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2.1 BF state The three constraints of canonical quantum gravity without a can be derived from the Holst action [10]

1 I J KL 2 I J SH = ǫIJKL e ∧ e ∧ Ω − e ∧ e ∧ ΩIJ , (1) 4k Z  β  where k =8πG, eI is the tetrad, and Ω = dω + ω ∧ ω is the curvature of the spin connection ωIJ . Capital Latin indices I, J, . . . are used as Lorentz indices. Performing the Legendre transformation, one obtains

1 4 a ˙ (β)i i b SH = d x Ei A a + λ Gi + N Vb + NH , (2) kβ Z   ˙ (β)i (β)i i a where A a = LtA a , λ , N , and N are Lagrange multipliers, and Gi,Vb, and H are Gauss, diffeomorphism, and Hamiltonian constraints respectively. Letters i, j, . . . and a, b, . . . de- (β)i note 3D internal and spatial indices, respectively. The configuration variable A a = i i i Γa + βKa is constructed from the Levi-Civita spin connection Γa and the extrinsic cur- i I vature Ka. Choosing the time gauge ea|I=0 = 0, the canonical momentum variable can be a j a written as Ei = det(eb)ei . For β = i, the action (1) can be written only with the left-handed variables: i S(+) = Σ(+)IJ ∧ Ω(+) , (3) H k Z IJ where (+)IJ 1 I J i IJ K L Σ = e ∧ e − ǫ KLe ∧ e , (4) 2  2  and 1 i Ω(+) = Ω − ǫ KLΩ . (5) IJ 2  IJ 2 IJ KL The sign (+) explicitly denotes that the variable is left-handed, namely, β = i. The three constraints are

(+) (+) (+)a (+)a k (+)j (+)a Gi = (Da E )i = ∂aE i + ǫij A aE k , (6) (+) (+)a (+)i Vb = E i F ab , (7) (+) i ijk (+)a (+)b (+) H = − ǫ E i E jFab k , (8) 2 | det(E(+))| p (+)a abc (+)jk (+)i (+)i i i where E i = ǫ ǫijkΣ bc , and F ab is the curvature of the connection A a =Γa + iKa. The wave function has to satisfy the quantized constraints, which are formally written as ˆ(+) ˆ (+) ˆ (+) Gi Ψ= Vb Ψ= H Ψ=0 . (9) In Ref. [8], it is suggested that the product of the group delta functions

(+) F (+)(x) ΨBF(A )= δ e ab (10) xY∈Σ Ya,b  

2 is a solution of the constraints. This state is originally derived from the formal integral F DB exp[ iSBF ]= δ(e ), where SBF = Σ Tr (B ∧F ) is the SU(2) BF action in 3D Euclidean Rspace Σ. Thus let us call state (10) theR BF state. The group delta function has the following properties:

F (+) −1 F (+) δ g e ab g = δ e ab , (11)     (+) (+) Fab Fab δ e = 0 , (12)   (+) where g is an element of the gauge group. Therefore the state ΨBF(A ) is gauge invariant ˆ (+) (+) ˆ (+) (+) and Vb ΨBF(A ) = H ΨBF(A ) = 0. This state is proposed as a tool to construct a flat vacuum state [9].

2.2 Chiral asymmetric extension Following the strategy of Ref. [6], we first consider the chiral asymmetric model with purely imaginary values of β. The left-handed action (3) is extended to the chiral asymmetric one as follows:

(+) (+) (−) (−) S = α SH + α SH

1 (+) (−) I J KL (+) (−) I J = α + α ǫIJKL e ∧ e ∧ Ω + 2i α − α e ∧ e ∧ ΩIJ . (13) 4k Z     Here α(+) and α(−) are mixing parameters of the left- and right-handed components. The sign (−) means right-handed, i.e., β = −i. To identify the action (13) with (1), the following identities are obtained: i α(+) + α(−) =1 , α(+) − α(−) = . (14) β

Note that, in the case of the left-handed action, these parameters take α(+) = 1 and α(−) = 0. One can find that imaginary β controls the degree of the chiral asymmetry. In this model, the Poisson brackets of the canonical variables (A(+), E(+)) and (A(−), E(−)) are

ik A(±)i (x), E(±)b(y) = ± δiδbδ3(x − y) . (15) a j α(±) j a  (+)a (−)a a 0 Here E i = E i = det(e)ei in the time gauge ea = 0; nevertheless, these variables are treated independently of each other. Each of the three constraints separates into left- and right-handed components independently. Therefore, the extended wave function is given by

