ISSN 0202-2893, Gravitation and Cosmology, 2014, Vol. 20, No. 2, pp. 132–137. c© Pleiades Publishing, Ltd., 2014.

Noether’s Charge in the Super-Group Field Cosmology

Mir Faizal*

Mathematical Institute, University of Oxford, Oxford OX1 3LB, United KingdomReceived January 20, 2013; in final form, November 20, 2013

Abstract—We analyze the third quantization of a model of supergroup field cosmology with gaugesymmetry. The effect of creation and annihilation of bosonic and fermionic universes in the multiverse isalso analyzed. We also construct a third-quantized Noether charge which is conserved even if the numberof universes is not conserved. Finally, we construct both the third-quantized BRST and the third-quantizedanti-BRST charges for this theory and use them to show that creation and annihilation of universes is aunitary process.

DOI: 10.1134/S0202289314020030

1. INTRODUCTION

In loop quantum gravity, the Hamiltonian con-straint is expressed in terms of the Ashtekar-Barberoconnection and a densitized triad, and the curvatureof the connection is expressed through the holonomyaround a loop [1–9]. The area operator in loop quan-tum gravity has a discrete spectrum and so it cannottake arbitrarily small nonzero values. Furthermore,loop quantum gravity has been studied using spinnetworks [10–13] and spin foam models [14–18].Spin foam models are generated by time evolution ofthese spin networks.

Despite the success of loop quantum gravity, theprocesses that change the topology of spacetime can-not be studied in it. This is because loop quantumgravity is a second-quantized formalism, and justlike the first-quantized formalism cannot be used toanalyze a situation where the particle number is notconserved, the second-quantized formalism cannotbe used to analyze a situation where the topologychanges dynamically. To analyze such processes, wehave to resort to a third-quantized formalism [19–22].

Third quantization of loop quantum gravity iscalled group field theory, and it is a field theory ona group manifold [23–26]. However, at present thistheory is not fully developed and understood. Onthe other hand, the minisuperspace approximationsto loop quantum gravity, called loop quantum cos-mology, are better understood [27–31]. The thirdquantization of such models is called group fieldcosmology [32–34]. Recently, a supersymmetricgeneralization of this loop quantum cosmology hasalso been made [35]. In the model thus developed,gauge symmetry was also incorporated. The BRST

*E-mail: mirfaizalmir@googlemail.com

symmetry of this gauge-invariant supergroup fieldcosmology was also studied.

It is known that in any theory with a gauge sym-metry, there are nonphysical degrees of freedom. Theymust be removed before quantizing the theory, andthis is done by fixing a gauge. Gauge fixing is in-corporated at the quantum level by adding gaugefixing and a ghost term. The action thus obtainedis invariant under a new symmetry called the BRSTsymmetry [36–39]. It is also invariant under an-other symmetry called the anti-BRST symmetry [40–43]. In conventional gauge theories, the BRST andanti-BRST symmetries have been studied in variousgauges [44–47]. Since the super-group field cosmol-ogy has been found to possess a gauge symmetry, inthis paper we will analyze the BRST and anti-BRSTsymmetries for it in the Landau gauge. We will alsoconstruct a third-quantized Noether charge and showthat this charge is conserved in the multiverse, eventhough the number of universes is not conserved.

2. THIRD-QUANTIZED NOETHERTHEOREM

In this section we will construct a third-quantizedNoether theorem. In a multiverse, different uni-verses can collide and get annihilated. Similarly, newuniverses can form from collisions of the previousuniverse. So, just as in a second-quantized inter-acting field theory the particle number is not con-served, in third-quantized loop quantum gravity, thenumber of universes is not conserved. However, ina second-quantized field theory, conserved chargescan be constructed by using the Noether theorem.These charges correspond to symmetries under whicha second-quantized field theory action is invariant.In a third-quantized theory, we can also construct a

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NOETHER’S CHARGE 133

third-quantized version of the Noether theorem. Thiswill also correspond to symmetries under which athird-quantized field theory action is invariant. Thus,in a multiverse, even after creation of new universes,these quantities will be conserved. In fact, thesequantities will be conserved in any third-quantizedprocess, including the formation and evaporation ofa black hole. Thus this formalism can also be used tosolve the information paradox for black holes.

