Supergravity on a 3-torus: quantum linearization instabilities with a supergroup

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 $\mathcal{N}=1$ simple supergravity in this spacetime. In addition to the linearization stability 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.


Introduction
In physics, the equations of interest are frequently non-linear and difficult to solve. Typically 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) ∈ R 2 , consider the algebraic system x(x 2 + y 2 ) = 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 background (x 0 , y 0 ), these satisfy δx(x 2 0 + y 2 0 ) + x 0 (2x 0 δx + 2y 0 δy) = 0. If we let the background be (x 0 , y 0 ) = (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(x 2 + y 2 ) = 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, where E 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 Q e vanishes. The linearization stability condition (LSC) in this case is Q e = 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 Q e 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 [2,4,5,6,7,8,9,10,11,12]. 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. 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 3-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 SO 0 (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 SO 0 (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 SO 0 (4, 1). This suggestion was taken up in Ref. [13]. In that work a new inner product for SO 0 (4, 1)-invariant states was defined by what would later be termed group-averaging, which is an important ingredient in the refined algebraic quantization [14] 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 a unimodular group such as SO 0 (4, 1), to obtain |Ψ = G dg U(g)|ψ , (1.2) 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 dg. If the volume of the symmetry group is finite, the inner product of the invariant states |Ψ 1 and |Ψ 2 obtained from |ψ 1 and |ψ 2 as in (1.2) is where V G is the volume of the group G. If G has infinite volume, e.g. if it is SO 0 (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 By this group-averaging procedure one obtains an infinite-dimensional Hilbert space of SO 0 (4, 1) invariant states for linearized gravity in de Sitter space [3,13]. The groupaveraging procedure was carried out for this spacetime also for other free fields [16,17] 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. [20,21,22,23]). 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 × T 3 , where the spatial Cauchy surfaces are copies of T 3 , the 3-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 R ×U(1) 3 symmetry group of the background. Now, if one considers 4-dimensional N = 1 simple supergravity [26,27] on this background spacetime, it is not difficult to see that 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 2-dimensional surface at infinity of a Cauchy 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 as a surface integral at infinity of perturbations of the next order. Then, one expects that in perturbation theory in the flat R × T 3 background, the total energy and momentum of the linearized fields vanish because there is no spatial 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 Q α , 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 2-dimensional surface at infinity of a Cauchy surface in asymptotically-flat spacetime [28]. This fact again implies that in static 3-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 N = 1 simple supergravity in the background of flat R × T 3 spacetime. We show that these conditions 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 physcal 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 × T 3 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 Ref. [29] throughout this paper.
Here, e a µ are the vierbein fields, e := det(e a µ ), and Ψ µα , α = 1, 2, 3, 4, is the gravitino field, which is a Majorana spinor. The 4 × 4 gamma matrices γ a , a = 0, 1, 2, 3, satisfy the Clifford relation {γ a , γ b } = 2η ab , where η ab = diag(−1, 1, 1, 1). The indices a, b, c, . . . are raised and lowered by the flat metric η ab whereas the spacetime indices µ, ν, σ, . . . are raised and lowered by the spacetime metric g µν = e a µ e aν . The matrices γ a have the properties γ 0 † = −γ 0 and γ i † = γ i , i = 1, 2, 3. One defines γ µ := e µ a γ a and γ µνρ := γ [µ γ ν γ ρ] , where [· · ·] indicates total anti-symmetrization. The Riemann tensor is given by One also defines Ψ = Ψ T C, where C is a unitary matrix satisfying C T = −C and γ µT = −Cγ µ C −1 . The Majorana condition satisfied by Ψ µ reads (Ψ † µ ) α = i(Cγ 0 Ψ µ ) α [30]. We later choose a Majorana representation for γ a in which C = iγ 0 . In this representation we have (Ψ † µ ) α = (Ψ µ ) α . The gravitino covariant derivative in (2.1) is given by The action (2.1) is invariant under a local supersymmetry transformation of the form where Y µab is quadratic in Ψ µ and where ǫ is a spacetime dependent Grassmann variable with four components. (See, e.g. [30] for the explicit form of Y µab .) We consider this theory on a purely bosonic static background whose spatial sections are flat 3-dimensional tori. For the background geometry we therefore take where the spatial coordinates x, y and z