− (+) (−) α(+)F (+)(x) −α(−)F ( )(x) Ψ(A , A )= δ e ab δ e ab . (16) xY∈Σ Ya,b    

3 3 Generalized BF state

3.1 Real values of β To consider the extended BF state for generic real values of β, new configuration variables are introduced:

(− 1 )i − − i 1 i A β = α(+)A(+) + α( )A( ) =Γ − K , (17) a a β a 1 A(β)i = α(+)A(+) − α(−)A(−) =Γi + βKi . (18) a α(+) − α(−) a a  The corresponding momentum variables are 1 Ca = E(+)a − E(−)a = ǫabce e0 , (19) i 2i i i bi c 1  Ea = E(+)a + E(−)a = det(e)ea . (20) i 2 i i i  One can obtain the Poisson bracket relations as follows:

1 (− β )i b i b 3 A a(x),Cj (y) = kδjδaδ (x − y) , (21) n o (β)i b i b 3 A a(x), Ej (y) = kβδjδaδ (x − y) . (22)  To construct the generalized BF state, we attempt to define the extended BF (EBF) action

(+) (+) (+) (−) (−) (−) SEBF = Tr α e ∧ F − α e ∧ F , (23) Z   where e(±) are triads playing the role of the B field of the BF action and are written as

(±)i i 1 ijk b c e a = ea = ǫabcǫ Ej Ek . (24) 2 | det(E)| p (− 1 ) (β) The action SEBF is expressed in terms of the variables A β and A as

i 2 SEBF = Tr e ∧ F − 1+ β K ∧ K β Z    i 1 1 = Tr e ∧ 1+ R − F − βdΓK , (25) β Z   β2  β2 

(β) where F and R are the curvatures of the connections A and Γ respectively, and dΓK = dK + [Γ,K]. The last term vanishes for the torsion-free condition dΓe = 0. One can propose an extended BF state defined in the following form:

1 i (β) (− β ) Ψ(A , A ) = De exp SEBF Z α(+) − α(−)  1 1 = δ exp 1+ R (x) − F (x) . (26)   β2  ab β2 ab  xY∈Σ Ya,b

4 Note that when β = i, state (26) keeps the ordinary form (10). This state has a problem. 0 Due to the gauge fixing ea = 0, the wave function should satisfy the additional constraint:

ˆa δ Ci Ψ= −ik 1 Ψ=0 . (27) (− β )i δA a

(− 1 ) This equation implies that the wave function does not depend on the variable A β . How- ever, the connection Γ included in the curvature R is the explicit function of both A(β) and (− 1 ) A β , namely, (− 1 ) A(β)i + β2A β i Γi = a a . (28) a 1+ β2 To avoid this problem, we regard the connection Γ as the explicit variable of E, instead (β) (− 1 ) of A and A β . This can be done via the torsion-free condition dΓe = 0. Taking this modification into account, the extended BF state is redefined:

(β) 1 1 ΨR(A )= δ exp 1+ Rab(x) − Fab(x) . (29)   β2  β2  xY∈Σ Ya,b

(β) (β) Although the state ΨR(A ) has the same form as (26), it is the explicit function of A only, and is parameterized by the Levi-Civita curvature R. It is an analog of the fact that the ordinary wave function Ψp(x) = exp[ −i(Et − p · x)] can be regarded as the position function parameterized by the momentum.

3.2 Constraints and inner products Here, we confirm whether state (29) satisfies the three constraints with real values of β. The Gauss constraint requires the wave function to be invariant under the SU(2) gauge (β) transformation. The state ΨR(A ) is gauge invariant because of the property of the group delta function (11). The simple inner product between two states can be supposed as

† (β) (β) h ΨR′ |ΨR i = DA Ψ ′ (A )ΨR(A ) Z R

1 ′ = δ exp 1+ (Rab − R )   β2  ab  Yx Ya,b ≡ δ (R − R′) . (30)

Here DA is the appropriate measure of the connection A(β) normalized such that DA = 1. This inner product is too sensitive. When R′ takes a different value from R, it vanishes,R even if R and R′ are in the equivalence class of SU(2) gauge and diffeomorphism transformations. To make the inner product more convenient, the following new inner product is introduced:

(ΨR′ |ΨR ) = Dφ h ΨR′ | U(φ) |ΨR i Z = Dφ δ (R − φR′) . (31) Z

5 Here U(φ) is an operator of the gauge and diffeomorphism transformations. The integral Dφ is over all of both transformations. This construction of the inner product is an analogy Rof LQG [11]. One can find that the dual state

