Supergravity on a 3-Torus : Quantum Linearization Instabilities with a Supergroup

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Article: Higuchi, Atsushi orcid.org/0000-0002-3703-7021 and Schmieding, Lasse (2020) Supergravity on a 3-torus : quantum linearization instabilities with a supergroup. Classical and Quantum Gravity. pp. 1-36. ISSN 1361-6382 https://doi.org/10.1088/1361-6382/ab90a4

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Supergravity on a three-torus: quantum linearization instabilities with a supergroup

To cite this article: Atsushi Higuchi and Lasse Schmieding 2020 Class. Quantum Grav. 37 165009

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This content was downloaded from IP address 85.210.238.88 on 26/08/2020 at 12:29 Classical and Quantum Gravity Class. Quantum Grav. 37 (2020) 165009 (35pp) https://doi.org/10.1088/1361-6382/ab90a4

Supergravity on a three-torus: quantum linearization instabilities with a supergroup

Atsushi Higuchi and Lasse Schmieding1

Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom

E-mail: [email protected]

Received 8 January 2020, revised 1 May 2020 Accepted for publication 6 May 2020 Published 28 July 2020

Abstract It is well known that linearized gravity in spacetimes with compact Cauchy surfaces and continuous symmetries suffers from linearization instabilities: solutions to classical linearized gravity in such a spacetime must satisfy so-called linearization stability conditions (or constraints) for them to extend to solutions in the full non-linear theory. Moncrief investigated implications of these conditions in linearized quantum gravity in such background spacetimes and found that the quantum linearization stability constraints lead to the requirement that all physical states must be invariant under the symmetries generated by these constraints. He studied these constraints for linearized quantum gravity in flat spacetime with the spatial sections of toroidal topology in detail. Subsequently, his result was reproduced by the method of group-averaging. In this paper the quantum linearization stability conditions are studied for = 1 simple supergravity in this spacetime. In addition to the linearization stabilityN conditions corresponding to the spacetime symmetries, i.e. spacetime translations, there are also fermionic linearization stability conditions corresponding to the background supersymmetry. We construct all states satisfying these quantum linearization stability conditions, including the fermionic ones, and show that they are obtained by group-averaging over the supergroup of the global supersymmetry of this theory.

Keywords: linearization stability conditions, group averaging, supergravity, quantum gravity

1Author to whom any correspondence should be addressed. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

1. Introduction

In physics, the equations of interest are frequently non-linear and difficult to solve. Typi- cally only a small number of solutions are known and these exploit special symmetries. To extract further physics from the known solutions, a common strategy is to perturbatively expand small deviations around the known background solutions. At lowest order in perturbation one obtains linear equations for the perturbations, which are typically easier to analyse. However, it is not guaranteed that all solutions to the linearized equations actually arise as first approximations to solutions of the non-linear equations of the system. Let us illustrate this point in a simple example [1]: for (x, y) R2, consider the algebraic system x(x2 + y2) = 0, of which the exact solutions are (0, y),∈ where y is any real number. On the other hand, if we consider the linearized equations for perturbations δx and δy around a back- 2 2 ground (x0, y0), these satisfy δx(x0 + y0) + x0(2x0δx + 2y0δy) = 0. If we let the background be (x0, y0) = (0, 0), then any pairs (δx, δy) satisfy the linearized equation of motion, but those with δx = 0 cannot have arisen as linearizations of solutions to the non-linear equation. One characterizes this phenomenon as the equation x(x2 + y2) = 0 being linearization unstable at (0, 0). A well-known field-theoretic system with linearization instabilities is electrodynamics in a ‘closed Universe’, i.e. in a spacetime with compact Cauchy surfaces [2, 3]. Consider, for example, the electromagnetic field coupled to a charged matter field in such a spacetime. Note that the conserved total charge of the matter field must vanish in this system. This fact is a simple consequence of Gauss’s law E ρ,whereE and ρ are the electric field and charge density, respectively. The integral over∇· a∝ Cauchy surface of the charge density ρ gives the total charge but the integral of E vanishes because the Cauchy surface is compact. At the level of the linearized theory about∇· vanishing background electromagnetic and matter field the theory is non-interacting. Since the matter field is non-interacting, there is no constraint on the total charge in the linearized theory, but a solution to the linearized (i.e. free) matter field equation does not extend to an exact solution to the full interacting theory unless its total charge Qe vanishes. The linearization stability condition (LSC) in this case is Qe = 0. If a solution to the linearized equations satisfies this condition, then it extends to an exact solution to the full theory. It is useful to note here that the charge Qe generates the global gauge symmetry of the free charged matter field. In gravitational systems, it is known that such linearization instabilities occur for any perturbations around a background which has both Killing symmetries and compact Cauchy surfaces [2, 4–12]. The LSCs which need to be imposed on such a closed Universe are that the generators Q of the background Killing symmetries must vanish when evaluated on any linearized solution. For example, if the spacetime possesses time and space translation symmetries, then the corresponding conserved charges are the energy and momentum, respectively. In the late seventies Moncrief studied the rôle of these conditions in linearized quantum gravity in spacetimes with compact Cauchy surfaces. He proposed that they should be imposed as physical-state conditions as in the Dirac quantization [11, 12], i.e.

Q phys = 0 . (1.1) | Since the conserved charges Q generate the spacetime symmetries (in the component of the identity), he concluded that quantum linearization stability conditions (QLSCs) imply that all physical states must be invariant under the spacetime Killing symmetries. He argued that these constraints can be viewed as a remnant of the diffeomorphism invariance of the non-linear theory.

2 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

Imposing the invariance of the physical Hilbert space under the full background symmetries (in the component of the identity) as required by the QLSCs would appear too restrictive. This problem is exemplified by de Sitter space, a spacetime which is physically relevant for inflationary cosmology. This spacetime has Cauchy surfaces with the topology of the three- dimensional sphere, which is compact. Therefore, the QLSCs imply that all physical states of linearized quantum gravity on a de Sitter background ought to be invariant under the full SO0(4, 1) symmetry group, i.e. the component of the identity of SO(4, 1), of the spacetime. However, this would appear to exclude all states except the vacuum state, which would make the Hilbert space for the theory quite empty [3, 12]. However, there are non-trivial SO0(4, 1) invariant states that have infinite norm and, hence, are not in the Hilbert space. Moncrief suggested that a Hilbert space consisting of these invariant states could be constructed by dividing the infinite inner product by the infinite volume of the group SO0(4, 1). This suggestion was taken up in reference [13]. In that work a new inner product for SO0(4, 1)-invariant states was defined by what would later be termed group-averaging, which is an important ingredient in the refined algebraic quantization14 [ ] and has been well studied in the context of loop quantum gravity. (The group-averaging procedure was also proposed in [15].) In this approach, one defines the invariant states Ψ by starting with a non-invariant state ψ and averaging against the symmetry group G, assumed| | here to be an unimodular group such as SO0(4, 1), to obtain

Ψ = dgU(g) ψ , (1.2) | | G where U is the unitary operator implementing the symmetry on the states. The state Ψ can readily be shown to be invariant, i.e. φ U(g) Ψ = φ Ψ for any state φ in the| Hilbert space by the invariance of the measure d|g. If| the volume | of the symmetry| group is finite, the inner product of the invariant states Ψ1 and Ψ2 obtained from ψ1 and ψ2 as in (1.2)is | | | |

Ψ1 Ψ2 = VG dg ψ1 U(g) ψ2 , (1.3) | | | G where VG is the volume of the group G.IfG has infinite volume, e.g. if it is SO0(4, 1), then one needs to redefine the inner product on the invariant states by removing a factor of the group volume to make them normalizable. Thus, one defines the inner product on the new Hilbert space by

Ψ1Ψ2 ga = dg ψ1 U(g) ψ2 . (1.4) | | | G By this group-averaging procedure one obtains an infinite-dimensional Hilbert space of SO0(4, 1) invariant states for linearized gravity in de Sitter space [3, 13]. The group-averaging procedure was carried out for this spacetime also for other free fields16 [ , 17] to obtain Hilbert spaces of invariant states. The QLSCs in de Sitter space were also studied in the context of cosmological perturbation [18, 19]. The group-averaging procedure has also been studied extensively in the context of constrained dynamical systems (see e.g. [20–23]). This method was extended to non-unimodular groups in [24]. The group-averaging procedure can also be explicitly carried out for perturbative quantum gravity in static space with topology of R T3, where the spatial Cauchy surfaces are copies of T3, the three-dimensional torus, to find× the states satisfying the QLSCs. The classical [2] and quantum theory [11, 25] of this model have been studied and the group-averaging procedure can be carried out to obtain a physical Hilbert space of states invariant under the

3 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

R U(1)3 symmetry group of the background. Now, if one considers four-dimensional = 1 simple× supergravity [26, 27] on this background spacetime, it is not difficult to see Nthat there are additional fermionic LSCs. The purpose of this paper is to find all states satisfying the bosonic and fermionic QLSCs and show that a Hilbert space of these states can be constructed using the group-averaging procedure over the supergroup of symmetries of the linearized theory. Let us describe how the LSCs arise for supergravity in this spacetime. Recall that the energy and momentum of a system in general relativity can be expressed as an integral over a two-dimensional surface at infinity ofauchy aC surface (the ADM mass and momentum) in asymptotically-flat spacetime. In classical perturbation theory about Minkowski space, this fact implies that the total energy and momentum of the linearized fields can be expressed asa surface integral at infinity of perturbations of the next order. Then, one expects that in perturbation theory in the flat R T3 background, the total energy and momentum of the linearized fields vanish because there× is nopatial s infinity. This is indeed the case, and the vanishing of the total energy and momentum of the linearized field is expressed as the LSCs. Note here that the expression for the total energy, or the Hamiltonian, in this spacetime is not positive definite and, therefore, can vanish for non-trivial field configurations. Now, in supergravity there is a spinor supercharge Qα, α = 1, 2, 3, 4, associated with a global supersymmetry variation, and it is known that this supercharge can be written as an integral over two-dimensional surface at infinity of a Cauchy surface in asymptotically-flat spacetime [28]. This fact again implies that in static three-torus space there are quadratic constraints on the linearized theory corresponding to the vanishing of the supercharge, which are the fermionic LSCs. In this paper we study these fermionic LSCs together with the bosonic ones in linearized quantum supergravity. The remainder of this paper is organized as follows. In section 2 we present a derivation of the (classical) bosonic and fermionic LSCs for = 1 simple supergravity in the background of flat R T3 spacetime. We show that these conditionsN are of the form that the conserved Noether charges× of the linearized theory vanish. In section 3 we discuss linearized supergravity in this spacetime and express the LSCs in terms of the classical analogues of annihilation and creation operators. In section 4 we impose the bosonic QLSCs on the states and recall how this can be understood in the context of group-averaging over the bosonic symmetry group. In section 5 we describe how all physical states satisfying both the bosonic and fermionic QLSCs are found and how a Hilbert space of physical states is constructed. Then we show that the procedure of finding the physical Hilbert space can be interpreted as group-averaging over the supergroup of global supersymmetry. We summarize and discuss our results in section 6.In appendix A we present a gauge transformation of the vierbein field that is proportional to the Lie derivative of a vector field, though it is not necessary for this work. In appendix B we illustrate our derivation of the LSCs in the simple example of electrodynamics in flat R T3 spacetime. In appendix C we present the proof of some identities used in this paper. In appendix× D we discuss some aspects of the zero-momentum sector of the gravitino field. Appendix E presents an example of a two-particle state satisfying all QLSCs. We follow the conventions of reference [29] throughout this paper.