are periodic with periods L 1 , L 2 and L 3 respectively, and we let V = L 1 L 2 L 3 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 ζ µ of the vierbein field e a µ and gravitino field Ψ µ is ‡ δ ζ e a µ = ζ ν ∂ ν e a µ + e a ν ∂ µ ζ ν , (2.8) Now, we write the vierbein field as e a µ = δ a µ +ẽ a µ . (2.10) That is, we write e a µ as the sum of its background value δ a µ and perturbationẽ a µ . Then, δ ζẽ a µ = δ a ν ∂ µ ζ ν + ζ ν ∂ νẽ a µ +ẽ a ν ∂ µ ζ ν . (2.11) It is important to note here that the part of δ ζẽ a µ that is independent ofẽ a µ vanishes if ζ µ is a constant vector, i.e. a Killing vector of the flat background. § Since the action (2.1) is invariant under the diffeomorphism transformation given by (2.9) and (2.11), we have 12) ‡ The transformation obtained as the commutator of two supersymmetry transformations is different from the diffeomorphism transformation presented here (see, e.g. [29]). The former takes the form to first order in the fieldsẽ a µ defined by (2.10) and Ψ µ . It can be shown that the LSCs corresponding to these two transformations are identical. § If the background has non-zero curvature, δ ξẽ a µ 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 transformation in this case to make δ ξẽ a µ vanish at lowest order. This is done in Appendix A.
We adopt left-variations for fermionic fields throughout this paper. We then expand δ ζẽ a µ , δ ζ Ψ µ , δS/δẽ a µ , δS/δΨ µ and √ −gJ µ (ζ) according to the order in the fieldsẽ a µ and Ψ µ , i.e. the number of these fields in the product, as (2.14) Recall that the background metric is flat. We note that E (0)µ a = 0 and E (0)µ = 0 because the flat spacetime with Ψ µ = 0 satisfies the field equations. That is, δS/δẽ a µ = 0 and δS/δΨ µ = 0 ifẽ a µ = 0 and Ψ µ = 0. The identity (2.12) must be satisfied order by order. The first-and second-order equalities read We emphasize here that these identities hold for any vector ζ µ and for any field configuration. Now, if the vector field ξ µ is a constant vector, then δ (0) ξẽ a µ = 0 as can readily be seen from (2.11). Hence, by (2.19), the current J  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 integral is evaluated. Since P (ξ) = 0 if the fields vanish, the conservation of this charge implies that for any field configuration if ξ µ 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. Now, suppose one attempts to solve the field equations forẽ a µ and Ψ µ order by order. Let (ẽ µ ) be the second-order correction. Then, µ ), and similarly for the others. The identities (2.19) and (2.20) imply for any vector field ζ µ as long as field equations (2.23)-(2.25) are satisfied. The charge corresponding to the conserved current J ] must vanish for any ζ µ because this charge is conserved even if the field ζ µ is smoothly deformed to zero in the past or future and is evaluated there. Hence, if we define for any vector field ζ µ as long as the fields (ẽ ν ) are perturbative solutions to the field equations. In particular, if ζ µ = ξ µ is a Killing vector, then because of (2.22). Equation (2.30) is a consequence of the requirement that the solution (ẽ ν ) extend to an exact solution and does not follow from the linearized field equations. 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 It is clear that ∆ ǫẽ a µ = ∆ ǫ e a µ given by (2.5) has no field independent contribution. That is, ∆ From this equation one finds for any spinor field ǫ and any field configuration. Since ∆ 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 for any spinor field ǫ if (ẽ ν ) gives a solution to the linearized field equations, E These are the LSCs arising from local supersymmetry on static 3-torus space. Next, we shall derive the conserved currents J given by (2.35) and the corresponding conserved charges P (1)µ a forẽ a µ , it is useful to note that eR in the Lagrangian density in terms of the vierbein fields equals √ −g R given in terms of the metric tensor g µν . Thus, we may vary √ −g R with respect to the metric tensor and then vary the metric tensor with respect to the vierbein fields. Writing g µν = η µν + h µν , we find the perturbation h µν in terms ofẽ a µ at first order as h µν = δ a µẽ aν +ẽ aµ δ a ν . (2.40) Then we find where h := η µν h µν and := ∂ λ ∂ λ , with indices raised and lowered by η µν . We readily find To find the bosonic conserved current J , which depends onẽ a µ only through h µν , is identified with the Noether current for the spacetime translation in the direction of ξ µ of the decoupled theory consisting of a Fierz-Pauli Lagrangian for the graviton and a Rarita-Schwinger Lagrangian for the Majorana gravitino [30] It can be given explicitly as Next we find the fermionic conserved current J (2)µ (ε) . We note that the first-order part of the global supersymmetry transformation is given by Note that the second term vanishes if the (linear) local Lorentz invariance is fixed by requiringẽ aν = δ µ a δ b νẽ bµ . (It vanishes by the linearized field equation for Ψ µ as well.) With this condition imposed, the term which explicitly depends onẽ bν in the supersymmetry transformation ∆   .37)). In the next section we discuss linearized supergravity on static 3-torus space at the classical level and express the LSCs in a form suitable for quantization.

