAVITY

We set up the standard theory of pure (i.e., no matter couplings) gravity in terms of the 24 SO(1, 3) gauge connections ωµIJ and 16 tetrad fields eIµ as the independent fields described by Hilbert-Palatini (HP) Lagrangian density: LHP =

1 IJ e Σµν (ω) IJ Rµν 2

(1)

where e ≡ det(eIµ ) ,

Σµν IJ ≡

Rµν IJ (ω) ≡ ∂[µ ων]IJ

1 µ ν 1 µ ν e[I eJ] ≡ (eI eJ − eµJ eνI ) , 2 2 IK J + ω[µ ων]K

and eµI is the inverse of the tetrad field, eµI eIν = δ µν , eIµ eµJ = δ IJ .

5

(2)

Modifications of the gravity Lagrangian density by terms which are quadratic in curvature and particularly also include torsion, without altering the field equations, have a long history, see for example [16]. In (1 + 3) dimensions, there are three possible topological terms that can be added to the HP Lagrangian density (1). These are: (i) Nieh-Yan class:[3] ˜ IJ (ω) + ǫµναβ Dµ (ω)eIν Dα (ω)eI IN Y = eΣµν β IJ Rµν

(3)

where the dual in the internal space is defined as: ˜ IJ ≡ 1 ǫIJKL X X KL 2 and the SO(1, 3) covariant derivative is: Dµ (ω)eIν = ∂µ eIν + ωµ I J eJ . This topological density involves torsion. It can be explicitly written as a total divergence as: IN Y ≡ ∂µ ǫµναβ eIν Dα (ω)eIβ

(4)

In the Euclidean theory, as discussed in [17], this topological density, properly normalized, characterizes the winding numbers given by three integers associated with the homotopy groups Π3 (SO(5)) = Z and Π3 (SO(4)) = (Z, Z). (ii) Pontryagin class: IP = ǫµναβ RµνIJ (ω)Rαβ IJ (ω)

(5)

This is the same topological density as in the case of QCD except that the gauge group here is SO(1, 3) instead of SU(3). Again, it is a total divergence, given in terms of the SO(1, 3) Chern-Simons three-form:

µναβ

IP ≡ 4∂µ ǫ

ων

IJ

2 K ∂α ωβIJ + ωαI ωβKJ 3

(6)

For the Euclidean theory, this topological density, properly normalized, characterizes the winding numbers given by two integers corresponding to the homotopy group Π3 (SO(4)) = (Z, Z). (iii) Euler class: ˜ αβ IJ (ω) IE = ǫµναβ RµνIJ (ω)R 6

(7)

which again is a total divergence which can be explicitly written as: 2 µναβ IJ K IE ≡ 4∂µ ǫ ω ˜ν ∂α ωβIJ + ωαI ωβKJ 3

(8)

For the Euclidean theory, integral of this topological density, properly normalized, over a compact four-manifold is an alternating sum of Betti numbers b0 −b1 +b2 −b3 , characterizing the manifold. Now we may construct the most general Lagrangian density by adding these topological terms (3), (5) and (7), with the coefficients η, θ and φ respectively, to the Hilbert-Palatini Lagrangian density (1). Since all the topological terms are total divergences, the classical equations of motion are independent of the parameters η, θ and φ. However, the Hamiltonian formulation and the symplectic structure do see these parameters. Yet, classical dynamics are independent of them. But, quantum theory may depend on them. All these topological terms in the action are functionals of local geometric quantities, yet they represent only the topological properties of the four-manifolds. These do not change under continous deformations of the four-manifold geometry. Notice that, while the Nieh-Yan IN Y and Pontryagin IP densities are P and T violating, the Euler density IE is not. So in a quantum theory of gravity including these terms, besides the Newton’s coupling constant, we can have three additional dimensionless coupling constants, two P and T violating (η, θ) and one P and T preserving (φ).

III.

HAMILTONIAN FORMULATION OF GRAVITY WITH NIEH-YAN, PON-

TRYAGIN AND EULER DENSITIES

Here we shall carry out the Hamiltonian analysis for the most general Lagrangian density containing all three topological terms besides the Hilbert-Palatini term: L =

θ φ η 1 IJ e Σµν IN Y + IP + I (ω) + IJ Rµν 2 2 4 4 E

(9)

where the Nieh-Yan IN Y , Pontryagin IP and Euler IE densities are given by (3), (5) and (7) respectively.

7

We shall use the following parametrization for tetrad fields1 : eIt = NM I + N a VaI , MI VaI = 0,

eIa = VaI ;

MI M I = −1

(10)

with N and N a as the lapse and shift fields. The inverse tetrads are: N a MI MI , eaI = VIa + ; N N = 0 , VaI VIb = δab , VaI VJa = δJI + M I MJ

etI = − M I VIa

(11)

The internal space metric is η IJ ≡ dia(−1, 1, 1, 1). The three-space metric is qab ≡ VaI VbI √ with q = det(qab ) which leads to e ≡ det(eIµ ) = N q. The inverse three-space metric is

q ab = VIa V bI , q ab qbc = δca . Two useful identities are: 2eΣta IJ

= −

√

q

M[I VJ]a

,

eΣab IJ

2Ne2 t[a b]t KL b]t = √ ΣIK ΣJL η + e N [a ΣIJ q

(12)

In this parametrization, we have, instead of the 16 tetrad components eIµ , the following 16 fields: 9 VIa (M I VIa = 0), 3 M I (M I MI = −1) and 4 lapse and shift vector fields N, N a . From these, instead of the variables VIa and M I , we define a convenient set of 12 variables,

as: √ t a t a q M[0 Vi]a , Eia = 2eΣta 0i ≡ e e0 ei − ei e0 = −

χi = −Mi /M 0

(13)

which further imply:

2eΣta ij = −

√

q M[i Vj]a = −E[ia χj]

(14)

Now, using the parametrization (10, 11) for the tetrads, and the second identity in (12), we expand the various terms to write: 1 η 1 IJ (η)IJ e Σµν (ω) + IN Y = eΣta + taI ∂t VaI − NH − N a Ha − ωtIJ GIJ IJ ∂t ωa IJ Rµν 2 2 2

(15)

where we have dropped the total space derivative terms. Here taI ≡ ηǫabc Db (ω)VcI with (η) ˜ =X + ǫabc ≡ ǫtabc and, for any internal space antisymmetric tensor, X ≡ X + η X IJ

1

IJ

IJ

IJ

This parametrization differs from the one used earlier in [2]. To obtain the present parametrization replace eN by N 2 in the earlier parametrization.