2. The linearization stability conditions

The action for the four-dimensional = 1 simple supergravity [29, 30] with 8πG = 1is N

1 4 μνρ S = d xe R Ψ γ D Ψ + X(4Ψ) , (2.1) 2 − μ ν ρ 4 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

where X(4Ψ) consists of terms quartic in Ψμ (see e.g. [30] for the explicit form of X(4Ψ)). a a Here, eμ are the vierbein fields, e := det(eμ), and Ψμα, α = 1, 2, 3, 4, is the gravitino field, which is a Majorana spinor. The 4 4 gamma matrices γa, a = 0, 1, 2, 3, satisfy the Clifford relation γa, γb = 2ηab,whereηab×= diag( 1, 1, 1, 1). The indices a, b, c, ....are raised and { } − loweredbytheflatmetricηab whereas the spacetime indices μ, ν, σ, ....are raised and low- a a 0 0 ered by the spacetime metric gμν = eμeaν . The matrices γ have the properties γ † = γ and i i μ μ a μνρ [μ ν ρ] − γ † = γ , i = 1, 2, 3. One defines γ := ea γ and γ := γ γ γ ,where[ ] indicates total anti-symmetrization. The Riemann tensor is given by ···

R ab := ∂ ω ab ∂ ω ab + ω acω b ω acω b , (2.2) μν μ ν − ν μ μ νc − ν μc μ ν ab ab [ab] and the curvature scalar is R := ea ebRμν . The spin connection ωμ = ωμ is expressed in a terms of eμ as

ab aν b bν a 1 aν bσ bν aσ c ω = e ∂ e e ∂ e (e e e e )e ∂ ec . (2.3) μ [μ ν] − [μ ν] − 2 − μ ν σ One also defines Ψ=ΨT C,whereC is a unitary matrix satisfying CT = C and γμT = μ 1 0 − Cγ C− . The Majorana condition satisfied by Ψ reads (Ψ† ) = i(Cγ Ψ ) [30]. We later − μ μ α μ α choose a Majorana representation for γa in which C = iγ0. In this representation we have (Ψμ† )α = (Ψμ)α. The gravitino covariant derivative in (2.1)isgivenby 1 D Ψ = ∂ Ψ + ω γabΨ . (2.4) μ ν μ ν 4 μab ν The action (2.1) is invariant under a local supersymmetry transformation of the form 1 ∆ ea = ǫγaΨ , (2.5) ǫ μ 2 μ ab ∆ǫΨμ = Dμǫ + Yμabγ ǫ , (2.6) where Yμab is quadratic in Ψμ and where ǫ is a spacetime dependent Grassmann variable with four components. (See, e.g. [30] for the explicit form of Yμab). We consider this theory on a purely bosonic static background whose spatial sections are flat three-dimensional tori. For the background geometry we therefore take

ds2 = η dxμdxν = dt2 + dx2 + dy2 + dz2 , (2.7) μν − where the spatial coordinates x, y and z are periodic with periods L1, L2 and L3 respectively, and we let V = L1L2L3 denote the spatial volume. Let us first discuss the bosonic LSCs for this theory, which are well known as we described in section 1. The diffeomorphism transformation in the direction of a vector ζμ of the vierbein a 2 field eμ and gravitino field Ψμ is

a ν a a ν δζ eμ = ζ ∂ν eμ + eν ∂μζ , (2.8) ν ν δζ Ψμ = ζ ∂ν Ψμ +Ψν ∂μζ . (2.9)

2 The transformation obtained as the commutator of two supersymmetry transformations is different from the dif- a = a ρ ab feomorphism transformation presented here (see, e.g. [29]). The former takes the form δζ′ eμ δζ eμ ζ ωρ eb μ, ρ a − δ′ζ Ψμ = δζ Ψμ ∂μ(ξ Ψρ) to first order in the fields ˜eμ defined by (2.10)andΨμ. It can be shown that the LSCs corresponding to− these two transformations are identical.

5 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

Now, we write the vierbein field as

a a a eμ = δμ +˜eμ . (2.10)

a a a That is, we write eμ as the sum of its background value δμ and perturbation ˜eμ. Then,

a a ν ν a a ν δζ˜eμ = δν ∂μζ + ζ ∂ν˜eμ +˜eν ∂μζ . (2.11)

a a μ It is important to note here that the part of δζ˜eμ that is independent of ˜eμ vanishes if ζ is a constant vector, i.e. a Killing vector of the flat background3. Since the action (2.1) is invariant under the diffeomorphism transformation given by (2.9) and (2.11), we have

δS δS ˜a + Ψ = μ δζ eμ a δζ μ ∂μ(√ gJ(ζ)) , (2.12) δ˜eμ δΨμ −

μ μ Ψ for some vector field J(ζ), which is linear in ζ . Here the variation δS/δ μ is a left-variation, i.e. δS δS = d4x δΨ . (2.13) μ δΨ μ a We adopt left-variations for fermionic fields throughout this paper. We then expand δζ˜eμ, δζ Ψμ, ˜a Ψ μ ˜a Ψ δS/δeμ, δS/δ μ and √ gJ(ζ) according to the order in the fields eμ and μ, i.e. the number of these fields in the product,− as

ã = (0)ã + (1)ã δζ eμ δζ eμ δζ eμ , (2.14) Ψ = (1)Ψ δζ μ δζ μ , (2.15) δS = (1)μ + (2)μ + a Ea Ea , (2.16) δ˜eμ ··· δS = (1)μ + (2)μ + , (2.17) δΨμ E E ··· √ gJμ = J(0)μ + J(1)μ + J(2)μ + , (2.18) − (ζ) (ζ) (ζ) (ζ) ··· (0)μ (0)μ Recall that the background metric is flat. We note that Ea = 0and = 0 because the flat aE spacetime with Ψμ = 0 satisfies the field equations. That is, δS/δ˜eμ = 0andδS/δΨμ = 0if a ˜eμ = 0andΨμ = 0. The identity (2.12) must be satisfied order by order. The first- and second-order equalities read

(0)˜a (1)μ = (1)μ (δζ eμ)Ea ∂μJ(ζ) , (2.19) (δ(0)˜ea )E(2)μ + (δ(1)˜ea )E(1)μ + (δ(1)Ψ ) (1)μ = ∂ J(2)μ . (2.20) ζ μ a ζ μ a ζ μ E μ (ζ)

3 a If the background has non-zero curvature, δξ˜eμ does not necessarily vanish at lowest order even if ξ is a Killing vector of the background metric. One needs to consider a transformation modified by an infinitesimal local Lorentz a transformation in this case to make δξ˜eμ vanish at lowest order. This is done in appendix A.

6 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

We emphasize here that these identities hold for any vector ζμ and for any field configuration. μ (0)˜a = Now, if the vector field ξ is a constant vector, then δξ eμ 0 as can readily be seen from (1)μ (2.11). Hence, by (2.19), the current J(ξ) is conserved. Then the charge (1) = 3 (1)0 P(ξ) : d xJ(ξ) , (2.21) is conserved for any field configuration. In particular, it is conserved even if the fields are smoothly deformed to 0 in the past or future of the t = constant Cauchy surface where the (1) = integral is evaluated. Since P(ξ) 0 if the fields vanish, the conservation of this charge implies that (1) = P(ξ) 0 , (2.22) for any field configuration if ξμ is a constant vector. The compactness of the Cauchy surfaces is crucial for this conclusion because this charge is not necessarily conserved if the field configuration is time-dependent at infinity for the case where the Cauchy surfaces are non-compact. a Now, suppose one attempts to solve the field equations for ˜eμ and Ψμ order by order. (1)a (1) (2)a (2) Let (˜eμ , Ψμ ) be a solution to the linearized equations and (˜eμ , Ψμ ) be the second-order correction. Then,

(1)μ (1) Ea [˜e ] = 0 , (2.23) (1)μ[Ψ(1)] = 0 , (2.24) E (1)μ (2) (2)μ (1) (1) Ea [˜e ] + Ea [˜e , Ψ ] = 0 , (2.25) (1)μ[Ψ(2)] + (2)μ[˜e(1), Ψ(1)] = 0 . (2.26) E E (2)μ (1) (1) (2)μ a (1)a (1) Here [˜e , Ψ ] is the vector-spinor evaluated with (˜eμ, Ψμ) = (˜eμ , Ψμ ), and similarlyE for the others. The identities (2.19E)and(2.20)imply

(1)μ ˜(2) + (2)μ ˜(1) Ψ(1) = ∂μ(J(ζ) [e ] J(ζ) [e , ]) 0 , (2.27) for any vector field ζμ as long as field equations (2.23)–(2.25) are satisfied. The charge corre- (1)μ ˜(2) + (2)μ ˜(1) Ψ(1) μ sponding to the conserved current J(ζ) [e ] J(ζ) [e , ] must vanish for any ζ because this charge is conserved even if the field ζμ is smoothly deformed to zero in the past or future and is evaluated there. Hence, if we define

(2) ˜(1) Ψ(1) = 3 (2)0 ˜(1) Ψ(1) P(ζ)[e , ]: d xJ(ζ) [e , ] , (2.28) then P(2)[˜e(1), Ψ(1)] = P(1)[˜e(2)], (2.29) (ζ) − (ζ) μ (1)b (1) (2)b (2) for any vector field ζ as long as the fields˜e ( ν , Ψν )and(˜eν , Ψν ) are perturbative solutions to the field equations. In particular, if ζμ = ξμ is a Killing vector, then

(2) ˜(1) Ψ(1) = P(ξ)[e , ] 0 , (2.30) because of (2.22). Equation (2.30) is a consequence of the requirement that the solution (1)b (1) (˜eν , Ψν ) extend to an exact solution and does not follow from the linearized field equations.

7 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

This equation is called a linearization stability condition (LSC) and has to be imposed on the solutions to the linearized equations. In appendix B we illustrate the argument leading to (2.30) for electrodynamics. The total charge of the linearized charged field must vanish in this model. The LSCs from local supersymmetry can be derived in a similar manner. We expand the supersymmetry transformation given by (2.5)and(2.6) according to the number of fields in the product as a (1) a ∆ǫ˜eμ =∆ǫ ˜eμ , (2.31) ∆ Ψ =∆(0)Ψ +∆(1)Ψ +∆(2)Ψ + . (2.32) ǫ μ ǫ μ ǫ μ ǫ μ ··· a a It is clear that ∆ǫ˜eμ =∆ǫeμ givenby(2.5) has no field independent contribution. That is, (0) a ∆ǫ ˜eμ = 0. The analogue of the identity (2.12)is δS δS ∆ ˜a +∆Ψ = μ ǫeμ a ǫ μ ∂μ(√ g (ǫ)) . (2.33) δ˜eμ δΨμ − J From this equation one finds

(∆(0)Ψ ) (1)μ = ∂ (1)μ , (2.34) ǫ μ E μJ(ǫ) (∆(0)Ψ ) (2)μ + (∆(1)˜ea )E(1)μ + (∆(1)Ψ ) (1)μ = ∂ (2)μ , (2.35) ǫ μ E ǫ μ a ǫ μ E μJ(ǫ) (0) for any spinor field ǫ and any field configuration. Since ∆ε Ψμ = ∂με = 0ifε is a constant spinor, the charge defined by

Q(1):= d3x (1)0 , (2.36) (ε) J(ε) vanishes for any field configuration if ǫ = ε is a constant spinor by the argument which led to (2.22). By the same argument as that led to (2.29), if we define

Q(2)[˜e(1), Ψ(1)]:= d3x (2)μ[˜e(1), Ψ(1)] , (2.37) (ǫ) J(ǫ) then Q(2)[˜e(1), Ψ(1)] = Q(1)[Ψ(2)], (2.38) (ǫ) − (ǫ) (1)b (1) (1)μ (1) for any spinor field ǫ if (˜eν , Ψν ) gives a solution to the linearized field equations, Ea [˜e ] = (1)μ Ψ(1) = = (1) Ψ(2) = [ ] 0. In particular, if ǫ ε is a constant spinor, since Q(ε)[ ] 0 in this case, we Emust have

