Hamiltonian analysis of SO(4,1) constrained BF theory
aa r X i v : . [ g r- q c ] M a r Hamiltonian analysis of SO (4 , constrained BF theory R. Durka ∗ and J. Kowalski-Glikman † Institute for Theoretical Physics, University of Wroc law, Pl. Maxa Borna 9, Pl–50-204 Wroc law, Poland (Dated: October 18, 2018)In this paper we discuss canonical analysis of SO (4 ,
1) constrained BF theory. The action of thistheory contains topological terms appended by a term that breaks the gauge symmetry down to theLorentz subgroup of SO (3 , B field equations one finds that the action of this theory containsnot only the standard Einstein-Cartan term, but also the Holst term proportional to the inverseof the Immirzi parameter, as well as a combination of topological invariants. We show that thestructure of the constraints of a SO (4 ,
1) constrained BF theory is exactly that of gravity in Holstformulation. We also briefly discuss quantization of the theory.
I. INTRODUCTION
One of the most remarkable developments in generalrelativity of the last decades was Ashtekar’s discoverythat the phase space of gravity can be described withthe help of a background independent theory of self-dual SU (2) connection [1]. This discovery became a founda-tion of the research program of Loop Quantum Gravity[2], [3]. The original Ashtekar’s formulation was gen-eralized few years later by Barbero to the case of realconnections [4], parametrized by a single real number γ ,called the Immirzi parameter [5]. It turns out that thisparameter is in fact an additional dimensionless couplingconstant of the gravitational action, which takes the sym-bolic form [6] S grav = 1 G Z e αµ e βν R ρσγδ (cid:18) ǫ αβγδ + 1 γ δ γδαβ (cid:19) ǫ µνρσ − Λ3 G e , (1.1) e ≡ ǫ µνρσ ǫ αβγδ e αµ e βν e γρ e δσ . In the action above e α is the tetrad one-form and R αβ is the curvature two-form of the Lorentz connection ω αβ ,where the Lorentz algebra indices α, β, . . . run from 0to 3. Normally the second term, called the Holst term,regardless of not being a total derivative, does not af-fect field equations, because its contribution vanishes onshell (for zero torsion) by virtue of the Bianchi identity.In spite of this, its presence is not completely innocent:it affects canonical structure of the classical theory, andquantum theories for different γ lead to different physi-cal predictions (for example the expression for black holeentropy calculated in this framework depends on γ [7]).It has been noticed in [8] that from Wilsonian perspec-tive it would be quite unnatural not to append the action(1.1) with all possible terms that are compatible withthe field content ( e and ω ) and (local Lorentz and diffeo-morphism) symmetries of the theory. It turns out that ∗ [email protected] † [email protected] there are only three such terms corresponding to threetopological invariants (Pontryagin, Euler and Nieh-Yanclasses, see (4.8)–(4.10) below). Again, the presence ofthese terms does not influence the classical field equa-tion when the constant time slices of the spacetime arecompact without boundaries. However, they may playan important role in quantum theory and/or in the casewhen boundaries are present. In the formulation of [8]all these terms come with a priori independent couplingconstants and one wonders if it would be possible to finda formulation of the theory so as to organize them in aunified way.Such a formulation is known for quite some time andis dubbed constrained BF theory. The idea that gravitycan be formulated as a constrained topological BF theoryhas its roots in works of MacDowell and Mansouri [9] andof Plebanski [10]. The starting point of the present workwill be the following action, proposed and discussed in[11] (see also [12]), S = Z d x ǫ µνλρ (cid:16) B µνIJ F IJλρ − β B µνIJ B IJλρ − α ǫ IJKL B IJµν B KLλρ (cid:17) . (1.2)In this action F µν IJ = ∂ µ A ν IJ − ∂ ν A µIJ + A µI K A ν KJ − A νI K A µKJ is the field strength of the SO (4 ,
1) (or SO (3 , A µIJ , while B µν IJ is a two-form field valued in thealgebra of the same gauge group. The capital Latin in-dices I, J, K, . . . are the algebra ones and run from 0 to4, when the Lorentz subalgebra of the gauge algebra islabeled by Greek indices from the beginning of the alpha-bet α, β, γ, . . . running from 0 to 3. We will decomposethem into timelike 0 and spacelike a, b, c, . . . . Below, inthe course of Hamiltonian analysis we also decomposethe spacetime indices µ, ν into time and space denotingthe space indices by letters from the middle of the Latinalphabet i, j, k, . . . .As we will show in the next section, the theory definedby the action (1.2) is equivalent to Einstein–Cartan the-ory with action accompanied with the Holst term andthe topological terms described above. The six couplingconstants of [8] are then replaced by two dimensionlesscouplings α and β of (1.2) and one dimensionful scale ℓ .In the Sec. II we will discuss the canonical formulationof this theory, while in Secs. IV and V we will showhow these constraints can be simplified and recast intothe form proposed by Holst. In the final section we willmake some comments concerning perturbative quantiza-tion of the theory around Kodama state. II. GRAVITY AS A CONSTRAINED BFTHEORY