8

η ǫ X KL . 2 IJKL

Further,

2e2 2e2 ta tb KL (η)IJ tb KL IJ H = √ Σta Σ η R (ω) = Σ Σ η Rab (ω) − M I Da (ω)taI √ ab q IK JL q IK JL (η)IJ

IJ tb Ha = eΣtb (ω) − VaI Db (ω)tbI IJ Rab (ω) = eΣIJ Rab o n (η)ta − ta[I VJ]a GIJ = − 2Da (ω) eΣta = − 2D (ω) eΣ a IJ IJ

(16)

where we have used the following identities:

2ηe2 tb KL ˜ IJ M I Da (ω)taI ≡ √ Σta Rab (ω) IK ΣJL η q n o ˜ ta ta[I VJ]a ≡ −2ηDa (ω) eΣ IJ

˜ IJ VaI Db (ω)tbI ≡ ηeΣtb IJ Rab (ω) ,

Next notice that, dropping the total space derivative terms and using the Bianchi identity, ǫabc Da (ω)RbcIJ ≡ 0, we can write φ θ IP + I = eaIJ ∂t ωa(η)IJ 4 4 E

(17)

where eaIJ are given by n o ˜ bcIJ (ω) 1 + η 2 eaIJ = ǫabc (θ + ηφ) RbcIJ (ω) + (φ − ηθ) R

(18)

Thus, collecting terms from (15) and (17), full Lagrangian density (9) assumes the following form: a L = πIJ ∂t ωa(η)IJ + taI ∂t VaI − NH − N a Ha −

1 IJ ω GIJ 2 t

(19)

with a a πIJ = e Σta IJ + eIJ (η)IJ

In this Lagrangian density, the fields ωa

(20)

a and πIJ form canonical pairs. Then, H, Ha and

GIJ of (16) can be expressed in terms of these fields as: GIJ =

a(η)

− 2Da (ω)πIJ − ta[I VJ]a

(21)

(η)IJ

b Ha = πIJ Rab (ω) − VaI Db (ω)tbI 2 a(η) b(η) b(η) a(η) H = √ πJL − eJL η KL RabIJ (ω) − M I Da (ω)taI πIK − eIK q

(22) (23)

where we have used the relations: Da (ω)eaIJ = 0 and Da (ω)˜ eaIJ = 0 which result from the Bianchi identity ǫabc Da (ω)RbcIJ (ω) = 0, and also used ebIJ RabIJ (ω) = 0 and e˜bIJ RabIJ (ω) = 0 which follow from the fact that 2q (θ2 + φ2 ) RabIJ = ǫabc {(θ + ηφ) ecIJ − (φ − ηθ) e˜cIJ }. 9

Now, in order to unravel the SU(2) gauge theoretic framework for the Hamiltonian formulation, from the 24 SO(1, 3) gauge fields ωµIJ , we define, in addition to 6 field variables ωtIJ , the following suitable set of 18 field variables: Aia ≡ ωa(η)0i = ωa0i + η ω ˜ a0i ,

Kai ≡ ωa0i

(24)

The fields Aia transform as the connection and the extrinsic curvature Kai as adjoint representations under the SU(2) gauge transformations. In terms of these, it is straight forward to check that: a a ˆ a ∂t Ai + Fˆ a ∂t K i πIJ ∂t ωa(η)IJ = 2π0i ∂t ωa(η)0i + πija ∂t ωa(η)ij = E i a i a

(25)

with 2 a(η) 2 2 a(η) 1 a a − π ˜0i ≡ − (˜ π0i − ηπ0i ) = Eia − e˜0i (A, K) + ǫijk Eja χk (26) η η η η 1 1 a ≡ 2 η+ π ˜0i = η+ −ǫijk Eja χk + 2˜ ea0i (A, K) (27) η η

î ≡ E a Fîa

a(η)

where ea0i and eãoi ≡ 12 ǫijk eajk as defined in (18) and e˜0i

≡ eã0i − ηea0i are written as functions

of the gauge field Aia and the extrinsic curvature Kai using

2 ijk j j ǫ Ka K b η 2 1 ijk k 1 ijk η −1 ij i k Rab (ω) = − ǫ Fab (A) + ǫ D[a (A)Kb] − K[a Kb]j 2 η η η

Rab0i (ω) = D[a (A)Kb]i −

(28)

with the SU(2) field strength and covariant derivative respectively as: 1 ijk j k ǫ Aa Ab , η

1 ijk j k ǫ Aa Kb η

(29)

1 IJ ω GIJ L = Eîa ∂t Aia + Fîa ∂t Kai + taI ∂t VaI − NH − N a Ha − 2 t

(30)

i Fab (A) ≡ ∂[a Aib] +

Da (A)Kbi ≡ ∂a Kbi +

Now, using (25), the Lagrangian density (19) can be written as:

Thus, we have the canonically conjugate pairs (Aia , Eîa ), (Kai , Fîa ) and (VaI , taI ). We may write GIJ , Ha and H of (21)-(23) in terms of these fields. For example, from (21): 1 j â a k â + ǫ Grot ≡ ǫ G = ηD (A) E K F − t V (31) jk i a i ijk a k j a 2 ijk 1 â 1 â ˆ a + Fˆ a + ǫijk K j Gboost ≡ G = −D (A) E η+ Ek + Fk − ta[0 Vi]a (32) i 0i a i i a η η 2 1 a a a ijk j ˆ + Fˆ − 1 ǫijk ta V − ta V − 1 Grot E η+ = −Da (A)Fî + ǫ Ka k j ak [0 i]a η η k η η i 10

where the covariant derivatives are: Da (A)Eîb = ∂a Eîb + η −1 ǫijk Aja Eˆkb and Da (A)Fîb = ∂a Fîb + η −1 ǫijk Aja Fˆkb . Next, for the generators of spatial diffeomorphisms Ha from (22): i Ha = Eîb Fab (A) + Fîb D[a (A)Kb]i − Kai Db (A)Fîb + tbi D[a (A)Vb]i − Vai Db (A)tbi i 1 boost + tb0 ∂[a Vb]0 − Va0 ∂b tb0 − Grot Ka i + ηGi η = Eîb ∂[a Aib] − Aia ∂b Eîb + Fîb ∂[a Kb]i − Kai ∂b Fîb + tbi ∂[a Vb]i − Vai ∂b tbi i 1 1 boost i Grot Ka (33) + tb0 ∂[a Vb]0 − Va0 ∂b tb0 + Grot i + ηGi i Aa − η η where we have used −VaI Db (ω)tbI ≡ −V I ∂b tbI + tbI ∂[a Vb]I + tbI VJb ωaIJ . Similarly we can express

H of (23) in terms of these fields. ˆ a ), (K i , Fˆ a ) and (V I , ta ) in the Lagrangian density Now, notice that all the fields (Aia , E i a i a I (30) are not independent. Of these, the fields VaI and taI are given in terms of others as: VaI = vaI and taI = τIa with 1 1 va0 ≡ − √ Eai χi (34) vai ≡ √ Eai , E E where Eai is inverse of Eia , i.e., Eai Eib = δab , Eai Eja = δji and E ≡ det(Eai ) = q −1 (M 0 )−2 and τia ≡ ηǫabc Db (ω)vci = ǫabc ηDb (A)vci − ǫijk Kbj vck + Kbi vc0 , τ0a ≡ −ηǫabc Db (ω)vc0 = −ηǫabc ∂b vc0 + Kbj vcj (35) In addition, the fields Fîa , which are conjugate to the extrinsic curvature Kai , are also not

independent; these are given in terms of other fields by (27) . In the Lagrangian density (30), there are no velocity terms associated with SO(1, 3) gauge fields ωtIJ , shift vector field Na and lapse field N. Hence these fields are Lagrange multipliers. Associated with these are as many constraints:

GIJ ≈ 0, Ha ≈ 0, and

H ≈ 0 where the weak equality ≈ is in the sense of Dirac theory of constrained Hamiltonian systems. Here from the form of Grot = i

1 2

ǫijk Gjk in (31), it is clear that these generate SU(2)

rotations on various fields. The boost transformations are generated by Gboost = G0i , spatial i diffeomorphisms by Ha and H ≈ 0 is the Hamiltonian constraint. This, thus can already be viewed, without fixing the boost degrees of freedom and without solving the second class constraints (34) and (35), as a SU(2) gauge theoretic framework. Here, besides the three boost SU(2) generators Grot , Ha and H. We may, however, fix i , we have seven constraints, Gi

the boost gauge invariance by choosing a time gauge. Then we are left with only the SU(2) gauge invariance besides the diffeomorphism Ha and Hamiltonian H constraints. This we do in the next section. 11

IV.

TIME GAUGE

We work in the time (boost) gauge by choosing the gauge condition χi = 0 which then implies for the tetrad components e0a ≡ Va0 = 0. Correspondingly the boost generators (32)

are also set equal to zero strongly, Gboost = 0. In this gauge, the Lagrangian density (30) i takes the simple form: L = Eîa ∂t Aia + Fîa ∂t Kai + tai ∂t Vai − H

(36)

with the Hamiltonian density as: 1 ijk ij rot ǫ ωt Gk + ξia Vai − vai 2 1 a i a a i a ˆ + φa (ti − τi ) + λa Fi − 2 η + e˜ (A, K) η 0i

H = NH + N a Ha +

(37)

where all the fields involved are not independent. In particular, the fields Vai , tai and Fîa depend on other fields. This fact is reflected in H above through terms with Lagrange multiplier fields ξia , φia and λia . Now, in this time gauge, expressions for Grot i , Ha and H are: Grot ≡ ηDa (A)Eîa + ǫijk Kaj Fˆka − taj Vak i

i i Ha ≡ Eîb Fab (A) + Fîb D[a (A)Kb]i − Kai Db (A)Fîb + tbi D[a (A)Vb]i − Vai Db (A)tbi − η −1 Grot i Ka = Eîb ∂[a Aib] − Aia ∂b Eîb + Fîb ∂[a Kb]i − Kai ∂b Fîb + tbi ∂[a Vb]i − Vai ∂b tbi + η −1 Grot Aia − Kai i √ E ijk a b k ǫ Ei Ej Fab (A) − 1 + η 2 D[a (A)Kb]k − η −1 ǫkmn Kam Kbn H ≡ 2η √ rot a i a EGk Ek (38) + Ka ti − η ∂a

where Da (A) is the SU(2) gauge covariant derivative. In the last line, we have used the √ a a i â i time-gauge identity: ta0 = τ0a = η EGrot k Ek . Also Ei are functions of Ei , Aa and Ka : ˆ A, K) ≡ Eˆ a + 2 eã(η) (A, K) Eia = Eia (E, i η 0i

(39)

. Associated with the Lagrange multiplier fields ωtij , N a and N in (37), we have the constraints: Grot ≈ 0, i

Ha ≈ 0 ,

H ≈ 0

(40)

In addition, corresponding to Lagrange multiplier fields ξia and φia , we have more constraints: Vai − vai (E) ≈ 0 ,

tai − τia (A, K, E) ≈ 0 12

(41)

where, from (34) and (35), in the time gauge: 1 vai ≡ √ Eai , E

τia ≡ ηǫabc Db (ω)vci = ǫabc ηDb (A)vci − ǫijk Kbj vck

Similarly, from the last term in (37), there are the additional constraints: 1 a a a ˆ e˜ (A, K) ≈ 0 χi ≡ Fi − 2 η + η 0i

(42)

(43)

Here ea0i and eã0i of (18), with the help of Eqn.(28), are written as functions of the gauge

fields Aia , extrinsic curvature Kai and the topological parameters θ, φ besides η as follows: η 2 1 + η 2 ea0i (A, K) ≡ −ǫabc η (φ − ηθ) Fbci (A) − 2η 1 − η 2 φ − 2ηθ Db (A)Kci − η 3 − η 2 θ + 3η 2 − 1 φ ǫijk Kbj Kck η 2 1 + η 2 eã0i (A, K) ≡ −ǫabc η (θ + ηφ) Fbci (A) − 2η 1 − η 2 θ + 2ηφ Db (A)Kci − 3η 2 − 1 θ − η 3 − η 2 φ ǫijk Kbj Kck (44) a(η)

a(η)

From these we can construct for e0i ≡ ea0i + η˜ ea0i and e˜0i ≡ eã0i − ηea0i : 2 1 abc (η − 1)φ + 2ηθ ijk j k a(η) i i e0i = − ǫ ǫ K b Kc φFbc (A) − (φ − ηθ) D[b (A)Kc] − η η 2 (η − 1)θ − 2ηφ ijk j k 1 abc a(η) i i θFbc (A) − (θ + ηφ) D[b (A)Kc] − ǫ K b Kc (45) e˜0i = − ǫ η η

The χai constraints (43) are of particular interest. To study their effect, we note that (Aia , Eˆjb ) and (Kai , Fˆjb ) are canonically conjugate pairs. They have accordingly the standard Poisson brackets. From these, using the relation (39) expressing E a in terms of Eˆ a , Ai and i

i

a

Kai , as indicated in the Appendix, the following Poisson brackets can be calculated with ˆ a ) and (K i , Fˆ a ): respect to phase variables (Ai , E a

i

a

i

i i h Aa (x), Ejb (y) = Aia (x), Eˆjb (y) = δji δab δ (3) (x, y), i a Ka (x), Ejb (y) = 0 , Ei (x), Ejb (y) = 0

These then imply the Poisson bracket relations: a χi (x), Ajb (y) = 0 , a χi (x), Ejb (y) = 0 ,

a χi (x), Kbj (y) = − δji δba δ (3) (x, y) , a χi (x), χbj (y) = 0

(46)

Using these, we notice that the Poisson brackets of Hamiltonian constraint H and χai are non-zero. Requiring [χai (x), H(y)] ≈ 0 leads us to the secondary constraints as: √ o √ 1 + η 2 n ijk [a b] j a a b ti − ηǫ D (A) ≈ 0 + EE E EE E K b j k j i b η2 13

which can be rewritten as: 1 + η 2 abc a ti − ǫ ηDb (A)vci − ǫijk Kbj vck ≈ 0 2 η Next, since from (41) and (42), tai ≈ τia ≡ ǫabc ηDb (A)vci − ǫijk Kbj vck , this implies tai ≈ 0. Thus we have the constraints:

ǫabc ηDb (A)vci − ǫijk Kbj vck ≈ 0

These can be solved for the extrinsic curvature Kai and recast as the following secondary constraints: ψai ≡ Kai − κia (A, E) ≈ 0 , η ijk j ǫ Ea Db (A)Ekb κia (A, E) ≡ 2 η k bcd k − Ea ǫ Eb Dc (A)Edi + Ebi Dc (A)Edk − δ ik Ebm Dc (A)Edm 2E

(47)

These are additional constraints and have the important property that these form second class pairs with the constraints χai of (43): a χi (x), ψbj (y) = − δba δij δ (3) (x, y)

(48)

To implement these second class constraints, χai and ψai , we need to go over from Poisson brackets to the corresponding Dirac brackets and then impose the constraints strongly, ψai = 0 (which also implies tai = 0) and χai = 0, in accordance with Dirac theory of constrained Hamiltonian systems. As outlined in the Appendix, the Dirac brackets of fields Aia and Eia turn out to be the same as their Poisson brackets; these are displayed in (A.20). On the other hand, those for (Ai , Eˆ a ; K i , Fˆ a ) are different; these have been listed in (A.22) and a

i

a

i

(A.23). Finally, after implementing these second class constraints, we have the Lagrangian density in the time-gauge as: L = Eîa ∂t Aia + Fîa ∂t Kai − H

(49)

with the Hamiltonian density H = NH + N a Ha + 14

1 ijk jk rot ǫ ω t Gk 2

(50)

and a set of seven first class constraints: ˆ a + ǫijk K j Fˆ a ≈ 0 Grot ≡ η Da (A)E i i a k ˆ b F i (A) + Fˆ b D (A)K i − K i D (A)Fˆ b − η −1 Grot K i ≈ 0 Ha ≡ E i ab i [a b] a b i i a √ √ 2 E ijk a b k 1+η √ 1 j a b i rot a H ≡ ǫ Ei Ej Fab (A) − ∂ EE E K K EG E + i j [a b] k k ≈ 0 2η 2η 2 η a

(51)

ˆ A, K) ≡ Eîa + 2 e˜ (A, K). The fields with Eia in the last equation given by: Eia = Eia (E, η 0i i a i a (Aa , Eî , Ka , Fî ) have non-trivial Dirac brackets as listed in (A.22) and (A.23). The second a(η)

class constraints χai and ψai are now set strongly equal to zero: Kai = κia (A, E) ≡ Fîa

η ijk j η ǫ Ea Db (A)Ekb − Eak ǫbcd Ebk Dc (A)Edi + Ebi Dc (A)Edk 2 2E − δ ik Ebm Dc (A)Edm

1 a = 2 η+ e˜ (A, K) η 0i

(52)

In writing the Hamiltonian constraint H in (51) from (38), we have used the identity: √ √ 1 kmn m n ijk b c k a rot E ǫ Ei Ej Db (A)Kc − ǫ Kb Kc = − ∂a EEi Gi (53) η which holds due to the time gauge relation EEia Grot = ǫabc Ebi Kci with the constraints i Kai = κia (A, E) imposed strongly. To evaluate the effect of generators (51) on various fields, we need to use the Dirac brackets instead of the Poisson brackets. For example, for the SU(2) gauge generators, using the results listed in the Appendix, we obtain: h i ˆ a (y) Grot (x), E = ǫijk Eˆka δ (3) (x, y) , i j D rot Gi (x), Aja (y) D = −η δ ij ∂a + η −1 ǫikj Aka δ (3) (x, y)

(54)

reflecting the fact Grot are generators of SU(2) transformations: Aia transform as the SU(2) i connection and fields Eˆ a as adjoint representations. Besides, the fields Fˆ a , K i and E a also i

i

a

i

behave as adjoint representations under SU(2) rotations: h i ˆ a (y) Grot (x), F = ǫijk Fˆka δ (3) (x, y) i j D rot Gi (x), Kaj (y) D = ǫijk Kak δ (3) (x, y) rot Gi (x), Eja (y) D = ǫijk Eka δ (3) (x, y) 15

(55)

Similar discussion is valid for the spatial diffeomorphism generators Ha . The Dirac brackets of Ha with various fields yield the Lie derivatives of these fields respectively, modulo SU(2) gauge transformations. As stated earlier and demonstrated in the Appendix, Dirac brackets for the fields (Aia , Eîa ; Kai , Fîa ) are different from their Poisson brackets (see (A.21), (A.22) and (A.23)). This is so because the transition from Poisson brackets to Dirac brackets, except for some special cases, in general, does not preserve canonical structure of the algebra [18]. When the second class constraints are imposed strongly, the algebraic structure of the Dirac brackets of phase variables (Aia , Eîa ) of the final theory is different from those of the phase variables (Ai , E a ) of the standard canonical theory. Thus the variables (Ai , Eˆ a ) are not related to a

i

(Aia ,

Eia )

a

i

through a canonical transformation. However, it is possible to construct a set of

new phase space field variables whose Dirac bracket algebra has the same structure as that of the standard canonical variables (Aia , Eia ). In fact, in general, for theories with second class constraints as is the case here, instead of the ordinary canonical transformations, what is relevant are the Gitman D-transformations, which preserve the form invariance of Dirac brackets and equations of motion [19]. Thus, in the present context also, new phase variables can be constructed through these Dtransformations. These transformations change both the gauge fields as well as their conjugate momentum fields. This procedure finally leads to the phase variables: Ei′a (x)

∞ X 1 ′(n)a Di (x) , = n! n=0

A′ia (x)

∞ X 1 ′(n)i = Ca (x) n! n=0

(56)

where ′(0)a

Di

Ca′(0)i (x) ≡ Aia (x)

(x) ≡ Eia (x) ,

and other D ′(n) and C ′(n) are recursively constructed Z h i ′(n+1)a ′(n)a Di (x) = d3 z Fˆlb (z) Kbl (z), Di (x) − D Z Ca′(n+1)i (x) = d3 z Fˆlb (z) Kbl (z), Ca′(n)i (x) D −

(57)

using Dirac brackets as follows: Z h i 2 (η)a ′(n)a 3 l d z e˜0l (z) Ab (z), Di (x) η D Z 2 (η)a d3 z e˜0l (z) Alb (z), Ca′(n)i (x) D η n = 0, 1, 2, 3, ..... (58)

In particular, ′(1)a Di (x)

=

Fîa (x)

2 (η)a − e˜ (A, K; x), η 0i

Ca′(1)i (x) 16

=−

Z

δκl (A, E; z) d3 z Fˆlb (z) b a δEi (x)

(59)

The new variables (A′ia , Ei′a ) are functions of the phase variables (Aia , Eîa ; Kai , Fîa ) of the theory described above and can be checked to satisfy the Dirac bracket relations:

A′ia (x), Ej′b (y)

D

= δij δab δ (3) (x, y),

A′ia (x), A′jb (y)

D

= 0,

Ei′a (x), Ej′b (y)

D

=0

(60)

As is expected under D-transformations, these relations reflect the fact that the algebraic structure of Dirac brackets for the fields (Aia , Eia ) as represented by (A.20) has been preserved. After the second class constraints, χai and ψai , are implemented, (A′ia , Ei′a ) are related to the phase variables (Aia , Eia ) through an ordinary canonical transformation. We have not presented many details of the construction of these new phase variables above. Instead, in the next section, we shall present, through an equivalent procedure, an elaborate construction of the new phase variables in the theory where second class constraints are already imposed strongly. This will be done by a direct canonical transformation of the phase variables (Aia , Eia ) of the standard canonical theory. The new canonical variables so obtained will be shown to be equal to the fields A′ia and Ei′a above, when the second class constraints χai and ψai are imposed.