(2) ˜(1) Ψ(1) = Q(ε)[e , ] 0 . (2.39) These are the LSCs arising from local supersymmetry on static three-torus space. (2)μ ˜(1) Ψ(1) Next, we shall derive the conserved currents J(ξ) [e , ]givenby(2.20)and (2)μ[˜e(1), Ψ(1)]givenby(2.35) and the corresponding conserved charges P(2)[˜e(1), Ψ(1)]and J(ε) (ξ) (2) ˜(1) Ψ(1) μ ˜(1)a =˜a Q(ε)[e , ] for a constant vector ξ and a constant spinor ε. From now on we write eμ eμ (1) and Ψμ =Ψμ. (1)μ a For either charge we only need the linearized field equations. To find Ea for ˜eμ,itisuseful to note that eR in the Lagrangian density in terms of the vierbein fields equals √ gRgiven − 8 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding in terms of the metric tensor g . Thus, we may vary √ gRwith respect to the metric tensor μν − and then vary the metric tensor with respect to the vierbein fields. Writing gμν = ημν + hμν, a we find the perturbation hμν in terms of ˜eμ at first order as

a a hμν = δμ˜eaν +˜eaμδν . (2.40)

Then we find 1 E(1)μ [˜e] = δν ∂μ∂λh ∂ ∂λh μ + ∂μ∂ h + hμ + δμ∂λ∂σh δμh , (2.41) a 2 a − λν − ν λ ν ν ν λσ − ν

μν λ where h := η hμν and := ∂λ∂ , with indices raised and lowered by ημν. We readily find

(1)μ [Ψ] = Cγμνρ∂ Ψ . (2.42) E − ν ρ (2)μ To find the bosonic conserved current J(ξ) ,weuse(2.40)–(2.42) together with

(1)˜a = ν ˜a δ(ξ) eμ ξ ∂ν eμ , (2.43) (1)Ψ = ν Ψ δ(ξ) μ ξ ∂ν μ , (2.44)

(0)˜a = in (2.20), recalling δ(ξ) eμ 0, to find 1 ∂ J(2)μ [˜e, Ψ] = (ξλ∂ h ) ∂μ∂λh ν ∂ν ∂λh μ + ∂μ∂ν h + hμν μ (ξ) 4 λ μν − λ − λ + ημν∂λ∂σh ημνh (ξλ∂ Ψ )γμνρ∂ Ψ . (2.45) λσ − − λ μ ν ρ (2)μ Ψ ˜a The current J(ξ) [h, ], which depends on eμ only through hμν, is identified with the Noether current for the spacetime translation in the direction of ξμ of the decoupled theory consisting of a Fierz–Pauli Lagrangian for the graviton and a Rarita–Schwinger Lagrangian for the Majorana gravitino [30]

1 1 1 1 1 = ∂μh∂ν h + ∂ h∂μh + ∂ hμσ∂ρh ∂ hμν∂ρh Ψ γμνρ∂ Ψ . (2.46) L −4 μν 8 μ 4 σ μρ − 8 ρ μν − 2 μ ν ρ It can be given explicitly as

∂ ∂ (2)μ Ψ = λ λ Ψ + μ J(ξ) [h, ] (ξ ∂λhρσ) L (ξ ∂λ ρ) L ξ − ∂(∂μhρσ) − ∂(∂μΨρ) L 1 = ξλ∂ h∂ hμν + ξλ∂ hμρ∂ h ξλ∂ h∂μh 4 λ ν λ ρ − λ 2ξλ∂ hμρ∂σh + ξλ∂ h ∂μhρσ − λ ρσ λ ρσ 1 1 1 + ξμ ∂ρh∂σh + ∂ h∂ρh + ∂ hρσ∂λh ∂ h ∂λhρσ 4 − ρσ 2 ρ σ ρλ − 2 λ ρσ 1 + ξσΨ γνμρ∂ Ψ ξμΨ γνσρ∂ Ψ . (2.47) 2 ν σ ρ − ν σ ρ

(2)μ Next we find the fermionic conserved current (ε) . We note that the first-order part of the global supersymmetry transformation is given byJ

9 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

1 ∆(1)˜ea = εγaΨ , (2.48) (ε) μ 2 μ 1 ∆(1)Ψ = ∂ ˜e γνa + ∂ h γνρ ε. (2.49) (ε) μ 4 μ aν ρ μν By substituting these formulas and equations (2.41)and(2.42)into(2.35), and using the identity [29]

γρνγσμκ = 3(ην[σγμκ]ρ ηρ[σγμκ]ν + ην[σγμηκ]ρ ηρ[σγμηκ]ν ) , (2.50) − − we find

(2)μ 1 ρν μσκ a ρν μσκ [h, Ψ] = ∂ h εγ γ Ψ δ ˜ea εγ γ ∂ Ψ . (2.51) J(ε) 4 ρ σν κ − ρ ν σ κ Note that the second term vanishes if the (linear) local Lorentz invariance is fixed by requir- μ b ing ˜eaν = δa δν˜ebμ. (It vanishes by the linearized field equation for Ψμ as well.) With this condition imposed, the term which explicitly depends on e˜bν in the supersymmetry transformation ∆(1)Ψ givenby(2.49) vanishes. With this choice the current (2)μ[h, Ψ]andthe (ε) μ J(ε) (2) Ψ charge Q(ε)[h, ] are the Noether current and charge, respectively, for the supersymmetry transformation, 1 ∆(1)h = (εγ Ψ + εγ Ψ ) , (2.52) (ε) μν 2 μ ν ν μ 1 ∆(1)Ψ = ∂ h γνρε , (2.53) (ε) μ 4 ρ μν of the linearized supergravity Lagrangian given by (2.46). (2) Ψ = (2) Ψ = (2) Ψ Thus, the classical LSCs are P(ξ)[h, ] 0andQ(ε)[h, ] 0, where P(ξ)[h, ]and (2) Ψ (2)μ Ψ Q(ε)[h, ] are the conserved charges corresponding to the conserved currents J(ξ) [h, ]in (2)μ Ψ (2.47)and (ε) [h, ]in(2.51) (without the second term) respectively (see (2.28)and(2.37)). In the nextJ section we discuss linearized supergravity on static three-torus space at the classical level and express the LSCs in a form suitable for quantization.

3. Linearized supergravity on static three-torus space

The results of the previous section allow us to describe the LSCs entirely in terms of the linearized theory. By linearizing = 1 simple supergravity in 4 dimensions about static three-torus background spacetime,N we have the Lagrangian, which is given here again for convenience: 1 1 1 1 1 = ∂μh∂νh + ∂ h∂μh + ∂ hμσ∂ρh ∂ hμν∂ρh Ψ γμνρ∂ Ψ . (3.1) L −4 μν 8 μ 4 σ μρ − 8 ρ μν − 2 μ ν ρ A suitable real Majorana representation for the γ-matrices is given by

01 10 γ0 = , γ1 = , 10 0 1 − − (3.2) 0 σ1 0 σ3 γ2 = , γ3 = , σ1 0 σ3 0

10 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding where σ1, σ2 and σ3 are the standard Pauli matrices. In this representation the charge conjuga- tion matrix takes the form C = iγ0. The action with the Lagrangian density (3.1)isinvariant under the following gauge transformations:

h h + ∂ ζ + ∂ ζ , (3.3) μν → μν μ ν ν μ Ψ Ψ + ∂ ǫ , (3.4) μ → μ μ μ where ζ is an arbitrary vector field and ǫα is an arbitrary Majorana spinor field satisfying ǫα† = ǫα. This invariance is a remnant of the diffeomorphism invariance and local supersymmetry of the full theory. We shall fix this gauge freedom completely. Working on the spatial torus and imposing periodic boundary conditions on the fields allows us to decompose each field as a Fourier series4. Due to the periodic boundary conditions, each field has a spatially constant zero-momentum component. The zero-momentum sector ofthe linearized theory will be seen to contain six bosonic and six fermionic degrees freedom and this sector forms an important ingredient in the QLSCs. Meanwhile, the non-zero momentum sector of the theory contains the usual gravitons and gravitinos, which each have two polarization states. Explicitly we expand the fields as

1 (0) 1 ˜ ik x h (t,x) = h (t) + h (t,k)e · , μν √V μν √V μν k=0 1 1 ˜ ik x Ψ (t,x) = ψ (t) + Ψ (t,k)e · , μ √V μ √V μ k=0 where the volume of the torus is V = L1L2L3 and the L1, L2 and L3 are the periods of the torus in the x-, y-andz-directions respectively. The periodic boundary condition implies k = 2π 2π 2π (0) ˜ n1, n2, n3 with n1, n2 and n3 integers, and reality restricts h to be real, hμν(t, k)to L1 L2 L3 μν − ˜ ˜ be equal to hμν† (t, k ), and similarly for ψμ and Ψμ(t, k). The Lagrangian for the theory does not provide dynamical coupling between the modes with zero momentum and those with non-zero momentumk. Therefore, it is possible to analyse (0) these separately. We begin with the zero-momentum sector of the theory. We write hμν(t) = hμν(t), dropping the superscript ‘(0)’, in the rest of this section. The Lagrangian, i.e. the space integral of the Lagrangian density, for this sector reads

1 i j 1 ij 1 ¯ 0 ij L0 = ∂0h i∂0h j + ∂0h ∂0hij + ψiγ γ ∂0ψ j . (3.5) −8 8 2 The classical [31] and quantum [25] theory of the graviton contributions have been studied previously using the ADM formalism without fixing the gauge. Here we fix the gauge to extract the physical degrees of freedom in the linearized theory. (This gauge fixing has little to do with the imposition of LSCs discussed later). The linearized Lagrangian does not provide equations of motion for hμ0 and ψ0α,andthese are precisely the components which carry the gauge degrees of freedom. It is therefore possible to gauge-fix these components to vanish by solving h00(t) = 2∂0ξ0(t), hi0(t) = ∂0ξi(t),

4 It is possible to choose other boundary conditions also compatible with local supersymmetry, such as anti-periodic boundary conditions for both the gravitino Ψμ and the local supersymmetry parameter ǫ. However, only the periodic boundary conditions lead to fermionic LSCs.