In this section we will recall some properties of theaction (1.2). It has been shown in [11] that this actionis equivalent to the standard action of Einstein-Cartangravity. To see this one first decomposes the connection A µIJ into tetrad and Lorentz connection A µα = 1 ℓ e αµ , A µαβ = ω µαβ , (2.1)with ℓ being a length scale, necessary for dimensionalreasons since the connection on the left hand side hasthe dimension of inverse length, while tetrad is dimen-sionless associated with the cosmological constant1 ℓ = Λ3Then one solves equations of motion for B and substi-tutes the result back into the action. As a result onefinds Einstein action appended with a number of topo-logical invariants. To find its canonical form one has toassociate the dimensionless coupling constants α and β of (1.2) with the physical ones: Newton’s constant G , thecosmological constant Λ, and the Immirzi parameter γ : α = G Λ3 1(1 + γ ) , β = G Λ3 γ (1 + γ ) , γ = βα . (2.2)Instead of repeating this derivation here, let us showthat field equations resulting from the action (1.2) are thestandard vacuum Einstein equations. The field equationsread ǫ µνρσ ( D Aµ B νρ ) IJ = 0 , (2.3) ǫ µνρσ (cid:16) F µν IJ − β B µνIJ − α ǫ IJKL B µνKL (cid:17) = 0 . (2.4) In our approach all generators of the gauge algebra are dimen-sionless. Alternatively, one can use dimensionful generators ofthe translational part of the algebra (as it is usually done whenone wants eventually to make the algebra contraction). Thenmomentum generators have canonical dimension of inverse lengthand ℓ shows up in the algebra as well. In (2.3) D Aµ is the covariant derivative defined by connec-tion A , so that( D Aµ B νρ ) IJ = ∂ µ B νρIJ + A IµK B νρKJ + A JµK B νρIK . The theory defined by (1.2) for non-zero α breaks theoriginal de Sitter SO (4 ,
1) gauge symmetry down toLorentz SO (3 , D Aµ into Lorentz so (3 , ω (2.1), to wit( D Aµ B νρ ) αβ = ( D ωµ B νρ ) αβ − ℓ e µα B νρβ + 1 ℓ e µβ B νρα , (2.5)( D Aµ B νρ ) α = ( D ωµ B νρ ) α − ℓ e µβ B νραβ , (2.6)where( D ωµ B νρ ) αβ = ∂ µ B νραβ + ω µαγ B νργβ + ω µβγ B νραγ (2.7)with an obvious generalization for another Lorentz ten-sors. Using this decomposition we rewrite the field equa-tions (2.4), (2.5) as ǫ µνρσ (cid:18) D ωµ B νραβ − ℓ e µα B νρβ + 1 ℓ e µβ B νρα (cid:19) = 0 , (2.8) ǫ µνρσ (cid:18) D ωµ B νρα − ℓ e µβ B νραβ (cid:19) = 0 , (2.9) F µν αβ − β B µν αβ − α ǫ αβγδ B µν γδ = 0 , (2.10) F µν α − β B µν α = 0 . (2.11)Notice that the curvature in (2.10) is the sum of Riemanntensor of ω and the cosmological curvature F µν αβ = R µναβ − ℓ (cid:0) e µα e νβ − e να e µβ (cid:1) , (2.12)while that in (2.11) is just the torsion F µν α = 1 ℓ (cid:0) D ωµ e να − D ων e µα (cid:1) = 1 ℓ T µν α . (2.13)Solving (2.10) and (2.11) for B we find B µνα = 1 β F µν α , B µν αβ = 12 M αβγδ F µν γδ , (2.14)where M αβγδ = 1( α + β ) ( βδ αβγδ − αǫ αβγδ ) , (2.15)with δ αβγδ ≡ δ αγ δ βδ − δ βγ δ αδ . The tensor M is a sum ofLorentz invariant tensors and, therefore, its covariantderivative D ωµ vanishes.Substituting (2.14) into (2.8) and using Bianchi iden-tity for Riemann curvature one can check that the result-ing equation forces torsion T µνα = ℓF µν α to vanish .Using this it is easy to see that (2.9) is equivalent toEinstein equations with cosmological constant Λ = 3 /ℓ .This completes the proof that field equations followingfrom the action (1.2) reproduce the standard Einsteinequations.It should be noticed that when the coupling constant α = 0 the theory becomes topological, so that the lastterm in the action (1.2) that explicitly breaks the gaugesymmetry from the topological SO (4 ,
1) down to physi-cal SO (3 ,
1) carries all the information about dynamicallocal degrees of freedom of gravity. As we will see belowthis fact is clearly reflected in the structure of constraintsalgebra.
III. CANONICAL ANALYSIS
In the first step of canonical analysis of the constrainedBF theory defined by (1.2) let us decompose the curva-ture F µν IJ into electric and magnetic parts F µν IJ → ( F iIJ , F ij IJ ) (3.1)with F iIJ = ˙ A iIJ − ∂ i A IJ + A I K A iKJ − A iI K A KJ = ˙ A iIJ − D i A IJ (3.2)where the dot denotes the time derivative, D i is the co-variant derivative for the connection A iIJ , and F ij IJ = ∂ i A j IJ + A iI K A j KJ − i ↔ j . (3.3)As usual the zero component of the connection becomesa Lagrange multiplier for Gauss law. Further we decom-pose B field into B µν IJ → (cid:0) B iIJ ≡ B iIJ , P iIJ ≡ ǫ ijk B jkIJ (cid:1) . (3.4)As we will see shortly, P iIJ turn out to be momenta as-sociated with spacial components of gauge field A , whilethe remaining components of B play a role of Lagrangemultipliers.Using these definitions and integrating by parts we canrewrite the action as follows S = Z dtL , (3.5) L = Z d x (cid:0) P iIJ ˙ A iIJ + B iIJ Π iIJ + A IJ Π IJ (cid:1) . (3.6) To prove this one has to assume invertibility of the tetrad.