V.

CANONICAL TRANSFORMATIONS AND NEW PHASE VARIABLES

Adding the Nieh-Yan term to Hilbert-Palatini Lagrangian density, in the time gauge, leads to a change of phase variables [2], from the ADM variables (κia , Eia ) to new variables (Aia , Eia ). This change is just a canonical transformation. Further inclusion of the Pontraygin and Euler densities results in a theory which can also be described in terms of canonically transformed phase variables. In the following, we shall develop such a description explicitly. We start with the standard canonical theory constructed from the Lagrangian density containing the Hilbert-Palatini term and the Nieh-Yan density as in (9) with θ = 0 and φ = 0. This is described, after partial gauge fixing (the time gauge), where the second class constraints are imposed, in terms the SU(2) gauge fields Aia and their conjugates, densitized triads Eia , by the Lagrangian density L1 = Eia ∂t Aia − H 1 ijk ij rot ǫ ωt Gk + N a Ha + NH H = 2 a i Grot Ha (A, E) = Eib Fab (A) − η −1 κia Grot i (A, E) = ηDa (A)Ei , i 17

H(A, E) =

√

√ 1 + η 2 kmn m n 1 E ijk a b k a + ǫ Ei Ej Fab (A) − ǫ κa κb ∂a EGrot E k k 2η η η

(61)

where the extrinsic curvature κia (A, E) is given in terms of Aia and Eia through (A.7). Canonical pairs of the phase variables (Aia , Eia ) obey the standard Poisson bracket relations:

Aia (x), Ejb (y) = δji δab δ (3) (x, y) ,

Aia (x), Ajb (y) = 0 ,

a Ei (x), Ejb (y) = 0

(62)

Next, we add the Pontraygin and Euler densities (6, 8), which are total divergences,

θ I 4 P

+ φ4 IE = ∂µ J µ , to L1 above. The resulting Lagrangian density, ignoring the spatial

derivative part, is L2 = Eia ∂t Aia + ∂t J t − H

(63)

Inclusion of the time derivative term here is equivalent to a canonical transformation on the phase space which can be constructed using J t . For this purpose, we first express J t as a function of the phase variables Aia and Eia : 1 ijk i j k θ abc i i t Aa Fbc − ǫ Aa Ab Ab J (A, κ(A, E)) = 2 ǫ η 3η 1 1 ijk j k i i i i abc − 2 (θ + ηφ) ǫ κa Fbc + Aa D[b (A)κc] − ǫ Ab κc η η 1 + 2 (1 − η 2 )θ + 2ηφ ǫabc κia D[b (A)κib] η 2 + 3 (3η 2 − 1)θ − η(3 − η 2 )φ ǫabc ǫijk κia κjb κkc 3η

Generating functional for the canonical transformation is: Z J (A, E) = d3 z J t (A(z), κ(A(z), E(z)))

(64)

(65)

which has functional dependence on both gauge fields Aia and their conjugates Eia . Following the standard procedure, J generates the canonical transformations, (Aia (x), Eia (x)) →

(Aia (x), Eia (x)), where the new phase variables are given in terms of Poisson bracket series as follows: 1 1 J , J , Aia (x) + J , J , J , Aia (x) + ..... Aia (x) = Aia (x) + J , Aia (x) + 2! 3! J i −J ≡ e Aa (x) e 1 1 [J , [J , [J , Eia (x)]]] + ..... Eia (x) = Eia (x) + [J , Eia (x)] + [J , [J , Eia (x)]] + 2! 3! ≡ eJ Eia (x) e−J 18

(66)

Alternately, these relations may be represented as: X 1 Ca(n)i (x) , n! n=0

Aia (x) =

Eia (x) =

X 1 (n)a Di (x) n! n=0

(67)

with Ca(0)i = Aia (x), Ca(n)i (x) = J , Ca(n−1)i (x) ,

(0)a

(x) = Eia (x), h i (n)a (n−1)a Di (x) = J , Di (x) , Di

n = 1, 2, 3, .....

The various terms can be evaluated recursively through the following formulae: Z (1)b (1)l (n)i Ca (x) = d3 z Dl (z) Alb (z), Ca(n−1)i (x) − Cb (z) Elb (z), Ca(n−1)i (x) Z h i h i (n)a (1)b (n−1)a (1)l (n−1)a 3 l b Di (x) = d z Dl (z) Ab (z), Di (x) − Cb (z) El (z), Di (x) n = 1, 2, 3........

(68)

(69)

where, using J from (64, 65), Ca(1)i (x) (1)a

Di

2(1 + η 2 ) = J , Aia (x) = − η (η)a

(x) = [J , Eia (x)] = 2e0i (κ; x)

Z

d3 u e˜b0l (κ; u)

δfbl (u) , δEia (x) (70)

(η)a

Here e0i (κ; x) ≡ ea0i (κ; x) + η˜ ea0i (κ; x) and the argument κ is to indicate that these functions are given by (44) with Kai replaced by κia (A, E) = Aia + fai (E) where fai (E) are as in (A.7).

By repeated use of Jacobi identity, it can be checked that the functions C (n) and D (n) of (68) satisfy the Poisson bracket relations: n X

h i 1 (n−l)j (l)i Ca (x), Cb (y) = 0 l!(n − l)! l=0 n h i X 1 (l)a (n−l)b Di (x), Dj (y) = 0 l!(n − l)! l=0 n X l=0

h i 1 (n−l)b Ca(l)i (x), Dj (y) = 0 , l!(n − l)!

n = 1, 2, 3, ......

(71)

The Poisson bracket relations (62) imply, by construction, same Poisson brackets for the new variables (66): i Aa (x), Ejb (y) = δab δji δ (3) (x, y) ,

Aia (x), Ajb (y) = 0 , 19

a Ei (x), Ejb (y) = 0

(72)

where the Poisson brackets are evaluated with respect to the phase variables (Aia , Eia ). This can be readily checked by using the identities (71). For a general analytic function P (A, E) of the phase variables Aia and Eia , the following relation holds: P (A, E) = eJ P (A, E)e−J ≡ P (A, E) + [J , P (A, E)] + +

1 [J , [J , P (A, E)]] 2!

1 [J , [J , [J , P (A, E)]]] + ..... 3!