Linearized Supergravity on static 3-torus space
The results of the previous section allow us to describe the LSCs entirely in terms of the linearized theory. By linearizing N = 1 simple supergravity in 4 dimensions about static 3-torus background spacetime, we have the Lagrangian, which is given here again for convenience: A suitable real Majorana representation for the γ-matrices is given by where σ 1 , σ 2 and σ 3 are the standard Pauli matrices. In this representation the charge conjugation matrix takes the form C = iγ 0 . The action with the Lagrangian density (3.1) is invariant under the following gauge transformations: where ζ µ is an arbitrary vector field and ǫ α is an arbitrary Majorana spinor field satisfying ǫ † α = ǫ α . 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 series. Due to the periodic boundary conditions, each field has a spatially constant zero-momentum component. The zeromomentum sector of the 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 where the volume of the torus is V = L 1 L 2 L 3 and the L 1 , L 2 and L 3 are the periods of the torus in the x-, y-and z-directions respectively. The periodic boundary condition implies k = 2π L 1 n 1 , 2π L 2 n 2 , 2π L 3 n 3 with n 1 , n 2 and n 3 integers, and reality restricts h µν to be real,h µν (t, − k) to be equal toh † µν (t, k), and similarly for ψ µ andΨ µ (t, k). The Lagrangian for the theory does not provide dynamical coupling between the modes with zero momentum and those with non-zero momentum k. Therefore, it is possible to analyse these separately. We begin with the zero-momentum sector of the theory. We write h 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 h µ0 and ψ 0α , and these are precisely the components which carry the gauge degrees of freedom. It is therefore possible to gauge-fix these components to vanish by solving h 00 (t) = 2∂ 0 ξ 0 (t), h i0 (t) = ∂ 0 ξ i (t), i = 1, 2, 3 and ψ 0 (t) = ∂ 0 ǫ(t). The elimination of h 0µ (t) and ψ 0 (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 h ij and vector-spinor ψ iα , 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 h ij . To analyse this tensor it is convenient to break it up into the trace and trace-free sectors. Thus, we let where the T A ij are a set of five trace-free tensors which satisfy the orthonormality condition T Aij T B ij = δ AB . We choose them as In terms of the six variables c and c A , the Lagrangian reads The momentum conjugate to h ij , i.e.
where c P = −∂ 0 c and c P A = ∂ 0 c A are the momenta conjugate to c and c A , respectively. The time derivative of the zero-momentum field h ij is then The canonical Poisson bracket relations for c, c A , c P and c P A , and equivalently for h ij and p kl , are The fermionic part of L 0 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 to be "promoted" to the anti-commutator upon quantization. The momentum variables π i α conjugate to ψ iα are where the derivative of L 0 with respect to ∂ 0 ψ iα is the left derivative. These momenta are not invertible in terms of the coordinates and velocities, so there are twelve primary constraints, where the λ i α 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 preserved under evolution by the primary Hamiltonian. This requirement leads to consistency conditions To evaluate the Poisson bracket between the constraints, we use the canonical Poisson bracket, which allows us to compute the Poisson bracket between the constraints (and therefore the primary Hamiltonian) as Hence, we find that the consistency conditions (3.16) imply λ i = 0. Thus, the primary Hamiltonian H P vanishes and, as a result, the fields ψ i 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 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 = iγ 0 , the Dirac bracket for ψ iα evaluates to To provide some insight into these relations we write where η and η A are Majorana spinors. The matrices T 1 and T 2 are given in (3.7). These symmetric and traceless matrices satisfy One can also show, by a component-by-component calculation, These relations can be used to show that the Dirac bracket relations for ψ iα are equivalent to 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 denoted byĥ µν andΨ µ , we can completely fix the