It is clear that B iIJ and A IJ are Lagrange multipliersenforcing the constraints Π iIJ and Π IJ , which explicitlyread: Π IJ ( x ) = (cid:0) D i P i (cid:1) IJ ( x )= (cid:16) ∂ i P iIJ + A iI K P iKJ + A iJ K P iIK (cid:17) ( x ) (3.7)which is the Gauss law for SO (4 ,
1) invariance (see be-low), andΠ iIJ ( x ) = (cid:16) ǫ ijk F jkIJ − β P iIJ − α ǫ IJKL P iKL (cid:17) ( x )(3.8)The Poisson bracket of the theory is (cid:8) A iIJ ( x ) , P jKL ( y ) (cid:9) = 12 δ ( x − y ) δ ji δ IJKL . (3.9)(The factor 1 / A defined as δL/δ ˙ A is 2 P ,not P .) The Lagrangian (3.6) contains just the standard( p ˙ q ) kinetic term appended with a combination of con-straints, reflecting the manifestation of diffeomorphisminvariance of the action (1.2) that we have started with.It is worth noticing that prior to taking care of the con-straints the dimension of phase space of the system is2 × ×
10 = 60 at each space point. As we will see thedimension of the physical phase space is going to be 4,as it should be.The Poisson brackets of the constraints can be straight-forwardly computed and read { Π IJ ( x ) , Π KL ( y ) } = δ ( x − y ) (cid:16) η IL Π JK ( x ) − η JL Π IK ( x ) − η IK Π JL ( x ) + η JK Π IL ( x ) (cid:17) ≈ IJ form a representation of the gaugegroup SO (4 ,
1) of the unconstrained theory ( α = 0), asexpected. Further { Π iIJ ( x ) , Π j KL ( y ) } = 2 αǫ ijk δ ( x − y ) (cid:16) ǫ KLIP A kP J ( x ) − ǫ KLJP A kP I ( x ) + ǫ IJKP A kP L ( x ) − ǫ IJLP A kP K ( x ) (cid:17) (3.11)and { Π IJ ( x ) , Π iKL ( y ) } = − α δ ( x − y ) (cid:16) ǫ KLIP P iP J ( x ) − ǫ KLJP P iP I ( x ) (cid:17) + α δ ( x − y ) (cid:16) η IL ǫ JKMN − η JL ǫ IKMN − η IK ǫ JLMN + η JK ǫ ILMN (cid:17) P iMN ( x )+ 12 δ ( x − y ) (cid:16) η IL Π iJK ( x ) − η JL Π iIK ( x ) − η IK Π iJL ( x ) + η JK Π iIL ( x ) (cid:17) . (3.12)It is worth noticing that in the topological limit α = 0 allthe constraints are first class. This observation leads tothe following, apparent puzzle. Namely, as we said abovethe kinematical phase space is 60 dimensional. On theother hand for α = 0 we have 10+30 first class constraintsthat remove from this phase space 80 degrees of freedom.How is this possible? To answer this let us notice that notall the constraints are independent. Indeed taking thecovariant divergence of the Π iIJ constraint and makinguse of the Bianchi identity we see that( D i Π i ) IJ = − β Π IJ (3.13)and thus the set of constraints is reducible. It followsthat we have only 30 independent first class constraintsΠ iIJ , which remove exactly 60 dimensions from the phasespace, as it should be since the theory with α = 0 istopological.Returning to the case α = 0 we notice that the action(1.2) is invariant under local gauge transformations thatbelong to the Lorentz subgroup SO (3 ,
1) of the initial deSitter group SO (4 , . It follows that it is natural toexpect that one can simplify the algebra of constraints(3.10)–(3.12) if one decomposes the constraints into thatbelonging to the Lorentz and the translational parts ofthe algebra. From (3.7) we getΠ α ( x ) ≡ Π α ( x ) = (cid:0) D ωi P i (cid:1) α ( x ) − ℓ e iβ ( x ) P iαβ ( x ) ≈ αβ ( x ) = (cid:0) D ωi P i (cid:1) αβ ( x ) − ℓ e iα ( x ) P iβ ( x ) + 1 ℓ e iβ ( x ) P iα ( x ) ≈ iα ( x ) ≡ Φ iα ( x ) = (cid:0) ǫ ijk F jkα ( x ) − β P iα ( x ) (cid:1) ≈ iαβ ( x ) ≡ Φ iαβ ( x ) == (cid:16) ǫ ijk F jkαβ ( x ) − β P iαβ ( x ) − α ǫ αβγδ P iγδ ( x ) (cid:17) ≈ (cid:8) Π α ( x ) , Φ iγδ ( y ) (cid:9) ≈ − α δ ( x − y ) ǫ γδαρ P iρ ( x ) (3.18) (cid:8) Π α ( x ) , Φ iγ ( y ) (cid:9) ≈ − α δ ( x − y ) ǫ αγρσ P iρσ ( x ) (3.19) In what follows we restrict ourself to the positive cosmologicalconstant case; the negative cosmological constant and the Antide Sitter group SO (3 ,