(73)

Further J of (64 , 65) written as a functional of (Aia , Eia ) and (Aia , Eia ) is form invariant: J (A, E) = J (A, E)

(74)

The converse relations expressing Aia and Eia in terms of the transformed variables are: 1 1 J , J , Aia (x) − J , J , J , Aia (x) + ..... Aia (x) = Aia (x) − J , Aia (x) + 2! 3! −J i J ≡ e Aa (x) e 1 1 Eia (x) = Eia (x) − [J , Eia (x)] + [J , [J , Eia (x)]] − [J , [J , [J , Eia (x)]]] + ..... 2! 3! −J a J ≡ e Ei (x) e

(75)

where J is written as a functional of Aia and Eia (refer (74)) and Poisson brackets are evaluated with respect to these new variables. Next, we evaluate the following: Z

3

dx

Eia (x)∂t Aia (x)

=

∞ X

F

(n)

,

F

(n)

n=0

≡

It is straight forward to check: Z (0) F = d3 x Eia (x)∂t Aia (x), F (n) =

1 ∂ G(n−1) , n! t

n X l=0

1 l!(n − l)!

Z

(l)a

d3 x Di (x)∂t Ca(n−l)i (x)

F (1) = ∂t G(0) + ∂t J

n = 2, 3, 4, .......

(76)

where G

(n)

n X ≡ J , G(n−1) = l=0

G(0) ≡

Z

n! l!(n − l)!

Z

(l)a

d3 x Di (x)Ca(n+1−l)i (x) ,

d3 x Eia (x)Ca(1)i (x)

n = 1, 2, 3..... (77)

20

To obtain this result, the following helpful identities may be used: Z n−1 X n! (n−l)a (l)a d 3 x Di δCa(l)i − Ca(n−l)i δDi = 0, n = 2, 3, 4..... l!(n − 1 − l)! l=0

(78)

which can be derived recursively by taking Poisson brackets with J . (1)i

Further, using expression for Ca

from (70) and eqns. (A.9, A.10) of the Appendix, the

following relation can be obtained: Eia (x)Ca(1)i (x) = (1 + η 2 ) ǫijk ∂a e˜b0i (κ; x)Ebj (x)Eka (x) which in turn implies G(0) ≡

R

(1)i

d3 x Eia (x)Ca (x) = 0 and hence all the G(n) of (77) are zero,

thus leading to the result: Z Z 3 a i d x Ei (x)∂t Aa (x) = d3 x Eia (x)∂t Aia (x) + ∂t J

(79)

Since the generating functional J , as given by (64, 65), is invariant under small SU(2)

gauge transformations and spatial diffeomorphisms generated respectively by Grot i (A, E) and Ha (A, E) of (61): J , Grot i (A, E) = 0 ,

[J , Ha (A, E)] = 0

Consequently, Grot and Ha written in terms of the phase variable (Aia , Eia ) and (Aia , Eia ) are i form invariant: J rot −J Grot = Grot i (A, E) = e Gi (A, E) e i (A, E)

Ha (A, E) = eJ Ha (A, E) e−J = Ha (A, E)

(80)

On the other hand, for the Hamiltonian constraint we have: H(A, E) = e−J (A,E) H(A, E) eJ (A,E) = H(A, E) − [J , H(A, E)] + −

1 [J , [J , H(A, E)]] 2!

1 [J , [J , [J , H(A, E)]]] + ......... 3!

(81)

where the Poisson brackets are with respect to phase variables (Aia , Eia ). This detail discussion, finally allows us to write the theory based on the Lagrangian density (63) in terms of the new phase variables as: ˆ L2 = Eia ∂t Aia − H ˆ = 1 ǫijk ωtij Grot (A, E) + N a H (A, E) + N H(A, ˆ H E) k a 2 21

(82)

where a Grot i (A, E) = ηDa (A)Ei i i Ha (A, E) = Eib Fab (A) − η −1 Grot i (A, E)κa (A, E)

ˆ H(A, E) = e−J (A,E) H(A, E) eJ (A,E)

(83)

The new variables (Aia , Eia ) obtained here are related to the variables (A′ia , Ei′a ) of eqns. (56-60) of Sec.IV derived by the Gitman D-transformations. When the second class constraints χai and ψai there are implemented, (A′ia , Ei′a ) collapse to (Aia , Eia ): A′ia (χ = 0, ψ = 0) = Aia ,

Ei′a (χ = 0, ψ = 0) = Eia

(84)

This is so because each of the terms in (56) and (67) coincide: ′(n)a

Ca′(n)i (χ = 0, ψ = 0) = Ca(n)i ,

Di

(n)a

(χ = 0, ψ = 0) = Di

(85)

This completes our discussion of the canonical transformation to new variables (Aia , Eia ) obtained by adding the Pontraygin and Euler densities to the standard canonical theory of gravity described in terms of the phase variables (Aia , Eia ).

VI.

SUMMARY AND CONCLUDING REMARKS

We have developed the canonical Hamiltonian formulation of gravity theory with all the three topological terms of the Lagrangian density (9) as an SU(2) gauge theory with Barbero-Immirzi parameter γ = η −1 as its coupling constant. In time-gauge, the theory containing only the Nieh-Yan topological term (θ = 0, φ = 0) developed earlier in ref.[2], is described by real SU(2) gauge fields, Aia = ωa0i + η ω ˜ a0i , and densitized triads Eia as their conjugate momentum fields. This coincides with the standard SU(2) gauge theoretical canonical formulation of the theory of gravity [4]. When the Pontryagin and Euler terms are also included, there is a formulation of the theory which retains the gauge fields Aia (independent of the topological parameters θ and φ) as the canonical fields, but, their conjugate momentum fields are modified from Eia to Eîa ≡ Eia − 2η −1 eã0i (A, K) + 2ea0i (A, K) developing dependence on θ and φ. Further, for the case with θ = 0 and φ = 0, the momentum conjugate to extrinsic curvature Kai is zero. Here, in the most general case, it is non-zero, represented by Fîa which depends on other fields through the χai constraints 22

(43). In addition, it also depends on the topological parameters θ and φ. Associated with χai , we have a set of secondary constraints ψai of (47) which expresses the fact that extrinsic curvature Kai is not an independent field. These constraints, (χai , ψai ), form second class pairs which are implemented by going over to the Dirac brackets from Poisson brackets. The theory is described by seven first class constraints: the SU(2) gauge constraints Grot i , spatial diffeomorphism constraints Ha and Hamiltonian constraint H as listed in (51). In this formulation, however, the Dirac brackets for the phase variables (Ai (x), Eˆ a (x)) do not a

i

possess the same algebraic structure as those for the canonical variable (Aia (x), Eia (x)) of the standard theory. Even after the second class constraints, χai = 0, ψai = 0, are imposed, there is no canonical transformation that relates the set (Ai , Eˆ a ) to (Ai , E a ). However, it a

i

a

i

is possible to construct another Hamiltonian formulation in terms of new canonical variables (Aia , Eia ) which indeed are related to the standard variables (Aia , Eia ) through a canonical transformation. Here both the gauge fields as well as their conjugate momentum fields, as represented in (66), are changed and these depend on all the topological parameters, η, θ and φ. From this classical Hamiltonian formulation described in terms of (Aia , Eia ), we can go over to the quantum theory by replacing the Poisson brackets by commutators of corresponding operators in the usual fashion. We already have some evidence that Barbero-Immirzi parameter η −1 is relevant in the quantum theory. For example, it appears in the spectrum of area and volume operators [20] and also in the black hole entropy [21]. How other parameters, θ and φ, will be reflected in the quantum theory is an open question requiring deeper study. The analysis presented in the present article is for pure gravity without matter couplings. Inclusion of matter, such as fermions, spin 1/2 or spin 3/2 (supergravity), may be achieved through standard minimal couplings. All the topological densities in the Lagrangian are described in terms of geometric quantities only. Their presence does not change the classical equations of motion even with matter. A Hamiltonian formulation, in the time gauge, can again be set up in terms of a real SU(2) gauge theory with η −1 as its coupling constant.