11 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

i = 1, 2, 3 and ψ0(t) = ∂0ǫ(t). The elimination of h0μ(t)andψ0(t) exhausts the gauge freedom in the zero-momentum sector, and all other components are physical. Thus, in the zero- momentum sector the physical degrees of freedom are contained in the symmetric tensor hij and vector-spinor ψiα, i = 1, 2, 3, each of which corresponds to six classical degrees of freedom. The six bosonic degrees of freedom are contained in the symmetric tensor hij.Toanalyse this tensor it is convenient to break it up into the trace and trace-free sectors. Thus, we let

2 5 h (t) = δ c(t) + 2 TAc (t), (3.6) ij 3 ij ij A A=1

A where the Tij are a set of five trace-free tensors which satisfy the orthonormality condition Ai j B AB T Tij = δ . We choose them as

100 10 0 1 1 T1 = 0 10, T2 = 01 0 , √ √ 2 ⎛000− ⎞ 6 ⎛00 2⎞ − ⎝ ⎠ ⎝ ⎠ 010 001 1 1 T3 = 100, T4 = 000, √ √ 2 ⎛000⎞ 2 ⎛100⎞ ⎝ ⎠ ⎝ ⎠ 000 1 T5 = 001. (3.7) √ 2 ⎛010⎞ ⎝ ⎠ In terms of the six variables c and cA, the Lagrangian reads 5 1 2 1 2 1 ¯ 0 ij L0 = (∂0c) + (∂0cA) + ψiγ γ ∂0ψ j . (3.8) −2 2 2 A=1

ij The momentum conjugate to hij,i.e.p = ∂L0/∂(∂0hij), is

ij 1 k ij 1 ij p = (∂0h k)δ + ∂0h −4 4 1 2 1 = δijc + TAi j c , (3.9) 2 3 P 2 PA A where cP = ∂0c and cPA = ∂0cA are the momenta conjugate to c and cA, respectively. The − time derivative of the zero-momentum field hij is then 5 A 2 ∂0hij = 2 T cPA δijcP . (3.10) ij − 3 A=1

kl The canonical Poisson bracket relations for c, cA, cP and cPA , and equivalently for hij and p , are

c, cP = 1, cA, cPB = δAB , (3.11) { }P { }P 1 h , pkl = δkδl + δlδk . (3.12) ij P 2 i j i j 12 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

The fermionic part of L0 is of first order in time derivatives and therefore already defines a constrained dynamical system [32, 33], and we need to use the Dirac bracket as the bracket i to be ‘promoted’ to the anti-commutator upon quantization. The momentum variables πα conjugate to ψiα are

i ∂L0 πα = ∂(∂0ψiα)

1 ¯ 0 ji = (ψ jγ γ ) , (3.13) −2 α where the derivative of L0 with respect to ∂0ψiα is the left derivative. These momenta are not invertible in terms of the coordinates and velocities, so there are twelve primary constraints,

i i 1 ¯ 0 ji φ = π + (ψ jγ γ ) 0 . (3.14) α α 2 α ≈ We note that the Poisson and Dirac brackets for two fermionic fields are symmetric under the exchange of arguments. That is, if the fields A and B are fermionic, then A, B P = B, A P, and similarly for the Dirac bracket. { } { } For the associated primary Hamiltonian HP we add arbitrary linear combinations of the primary constraints to the canonical Hamiltonian,

i i i HP = ∂0ψiπ L0 + λ φi = λ φi , (3.15) − i where the λα are arbitrary. Notice in particular that the primary Hamiltonian for these modes weakly vanishes, i.e. it vanishes if the constraints are satisfied. The constraints are to be pre- served under evolution by the primary Hamiltonian. This requirement leads to consistency conditions i i ∂0φ φ , HP 0 . (3.16) ≈ P ≈ To evaluate the Poisson bracket between the constraints, we use the canonical Poisson bracket,

j = i ψiα, πβ δ jδαβ , (3.17) P − which allows us to compute the Poisson bracket between the constraints (and therefore the primary Hamiltonian) as

i j = 0 ij φα, φβ (Cγ γ )αβ . (3.18) P − Hence, we find that the consistency conditions3.16 ( )implyλi = 0. Thus, the primary Hamil- tonian HP vanishes and, as a result, the fields ψi are time independent. This fact can also be deduced from the Euler–Lagrange equation resulting from (3.8). There are no secondary constraints and all the constraints are of second class. To obtain the bracket structure for the theory suitable for subsequent quantization, we compute the Dirac bracket

j j k 1 l j = − ψiα, πβ ψiα, πβ ψiα, φγ φ, φ P φδ , πβ D P − P { } klγδ P 1 = j ψiα, πβ 2 P 1 = δ jδ . (3.19) −2 i αβ

13 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

On the Dirac bracket, we can impose the second-class constraints as strong conditions since the Dirac bracket between second-class constraints vanishes. Working explicitly in the Majorana representation, where C = iγ0, the Dirac bracket for ψiα evaluates to i ψi , ψ j = δijδ (γij) . (3.20) { α β}D −2 αβ − αβ To provide some insight into these relations we write

2 1 A j A ψiα = (γiη)α + T (γ η )α , (3.21) √6 ij A=1

1 2 where η and ηA are Majorana spinors. The matrices T and T are given in (3.7). These symmetric and traceless matrices satisfy

Ai j B k AB T γ jTikγ = δ . (3.22) One can also show, by a component-by-component calculation,

2 A k A ℓ 2 1 T γ T γ = δij γij. (3.23) ik jℓ 3 − 3 A=1

These relations can be used to show that the Dirac bracket relations for ψiα are equivalent to

η , η = iδ , ηA, ηB = iδABδ , η , ηA = 0 . (3.24) { α β}D αβ α β D − αβ α β D Next, we briefly describe the mode expansion of the non-zero-momentum sector of this theory, which is well known. Letting the non-zero-momentum components of the fields be ˆ ˆ denoted by hμν and Ψμ, we can completely fix the gauge freedom in this sector by imposing the following conditions (see, for instance, [29, 30]):

ˆ ˆ iˆ hμ0 = h = ∂ hij = 0 , (3.25) i ˆ ˆ i ˆ γ Ψi = Ψ0 = ∂ Ψi = 0 . (3.26) Then we can write the expansion of the graviton as follows:

ˆ 2 1 λ ik x λ ik x hij(x, t) = H (k)aλ(k)e · + H ∗(k)a† (k)e− · , (3.27) V √k ij ij λ k=0λ= ± = μ = where k : k and k (k, k). We have let the complex conjugate of aλ(k) be denoted by aλ† (k), anticipating| quantization.| The symmetric and traceless polarization tensors are given by λ λ λ Hij(k) = ǫi (k)ǫ j (k), (3.28)

λ where the polarization vectors ǫi (k)aregivenby

1 (1) (2) ǫ±(k) = (ê (k) iê (k)) . (3.29) i √2 i ± i (1) (2) (3) The unit spatial vectors êi (k), êi (k)andêi (k) = k/k form a right-handed orthonormal sys- λ λ λλ i λ tem in this order. The polarization vectors ǫ±(k) satisfy ǫ ∗(k) ǫ ′ (k) = δ ′ and k ǫ (k) = 0. i · i 14 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

λ i λ λ λ′ij λλ′ As a result, Hij(k) satisfy k Hij(k) = 0andHij∗(k)H (k) = δ . The Lagrangian density for the non-zero-momentum sector of the graviton field in this gauge can be found from (3.1)as

1 ˆij ρˆ TT = ∂ h ∂ hij. (3.30) L −8 ρ The standard procedure to find the Poisson bracket relations for the coefficients aλ(k)andaλ† (k) leads to

= aλ(k), aλ† (k′) iδk,k δλλ , (3.31) P − ′ ′ with all other brackets among aλ(k)andaλ† (k) vanishing. The non-zero-momentum sector of the gravitino field can similarly be expanded into modes as

ˆ 1 1 λ λ ik x λ λ ik x Ψiα = ǫ (k)u (k)bλ(k)e · + ǫ ∗(k)u ∗(k)b† (k)e− · , (3.32) √V √2k i α i α λ k=0λ= ±

0 1 2 3 0ˆ ˆ where u±(k) are eigenspinors of γ5 = iγ γ γ γ and γ k γ,wherek :=k/k, with eigenval- · λ λ λλ ues 1and 1, respectively. We normalize them by requiring u †(k)u ′ (k′) = 2kδ ′ .The ± − fermionic coefficients bλ(k)andbλ† (k) satisfy the classical analogues of the anti-commutation relations for the annihilation and creation operators:

= bλ(k), b† (k′) iδk,k δλλ , (3.33) λ′ D − ′ ′ with all other brackets among bλ(k)andbλ† (k) vanishing.

4. Imposing the bosonic linearization stability conditions

We quantize the linearized theory by promoting the physical degrees of freedom to operators acting on some Hilbert space, and we impose

[(anti) commutator] = i Poisson/Dirac Bracket , (4.1) − ± { } as the algebraic relations between the operators, where [A, B] := AB BA. We work in units with = 1. Thus, equations (3.11)and(3.31) become ± ±

[c, cP] = i, [cA, cPB] = iδAB , (4.2) − −

aλ(k), a† (k′) = δλλ δ , (4.3) λ′ ′ k,k′ − respectively. On the other hand, the relations (3.24)and(3.33) become

η , η = δ , ηA, ηB = δABδ , η , ηA = 0, (4.4) α β + − αβ α β + αβ α β + bλ(k), b† (k′) = δλλ δ , (4.5) λ′ + ′ k,k′ respectively.

15 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

After imposing our gauge conditions and using the field equations, the time component of (2)μ Ψ the conserved Noether current J(ξ) [h, ] for the spacetime translation symmetry becomes

(2)0 √V ij j i 1 ν ˆij ˆ 1 ˆ ν ˆij i ˆ T ν ˆ i J = ξ0 ∂0hij∂0h ∂0h j∂0h i ξ ∂ h ∂0hij + ξ0∂ hij∂ h ξ Ψ ∂ Ψ , (ξ) 8 − − 4 ν 8 ν − 2 ν i (4.6) where we have dropped the cross terms between the zero-momentum and non-zero-momentum (2)0 sectors because they do not contribute to the space integral of J(ξ) , which gives the conserved charge. Let us write + i = 3 (2)0 ξ0H ξiP d xJ(ξ) . (4.7) By substituting (3.9), (3.27)and(3.32)into(4.6) and integrating the result over space, one finds 5 1 2 1 2 H = c + c + k a† (k)a (k) + b† (k)b (k) , (4.8) −2 P 2 PA | | λ λ λ λ A=1 k=0λ= ± = + P k aλ† (k)aλ(k) bλ† (k)bλ(k) , (4.9) k=0λ= ± the bosonic sector of which agrees with the conserved charges given in [25]. Note that the zero-point energy which had to be renormalized away in the pure-gravity case is absent here because of supersymmetry. Note also that the Hamiltonian H is not positive definite. The Hilbert space for the theory can be constructed as the tensor product of a non-zero momentum sector and a sector with zero momentum. In the sector with non-zero momentum, we have the usual Fock spaces for the gravitons and gravitinos with the number operators for the gravitons and gravitinos (with a given momentum and a polarization) being aλ† (k)aλ(k)and bλ† (k)bλ(k), respectively. A suitable basis of states is given by states with definite number of particles in each momentum and polarization. Thus, the normalized state with nBλ(k) gravitons and nFλ(k) gravitinos with momentum k and polarization λ can be given as

1 nB (k) nF (k) nB nF = (a† (k)) λ (b† (k)) λ 0 , (4.10) |{ } ⊗ |{ } ⎡ λ λ ⎤ | k,λ= nBλ(k)! ± ⎣ ⎦ where the vacuum state 0 satisfies a (k) 0 = b (k) 0 = 0forallk and λ. (For the gravitino | λ | λ | nFλ(k) = 0or1,ofcourse). For the graviton zero-modes, we represent the commutation rules [c, cP] = iand − [cA, cPB] = iδAB on wave functions which are functions of the variables c and cA by − ∂ c multiply by c, cP i , (4.11) → →−∂c ∂ cA multiply by cA, cPA i . (4.12) → →−∂cA Therefore, if we take some state which is proportional to a single eigenstate of the number operators,

state =Ψ nB nF , (4.13) | ⊗|{ } ⊗ |{ } 16 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

where Ψ is some wave function of c and cA, then the Hamiltonian and momentum operators, (4.8)and(4.9), acting on such a state take the form

5 1 ∂2 ∂2 H state = + + M2 state , (4.14) | 2 ∂c2 − ∂c2 | A=1 A

P state = k nB (k) + nF (k) state , (4.15) | λ λ | k=0λ= ± where we have defined a ‘squared mass’ for these states by

2 M = 2 k nB (k) + nF (k) . (4.16) | | λ λ k=0λ= ±

The operators H and P do not contain the zero-momentum sector of the gravitino field. They appear in the global supercharges as we shall see in the next section. As discussed in section 2, the conserved charges H and P must vanish classically for the classical solutions to the linearized equations if they were to be extendible to exact solutions. Moncrief proposed that in quantum theory these linearization stability conditions (LSCs) should be imposed as constraints on the physical states. Thus, the constraints on the physical states phys are | H phys = 0, P phys = 0 . (4.17) | | In [25] the group-averaging procedure was used to find all states satisfying these constraints and define an inner product among these statesor f pure gravity on static three-torus space. We apply this procedure to = 1 simple supergravity in this section. N We start with the Hilbert space 0 of superpositions of the states defined by4.13 ( ). Take two H states in this Hilbert space of the form ϕ1 =Ψ1(c, cA) Φ1 and ϕ2 =Ψ2(c, cA) Φ2 , | ⊗| | ⊗| where Φ1 and Φ2 are states in the Fock space of non-zero-momentum gravitons and gravitinos.| The inner| product between these states isF