2) can be analyzed analogously. { Φ iα ( x ) , Φ j βγ ( y ) } ≈ α ǫ ijk δ ( x − y ) ǫ αβγδ A kδ ( x ) =2 αℓ ǫ ijk δ ( x − y ) ǫ αβγδ e kδ ( x ) (3.20)Now we can turn to the next step of canonical analysis,i.e., to checking if there are any tertiary constraints. TheHamiltonian, being a combination of constraints reads H = − A α Π α − A αβ Π αβ − B iα Φ iα − B iαβ Φ iαβ (3.21)It follows from (3.18-3.20) that we have to satisfy thefollowing conditions to ensure that the constraints arepreserved by time evolution, generated by hamiltonian(3.21)˙Π α = α (cid:0) B iβ P iγδ + B iβγ P iδ (cid:1) ǫ αβγδ ≈ iα = − α (cid:0) ℓ ǫ ijk B jβγ e kδ − A β P i γδ (cid:1) ǫ αβγδ ≈ iαβ = − α (cid:0) ℓ ǫ ijk B jγ e kδ + A γ P i δ (cid:1) ǫ αβγδ ≈ B iβ , B iβγ , A γ with arbitrary coefficients) and thus there are no tertiaryconstraints.Notice however that there is an ambiguity in Diracprocedure in the case of diff-invariant systems, i.e., suchthat hamiltonian is a combination of constraints. Theusual approach is to check if one can solve the vanishingof time derivative of the constraints condition for La-grange multipliers, as we did above. But this is, clearly,not a general solution of these conditions. In general onemay look for the solutions with arbitrary values of theLagrange multipliers, but instead restricting the phasespace (for example if we impose the condition that allthe Lagrange multipliers in (3.22)–(3.24) are arbitrarythere would be additional constraints saying that com-ponents of tetrad and momenta are to be equal zero.)Notice that this problem does not arise in the case ofthe hamiltonian not being weakly zero, because then theresulting equations pertaining to the time invariance ofthe constraints are non-homogeneous. Thus the proce-dure that is usually employed does not seem to provide acomplete characterization of the phase space, but we willadopt it here, leaving the discussion of this subtle pointto the future work. IV. SIMPLIFYING THE CONSTRAINTS
The aim of this section is to rewrite the system of con-straints (3.10)–(3.12) in a form that makes comparisonwith constraints of General Relativity with Holst term,discussed in [6]. In what follows we will borrow someideas from the paper of Perez and Rezende [8]. (Similarideas, albeit in more restricted setting, were discussed,e.g., in [13] and [14].)In the first step let us rearrange the constraints (3.10)–(3.12) to write them in the following formΦ iα = P iα − ℓβ ǫ ijk D ωj e k α ≈ iαβ = P iαβ − M αβγδ F jk γδ ǫ ijk ≈ αβ = 2 ℓ ǫ ijk D ωi (cid:16) K αβγδ e j γ e k δ (cid:17) ≈ α = 1 ℓ ǫ ijk K αβγδ e βi R jk γδ − α ( α + β ) ℓ ǫ ijk ǫ αβγδ e βi e γj e δk ≈ α and β satisfy theidentity α/ ( α + β ) = ℓ /G , while the operators M and K are defined to be M αβγδ ≡ α ( α + β ) ( γ δ αβγδ − ǫ αβγδ ) , (4.5) K αβγδ ≡ α ( α + β ) ( 1 γ δ αβγδ + ǫ αβγδ ) . (4.6)Also recall that the action (1.2), after solving for B andexpressing the resulting action in terms of the SO (3 , ω and tetrad e , has the form [11] S = 1 G Z ǫ αβγδ ( R µν αβ e ρ γ e σ δ − Λ3 e µ α e ν β e ρ γ e σ δ ) ǫ µνρσ + 2 Gγ Z R µν αβ e αν e βρ ǫ µνρσ + γ + 1 γ G N Y + 3 γ G Λ P − G Λ E . (4.7)One immediately recognizes here the standard gravita-tional action in the first line, and the Holst term, whosestrength is governed by the Immirzi parameter γ = β/α in the second. The last three terms are proportional totopological invariants (Nieh-Yan, Ponryagin, and Euler): N Y = Z ( T µν α T αρσ − R µν αβ e αν e βρ ) ǫ µνρσ , (4.8) P = Z R µν αβ R αβρσ ǫ µνρσ , (4.9) E = Z R µν αβ R ρσ γδ ǫ αβγδ ǫ µνρσ . (4.10)As we will show, in the case when the constant timesurface is without boundaries ∂ Σ = 0, the topologicalterms play the role of the generating functional for canon-ical transformations, which simplify the constraints con-siderably [8]. The key observation is that Pontryagin andNieh-Yan invariants can be expressed as total derivatives