(ΨR′ | = Dφ h ΨR′ | U(φ)= Dφ h ΨφR′ | (32) Z Z is a solution of the Gauss and diffeomorphism constraints. Finally, we consider the Hamiltonian constraint

β ijk a b 2 l m H = − ǫ E E Fabk − 1+ β ǫklmK K . (33) E i j a b 2 | det( )|    p Performing a similar calculation to (25), the smeared Hamiltonian constraint is deformed as

H(N) = d3x NH Z

3 Nβ ijk a b 1 1 = − d x ǫ Ei Ej 1+ Rabk − Fabk . (34) Z 2 | det(E)|  β2  β2  p (β) Therefore, if the Levi-Civita curvature operator Rˆ can be well defined, i.e., RˆΨR(A ) = (β) (β) RΨR(A ), then the state ΨR(A ) will satisfy

3 abk 1 1 (β) d x χ 1+ Rˆabk − Fˆabk ΨR(A )=0 , (35) Z  β2  β2 

where χ is a test function. According to Ref. [6], the Levi-Civita curvature operator Rˆ is defined as follows:

3 abk ′ 3 abk ′ d x χ Rˆabk = DφDR d x χ (φR ) |ΨφR′ ih ΨφR′ | , (36) Z Z Z abk 

where the integral DR′ is over the Levi-Civita curvature R′ modulo the equivalence class of the gauge and diffeomorphismR transformations. The action of this operator on the state |ΨR i becomes

3 abk ′ ′ 3 abk ′ d x χ Rˆabk|ΨR i = DφDR δ(R − φR ) d x χ (φR ) |ΨφR′ i Z Z Z abk 

3 abk = d x χ Rabk|ΨR i . (37) Z With this operator, one obtains

(β) Hˆ (N)ΨR(A )=0 . (38)

(β) Consequently, the state ΨR(A ) satisfies all three constraints.

6 4 Conclusions and discussion

In this paper, we have constructed the generalized BF state for real values of β. This has been done via the chiral asymmetric extension. The generalized state is an explicit function of the connection A(β), and is parameterized by the Levi-Civita curvature R as well as in the generalized CS state. It is gauge invariant and solves all constraints with the appropriate inner product and the operator. 2 This state would be associated with the space such that (1+β )Rab −Fab = 0. It contains a special case, i.e., a flat 3D space R = F = 0. More discussions are necessary to obtain further specific interpretations. Problems with the connection with generic β may arise, because this connection is not a pull-back of a space-time connection [12]. (β) It would be interesting to consider a loop representation of the state ΨR(A ):

(β) (β) ΨR(γ) = h γ|ΨR i = DA h γ|A ih A |ΨR i Z (β) 2 ∼ DA W (A ,γ) δ exp Fab − (1 + β )Rab . (39) Z Yx Ya,b   Here W (A(β),γ) is a spin network with a graph γ, which is a generalized Wilson loop con- structed from holonomy edges and invariant tensors. The part

2 DA δ exp Fab − (1 + β )Rab (40) Z Yx Ya,b   looks like a generating functional of the spin foam model with a source term, which is known as the Freidel–Krasnov (FK) model [13]. Let us consider a discretized 3D space with a triangulation ∆. The corresponding dual cell ∆∗ has vertices v, edges l, and faces f. In the FK model, the discretized generating functional is given by

ZFK[J] = DADB exp i Tr (B ∧ F + B ∧ J) Z  Z 

(Λf ) Jv Jvn = dgl dim(Λf ) Tr R gl e 1 ··· gl e , (41) Z 1 n Yl Yf ΛfX∈Irrep  

(Λ) where J is a source term for B, R (gl) is a representation of the group element gl = exp[ l A] with a spin label Λ, vertices v1, ··· , vn and edges l1, ··· ,ln are associated with the n-polygonalR ∂f, and the sum is taken over all irreducible representations. Thus the loop representation (39) is expressed as

(β) (Λf ) rv rvn ΨR(γ) = dgl W (A ,γ) dim(Λf ) Tr R (gl e 1 ··· gl e ) , (42) Z l 1 n Yl Yf ΛfX∈Irrep   where r = −(1 + β2)R. The limit R → 0 is consistent with the spin network invariant ΨR=0(γ) in Ref. [8].

Acknowledgements

The authors would like to thank H. Taira, T. Oka, and K. Eguchi for helpful discussions.

7 References

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[11] A. Perez, Introduction to loop quantum gravity and spin foams, arXiv:gr-qc/0409061.

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