gauge freedom in this sector by imposing the following conditions (see, for instance, [29,30]): Then we can write the expansion of the graviton as follows: (3.27) where k := | k| and k µ = (k, k). We have let the complex conjugate of a λ ( k) be denoted by a † λ ( k), anticipating quantization. The symmetric and traceless polarization tensors are given by where the polarization vectors ǫ λ i ( k) are given by The unit spatial vectorsê i ( k) = k/k form a right-handed orthonormal system in this order. The polarization vectors The Lagrangian density for the non-zero-momentum sector of the graviton field in this gauge can be found from (3.1) as The standard procedure to find the Poisson bracket relations for the coefficients a λ ( k) and a † λ ( k) leads to with all other brackets among a λ ( k) and a † λ ( k) vanishing. The non-zero-momentum sector of the gravitino field can similarly be expanded into modes asΨ where u ± ( k) are eigenspinors of γ 5 = iγ 0 γ 1 γ 2 γ 3 and γ 0k · γ, wherek := k/k, with eigenvalues ±1 and −1, respectively. We normalize them by requiring u λ † ( k)u λ ′ ( k ′ ) = 2kδ λλ ′ . The fermionic coefficients b λ ( k) and b † λ ( k) satisfy the classical analogues of the anti-commutation relations for the annihilation and creation operators: with all other brackets among b λ ( k) and b † λ ( k) vanishing.

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 respectively. On the other hand, the relations (3.24) and (3.33) become respectively. After imposing our gauge conditions and using the field equations, the time component of the conserved Noether current J where we have dropped the cross terms between the zero-momentum and non-zeromomentum sectors because they do not contribute to the space integral of J (4.7) By substituting (3.9), (3.27) and (3.32) into (4.6) and integrating the result over space, one finds 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 nonzero 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 a † λ ( k)a λ ( k) and b † λ ( k)b λ ( 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 n Bλ ( k) gravitons and n F λ ( k) gravitinos with momentum k and polarization λ can be given as Therefore, if we take some state which is proportional to a single eigenstate of the number operators, where Ψ is some wave function of c and c A , then the Hamiltonian and momentum operators, (4.8) and (4.9), acting on such a state take the form 14) where we have defined a "squared mass" for these states by 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 states for pure gravity on static 3-torus space. We apply this procedure to N = 1 simple supergravity in this section. We start with the Hilbert space H 0 of superpositions of the states defined by (4.13). Take two states in this Hilbert space of the form |ϕ 1 = Ψ 1 (c, c A ) ⊗ |Φ 1 and |ϕ 2 = Ψ 2 (c, c A ) ⊗ |Φ 2 , where |Φ 1 and |Φ 2 are states in the Fock space F of non-zero-momentum gravitons and gravitinos. The inner product between these states is where p is also a 5-dimensional vector and p · c = p A c A . The inner product for these states is Since the operators H and P are the generators of spacetime translations, we can construct states satisfying (4.17) by averaging the states |ϕ I over this translation group as follows: The integral over space acts as the projector onto the sector of the Fock space with zero total momentum: The integral over α 0 can readily be evaluated using the representation (4.19) since Thus we find, with the notation |Φ where E( p) = p 2 + M 2 and where f (±) Although we obtained the invariant state |ϕ Notice that, although this inner product is defined in terms of the "seed states" |ϕ I , it depends only on the invariant states |ϕ . This inner product is equivalent to that proposed in [34] in the context of quantum cosmology.
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 H B : This inner product can readily be evaluated as which is identical with the group-averaging inner product, ϕ H B , given by (4.26) except for the minus sign in the second term. Although the Klein-Gordon 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 6-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 can be found by group-averaging over the relevant supergroup.