N Y = 4 Z ∂ µ (cid:16) e ν α D ωρ e ασ (cid:17) ǫ µνρσ (4.11) P = 4 Z ∂ µ (cid:16) ω ν ab ∂ ρ ω abσ + 23 ω ν ab ω aρ c ω cbσ (cid:17) ǫ µνρσ (4.12) The same holds for Euler class. However in this case onehas to make use of self and anti-self dual combinations ofLorentz connection ± ω αβi = 12 ( ω αβi ∓ i ǫ αβγδ ω γδi ) , ± ω γδi ǫ αβγδ = ± i ± ω αβi (4.13)and curvature (see e.g., [15]) ± R αβµν = 12 (cid:16) R αβµν ∓ i ǫ αβγδ R γδµν (cid:17) . (4.14)It can be checked that both Pontryagin and Euler classcan be rewritten with the help of ± R αβµν as follows P = Z ǫ µνσρ ( + R αβµν + R ρσ αβ + − R αβµν − R ρσ αβ ) (4.15) E = 2 i Z ǫ µνσρ ( + R αβµν + R ρσ αβ − − R αβµν − R ρσ αβ ) (4.16)Introducing C µ ( ω ) = (cid:16) ω ν αβ ∂ ρ ω abσ + 23 ω ν ab ω aρ c ω cbσ (cid:17) ǫ µνρσ (4.17)we write Pontryagin and Euler classes as total derivatives P = 4 Z (cid:16) ∂ µ C µ ( + ω ) + ∂ µ C µ ( − ω ) (cid:17) (4.18) E = 8 i Z (cid:16) ∂ µ C µ ( + ω ) − ∂ µ C µ ( − ω ) (cid:17) (4.19)Therefore the topological part of action (4.7) takes theform S T = 4 βℓ Z ∂ µ (cid:0) e ν α D ωρ e ασ (cid:1) ǫ µνρσ + 2 α ( α + β ) βα Z ∂ µ (cid:16) C µ ( + ω ) + C µ ( − ω ) (cid:17) − i α ( α + β ) Z ∂ µ (cid:16) C µ ( + ω ) − C µ ( − ω ) (cid:17) . (4.20)It is worth noticing that in spite of the presence of theimaginary i here, the action S T is real (for real γ .)For constant time surfaces, being a manifold withoutboundary ( ∂ Σ = 0), all total spacial derivatives termsdrop out and only the ones with total time derivativesurvive S T = Z ∂ W ( e, ω ) , where W ( ω, e ) is a functional of torsion and self and anti-self dual Chern-Simons forms L CS ≡ C W ( e, ω ) = 4 βℓ Z Σ ǫ ijk (cid:0) e i α D ωj e αk (cid:1) + (4.21)+ 2 α ( α + β ) Z Σ (cid:16) ( γ − i ) L CS ( + ω ) + ( γ + i ) L CS ( − ω ) (cid:17) . Having the functional W we can make canonical trans-formation, which defines new momenta P ia , P iab of thetetrad e and the connection ω , respectively P iα = P iα + {P iα , W ( ω, e ) } , P iαβ = P iαβ + {P iαβ , W ( ω, e ) } (4.22)with n e αi , P jβ o = 12 ℓ δ ji δ αβ and n ω αβi , P jγδ o = 12 δ ji δ γδαβ . (4.23)Since the variations of the functional W ( ω, e ) are12 δWδω αβi = M αβγδ R jk γδ ǫ ijk − βℓ e j α e k β ǫ ijk (4.24)12 δWδe αi = 4 ℓβ ǫ ijk D ωj e k α (4.25)we find that the resulting constraints, expressed in termsof new momenta (4.22) take the formΦ iα = P iα ≈ , (4.26)Φ iαβ = P iαβ − ℓ K γδαβ e j γ e k δ ǫ ijk ≈ αβ = 2 ℓ ǫ ijk K αβγδ D ωi (cid:16) e j γ e k δ (cid:17) ≈ α = 1 ℓ ǫ ijk K αβγδ e βi F jk γδ ≈ V. TIME GAUGE
In order to make contact with the Hamiltonian analysisof Holst, we have to fix the gauge so as to remove thetime component of the tetrad and then to relate momentaassociated with Lorentz connection with an appropriatecombination of the remaining tetrad components.To this end, let us introduce the gauge condition whichmust be added to the list of constraints e i ≈ P i ≈ P ia ≈ P i a + 2 αℓ ( α + β ) ǫ ijk ǫ abc e bj e ck ≈ P iab − αℓ ( α + β ) 1 γ ǫ ijk δ cdab e j c e k d ≈ ǫ abc = ǫ abc and ǫ abc = − ǫ abc .Combining the last two equations we find constraintsfor generalized self and anti self-dual parts of P + P ia ≈ − P ia + 4 α ( α + β ) ℓ ǫ ijk ǫ abc e bj e ck ≈ ± P ia = P i a ± γ ǫ abc P i bc (5.7)It can be easily checked that ± P are momenta associatedwith generalized (anti) self-dual combinations of Lorentzconnection (which for γ = ± i become usual self and antiself dual ones) ± w ai = ω ai ± γ ǫ abc ω i bc (5.8)with the Poisson brackets being { ∓ w ai , ± P jd } = 0 , { ± w ai , ± P jd } = δ ji δ ad . (5.9)Let us now turn to the constraint Π αβ (4.28). Decom-posing it into components we findΠ ab = 2 α ( α + β ) ℓ ǫ ijk (cid:16) − ǫ abc ω di e j d e ck ++ 2 γ (cid:0) ∂ i ( e j a e k b ) + ω ci a e j c e k b − ω ci b e j c e k a ) (cid:17) (5.10)Π a = − α ( α + β ) ℓ ǫ ijk (cid:16) ǫ abc (cid:0) ∂ i ( e bj e ck ) + 2 ω bdi e j d e ck ) + 2 γ ω bi e j b e k a (cid:17) (5.11)Taking the combination Π a ± γ ǫ abc Π bc we get4 α ( α + β ) ℓ ǫ ijk (cid:16) γ γ (cid:17) ω bi e j a e k b ≈ α ( α + β ) ℓ ǫ ijk (cid:16)(cid:0) − γ γ (cid:1) ω bi e j a e k b − ǫ abc (cid:0) ∂ i ( e bj e ck ) + 2 ω bdi e j d e ck ) (cid:17) ≈ . (5.13)From these two equations it follows that ω i b e j a e bk ǫ ijk ≈ (cid:16) ∂ i ( e bj e ck ) + 2 ω bi d e dj e ck (cid:17) ǫ ijk ǫ abc ≈ ω ai = 12 (cid:0) + w ai + − w ai (cid:1) , ω abi = γ ǫ abc (cid:0) + w i c − − w i c (cid:1) (5.16)take the form of the Gauss and the boost constraints G a ≡ ( + w bi + − w bi ) e j a e k b ǫ ijk ≈ , (5.17) B a ≡ (cid:16) ∂ i ( e bj e ck ) ǫ abc − γ ( + w bi − − w bi ) e j a e k b (cid:17) ǫ ijk ≈ . (5.18)We