To construct a third-quantized Noether theorem,we first mention some results in loop quantum cos-mology [1–9]. In this theory, the curvature of Ai

μ isexpressed through the holonomy around a loop, andthe area of such a loop cannot be smaller than a fixedminimum area because the smallest eigenvalue of thearea operator in loop quantum gravity is nonzero.Furthermore, the eigenstates of the volume operatorV are V|ν〉 = 2πγG|ν||ν〉, where ν = ±a2V0/(2πγG)has the dimensions of length. So, in Planck units, theHamiltonian constraint for a homogeneous isotropicuniverse with a massless scalar field φ can be writtenas [48, 49]

K2Φ(ν, φ) = [E2 − ∂2φ]Φ(ν, φ) = 0, (1)

where ν0 = 4 and

E2Φ(ν, φ) = −[B(ν)]−1C+(ν)Φ(ν + 4, φ)

− [B(ν)]−1C0(ν)Φ(ν, φ)

− [B(ν)]−1C−(ν)Φ(ν − 4, φ). (2)

Now, K1 = E, K2 = ∂φ, and ημν = (1,−1), so wehave ημνKμKν = K2. Just as the wave function offirst-quantized theories is viewed as a classical field insecond-quantized formalism, the wave function of thesecond-quantized theory is viewed as a classical fieldin third-quantized formalism. Hence, the wave func-tion of loop quantum cosmology will now be viewedas a classical field of group field cosmology. So, wecan write the free part of the group field action as

Sb =∑

ν

∫dφ Φ(ν, φ)K2Φ(ν, φ). (3)

Thus, if the third-quantized action is invariant undersome symmetry transformation δ, then we will have aconserved current corresponding to this symmetry

Jμ(ν, φ) =[

∂Lb

∂KμΦ(ν, φ)δ Φ(ν, φ)

], (4)

where

Sb =∑

ν

∫dφ Lb. (5)

Furthermore, we have KμJμ(ν, φ) = 0. We can nowuse the fact that the matter field acts like a time

variable and define a Noether charge as

QN (φ) =∑

ν

[∂Lb

∂φΦ(ν, φ)δ Φ(ν, φ)

]. (6)

Thus even if new universes form in a multiverse, thisthird-quantized Noether charge is conserved. Fur-thermore, this charge is also conserved at any topo-logical change, like formation and evaporation of ablack hole. Thus it seems that while quantum grav-itational second-quantized quantities are not con-served, third-quantized quantities are.

Now we can also construct a fermionic group fieldcosmology by using the analogy with a regular two-dimensional fermionic theory [53]. Thus, we definea fermionic field as Ψa(ν, φ) = (Ψ1(ν, φ),Ψ2(ν, φ)).The spinor indices are raised and lowered by thesecond-rank antisymmetric tensors Cab and Cab,respectively, These second-rank antisymmetric ten-sors also satisfy CabC

cb = δca. So, we have (γμ)ab =

(γμ)caCcb = (γμ)ba [54]. Now we define Kμ = (E, ∂φ),ημν = (1,−1), Kab = (γμ)abKμ, and K2 = [E2 −∂2

φ]. We can write the action for the fermionic groupfield cosmology as [35]

Sf =∑

ν

∫dφ Ψb(ν, φ)Ka

b Ψa(ν, φ). (7)

The classical equations of motion obtained from thisaction again generate the Hamiltonian constraint forloop quantum gravity. We can now define a Noethercurrent for a fermionic field theory as

Jμ(ν, φ) =[

∂Lf

∂KμΨa(ν, φ)δ Ψa(ν, φ)

], (8)

where

Sf =∑

ν

∫dφ Lf . (9)

We again have KμJμ(ν, φ) = 0. So, we can again usethe fact that the matter field acts like the time variableand define the Noether charge as

QN (φ) =∑

ν

[∂Lf

∂φΨa(ν, φ)δ Ψa(ν, φ)

]. (10)

These charges are conserved again if in the course ofinteraction of different universes in the multiverse.