23

Acknowledgments

Discussions with Ghanashyam Date are gratefully acknowledged. R.K.K also acknowledges the support of Department of Science and Technology, Government of India through a J.C. Bose Fellowship.

Appendix: Poisson and Dirac brackets

In the time-gauge Lagrangian density (49), the fields (Aia , Eîa ) and (Kai , Fîa ) are canonical pairs which have the standard Poisson bracket relations: [Aia (t, ~x), Eˆjb (t, ~y )] = δji δab δ (3) (~x, ~y ) ,

[Kai (t, ~x), Fˆjb (t, ~y )] = δji δab δ (3) (~x, ~y )

(A.1)

and all other brackets amongst these fields are zero. Thus the Poisson bracket for any two arbitrary fields P and Q is given by: ! Z δP (x) δP (x) δQ(y) δQ(y) [P (x), Q(y)] = d3 z − δAia (z) δ Eîa (z) δ Eîa (z) δAia (z) ! Z δP (x) δQ(y) δQ(y) δP (x) − + d3 z δKai (z) δ Fîa (z) δ Fîa (z) δKai (z)

(A.2)

ˆ A, K) ≡ E ˆ a + 2η −1 eã(η) (A, K), we have the Poisson ¿From these, using Eia = Eia (E, i 0i bracket relations

Aia (x), Ejb (y)

=

h i Aia (x), Eˆjb (y) = δji δab δ (3) (x, y),

Kai (x), Ejb (y)

= 0

(A.3)

Using the expressions for ea0i (A, K) and eã0i (A, K) as functions of Aia and Kai as in (44), the following relations obtain: h i 2h i 4 abc θ + ηφ ikj k (3) b(η) a b a ij ˆ ˆ Ei (x), Ej (y) = Ei (x), e˜0j (y) = − 2 ǫ ǫ Kc δ (x, y) θDc − η η η h i 2h i b(η) a b a ˆ ˆ Fi (x), Ej (y) = F (x), e˜0j (y) η i 4 abc (1 − η 2 )θ + 2ηφ ikj k (3) ij = 2 ǫ (θ + ηφ)Dc − ǫ Kc δ (x, y) η η i h i h θ + ηφ ikj k (3) 2 abc a(η) a(η) ij b b ˆ ǫ Kc δ (x, y) θDc − e˜0i (x), Ej (y) = e˜i (x), Ej (y) = ǫ η η h i (1 + η 2 ) eã0i (x), Ejb (y) = (1 + η 2 ) eã0i (x), Eˆjb (y) 2 abc (1 − η 2 )θ + 2ηφ ikj k (3) ij (A.4) ǫ Kc δ (x, y) = ǫ (θ + ηφ)Dc − η η 24

where the SU(2) gauge covariant derivative is: Dcij ≡ δ ij ∂c + η −1 ǫikj Akc . These Poisson ˆ A, K) ≡ Eˆ a + 2η −1 eã(η) (A, K): bracket relations imply for E a = E a (E, i

i

i

Eia (x), Ejb (y)

0i

= 0

(A.5)

Now, using these Poisson bracket relations along with (A.3), yields: i κa (x), Ejb (y) = Aia (x), Ejb (y) = δab δji δ 3 (x, y)

(A.6)

where κia (E, A) is given by (47) and can be rewritten explicitly as:

κia (A, E) = Aia + fai (E) η ijk j η fai (E) = ǫ Ea ∂b Ekb − Eak ǫbcd Ebi ∂c Edk + Ebk ∂c Edi − δ ik Ebl ∂c Edl 2 2E 1 ij r r j i j bcd δ vb ∂c vd vb ∂c vd − = − η Ea ǫ 2 √ with vai ≡ Eai / E. It is straight forward to check that fai satisfy the identity: n o √ ǫabc ∂b Eci − ∂b (ln E) Eci − η −1 ǫijk fbj Eck = 0

(A.7)

Equivalently, this relation can also be written as:

∂a Eia − η −1 ǫijk faj Eka ≡ Da (A)Eia − η −1 ǫijk κja Eka = 0

(A.8)

These relations can be used to calculate the variation δfai to be: η il ∂c Aab cil δElb δElb − η ∂c Aab cil δElb + δfai = Sab 2

with

3 1 m m c l Eai fbl + Ebl fai − Ea Eb Ei fc + Elc fci 4 2 c m 1 + Eam Ebl Eic + Ebm Eai Elc fcm + Ebi Eal − Eai Ebl + δ li Ean Ebn Em fc 4 η lmk η ǫ ∂c Ebm Eai − ∂c Eai Ebm Ekc − ǫimk ∂c Eam Ebl − ∂c Ebl Eam Ekc − 4 4 η ilk m m m m c + ǫ (∂c Ea Eb − ∂c Eb Ea ) Ek 2 1 imk m l 1 lmk m i ilk m m = ǫ Ea Eb − ǫ Ea Eb + ǫ Eb Ea Ekc (A.10) 2 2

il Sab = − Eal fbi + Ebi fal

Aab cil

(A.9)

+

il Notice that Sab and Aab cil are respectively symmetric and antisymmetric under the inter-

change of the pair of indices (a, i) and (b, l): il li Sab = Sba ,

Aab cil = − Aba cli 25

(A.11)

These properties, immediately, lead to the relation: δfai (x) δfbl (y) = δEia (x) δElb (y) Next, using χai (x) ≡ Fîa (x) −

2(1+η2 ) η

(A.12)

eã0i (x) from (43), equations (A.1) also imply the

following: h i i 2(1 + η 2 ) h a ˆ b (y) χai (x), Eˆjb (y) = − e˜0i (x), E j η (1 − η 2 )θ + 2ηφ ikj k (3) 4 abc ij ǫ Kc δ (x, y) (θ + ηφ)Dc − = − 2ǫ η η h i i 2(1 + η 2 ) h a χai (x), Fˆjb (y) = − e˜0i (x), Fˆjb (y) η 2 ij (3η − 1)θ − η(3 − η 2 )φ ikj k (3) 4 abc 2 (1 − η )θ + 2ηφ Dc + ǫ Kc δ (x, y) = 2ǫ η η a a χi (x), Ajb (y) = 0 , χi (x), Kbj (y) = − δij δba δ (3) (x, y) h i a 2 b 2 a b ˆ (1 + η ) χi (x), e0j (y) = (1 + η ) Fi (x), e0j (y) ij η(3 − η 2 )θ + (3η 2 − 1)φ ikj k (3) 2 abc 2 ǫ Kc δ (x, y) ǫ (1 − η )φ − 2ηθ Dc + = η η h i (1 + η 2 ) χai (x), e˜b0j (y) = (1 + η 2 ) Fîa (x), eã0j (y) 2 ij 2 abc (3η − 1)θ − η(3 − η 2 )φ ikj k (3) 2 = ǫ ǫ Kc δ (x, y) (1 − η )θ + 2ηφ Dc + η η (η)b (η)b [χai (x), e˜j (y)] = [Fîa (x), e˜j (y)] 2 abc ((1 − η 2 )θ + 2ηφ ikj k (3) ij = ǫ (θ + ηφ)Dc − ǫ Kc δ (x, y) (A.13) η η which further imply:

χai (x), Ejb (y)

= 0,

a χi (x), κib (y) = 0,

a χi (x), χbj (y) = 0

(A.14)

For ψai ≡ Kai − κia (A, E) as given by (47), using (A.3) and (A.7), we have the following useful relations: i ψa (x), i ψa (x), i ψa (x), i ψa (x),

Ejb (y) Ajb (y) Ebj (y) E(y)

= − κia (x) , Ejb (y) = −δab δji δ 3 (x, y), δκia (x) = − κia (x), Ajb (y) = δEib (y) i = − κa (x), Ebj (y) = Eaj Ebi δ 3 (x, y), = − κia (x), E(y) = EEai δ 3 (x, y) 26

(A.15)

The Poisson bracket relations among χai and ψai , obtained by using the properties listed above, can be summarized as:

χai (x), χbj (y) = 0 ,

χai (x), ψbj (y) = −δba δij δ (3) (x, y) ,

where the last equation follows from the relation:

i ψa (x), ψjb (y) = 0

(A.16)

i i i δfbj (y) δfai (x) j j j κa (x), κb (y) = Aa (x), fb (y) + fa (x), Ab (y) = − = 0 (A.17) δEia (x) δEjb (y)

Here the Poisson brackets involving fai (E) are calculated by using their expressions as functions of Eai as given by (A.7). The identity (A.17) further implies the following Poisson bracket relations:

i Fab (x), κjd (y) + D[a (A)κib] (x), Ajd (y) = 0, h i j j i Fab (x), D[c (A)κd] (y) + D[a (A)κib] (x), Fcd (y) = 0, h i ikl k j i n l η D[a (A)κib] (x), D[c (A)κjd] (y) + Fab (x), ǫjmn κm c (y)κd (y) + ǫ κa (x)κb (x), Fcd (y) = 0 i h ikl k j n l D[a (A)κib] (x), ǫjmn κm (y)κ (y) + ǫ κ (x)κ (x), D (A)κ (y) = 0, (A.18) c d a b [c d] To implement the second-class constraints χai ≈ 0 and ψia ≈ 0, we need to go over to the

corresponding Dirac brackets and then put χai = 0 and ψia = 0 strongly. From the Poisson

bracket relations of these constraints (A.16), the Dirac bracket of any two fields C and D can be constructed to be: [C, D]D = [C, D] − [C, χ] [ψ, D] + [C, ψ] [χ, D]

(A.19)

Using the Poisson bracket relations listed above, it is straight forward to check that the Dirac brackets amongst Aia and Eia are the same as their Poisson brackets: a i Ei (x), Ejb (y) D = Eia (x), Ejb (y) = 0 , Aa (x), Ajb (y) D = Aia (x), Ajb (y) = 0 i Aa (x), Ejb (y) D = Aia (x), Ejb (y) = δab δji δ 3 (x, y) (A.20)

Also we note that,

Kai (x), Ejb (y) D = κia (x), Ejb (y) D = κia (x), Ejb (y) = [Aia (x), Ejb (y)] = δab δji δ 3 (x, y), i δf i (x) Ka (x), Ajb (y) D = κia (x), Ajb (y) D = κia (x), Ajb (y) = fai (x), Ajb (y) = − ab , δEj (y) i Ka (x), Kbj (y) D = κia (x), κjb (y) D = κia (x), κjb (y) = Aia (x), fbi (y) + fai (x), Ajb (y) = 0 (A.21)

27

where in the last terms of second and third equations, the Poisson brackets are to be evaluated using (A.7) which express fai (E) as functions of Eai . îa ) and (Eîa , E ˆjb ) are not same as their Poisson brackets: The Dirac brackets of (Aia , E i h i 2 h i 2 b(η) b(η) i b i b ˆ Aa (x), e˜0j (κ; y) , Aa (x), Ej (y) = Aa (x), Ej (y) − e˜0j (y) = δab δji δ 3 (x, y) − η η D D h i h i 4 a(η) b(η) Eîa (x), Eˆjb (y) = 2 e˜0i (κ; x), e˜0j (κ; y) η D 4(θ2 + φ2 ) acd bef i jmn m n ǫ ǫ F (x), ǫ κ (y)κ (y) = − cd e f η3 imn m j + ǫ κc (x)κnd (x), Fef (y) i 4(θ2 + φ2 ) acd bef h j i = D (A)κ (x), D (A)κ (y) (A.22) ǫ ǫ [c d] [e f] η2 Here the argument κ in ea0i (κ) and eã0i (κ) is to indicate that these are as in (44) with Kai

replaced by κia which in turn are given by (A.7) as functions of Aia and Eia . Further, here in the second equation, we have used: h i h i b(η) a(η) Eia (x), e˜0j (κ; y) + e˜0i (κ; x), Ejb (y) = 0

Also, h i 2(1 + η 2 ) i 2(1 + η 2 ) i Aa (x), e˜b0j (y) D = Aa (x), e˜b0j (κ; y) Aia (x), Fˆjb (y) = η η D h i 2 2(1 + η 2 ) a 2(1 + η ) a Ei (x), e˜b0j (κ; y) [Ei (x), e˜b0j (y)]D = Eia (x), Fˆjb (y) = η η D h i 2 2 4(1 + η ) 4(1 + η 2 )2 a a b b Fîa (x), Fˆjb (y) = = e ˜ (x), e ˜ (y) e ˜ (κ; x), e ˜ (κ; y) 0i 0j 0i 0j D D η2 η2 h i 2 acd bef 4(1 + η ) 2 j 2 i = θ +φ ǫ ǫ D[c (A)κd] (x), D[e (A)κf ] (y) η2 h i h i 8 (1 + η 2 ) a b a b a b ˆ ˆ ˆ ˆ e ˜ (κ; x), e ˜ (κ; y) Ei (x), Fj (y) + Fi (x), Ej (y) = − 0i 0j η2 D D i 8 (θ2 + φ2 ) acd bef h j i =− ǫ ǫ D (A)κ (x), D (A)κ (y) [c d] [e f] η2 h i 2(1 + η 2 ) i 2(1 + η 2 ) i Ka (x), e˜b0j (y) D = κa (x), e˜b0j (κ; y) (A.23) Kai (x), Fˆjb (y) = η η D

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