can handle the scalar part of (4.29) S = α ( α + β ) ℓ (cid:16) γ e ci R jk c − ǫ abc e ai F bcjk (cid:1) ǫ ijk ≈ h(cid:16) γ γ (cid:17) ǫ dbc ∂ j ( + w k d ) + (cid:16) − γ γ (cid:17) ǫ dbc ∂ j ( − w k d )+(1+ γ ) + w bj − w ck − (cid:16) γ (cid:17) + w bj + w ck − (cid:16) − γ (cid:17) − w bj − w ck − ℓ e bj e ck i ǫ abc e ai ǫ ijk α ( α + β ) ℓ ≈ . As for the vector part of (4.29) V a = α ( α + β ) ℓ (cid:16) γ e bi R jk ab + 2 ǫ cab e bi R jk c (cid:17) ǫ ijk ≈ , (5.20)we find h ǫ abc ∂ − j w ak − (cid:16) γ γ (cid:17) + w j b + w k c − (cid:16) γ − γ (cid:17) − w j b − w k c − (cid:16) γ γ (cid:17) ( + w j b − w k c + − w j b + w k c ) i e ci ǫ ijk ℓG ≈ . It should be noted that in the case γ = − + w i a cancel. Notice also that with the help ofthe Gauss constraint the vector constraint can be reducedto the form h ǫ abc ∂ − j w ak + (cid:16) − γ γ (cid:17) − w j b − w k c i e ci ǫ ijk ≈ . This is obvious for γ = −
1. For γ = − The form of the constraints that we have obtained sofar is still not the final one. At some point we will have toget rid of the constraint + P ia ≈ + w j b of all the remaining constraints. How-ever in order to be able to do that we must simplify theform of the boost constraint (5.18). To see how this canbe done we multiply the constraint (5.5) by tetrad anddecompose the resulting constraint C ab into symmetricand antisymmetric parts C ( ab ) = + P i ( a e ib ) , C [ ab ] = + P i [ a e ib ] . (5.21)Let us now calculate the Poisson bracket of C ab with thescalar constraint { C ab , S } =(1 + γ ) ℓγG ǫ ijk e j b (cid:16) ∂ k e i a − γ ( + w ck − − w ck ) e di ǫ acd (cid:17) . (5.22)The bracket of the antisymmetric part C [ ab ] gives exactlythe boost constraint (5.18). However, the bracket of thesymmetric part C ( ab ) leads to the secondary constraint B ab ≡ ǫ ijk ( e j b ∂ k e i a + e j a ∂ k e i b ) − γ ( + w ck − − w ck ) e di ( e j b ǫ acd + e j a ǫ bcd ) ≈ . (5.23)Clearly, this constraint would arise if we impose the re-quirement that all the constraints are to be preserved inthe time evolution. Therefore one has to add B ab to theset of constraints of the theory. But then the road sud-denly becomes sunny. It suffices to note that the boostconstraints B a and the newly derived constraints B ab arejust the antisymmetric and symmetric parts of the simpleconstraint (cid:16) ci − γ ( + w ci − − w ci ) (cid:17) ≈ , (5.24)where Γ ai is a (unique) solution of the Cartan first struc-tural equation (cid:16) ∂ [ i e j ] b ) + ǫ bcd Γ c [ i e dj ] (cid:17) = 0 . (5.25)Using (5.24) and (5.6) we get rid of both + P ia and + w j b , replacing + w j b in all the remaining constraintswith the solution of (5.24). Similarly using (5.2) and(5.6) we can identify the momentum of − w j b with − α ( α + β ) ℓ ǫ ijk ǫ abc e bj e ck . What remains are therefore 3 Gauss, 3 vector and 1 scalarconstraints, all of them first class, constraining the 18-dimensional phase space of − w j b and its momenta. Thusthe dimension of physical phase space is 18 −
14 = 4 asit should. Of course, the final set of constraints we haveobtained has exactly the form of the constraints describ-ing gravity, cf. [6]. This completes our analysis of thecanonical structure of SO (4 ,
1) constrained BF theory. Instead of C [ ab ] = + P i [ a e ib ] just take the expression C ab ǫ abc = + P i a e ib ǫ abc , so { C ab ǫ abc , S } = (1+ γ ) ℓγG B c . VI. COMMENTS ON QUANTIZATION
Let us conclude this paper with some comments con-cerning quantization. Clearly, one can take the first classGauss, vector, and scalar constraints as a starting pointin construction of the quantum theory, as it is done inLoop Quantum Gravity [2], [3]. However, the structureof constraints of the original theory opens another pos-sibility of devising a perturbative expansion in parame-ter α around topological vacuum. Here we will describebriefly this perturbative theory leaving details to a sepa-rate publication.Our starting point will be the set of constraints (4.1)–(4.4). Consider now the canonical transformation (4.22).Its quantum counterpart can be easily found. To seehow, take