3. SUPERSYMMETRY

In this section we will analyze supersymmetricgroup field cosmology. Bosonic and fermionic ac-tions describe bosonic and fermionic universes in themultiverse. Hence we can construct a supersymmet-ric gauge-invariant multiverse. To do so, we define

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134 FAIZAL

two complex scalar supergroup fields Ω(ν, φ, θ) andΩ†(ν, φ, θ) and a spinor supergroup field Γa(ν, φ, θ).All these fields are suitably contracted with genera-tors of a Lie algebra, [TA, TB ] = ifC

ABTC ,

Ω(ν, φ, θ) = ΩA(ν, φ, θ)TA,

Ω†(ν, φ, θ) = Ω†A(ν, φ, θ),

Γa(ν, φ, θ) = ΓAa (ν, φ, θ)TA. (11)

These superfields transform under infinitesimal gaugetransformations as

δΩA(ν, φ, θ) = ifACBΛC(ν, φ, θ)ΩB(ν, φ, θ),

δΩA†(ν, φ, θ) = −ifACBΩC†(ν, φ, θ)ΛB(ν, φ, θ),

δΓAa (ν, φ, θ) = ∇aΛA(ν, φ, θ). (12)

Now we define a super-covariant derivative of thesesuperfields as

∇aΩA(ν, φ, θ) = DaΩA(ν, φ, θ)

− ifACBΓC

a (ν, φ, θ)ΩB(ν, φ, θ),

∇aΩA†(ν, φ, θ) = DaΩA†(ν, φ, θ)

+ ifACBΩC†(ν, φ, θ)ΓB

a (ν, φ, θ), (13)

where Da = ∂a + Kbaθb. We also define ωa(ν, φ, θ) =

∇b∇aΓb(ν, φ, θ). We can write the action for thegauge theory as [35]

Sga =∑

ν

∫dφ [D2[Ω†(ν, φ, θ)∇2Ω(ν, φ, θ)

+ ωa(ν, φ, θ)ωa(ν, φ, θ)]]|. (14)

This theory has a gauge symmetry since it is invariantunder the supergauge transformations. We can definea Noether current for this supersymmetric field theoryby defining a Noether charge for it as

QN (φ) =∑

ν

[∂Lga

∂φΩ(ν, φ, θ)δ Ω(ν, φ, θ)

+∂Lga

∂φΩ†(ν, φ, θ)δ Ω†(ν, φ, θ)

+∂Lga

∂φΓa(ν, φ, θ)δ Γa(ν, φ, θ)

], (15)

where

Sga =∑

ν

∫dφ Lga. (16)

We need to fix a gauge before we can quantize thetheory. Thus we choose the following gauge [35]

DaΓa(ν, φ, θ) = 0. (17)

This can be incorporated at a quantum level by addingthe following gauge fixing term to the original classi-cal action:

Sgf =∑

ν

∫dφ [D2[B(ν, φ, θ)

× DaΓa(ν, φ, θ)]]|, (18)

where B(ν, φ, θ) = BA(ν, φ, θ)TA is an auxiliary su-perfield. This field can be integrated out, in the pathintegral formalism, to impose the gauge fixing condi-tion. Now we can write the ghost term by transform-ing the gauge fixing condition, replacing Λ(ν, φ, θ) =ΛA(ν, φ, θ)TA with the ghost superfield C(ν, φ, θ) =CA(ν, φ, θ)TA and then contracting it with the anti-

ghost superfield C(ν, φ, θ) = CA(ν, φ, θ)TA,

Sgh =∑

ν

∫dφ [D2[C(ν, φ, θ)

× Da∇aC(ν, φ, θ)]]|. (19)

Now the combined action S = Sga + Sgh + Sgf isinvariant under the third-quantized BRST transfor-mations given by

s ΩA(ν, φ, θ) = ifACBCC(ν, φ, θ)ΩB(ν, φ, θ),

s ΩA†(ν, φ, θ) = −ifACBΩC†(ν, φ, θ)CB(ν, φ, θ),

s CA(ν, φ, θ) = fACBCC†(ν, φ, θ)CB(ν, φ, θ),

s ΓAa (ν, φ, θ) = ∇aC

A(ν, φ, θ),

s CA(ν, φ, θ) = BA(ν, φ, θ),

s BA(ν, φ, θ) = 0. (20)