5 ϕ1 ϕ2 = Φ1 Φ2 dcd c Ψ1∗Ψ2 , (4.18) | H0 | F where c is the vector with components cA, A = 1, 2, 3, 4, 5. Let us choose both Φ1 and Φ2 to have a definite value of M2 defined by (4.16). (If these states have different| values of|M2, then Φ1 Φ2 = 0.) Then, by expressing ΨI(c,c), I = 1, 2, as a Fourier integral in the six- dimensional | spaceF with coordinates (c,c), these states can be given as

0 5 dp d p 0 ip0c+ipc ϕI = FI(p ,p)e− · ΦI , (4.19) | 2π (2π)5 ⊗|

A where p is also a five-dimensional vector and p c = p cA. The inner product for these states is ·

0 5 dp d p 0 0 ϕ1 ϕ2 = Φ1 Φ2 F1∗(p ,p)F2(p ,p) . (4.20) | H0 | F 2π (2π)5 17 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

Since the operators H and P are the generators of spacetime translations, we can construct states satisfying (4.17) by averaging the states ϕI over this translation group as follows: |

3 Li (B) 1 ∞ 0 0 ϕ = dα dαi exp(iα H iα P) ϕI . (4.21) | I 2V − · | i=1 0 −∞ The integral over space acts as the projector onto the sector of the Fock space with zero total momentum: 3 1 Li dαi exp( iα P) ΦI = ΦI . (4.22) V − · | PP=0| i=1 0 The integral over α0 can readily be evaluated using the representation (4.19)since

ip0c+ipc 1 0 2 2 2 ip0c+ipc He− · ΦI = (p ) + p + M e− · ΦI . (4.23) ⊗| 2 − ⊗| (P=0) Thus we find, with the notation Φ = ΦI , | I PP=0| 0 5 (B) dp d p 0 2 2 2 0 ip0c+ipc (P=0) ϕ = 2πδ((p ) p M )FI(p ,p)e− · Φ | I 2π (2π)5 − − ⊗| I 5 d p (+) iE(p)c+ipc ( ) iE(p)c+ipc (P=0) = f (p)e− · + f − (p)e · Φ , (4.24) (2π)5 I I ⊗| I where E(p) = p2 + M2 and where

!( ) FI( E(p),p) f ± (p) = ± . (4.25) I 2E(p)

(B) Although we obtained the invariant state ϕI by averaging ϕI over the spacetime translation group, it is easy to show that any state with| definite value of M| 2 satisfying the constraints (4.17) is of this form. (The zero-momentum sector is a solution to the six-dimensional Klein–Gordon equation with mass M). The states ϕ(B) are indeed invariant, i.e. they satisfy the QLSCs given by (4.17). How- | I ever, they have infinite norm and, hence, are not in the Hilbert space 0: the inner prod- (B) (B) H uct ϕI ϕI computed using (4.20) is infinite because of the δ-function in (4.24). The | H0 group-averaging inner product for the Hilbert space B of invariant states is defined by H 3 1 Li (B) (B) = ∞ 0 0 ϕ1 ϕ2 B dα dαi ϕ1 exp(iα H iα P) ϕ2 0 | H 2V | − · | H i=1 0 −∞ = (B) ϕ1 ϕ2 0 | H 5 d p (+) (+) ( ) ( ) (P=0) (P=0) = 2 E(p) f1 ∗(p) f2 (p) + f1− ∗(p) f2− (p) Φ1 Φ2 . (2π)5 | F (4.26)

Notice that, although this inner product is defined in terms of the ‘seed states’ ϕI , it depends (B) | only on the invariant states ϕI . This inner product is equivalent to that proposed in [34]in the context of quantum cosmology.|

18 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

Since the constraint H phys = 0 is a Klein–Gordon equation, it is tempting to use the | Klein–Gordon inner product for the Hilbert space B: H

∂Ψ ∂Ψ (B) (B) = 5 Ψ 2 1∗ Ψ Φ(P=0) Φ(P=0) ϕ1 ϕ2 KG i d c 1∗ 2 1 2 . (4.27) | ∂c − ∂c | F This inner product can readily be evaluated as

5 (B) (B) d p (+) (+) ( ) ( ) (P=0) (P=0) ϕ1 ϕ2 KG = 2 E(p) f1 ∗(p) f2 (p) f1− ∗(p) f2− (p) Φ1 Φ2 , | (2π)5 − | F (4.28)

(B) (B) which is identical with the group-averaging inner product, ϕ1 ϕ2 B ,givenby(4.26) except for the minus sign in the second term. Although the Klein–Gordon | H inner product is more widely used in quantum cosmology, it can be used only if the space of geometries considered has the structure of spacetime [35]. (In our example, it is the six-dimensional Minkowski space.) The group-averaging inner product, which we use in this paper, has wider applicability. For example, it can be used in recollapsing quantum cosmology as shown in [34]. In the next section we impose the fermionic QLSCs derived in the previous section. We shall find that the states satisfying these constraints and the inner product among them canbe found by group-averaging over the relevant supergroup.

5. Imposing the fermionic linearization stability conditions

The time component of the conserved fermionic current (2)μ[h, Ψ]is J(ε)

(2)0 1 i i j ˆ j ˆ i ˆ 0 ℓ j ˆ i [h, Ψ] = ∂0hεγ ψi ∂0hijεγ ψ ∂0hijεγ Ψ + ∂ hijεγ γ γ Ψ , (5.1) J(ε) 4 − − ℓ where we dropped the cross terms between the zero-momentum and non-zero-momentum sectors since they do not contribute to the space integral. By letting

εQ = d3x (2)0 , (5.2) − J(ε) and substituting (3.10)and(3.21)into(5.1) and integrating over space, we find

Q = Q(0) + Q , (5.3) with the zero- and non-zero-momentum" contributions respectively given by

1 5 2 Q(0) = c η + TAi TBγ jγkc ηB , (5.4) 2 P j ik PA A=1 B=1

1 3 ˆij ˆ ˆij ˆ Q = d x ∂0h γ jΨi h γ j∂0Ψi , (5.5) 4 − " j ˆ 0 ˆ where we have integrated by parts in the second term of Q and used γ ∂ jΨi = γ ∂0Ψi. − One is readily able to find the mode expansion for Q, which then yields " 19 " Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

1 5 2 Q = c η + TAi TBγ jγkc ηB 2 p j ik PA A=1 B=1

i λ λ λ λ ǫ (k) γu ∗(k)a (k)b† (k) ǫ ∗(k) γu (k)a† (k)b (k) . (5.6) − 2 · λ λ − · λ λ k=0λ= ± This charge and the bosonic charges, Pμ = (H, P), satisfy the supersymmetry algebra. That is, [Pμ, Q] = 0, [Pμ, Pν] = 0and − − 1 Q , Q = (γ γ0) Pμ . (5.7) α β + 2 μ αβ Since the contribution to Q with different k anti-commute, equation (5.7) holds for each k including k = 0. In deriving (5.7) we have used the following identities for k = 0andk = 0 proved in appendix C:

2 5 5 5 Ai B A′ Bi′ j k k′ j′ 2 T jT T T k γ γ γ γ cPA cPA = (cPA ) , (5.8) ik i′ j′ ′ ′ B=1 A=1 = A=1 A ′ 1 0 ǫ±∗(k) γu±(k) ǫ±(k) γu±∗(k) + (α β) = 2(γ kγ )αβ . (5.9) · α · β ↔ · μ We also recall that γ are real, γ5 is purely imaginary and that ǫ±∗(k) = ǫ∓(k)andu±∗(k) = u∓(k). One can represent the anti-commutation relations (4.4) satisfied by the twelve fermionic A operators ηα and ηα in the zero-momentum sector of the gravitino field by a 64-dimensional indefinite-metric Hilbert space (or Krein space) 0F as shown in appendix D. The basis vectors of this space can be chosen such that 32 ofH them are normalized with positive norm and the other 32 are normalized with negative norm. [We say here that v is normalized with | positive (negative) norm if v v = 1( v v = 1).] An important property of 0F used later | | − H is that the 16-dimensional subspace of 0F of states annihilated by the operators d1 and d2 defined by H

d1 := (η1 + iη2)/√2, d2 = (η3 + iη4)/√2 , (5.10) is a positive-norm subspace because this subspace is spanned by the states of the form (D.5) with n1 = n2 = 0 (with M = 2) [see (D.6)]. The Hilbert space of states satisfying the bosonic QLSCs is in fact the tensor product B 0F. We call this tensor product space also B from now on in order not to complicateH the⊗H notation. H Now we are in a position to find the physical states phys satisfying the fermionic QLSCs, Q phys = 0, as well as the bosonic ones, Pμ phys =|0. Since the operators Pμ commute with α| | Qα, we specialize to the Hilbert space B of states satisfying the bosonic QLSCs and iden- μ H = tify P with the null operator. Thus, we have Qα, Qβ + 0 in place of (5.7). Splitting the supercharge into zero- and non-zero-momentum contributions, these anti-commutation relations of the supercharge can be rewritten as

(0) (0) 1 2 [Q , Q ]+ = M δ = [Q , Q ]+ , (5.11) α β −4 αβ − α β where M2 is defined by (4.16). We specialize to" an eigenspace" of the operator M2 without loss of generality. Thus, we treat M as if it were a non-negative number.

20 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

Assume M > 0. Then, in each common eigenspace of the operators cP and cPA with eigen- 2 2 2 (0) values p0 and pA, respectively, satisfying p0 p = M , the operators Qα given by (5.4)isof the form − M 2 4 Q(0) = η cosh θ + C β ηB sinh θ , (5.12) α 2 ⎛± α αB β ⎞ B=1 =1 β ⎝ ⎠ where C β can be regarded as a 4 8 matrix with the rows and columns labelled αB × by α and (B, β), respectively. Here, cosh θ = cP /M and sinh θ = cP /M,wherecP = | | | | β (cP1, cP2, cP3, cP4, cP5). The four eight-dimensional column vectors of the matrix CαB are orthonormal, i.e. 2 4 β β = Cα1 BCα2 B δα1α2 . (5.13) B=1 = β 1 Then by appendix D there is a unitary operator U, i.e. an operator preserving the inner product, on 0F such that H (0) Qα = (M/2)UηαU† . (5.14)

(0) Instead of directly working with Qα and Qα, it is more convenient to combine them into annihilation- and creation-type operators [33]as " √2 √2 a = (Q(0) + iQ(0)), a = Q(0) + iQ(0) , (5.15) 1 M 1 2 2 M 3 4

√2 √2 b = (Q + iQ ), b = (Q + iQ ) , (5.16) 1 M 1 2 2 M 3 4 provided that M > 0.5 These" new" operators satisfy the anti-commutation" " relations of Fermi 2 2 2 2 oscillators (up to a sign), i.e. ai = bi = ai† = bi† = 0and

[ai, a†]+ = δij,[bi, b†]+ = δij. (5.17) j − j (BF) (BF) Now we construct all states ϕ B satisfying Qα ϕ = 0, α = 1, 2, 3, 4. These constraints can be organized as | ∈H |

(BF) (BF) (ai + bi) ϕ = (ai† + bi†) ϕ = 0, i = 1, 2 . (5.18) # $ # $ Note that all states are# linear combinations# of the states each annihilated by either b1 or b1† (B) (B) (B) (B) since ϕ = b†b1 ϕ + b1b† ϕ for any state ϕ B. The same is true for each pair | 1 | 1| | ∈H of operators, (b2, b2†), (a1, a1†)and(a2, a2†). This means that a general state is a superposition of states, each belonging to a 16-dimensional Fock space built on a state χ(B) satisfying (B) (B) | ai χ = bi χ = 0, i = 1, 2, by applying the creation-type operators a† and b†. Notice | | i i that equations (5.14)and(5.15)implya1 = Ud1U† and a2 = Ud2U†,whereU is a unitary

5 For M = 0, formally one can proceed in a similar manner by considering (δ-function normalisable) plane wave Ψ = ipc+ipc = p + = states (c,c) e± · . On such states the supercharge takes the form Q 2 ( η R), with [ηα, ηβ ]+ δαβ = = + √ ± = + √ = − + and Rα, Rβ + δαβ . Then, we define ladder-type operators a1 (η1 iη2)/ 2, a2 (η3 iη4)/ 2, b1 (R1 iR2)/√2andb2 = (R3 + iR4)/√2 and proceed in a manner similar to the case with M > 0.