the mechanical model in which one makes thetransformation (see [16]) p i → p ′ i = p i + { p i , f ( q ) } so that quantum mechanically we haveˆ p i → ˆ p ′ i = ˆ p i + i [ˆ p i , f (ˆ q )] . If we represent ˆ p i = i∂/∂q i then ˆ p ′ i = i∂/∂q i − ∂f ( q ) /∂q i .Therefore if we decompose the wave function ψ ( q ) =exp( − if ( q )) ψ ′ ( q ) thenˆ p ′ ψ ( q ) = exp( − if ( q )) ˆ p ′ ψ ′ ( q ) , which means that we just have to multiply the wavefunction with the phase exp( − if ( q )) and then use thestandard representation of the new momenta p ′ as thederivatives over positions. In the case at hands (4.22),it is therefore sufficient to multiply the wave function bythe prefactor exp ( − iW ( e, ω )) where W ( e, ω ) is given by(4.22), and replace all the momenta P with the new ones P . Then we can just use the constraints (4.26)–(4.29).When α = 0 these constraints reduce to the first classset P iα ≈ , P iαβ ≈ . (6.1)The wave function annihilated by them is just a con-stant, and thus the full physical wave function is a phaseexp ( − iW ( e, ω )). Clearly, and not surprisingly, in thiscase the wave function is the Kodama state [17] (strictlyspeaking this is the Kodama state for SO (4 ,
1) multipliedby the phase proportional to Euler class of a constanttime manifold.) Notice that here this state is delta func-tion normalizable, because all our constraints are real (cf.[18]). The simplicity of the zeroth order (in α ) solutionreflects the fact that to this order the theory is topologi-cal.Let us now turn to devising the α perturbative the-ory. The constraints (4.26)–(4.29) all have the formΦ = Φ (0) + α Φ (1) (for the last two Φ (0) = 0). We alsoexpand the wave function in the series in α , to witΨ = Ψ (0) + α Ψ (1) + . . . . (6.2) The problem we are facing now is that for non-zero α the constraints are no longer first class and therefore weneed a nonstandard procedure to handle them. One pos-sibility would be Gupta–Bleuler quantization [19], butthe required procedure of splitting the constraints intoholomorphic and anti-holomorphic parts is technicallycomplex and, presumably, leads to explicit breaking ofLorentz covariance (see [20] for discussion in a similarcontext.) Another possibility would be to make use ofthe master constraint program [21], [22], [23], and [20],but this is again technically involved.Instead we adopt the definition of physical wave func-tion Ψ such that the matrix elements of all the constraintsare zero h Ψ | Φ | Ψ i = 0 , (6.3)which is a weakened version of Gupta–Bleuler scheme.It should be stressed that the expression (6.3) is formal,because to make the precise sense of it we must specifythe inner product in the Hilbert space of states.Now we use (6.3) to define the perturbative theory in α . In the zeroth order we have D Ψ (0) (cid:12)(cid:12)(cid:12) Φ (0) (cid:12)(cid:12)(cid:12) Ψ (0) E = 0 , (6.4)while in the first order in α we find D Ψ (1) (cid:12)(cid:12)(cid:12) Φ (0) (cid:12)(cid:12)(cid:12) Ψ (0) E + D Ψ (0) (cid:12)(cid:12)(cid:12) Φ (0) (cid:12)(cid:12)(cid:12) Ψ (1) E + D Ψ (0) (cid:12)(cid:12)(cid:12) Φ (1) (cid:12)(cid:12)(cid:12) Ψ (0) E = 0 . (6.5)Inspecting (4.26)–(4.29) we find that it follows from (6.4),(6.5) that the zeroth order wave function has to satisfythe following four conditions0 = D Ψ (0) (cid:12)(cid:12)(cid:12) i δδe αi ( x ) (cid:12)(cid:12)(cid:12) Ψ (0) E (6.6)0 = D Ψ (0) (cid:12)(cid:12)(cid:12) i δδω αβi ( x ) (cid:12)(cid:12)(cid:12) Ψ (0) E (6.7)0 = D Ψ (0) (cid:12)(cid:12)(cid:12) ǫ ijk K αβγδ D ωi (cid:16) e j γ e k δ (cid:17) (cid:12)(cid:12)(cid:12) Ψ (0) E (6.8)0 = D Ψ (0) (cid:12)(cid:12)(cid:12) ǫ ijk K αβγδ e βi F jk γδ (cid:12)(cid:12)(cid:12) Ψ (0) E (6.9)Knowing Ψ (0) one can turn to the remaining first orderequation, resulting from (4.27), along with some of thesecond order ones, to find Ψ (1) , and then go to the nextorder analysis. We stop the discussion at this point leav-ing the details to another paper. VII. CONCLUSIONS
In this paper we have performed the canonical analy-sis of the constrained SO (4 ,