Imposing the fermionic linearization stability conditions
The time component of the conserved fermionic current J 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 and substituting (3.10) and (3.21) into (5.1) and integrating over space, we find with the zero-and non-zero-momentum contributions respectively given by where we have integrated by parts in the second term of Q and used γ j ∂ jΨi = −γ 0 ∂ 0Ψi . One is readily able to find the mode expansion for Q, which then yields

(5.6)
This charge and the bosonic charges, P µ = (H, P ), satisfy the supersymmetry algebra. That is, [P µ , Q] − = 0, [P µ , P ν ] − = 0 and 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 = 0 and k = 0 proved in Appendix C: We also recall that γ µ are real, γ 5 is purely imaginary and that ǫ ± * ( k) = ǫ ∓ ( k) and u ± * ( k) = u ∓ ( k). We call this tensor product space also H B from now on in order not to complicate the notation. Now we are in a position to find the physical states |phys satisfying the fermionic QLSCs, Q α |phys = 0, as well as the bosonic ones, P µ |phys = 0. Since the operators P µ commute with Q α , we specialize to the Hilbert space H B of states satisfying the bosonic QLSCs and identify P µ with the null operator. Thus, we have [Q α , Q β ] + = 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 where M 2 is defined by (4.16). We specialize to an eigenspace of the operator M 2 without loss of generality. Thus, we treat M as if it were a non-negative number. Assume M > 0. Then, in each common eigenspace of the operators c P and c P A with eigenvalues p 0 and p A , respectively, satisfying p 2 0 − p 2 = M 2 , the operators Q where C α β B can be regarded as a 4 × 8 matrix with the rows and columns labeled by α and (B, β), respectively. Here, cosh θ = |c P |/M and sinh θ = | c P |/M, where c P = (c P 1 , c P 2 , c P 3 , c P 4 , c P 5 ). The four 8-dimensional column vectors of the matrix C α β B are orthonormal, i.e.
Then by Appendix D there is a unitary operator U, i.e. an operator preserving the inner product, on H 0F such that Instead of directly working with Q (0) α and Q α , it is more convenient to combine them into annihilation-and creation-type operators [33] as provided that M > 0. These new operators satisfy the anti-commutation relations of Fermi oscillators (up to a sign), i.e. a 2 Now we construct all states |ϕ (BF ) ∈ H B satisfying Q α |ϕ (BF ) = 0, α = 1, 2, 3, 4. These constraints can be organized as Note that all states are linear combinations of the states each annihilated by either b 1 for any state |ϕ (B) ∈ H B . The same is true for each pair of operators, (b 2 , b † 2 ), (a 1 , a † 1 ) and (a 2 , a † 2 ). 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 a i |χ (B) = b i |χ (B) = 0, i = 1, 2, by applying the creation-type operators a † i and b † i . Notice that equations (5.14) and (5.15) imply a 1 = Ud 1 U † and a 2 = Ud 2 U † , where U is a unitary operator on H 0F and where d 1 and d 2 are defined by (5.10). Since the states annihilated by d 1 and d 2 have positive norm as we stated before, we have χ (B) |χ (B) H B > 0. We label the 16 possible states in such a Fock space as follows: with each m 1 , m 2 , n 1 , n 2 being either 0 or 1. For example, we define For M = 0, formally one can proceed in a similar manner by considering (δ-function normalisable) plane wave states Ψ(c, c) = e ±ipc+i p· c . On such states the supercharge takes the form Q = p 2 (±η + R), with [η α , η β ] + = −δ αβ and [R α , R β ] + = δ αβ . Then, we define ladder-type operators a 1 = (η 1 + iη 2 )/ √ 2, a 2 = (η 3 + iη 4 )/ √ 2, b 1 = (R 1 + iR 2 )/ √ 2 and b 2 = (R 3 + iR 4 )/ √ 2 and proceed in a manner similar to the case with M > 0.