21 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

operator on 0F and where d1 and d2 are defined by (5.10). Since the states annihilated by d1 H (B) (B) and d2 have positive norm as we stated before, we have χ χ B > 0. We label the 16 possible states in such a Fock space as follows:| H

m1 m2 n1 n2 (B) (a†) (a†) (b†) (b†) χ = m1m2n1n2 , (5.19) 1 2 1 2 | | with each m1, m2, n1, n2 being either 0 or 1. For example, we define

(B) (B) (B) χ = 0000 , a† χ = 1000 , a†a† b†b† χ = 1111 . (5.20) | 1 | 1 2 1 2 | # $ # $ # $ Thus, we look for# states satisfying the# fermionic QLSCs in the form# 1 1 1 1 (BF) ϕ = cm m n n m1m2n1n2 . (5.21) | 1 2 1 2 | m =0 m =0 n =0 n =0 1 2 1 2 We find

1 1 1 1 (BF) m1+m2 (a1 + b1) ϕ = cm m n n m1 0m2n1n2 + ( 1) n1 m1m20n2 . | 1 2 1 2 − | − | m =0 m =0 n =0 n =0 1 2 1 2 (5.22)

The coefficient of the term 0m21n2 in this equation is c1m 1n . Hence c1m1n = 0forallm | − 2 2 and n. We can conclude similarly that c0m0n = cm0n0 = cm1n1 = 0 by using the other constraints. = = = (BF) Thus, cm1m2n1n2 0ifm1 n1 or m2 n2. Hence the invariant state ϕ B,i.e.thestate satisfying the fermionic QLSCs, must be of the following form: | ∈H

ϕ(BF) = A 1100 + B 0110 + C 1001 + D 0011 , (5.23) | | | | # $ where A, B, C and# D are constants. Note that the four states 1100 , 0110 , 1001 and 0011 are mutually orthogonal and satisfy | | | | = = (B) (B) 1100 1100 0011 0011 χ χ B , | | | H (5.24) = = (B) (B) 0110 0110 1001 1001 χ χ B . | | − | H In particular, the two states 0110 and 1001 have negative norm. By applying the constraints| (5.18 ) we| find the following unique solution up to an overall normalization:

ϕ(BF) ϕ(BF) := 1100 0110 + 1001 + 0011 . (5.25) ∝| P | −| | | # $ Thus, there is# precisely one possible combination which satisfies the fermionic QLSCs in n1 n2 n1 n2 (B) (B) each Fock space spanned by (a1†) (a2†) (b1†) (b2†) χ ,where χ is any state with fixed positive M2, satisfying the bosonic QLSCs and the conditions| |

(Q(0) + iQ(0)) χ(B) = (Q(0) + iQ(0)) χ(B) = 0 , (5.26) 1 2 | 3 4 | (B) (B) (Q1 + iQ2) χ = (Q1 + iQ2) χ = 0 . (5.27) | | Now, in section 4"it was" shown that all" states" satisfying the bosonic QLSCs are obtained by group-averaging. Here we show that the same is true for the fermionic QLSCs. That is, the state ϕ(BF) in (5.23) is obtained as | 22 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

(BF) 4 θQ (B) ϕ = d θ e− ϕ , (5.28) | − | (B) for some linear combination ϕ of the states m1m2n1n2 ,whereθ = (θ1, θ2, θ3, θ4)are | T 0 4 | Grassmann numbers and where θ = iθ γ and d θ = dθ4dθ3dθ2dθ1. (The minus sign in (5.28) has been introduced for convenience.) By the usual rules, dθα = 0and dθαθα = 1, α = 1, 2, 3, 4, this equation becomes % % (BF) (B) ϕ = Q1Q2Q3Q4 ϕ | − | 1 (B) = (Q1 iQ2)(Q1 + iQ2)(Q3 iQ4)(Q3 + iQ4) ϕ 4 − − | 4 M (B) = (a† + b† )(a1 + b1)(a† + b†)(a2 + b2) ϕ . (5.29) 16 1 1 2 2 |

One readily finds that Q1Q2Q3Q4 m1m2n1n2 = 0 unless m1m2n1n2 = 1100 , 0110 , 1001 or 0011 and that | | | | | | (BF) (a† + b† )(a1 + b1)(a† + b† )(a2 + b2) 1100 = ϕ , (5.30) 1 1 2 2 | | P (BF) (a† + b† )(a1 + b1)(a† + b† )(a2 + b2) 0110 = ϕ , (5.31) 1 1 2 2 | | P (BF) (a† + b† )(a1 + b1)(a† + b† )(a2 + b2) 1001 = ϕ , (5.32) 1 1 2 2 | −| P (BF) (a† + b† )(a1 + b1)(a† + b† )(a2 + b2) 0011 = ϕ , (5.33) 1 1 2 2 | | P where the state ϕ(BF) is defined by (5.25). Thus, starting from any linear combination ϕ(B) | P | of m1m2n1n2 ,wefind |

4 θQ (B) (B) (BF) d θ e− ϕ = Q1Q2Q3Q4 ϕ = κ ϕ , (5.34) − − | P # $ # $ with κ C. # # ∈ (BF) (BF) = Now, an explicit computation shows that ϕP ϕP B 0. In fact this can be deduced 2 | H immediately from (5.34) because Qα = 0 on the space B of states satisfying the bosonic QLSCs. However, the group-averagingformula (5.34) suggestsH that one can proceed in analogy (B) (B) with the bosonic case to define a new inner product. Thus, for two states ϕ1 , ϕ2 B we obtain two states satisfying the fermionic QLSCs as | | ∈H

(BF) (B) ϕ = Q1Q2Q3Q4 ϕ , (5.35) | I − | I and define the new inner product by

(BF) (BF) = (B) (B) ϕ1 ϕ2 BF ϕ1 Q1Q2Q3Q4 ϕ2 B | H − | | H = (B) (BF) ϕ1 ϕ2 B . (5.36) | H Equations (5.30)–(5.33)and(5.25) imply that, if

ϕ(B) = κ(1) 1100 + κ(2) 0110 + κ(3) 1001 + κ(4) 0011 , (5.37) | I I | I | I | I | then

23 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

4 (BF) M (B) ϕ = λI ϕ , (5.38) | I 16 | P where (1) (2) (3) (4) λI = κ + κ κ + κ , (5.39) I I − I I and M4 (BF) (BF) = (B) (B) ϕ1 ϕ2 BF λ1∗λ2 χ χ . (5.40) | H 16 |

6 Thus, our new inner product BF is positive definite . Although it is defined in terms of the (B) ·|·H ‘seed states’ ϕI , which do not satisfy the fermionic QLSCs, the new inner product depends | (BF) only on the invariant states ϕI . We constructed the states| satisfying the bosonic and fermionic QLSCs step by step. We first found the states satisfying the bosonic QLSCs and an inner product among them inthe previous section, following [25]. Then, we found the states satisfying the fermionic QLSCs as well among these states and defined an inner product for these invariant states. Our con- struction can in fact be understood as group-averaging over the supergroup generated by Pμ and Qα. Starting from a state ϕI , I = 1, 2, in the original Hilbert space 0, we define a state satisfying all constraints by integrating| over the supergroup [37, 38]: H

(BF) 1 4 4 ϕ = d α,d θ exp iα P θQ ϕI , (5.41) | I −2V − · − | μ 0 where θ = (θ1, θ2, θ3, θ4) are Grassmann numbers and α = (α , α) are commuting numbers. μ As P and Qα commute, we can write this integral as

(BF) 1 4 4 ϕ = d θ exp( θQ) d α exp( iα P) ϕI | I −2V − − · | (B) = Q1Q2Q3Q4 ϕ , (5.42) − | I where ϕ(B) satisfies only the bosonic QLSCs: | I

(B) 1 4 ϕ = d α exp( iα P) ϕI . (5.43) | I 2V − · | We have shown that all states satisfying both the bosonic and fermionic QLSCs can be obtained in this manner. The inner product BF we have defined is ·|·H 1 (BF) (BF) = 4 4 ϕ1 ϕ2 BF d α d θ ϕ1 exp iα P θQ ϕ2 | H −2V | − · − | H0 = 4 (B) (B) d θ ϕ1 exp θQ ϕ2 B . (5.44) − | − | H In appendix E we present an example of a state with two particles with non-zero momenta which satisfies all QLSCs.

6 The definition of a positive-definite inner product used here is similar to that in appendix D of reference [36].

24 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

6. Summary and discussion

In this paper we pointed out that there are fermionic linearization stability conditions as well as bosonic ones in four-dimensional = 1 simple supergravity in the background of static three-torus space. Then we showed thatN states satisfying both fermionic and bosonic quantum linearization stability conditions (QLSCs) can be constructed by group-averaging over the supergroup of global supersymmetry and spacetime translation symmetry. States satisfying the bosonic QLSCs have infinite norm in the original Hilbert space. This infinity results from the infinite volume of the symmetry group generated by the LSCs. Roughly speaking, this infinite volume is factored out in the group-averaging inner product. It is interesting that the inner product of states satisfying all QLSCs have zero norm in the Hilbert space of states satisfying only the bosonic ones. The finite group-averaging inner product is obtained by factoring out zero in this case. As the bosonic QLSCs can be interpreted as a remnant of diffeomorphism invariance of the full generally covariant theory, one should be able to interpret the fermionic QLSCs as a remnant of full local supersymmetry in the context of canonical quantization [39, 40]. The bosonic QLSCs, H phys = 0andP phys = 0, have a natural physical picture. These conditions imply that all| physical states| are invar iant under spacetime translations. This means that there is no meaning in the position and time coordinates of an event relative to the background spacetime of static three-torus. However, the physical states still encode relative posi- tions and relative time differences between two or more events. Thus, the bosonic QLSCs can be seen as a manifestation of Mach’s principle in quantum general relativity. (See e.g. [41] for a discussion of Mach’s principle in general relativity.) On the other hand, it is not clear if there is a simple interpretation of the fermionic QLSCs. It would be interesting to find one. It would also be interesting to investigate whether there are analogues of LSCs in string theory. A preliminary investigation in this direction [31] did not find such analogues in Bosonic string theory, but since string theory contains general relativity, we believe there should be analogues of LSCs in (super) string theory on any background spacetime with compact Cauchy surfaces. The supergroup relevant to this work was a simple one with an abelian bosonic subgroup. It would be interesting to investigate the group-averaging procedure for general supergroups and establish general properties in analogy with the bosonic case studied in [24].

Acknowledgments

LS was supported by a PhD studentship from the Engineering and Physical Sciences Research Council (EPSRC) (EP/N509802/1).