1) BF theory. This analysis,although quite involved, seems to be significantly sim-pler than the analogous one of Plebanski theory reportedin [24], leading however to the slightly more general ef-fective description of the dynamical degrees of freedomprovided by Holst constraints that include Immirzi pa-rameter. This suggests that it might be not only simpler,but also more natural to consider spin foam model asso-ciated with this particular formulation of gravity. Unfor-tunately, not much work has been done till now on the SO (4 ,
1) spin foam models, which would require to han-dle somehow not only the quadratic B field term, butalso the representation theory of SO (4 ,
1) group, whichis more complicated than the one of SU (2) group, usuallyused in the spin foam context. ACKNOWLEDGMENTS
We would like to thank Alejandro Perez and MichalSzczachor for helpful discussion. For JKG this workis supported in part by Research Project No. N202081 32/1844 and No. NN202318534 and by Polish Min-istry of Science and Higher Education Grant 182/N-QGG/2008/0. [1] A. Ashtekar, “New Variables for Classical and QuantumGravity,” Phys. Rev. Lett. (1986) 2244.[2] C. Rovelli, “Quantum Gravity,” Cambridge, UK: Univ.Pr. (2004) 455 p [3] T. Thiemann, “Modern canonical quantum general rela-tivity,”
Cambridge, UK: Cambridge Univ. Pr. (2007) [4] J. F. Barbero G., “Real Ashtekar variables for Lorentziansignature space times,” Phys. Rev. D (1995) 5507[arXiv:gr-qc/9410014].[5] G. Immirzi, “Real and complex connections for canonicalgravity,” Class. Quant. Grav. (1997) L177 [arXiv:gr-qc/9612030].[6] S. Holst, “Barbero’s Hamiltonian derived from a gener-alized Hilbert-Palatini action,” Phys. Rev. D (1996)5966 [arXiv:gr-qc/9511026].[7] C. Rovelli, “Black hole entropy from loop quantumgravity,” Phys. Rev. Lett. (1996) 3288 [arXiv:gr-qc/9603063].[8] D. J. Rezende and A. Perez, “4d Lorentzian Holst actionwith topological terms,” Phys. Rev. D , 064026 (2009)[arXiv:0902.3416 [gr-qc]].[9] S. W. MacDowell and F. Mansouri, “Unified GeometricTheory Of Gravity And Supergravity,” Phys. Rev. Lett. (1977) 739 [Erratum-ibid. (1977) 1376].[10] J. F. Plebanski, “On the separation of Einsteinian sub-structures,” J. Math. Phys. , 2511 (1977).[11] L. Freidel and A. Starodubtsev, “Quantum gravity interms of topological observables,” arXiv:hep-th/0501191.[12] L. Smolin and A. Starodubtsev, “General relativity witha topological phase: An action principle,” arXiv:hep-th/0311163.[13] A. Randono, “A New Perspective on Covariant Canon-ical Gravity,” Class. Quant. Grav. , 235017 (2008)[arXiv:0805.3169 [gr-qc]].[14] G. Date, R. K. Kaul and S. Sengupta, “Topological In-terpretation of Barbero-Immirzi Parameter,” Phys. Rev. D (2009) 044008 [arXiv:0811.4496 [gr-qc]].[15] H. Garcia-Compean, O. Obregon, C. Ramirez andM. Sabido, “Remarks on 2+1 self-dual Chern-Simonsgravity,” Phys. Rev. D , 085022 (2000) [arXiv:hep-th/9906154].[16] S. Mercuri, “From the Einstein-Cartan to the Ashtekar-Barbero canonical constraints, passing through theNieh-Yan functional,” Phys. Rev. D (2008) 024036[arXiv:0708.0037 [gr-qc]].[17] H. Kodama, “Holomorphic Wave Function Of The Uni-verse,” Phys. Rev. D (1990) 2548.[18] L. Freidel and L. Smolin, “The linearization of theKodama state,” Class. Quant. Grav. , 3831 (2004)[arXiv:hep-th/0310224].[19] J. Kowalski-Glikman, “On The Gupta-Bleuler Quanti-zation Of The Hamiltonian Systems With Anomalies,”Annals Phys. (1994) 1 [arXiv:hep-th/9211028].[20] S. Sengupta, “Quantum realizations of Hilbert-Palatinisecond-class constraints,” arXiv:0911.0593 [gr-qc].[21] J. R. Klauder, “Coherent state quantization of constraintsystems,” Annals Phys. (1997) 419 [arXiv:quant-ph/9604033].[22] T. Thiemann, “The Phoenix project: Master constraintprogramme for loop quantum gravity,” Class. Quant.Grav. (2006) 2211 [arXiv:gr-qc/0305080].[23] B. Dittrich and T. Thiemann, “Testing the master con-straint programme for loop quantum gravity. II: Finite di-mensional systems,” Class. Quant. Grav. (2006) 1067[arXiv:gr-qc/0411139].[24] E. Buffenoir, M. Henneaux, K. Noui and Ph. Roche,“Hamiltonian analysis of Plebanski theory,” Class.Quant. Grav.21