Thus, we look for states satisfying the fermionic QLSCs in the form c m 1 m 2 n 1 n 2 |m 1 m 2 n 1 n 2 . (5.21) We find The coefficient of the term |0m 2 1n 2 in this equation is −c 1m 2 1n 2 . Hence c 1m1n = 0 for all m and n. We can conclude similarly that c 0m0n = c m0n0 = c m1n1 = 0 by using the other constraints. Thus, c m 1 m 2 n 1 n 2 = 0 if m 1 = n 1 or m 2 = n 2 . Hence the invariant state |ϕ (BF ) ∈ H B , i.e. the state satisfying the fermionic QLSCs, must be of the following form: where A, B, C and D are constants. Note that the four states |1100 , |0110 , |1001 and |0011 are mutually orthogonal and satisfy 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: Thus, there is precisely one possible combination which satisfies the fermionic QLSCs in each Fock space spanned by (a † 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 for some linear combination |ϕ (B) of the states |m 1 m 2 n 1 n 2 , where θ = (θ 1 , θ 2 , θ 3 , θ 4 ) are Grassmann numbers and where θ = iθ T γ 0 and d 4 θ = dθ 4 dθ 3 dθ 2 dθ 1 . (The minus sign in (5.28) has been introduced for convenience.) By the usual rules, dθ α = 0 and dθ α θ α = 1, α = 1, 2, 3, 4, this equation becomes One readily finds that Q 1 Q 2 Q 3 Q 4 |m 1 m 2 n 1 n 2 = 0 unless |m 1 m 2 n 1 n 2 = |1100 , |0110 , |1001 or |0011 and that where the state |ϕ (BF ) P is defined by (5.25). Thus, starting from any linear combination |ϕ (B) of |m 1 m 2 n 1 n 2 , we find with κ ∈ C. Now, an explicit computation shows that ϕ (1) 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 in the 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 construction can in fact be understood as group-averaging over the supergroup generated by P µ and Q α . Starting from a state |ϕ I , I = 1, 2, in the original Hilbert space H 0 , we define a state satisfying all constraints by integrating over the supergroup [37,38]: where θ = (θ 1 , θ 2 , θ 3 , θ 4 ) are Grassmann numbers and α µ = (α 0 , α) are commuting numbers. As P µ and Q α commute, we can write this integral as In Appendix E we present an example of a state with two particles with non-zero momenta which satisfies all QLSCs.

Summary and discussion
In this paper we pointed out that there are fermionic linearization stability conditions as well as bosonic ones in 4-dimensional N = 1 simple supergravity in the background of static 3-torus space. Then we showed that 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 = 0 and P |phys = 0, have a natural physical picture. These conditions imply that all physical states are invariant 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 3-torus. However, the physical states still encode relative positions 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]. connection Γ σ ρµ . Our task is to choose s ab so that δ (m) e a µ is proportional to ∇ µ ζ ν + ∇ ν ζ µ . To this end, we require Thus, Then

Appendix B. Linearization stability condition for electrodynamics
In this appendix we discuss the linearization stability condition for quantum electrodynamics in a static 3-torus space. The Lagrangian density is where F µν = ∂ µ A ν − ∂ ν A µ and ψ = iψ † γ 0 . The field equations are where S is the action obtained by integrating L over spacetime. We have defined the left-hand side of (B.3) and (B.4) as the left functional derivative of the action S. The field equations (B.3) and (B.4) imply that the current J µ (E) = ieψγ µ ψ is conserved. As a result, the total charge, which is the integral of J 0 (E) over the space, is time independent. This charge must vanish because the zeroth component of (B.2) implies ∂ i F i0 = eψ † ψ and hence which must vanish by Gauss's divergence theorem. (Recall our space is compact.) At linearized level, the fields A µ 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 δA µ = ∂ µ Λ, δψ = ieΛψ for any function Λ of the spacetime point. The argument in section 3 leads to the conclusion that for some current J µ (Λ) [A, ψ] for any Λ. We indeed find that this equation is satisfied with The identity (B.7) is satisfied order by order in the fields A µ and ψ, i.e.
Now, if Λ(x) = λ is constant, then the left-hand side of (B.9) vanishes. Hence the current J (1)µ (λ) [A] = λ∂ ν F νµ is conserved for any field configuration A µ . In particular, we can smoothly deform A µ to zero in the far future or past of any given Cauchy surface while keeping the value of the conserved charge unchanged. It follows that Q (E) [A] = 0 for any field A µ . 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 Then, equations (B.9) and (B.10) imply that the current is conserved for any Λ(x). This implies that the corresponding charge must vanish, which is rather obvious in this example because, in fact, J 19). In particular, if we define 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) We first note that the matrices T B ij , B = 1, 2, are diagonal with T Bi k = δ i k q B k , where q 1 1 = −q 1 2 = 1/ √ 2, q 1 3 = 0. q 2 1 = q 2 2 = 1/ √ 6 and q 2 3 = −2/ √ 6. Hence By a component-by-component calculation we find By substituting this formula into (C.1) we obtain The second sum vanishes because the matrices T A ij are symmetric and traceless so that T A kj γ j γ k = T A jk (δ jk + γ jk ) = 0. Since the first sum is of the form S jj ′ γ j γ j ′ where S jj ′ is symmetric, we may replace γ j γ j ′ by δ jj ′ . Hence which is equation (5.8).