This action is also invariant under the anti-BRSTtransformations

s ΩA(ν, φ, θ) = ifACBC

C(ν, φ, θ)ΩB(ν, φ, θ),

s ΩA†(ν, φ, θ) = −ifACBΩC†(ν, φ, θ)CB(ν, φ, θ),

s CA(ν, φ, θ) = fA

CBCC(ν, φ, θ)CB(ν, φ, θ),

s ΓAa (ν, φ, θ) = ∇aC

A(ν, φ, θ),

s CA(ν, φ, θ) = −BA(ν, φ, θ)

+ fACBC

C(ν, φ, θ)CB(ν, φ, θ),

s BA(ν, φ, θ) = fACBC

C(ν, φ, θ)BB(ν, φ, θ). (21)

Both these transformations are nilpotent, s2 =s2 = 0. In fact, they also satisfy ss + ss = 0. Thesum of the ghost term with the gauge fixing term canbe written as a total third-quantized BRST or a totalthird-quantized anti-BRST variation,

Sgh + Sgf

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NOETHER’S CHARGE 135

=∑

ν

∫dφ s [D2[C(ν, φ, θ)DaΓa(ν, φ, θ)]]|

= −∑

ν

∫dφ s [D2[C(ν, φ, θ)

× DaΓa(ν, φ, θ)]]|. (22)

This can also be written as

Sgh + Sgf

=12

∑

ν

∫dφ ss [D2[Γa(ν, φ, θ)Γa(ν, φ, θ)]]|

= −12

∑

ν

∫dφ ss [D2[Γa(ν, φ, θ)

× Γa(ν, φ, θ)]]|. (23)

Thus the sum of the gauge fixing term and the ghostterm can be expressed as a combination of third-quantized BRST and third-quantized anti-BRSTtransformations.

4. UNITARITY IN THE MULTIVERSE

Now, in the multiverse, universes can collide witheach other to form other universes. From the pointof view of a single universe, this collision will appearas a big bang in which it was formed. It is possiblethat even our own universe formed in a collision of twoprevious universes. In a second-quantized theory,it is possible to show that if BRST or anti-BRSTtransformations are nilpotent, the resulting quantumfield theory is unitary. We have seen that the third-quantized BRST and anti-BRST transformations arenilpotent. Thus, in this section we will show that cre-ation and annihilation of universes in the multiverse isdescribed by a unitary process.

We first note that it is possible to constructNoether charges corresponding to the BRST andanti-BRST symmetries of group field cosmology witha gauge symmetry. The Noether charge correspond-ing to BRST is given by

QB =∑

ν

[∂Leff

∂∂φΓa(ν, φ, θ)s Γa(ν, φ, θ)

+∂Leff

∂∂φC(ν, φ, θ)s C(ν, φ, θ)

+∂Leff

∂∂φC(ν, φ, θ)s C(ν, φ, θ)

+∂Leff

∂∂φB(ν, φ, θ)s B(ν, φ, θ)

+∂Leff

∂∂φΩ(ν, φ, θ)s Ω(ν, φ, θ)

+∂Leff

∂∂φΩ†(ν, φ, θ)s Ω†(ν, φ, θ)

], (24)

and the one corresponding to anti-BRST is given by

QB =∑

ν

[∂Leff

∂∂φΓa(ν, φ, θ)s Γa(ν, φ, θ)

+∂Leff

∂∂φC(ν, φ, θ)s C(ν, φ, θ)

+∂Leff

∂∂φC(ν, φ, θ)s C(ν, φ, θ)

+∂Leff

∂∂φB(ν, φ, θ)s B(ν, φ, θ)

+∂Leff

∂∂φΩ(ν, φ, θ)s Ω(ν, φ, θ)

+∂Leff

∂∂φΩ†(ν, φ, θ)s Ω†(ν, φ, θ)

], (25)

where

S =∑

ν

∫dφ Leff. (26)

It may be noted that we have again used the fact thatmatter fields act like a time variable in this formalism.

Now these charges associated with the BRST andanti-BRST symmetry transformations commute andare conserved. These charges are also nilpotent,

Q2B = Q

2B = 0. (27)

Now we define physical states as states annihilatedby QB, QB|φp〉 = 0, or as states annihilated by QB ,QB |φp〉 = 0. The inner product of those physicalstates which are obtained from nonphysical states bythe action of these charges, vanishes with all otherphysical states. Thus the whole physical informationlies in the physical states which are not obtained bythe action of these charges on nonphysical states.