Appendix A. Modified transformation of the vierbein in general spacetime

In this appendix we show that one can modify the infinitesimal diffeomorphism transformation a on eμ by an infinitesimal local Lorentz transformation so that the transform is proportional to a μ μζν + ν ζμ. If we transform eμ by diffeomorphism in the direction of ζ and by a local ∇Lorentz transformation,∇ we have

(m) a ρ a a ρ ab δ e = ζ e + e ζ s eb ,(A.1) μ ∇ρ μ ρ∇μ − μ 25 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

ab a a σ a where s is anti-symmetric and where ρeμ = ∂ρeμ Γρμeσ with the Levi-Civita connection σ ab (∇m) a − Γρμ. Our task is to choose s so that δ eμ is proportional to μζν + ν ζμ. To this end, we require ∇ ∇

(m) a ρ c c ρ a b ea[ δ e = (ζ ec[ )e + ec e ]ζ sabe e = 0 . (A.2) μ ν] ∇ρ ν μ] ρ [μ∇ν − [μ ν] Since c c cb D e = e + ω eb = 0, (A.3) ρ ν ∇ρ ν μ ν we find a b ρ c b sabe e = ζ ω cbe e + ζ ] . (A.4) [μ ν] − ρ [μ ν] ∇[ν μ Thus, ab ρ ab aμ bν s = ζ ω + ζ ]e e . (A.5) − ρ ∇[ν μ Then

(m) a ρ a a ρ ρ ab 1 aλ bν δ e = ζ e + e ζ + ζ ω + ( ζ ζ )e e eb μ ∇ρ μ ρ∇μ ρ 2 ∇λ ν −∇ν λ μ 1 = ζρD ea + ea ζλ + ( ζ ζ )eaλ ρ μ λ∇μ 2 ∇λ μ −∇μ λ 1 = ( ζ + ζ )eaλ . (A.6) 2 ∇λ μ ∇μ λ

Appendix B. Linearization stability condition for electrodynamics

In this appendix we discuss the linearization stability condition for quantum electrodynamics in a static three-torus space. The Lagrangian density is

1 = F Fμν + ψγμ(∂ ieA )ψ mψψ ,(B.1) L −4 μν μ − μ −

0 where F = ∂ A ∂ A and ψ = iψ†γ . The field equations are μν μ ν − ν μ

δS νμ μ = ∂ν F ieψγ ψ = 0, (B.2) δAμ −

δS 0 μ = iγ γ (∂μ ieAμ)ψ mψ = 0, (B.3) δψ† − − δS = (∂ + ieA )ψγμ + mψ = 0, (B.4) δψ μ μ where S is the action obtained by integrating over spacetime. We have defined the left-hand side of (B.3)and(B.4) as the left functional derivativeL of the action S. μ μ The field equations (B.3)and(B.4) imply that the current J(E) = ieψγ ψ is conserved. As 0 a result, the total charge, which is the integral of J(E) over the space,

3 Q(E) = e d x ψ†ψ ,(B.5) 26 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding is time independent. This charge must vanish because the zeroth component of (B.2) implies i0 ∂iF = eψ†ψ and hence

3 i0 Q(E) = d x ∂iF ,(B.6) which must vanish by Gauss’s divergence theorem. (Recall our space is compact.) At linearized level, the fields Aμ and ψ are decoupled, and the condition Q(E) = 0 does not automatically arise in the linearized theory. Hence this condition has to be imposed as a linearization stability condition on the linearized solution for it to extend to an exact solution. The aim of this appendix is to illustrate how the argument in section 3 works in this simple model. The action is invariant under the gauge transformation δAμ = ∂μΛ, δψ = ieΛψ for any function Λ of the spacetime point. The argument in section 3 leads to the conclusion that

∂ Λ(∂ Fνμ ieψγμψ) ieΛψ γμ(∂ ieA )ψ mψ ieΛ (∂ + ieA )ψ γμ + mψ ψ μ ν − − μ − μ − − μ μ = μ ∂μJ(Λ) [A, ψ] ,( B.7)

μ Λ for some current J(Λ) [A, ψ] for any . We indeed find that this equation is satisfied with

Jμ [A, ψ] =Λ(∂ Fνμ ieψγμψ) . (B.8) (Λ) ν −

The identity (B.7) is satisfied order by order in the fields Aμ and ψ,i.e.

Λ νμ = (1)μ ∂μ ∂ν F ∂μJ(Λ) [A] ,(B.9) ie∂ Λψγμψ ieΛψγμ∂ ψ ieΛ(∂ ψ)γμψ = ∂ J(2)μ [A, ψ] , (B.10) − μ − μ − μ μ (Λ) where (1)μ =Λ νμ J(Λ) [A] ∂ν F , (B.11) J(2)μ[A, ψ] = ieΛψγμψ. (B.12) (Λ) − Now, if Λ(x) = λ is constant, then the left-hand side of (B.9) vanishes. Hence the current (1)μ = νμ J(λ) [A] λ∂ν F is conserved for any field configuration Aμ. In particular, we can smoothly deform Aμ to zero in the far future or past of any given Cauchy surface while keeping the value of the conserved charge

(1) 3 i0 Q(E)[A] = d x ∂iF , (B.13) (1) unchanged. It follows that Q(E)[A] = 0 for any field Aμ. Of course, one can readily verify this fact by Gauss’s divergence theorem, as we observed before. Now, consider solving the field equations order by order by letting

A = A(1) + A(2) + , (B.14) μ μ μ ··· ψ = ψ(1) + ψ(2) + . (B.15) ··· (I) (I) (I) Then, with the definition Fμν := ∂μAν ∂ν Aμ , I = 1, 2, the field equationsB.2 ( )–(B.4) become −

27 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

(1)νμ ∂ν F = 0 , (B.16) iγ0 γμ∂ ψ(1) mψ(1) = 0 , (B.17) μ − (1) μ (1) ∂μψ γ + mψ = 0 , (B.18) ∂ F(2)νμ ieψ(1)γμψ(1) = 0 . (B.19) ν − Then, equations (B.9)and(B.10) imply that the current

J(1)μ[A(2)] + J(2)μ[A(1), ψ(1)] =Λ(∂ F(2)νμ ieψ(1)γμψ(1)) , (B.20) (Λ) (Λ) ν − is conserved for any Λ(x). This implies that the corresponding charge must vanish, which is (1)μ (2) + (2)μ (1) (1) = rather obvious in this example because, in fact, J(Λ) [A ] J(Λ) [A , ψ ] 0by(B.19). In particular, if we define

(2) (1) 3 (1) (1) Q [ψ ] = e d x ψ †ψ , (B.21) (E) − (1) (1) (1) 0 (1) (2)0 (1) (1) then, since eλψ †ψ = ieλψ γ ψ = J [A , ψ ], we find − (λ) Q(2) [ψ(1)] = Q(1) [A(2)] = 0 , (B.22) (E) − (E) which is the linearization stability condition for electrodynamics in static torus space. In this example, it is the linearization of the condition (B.6) in the exact theory.

Appendix C. Proof of identities (5.8)and(5.9)

B Bi i B 1 1 We first note that the matrices T , B = 1, 2, are diagonal with T k = δ q ,whereq = q = ij k k 1 − 2 1/√2, q1 = 0. q2 = q2 = 1/√6andq2 = 2/√6. Hence 3 1 2 3 − 2 5 5 A Bi A′ Bi′ j k k′ j′ X := T T kT T k γ γ γ γ cPA cPA ij i′ j′ ′ ′ B=1 A=1 = A ′ 1 3 3 5 5 2 A A′ B B j k k′ j′ = T T q q γ γ γ γ cPA cPA . (C.1) kj k′ j′ k k′ ′ k=1 = A=1 = B=1 k ′ 1 A ′ 1 By a component-by-component calculation we find 2 B B 1 q q = δkk . (C.2) k k′ ′ − 3 B=1

By substituting this formula into (C.1) we obtain 5 5 1 5 5 Ak A′ j j′ A A′ j k k′ j′ X = T jT γ γ cPA cPA T T γ γ γ γ cPA cPA . (C.3) kj′ ′ − 3 kj kj′ ′ A=1 = A=1 = A ′ 1 A ′ 1 A The second sum vanishes because the matrices Tij are symmetric and traceless so that TA j k = TA ( jk + jk) = 0. Since the first sum is of the form S j j where S is symmetric, kjγ γ jk δ γ jj′ γ γ ′ jj′ j j jj we may replace γ γ ′ by δ ′. Hence

28 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

5 5 5 5 Ak A jj′ AA′ 2 X = T jT δ cPA cPA = δ cPA cPA = (cPA ) ,(C.4) kj′ ′ ′ A=1 = A=1 A=1 A ′ 1 which is equation (5.8). 0ˆ To show (5.9) we note that, since u±(k) is a simultaneous eigenspinor of γ k γ and · γ5 with eigenvalues 1and 1, respectively, with the normalization u±†(k)u±(k) = 2k,the − ± matrix Mαβ = u±(k)αu±∗(k)β is the product of the projection operator onto the eigenspace of 0ˆ γ k γ with eigenvalue 1 and that of γ5 with eigenvalue 1 (multiplied by 2k) since these projection· operators commute.− Hence, we have ±

k 0ˆ u±(k)αu±∗(k)β = (1 γ k γ)(1 γ5) 2 − · ± αβ 1 0 = (1 γ5)k γγ ,(C.5) 2 ± · αβ where we have used k γ = k( γ0 + ˆk γ)sothatk(1 γ0ˆk γ) = k γγ0. Hence · − · − · ·

1 0 ǫ±∗(k) γu±(k) ǫ±(k) γu±∗(k) = (1 γ5)k γ(ǫ±∗(k) γ)(ǫ±(k) γ)γ ,(C.6) · α · β 2 ∓ · · · αβ 0 where we have used the fact that k γ, γ and γ5 all anti-commute with ǫ±(k) γ and ǫ±∗(k) · 0ˆ 0ˆ · · γ. Now the identities (ǫ±∗(k) γ)(ǫ±(k) γ) = 1 γ k γγ5 and k γγ k γ = k γ imply k · · ± · · · · · γ(ǫ±∗(k) γ)(ǫ±(k) γ) = k γ(1 γ5). By substituting this identity into (C.6)wefind · · · ±

0 ǫ±∗(k) γu±(k) ǫ±(k) γu±∗(k) = (1 γ5)k γγ . (C.7) · α · β ∓ · αβ 0 0 Finally, by noting that [(1 γ5)k γγ ] = [(1 γ5)k γγ ] we find (5.9). ∓ · βα ± · αβ

Appendix D. Zero-momentum sector of the gravitino field

In this appendix we provide some details of the (indefinite-metric) Hilbert space describing the zero-momentum sector of the gravitino field. Some of the material presented here can be found in [33]. We make the discussion general and treat the fermionic algebra where the self- (+) adjoint operators ηα, α = 1, 2, ...,2M and ηα , α = 1, 2, ...,2N satisfying

(+) (+) (+) [η , η ]+ = δ ,[η , η ]+ = δ ,[η , η ]+ = 0 . (D.1) α β − αβ α β αβ α β The zero-momentum sector of the gravitino field corresponds to M = 2andN = 4. We represent this algebra as follows. We define annihilation-type operators as 1 da := (η2a 1 + iη2a), a = 1, 2, ..., M ,(D.2) √2 −

(+) 1 (+) (+) da := (η2a 1 + iη2a ), a = 1, 2, ..., N . (D.3) √2 −

These operators and their adjoint, the creation-type operators, have the following non-zero anti-commutators:

29 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

(+) (+) [da, d†]+ = δab ,[d , d †]+ = δab . (D.4) b − a b (+) We define the state 00F by requiring da 00F = da 00F = 0foralla and 00F 00F = 1. Then the 2M+N-dimensional| (indefinite-metric)| Hilbert| space representing the algebra | (D.1 )is spanned by the following orthogonal states: M N (+) na (+) nb n , n = (d†) (d †) 00F ,(D.5) |{ } a b | a=1 b=1 (+) (+) where na and nb are either 0 or 1 and where n = n1, n2, ..., nM and n = (+) (+) (+) { } { } n1 , n2 , ..., nN . One readily finds 1ifn1 + n2 + + nM is even, n , n(+) n , n(+) = ··· (D.6) { } |{ } & 1ifn1 + n2 + + nN is odd. − ··· If Ω is an anti-self-adjoint operator satisfying Ω† = Ω, then the operator exp Ω,which can be defined as a power series, is unitary in the sense− that it preserves the inner product. Of particular interest are the following unitary operators: θ θ θ U (θ):= exp η η = cos + η η sin , α = β ,(D.7) αβ 2 α β 2 α β 2

θ θ θ V (θ):= exp η η(+) = cosh + η η(+) sinh ,(D.8) αβ 2 α β 2 α β 2

θ θ θ W (θ):= exp η(+)η(+) = cos + η(+)η(+) sin , α = β. (D.9) αβ 2 α β 2 α β 2

(+) These unitary operators act on ηα and ηα as follows:

Uαβ(θ)ηαUαβ(θ)† = ηα cos θ + ηβ sin θ , (D.10)

(+) (+) (+) W (θ)η W (θ)† = η cos θ η sin θ , (D.11) αβ α αβ α − β = + (+) Vαβ(θ)ηαVαβ(θ)† ηα cosh θ ηβ sinh θ , (D.12) (+) = (+) + Vαβ(θ)ηβ Vαβ(θ)† ηβ cosh θ ηα sinh θ. (D.13) = 2M 2N (+) = 1 Let us define Y : α=1 ηα β=1 ηβ . Then, Y is unitary, i.e. Y† Y− . The opera- (+) (+) (+) tor Wα := Yηα , α = 1, 2, ...,2N, is also unitary. We find Wαηα Wα† = ηα whereas (+) (+) ' ' − W η W† = η if β = α and W η W† = η for all β. α β α β α β α β Since any product of Uαβ(θ), Wαβ(θ), Vα(θ)andWα is unitary, we can conclude the following. Suppose that M N and let

X˜ = A βη cosh θ C β η(+) sinh θ , (D.14) α α β α − α β α β β where the index α = 1, 2, ...,2M, is not summed over, and where (Aα )and(Cα ) β1 β2 = area2M 2M matrix and a 2M 2N matrix, respectively, satisfying Aα1 Aα2 δβ1β2 β1 β×2 ×α 7 Cα Cα δβ β = δα α and det(A β) = 1. Then there is a unitary operator U such that ˜ 1 2 1 2 1 2 Xα = UηαU†.

7 The unitarity of the operators W allows us to have det(Cα ) = 1. α β − 30 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

Appendix E. Example of a state satisfying all quantum linearization stability conditions

In this appendix we present a concrete example of a state satisfying all fermionic as well as bosonic QLSCs. We consider an example where there are two particles, one going in the positive x-direction and the other in the negative x-direction with momentum k > 0 and both with positive helicity. In this sector of the theory M2 = 4k. To find the non-zero-momentum contribution to the supercharge in this sector weneed + to find the spinors u (k) with k = kˆe1,whereê1 is the unit vector in the x-direction. It is an eigenspinor of γ0γ1 and iγ2γ3 with± eigenvalue 1 for both and, hence, an eigenspinor 0 1 2 3 − + + of γ5 = iγ γ γ γ with eigenvalue 1. It is normalized as u †( kê1)u ( kê1) = 2k.Wefind from (3.2) ± ±

0 1 σ2 0 γ0γ1 = − ,iγ2γ3 = . (E.1) 10 0 σ2 − Hencewecantake 1 1 k i k i u+(kê ) = , u+ (kê ) = . (E.2) 1 ⎛−1 ⎞ ∗ 1 ⎛1⎞ 2 2 ⎜ i⎟ ⎜i⎟ ⎜− ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Now, ê2, ê3, ê1 form a right handed basis so that we can choose { } + 1 2 3 1 2 2 3 ǫ (kê1) γ = (γ + iγ ) = γ (1 + iγ γ ) . (E.3) · √2 √2

+ 2 3 Since u ∗(kê1) is an eigenspinor of iγ γ with eigenvalue +1, we have 1 i + + 2 i 1 ǫ (kê1) γu ∗(kê1) = √kγ ⎛ ⎞ = √k ⎛ ⎞ ,(E.4) · 1 i ⎜i⎟ ⎜1⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ and 1 i − + + 2 i 1 ǫ ∗(kê1) γu (kê1) = √kγ ⎛− ⎞ = √k ⎛ ⎞ . (E.5) · 1 i − ⎜ i⎟ ⎜ 1 ⎟ ⎜− ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + 0 1 2 3 The spinor u ( kê1) is an eigenspinor of γ γ and iγ γ with eigenvalue +1 for both. We can choose −

+ 1 2 2 3 ǫ ( kˆe1) γ = γ (1 iγ γ ) . (E.6) − · √2 −

Proceeding similarly as above, we find 1 i − + + 2 i 1 ǫ ( kê1) γu ∗( kê1) = √kγ ⎛ ⎞ = √k ⎛− ⎞ ,(E.7) − · − 1 i − ⎜ i⎟ ⎜ 1 ⎟ ⎜− ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 31 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding and 1 i + + 2 i 1 ǫ ∗( kê1) γu ( kê1) = √kγ ⎛− ⎞ = √k ⎛− ⎞ ,(E.8) − · − 1 i − − ⎜ i ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ Thus, the relevant part of the k = 0 contribution to the supercharge

i λ λ λ λ Q = ǫ (k) γu ∗(k)a (k)b† (k) ǫ ∗(k) γu (k)a† (k)b (k) ,(E.9) −2 · λ λ − · λ λ k=0λ= ± " is, with the notation a( ) = a+( kê1)andb( ) = b+( kê1), ± ± ± ±

a(+)b(†+) + a(†+)b(+) a( )b(† ) + a(† )b( ) − − − − − ⎛ + ⎞ √k i a(+)b(†+) a(†+)b(+) i a( )b(† ) a(† )b( ) Q = − − − − − − − . (E.10) ⎜ ⎟ 2 ⎜ a b† + a† b + a b† + a† b ⎟ ⎜ (+) (+) (+) (+) ( ) ( ) ( ) ( ) ⎟ ⎜ − − − − ⎟ " ⎜ ⎟ ⎜ i a(+)b(†+) a(†+)b(+) i a( )b(† ) a(† )b( ) ⎟ ⎜− − − − − − − − ⎟ ⎝ ⎠ Proceeding as in section 5 to construct a state satisfying the fermionic QLSCs we define

√2 √2 a = (Q(0) + iQ(0)), a = (Q(0) + iQ(0)) , (E.11) 1 M 1 2 2 M 3 4 as before and also 1 1 b1 = (Q1 + iQ2) = (a(+)b(†+) a( )b(† )) , (E.12) √2k √2 − − − 1 " " 1 b2 = (Q3 + iQ4) = (a(+)b(†+) + a( )b(† )) , (E.13) √2k √2 − − in the relevant subspace. Then" in" the two-particle sector, M2 = 4k, it is easy to see that

[ai, a†]+ = δij,[bi, b†]+ = δij. (E.14) j − j We define a state ϕ(B) obeying the bosonic QLSCs as follows. We take the bosonic zero- momentum sector wave function to be given by the Gaussian # $ # c2 c2 Ψ(c,c) = Ne− e− , (E.15) where N is a constant, and the non-zero-momentum part of the state contains two positive helicity gravitons with momentum kê1. Then the state obeying the bosonic QLSCs by group- averaging is given by (4.24), which± reads

5 (B) d p (+) iE(p)c+ipc ( ) iE(p)c+ipc = + † † ϕ 5 f (p)e− · f − (p)e · a(+)a( ) 0 , (E.16) (2π) ⊗ − | # $ # where the state 0 is the Fock vacuum in the non-zero-momentum sector, E(p) = p2 + M2 and | ! 32 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

3 M2/4 p2/2 ( ) π Ne− e− f ± (p) = . (E.17) 2 p2 + M2

The norm of this state in the Hilbert! space B is then given by (4.26), which reads H 3 4 p2 π 2 p e (B) (B) = 2 M /2 ∞ − ϕ ϕ B N e− dp < . (E.18) | H 12| | p2 + M2 ∞ 0 (B) !(B) = We choose the normalisation constant N so that ϕ ϕ B 1. Now we need to consider the fermionic zero-momentum | H sector. As we stated in section 5, we need to consider the tensor product of B and the 64-dimensional Hilbert space 0F H H describing the fermionic zero-momentum sector. We call this tensor product B as in section 5. H As in appendix D let 00F be the normalized state annihilated by d1 = (η1 + iη2)/√2, d2 = B | B B B B B (η3 + iη4)/√2, d1 = (η1 + iη2 )/√2andd2 = (η3 + iη4 )/√2, B = 1, 2. Let a¯1 and a¯2 be the operators a1 and a2, respectively, restricted to the eigenspace of cP and cPA with eigen- = ¯ ¯ values p0 and pA, respectively. Define p0,p 0F by p0,p : Cp0,p a1a2d2†d1† 00F ,where | ∈H |= ¯ =¯| = Cp0,p is a normalization constant such that p0,p p0,p 0F 1. Then, a1 p0,p a2 p0,p 0 2 2 | H | | because a¯ =¯a = 0anda¯1a¯2 = a¯2a¯1.Now,welet 1 2 − 5 (B) d p (+) iE(p)c+ipc ( ) iE(p)c+ipc ϕ = f (p)e− · E(p),p + f − (p)e · E(p),p (2π)5 ⊗| ⊗|− # $ # a(†+)a(† ) 0 . (E.19) ⊗ − |

(B) = (B) = (B) (B) = (B) Then, a1 ϕ a2 ϕ 0and ϕ ϕ B 1. Notice also that because ϕ contains H (B) |(B) only gravitons, it obeys b† = b† = 0. # $ # 1$ ϕ 2 ϕ # $ In the# notation of section# 5 we can thus take ϕ(B) 0011 . Indeed, by the group-averaging# # $ # $ ∝|(BF) procedure described in section# 5, we obtain# a state ϕ BF obeying the bosonic and # $ ∈H fermionic QLSCs by applying Q1Q2Q3Q4.Thatis,# − # $ # (BF) (B) ϕ = Q1Q2Q3Q4 ϕ (E.20) − # $ 1 # $ (B) # = (Q1 iQ2)(#Q1 + iQ2)(Q3 iQ4)(Q3 + iQ4) ϕ (E.21) 4 − − 4 # $ M (B) # = (a† + b†)(a + b )(a† + b†)(a + b ) ϕ . (E.22) 16 1 1 1 1 2 2 2 2 # $ (B) (B) # By writing ϕ = a(†+)a(† ) Ψ this can be evaluated as − # $ 4 # $ # (BF) M # (B) 1 (B) ϕ = a(†+)a(† ) Ψ (a1† a2†)a(†+)b(† ) Ψ 16 − | −√2 − − | # $ # 1 (B) (B) + (a1† + a2†)b(†+)a(† ) Ψ a1†a2† b(†+)b(† ) Ψ . (E.23) √2 − | − − | In particular, by the group-averaging procedure, the norm of this state is given by

M4 (BF) (BF) = (B) (BF) = ϕ ϕ BF ϕ ϕ B , (E.24) | H | H 16 as would be expected by the procedure presented in section 5.

33 Class. Quantum Grav. 37 (2020) 165009 A Higuchi and L Schmieding

ORCID iDs

Atsushi Higuchi https://orcid.org/0000-0002-3703-7021 Lasse Schmieding https://orcid.org/0000-0002-5198-1342

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