Now if the asymptotic physical states are |out, a〉and |in, b〉, then a typical S-matrix element can bewritten as

〈out|in〉 = 〈p a|S†S|p b〉. (28)

Since the states S|φpb〉 must be linear combinationsof physical states, we can write

〈p a|S†S|p b〉 =∑

i

〈pa|S†|0, i〉〈0, i|S|p b〉. (29)

Since the full S-matrix is unitary, this relation im-plies that the S-matrix restricted to the physical sub-space is also unitary. Thus scattering of universes ina multiverse is a unitary process.

GRAVITATION AND COSMOLOGY Vol. 20 No. 2 2014

136 FAIZAL

5. CONCLUSION

In this paper we have first reviewed the groupfield theory as third-quantized loop quantum gravity.We have analyzed the properties of a supersymmetricgroup field theory with a gauge symmetry. It has beendemonstrated that in a multiverse different universescan collide and get annihilated. Furthermore, newuniverses can form from collisions of these previousuniverses. So, a third-quantized Noether charge isconstructed. This charge is conserved in the mul-tiverse even if the number of universes is not con-served. We have also analyzed a model of group fieldcosmology with gauge symmetry. This gauge wasfixed to give rise to third-quantized BRST and anti-BRST symmetries. The total Lagrangian obtainedas a sum of the original Lagrangians with a gaugefixing term and the ghost term is invariant under boththese third-quantized transformations. Furthermore,third-quantized Noether charges, corresponding tothe invariance of group field cosmology under third-quantized BRST and anti-BRST transformations,was also constructed. These charges were used todemonstrated that the formation of universes in amultiverse is a unitarity process.

It would be interesting to carry out this analysisfor supergroup field cosmology in nonlinear gauges.It is known that in these gauges a sum of the gaugefixing term and the ghost term can be expressed asa total BRST or total anti-BRST variation in gaugetheories [52]. It will be interesting to find out if a sim-ilar result holds there. It may be noted that both thefermionic and bosonic group field theories have beenstudied, and color has also been incorporated in groupfield theory [50, 51]. It would thus be interestingto analyze a supersymmetric group field theory withgauge symmetry. This way we will be symmetrizingthe full group field theory. We will be able to developthat theory by analogy with the usual gauge theory. Itwill be possible to derive this theory in the superspaceformalism with manifest supersymmetry.

Appendix

Whenever the sum of the gauge fixing term andthe ghost term can be written as a combination ofthe total BRST and anti-BRST variations, the totalLagrangian density is invariant under a set of symme-try transformations which obey the SL(2, R) algebracalled the Nakanishi-Ojima algebra [52]. We willshow that this algebra also holds for this group fieldcosmology. To do so we first define the followingtransformations:

δ1 CA(ν, φ, θ) = 0,

δ1 BA(ν, φ, θ) = fABCCB(ν, φ, θ)CC(ν, φ, θ),

δ1 Γa(ν, φ, θ) = 0, δ1 CA(ν, φ, θ) = CA(ν, φ, θ),

δ1 ΩA(ν, φ, θ) = 0, δ1 ΩA†(ν, φ, θ) = 0,

δ2 CA(ν, φ, θ) = 0,

δ2 BA(ν, φ, θ) = fABCC

B(ν, φ, θ)CC(ν, φ, θ),

δ2 Γa(ν, φ, θ) = 0, δ2 CA(ν, φ, θ) = CA(ν, φ, θ),

δ2 ΩA(ν, φ, θ) = 0, δ2 ΩA†(ν, φ, θ) = 0. (A.1)

Now we can see that these transformations, theBRST and anti-BRST transformation along with theFP-conjugation denoted by δFP , form the Nakani-shi–Ojima SL(2, R) algebra,

[s, s]� = 0, [s, s]� = 0,[s, s]� = 0, [δ1, δ2]� = −2δFP ,

[δ1, δFP ]� = −4δ1, [δ2, δFP ]� = 4δ2,

[s, δFP ]� = −2s, [s, δFP ]� = 2s,[s, δ1]� = 0, [s, δ1]� = −2s,[s, δ2]� = 2s, [s, δ2]� = 0. (A.2)

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