Quantum isotropy and the reduction of dynamics in Bianchi I
Christopher Beetle, Jonathan Steven Engle, Matthew Ernest Hogan, Phillip Mendonça
aa r X i v : . [ g r- q c ] F e b Quantum isotropy and the reduction of dynamics in Bianchi I
C Beetle , J S Engle , M E Hogan , , , , and P Mendon¸ca Department of Physics, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA Department of Physics and Astronomy, Texas Tech University – Costa Rica, Avenida Escaz´u, EdificioAE205, San Rafael de Escaz´u, San Jos´e 10201, Costa Rica Department of Physics and Astronomy, Texas Tech University, Box 41051, Lubbock, TX 79409, USA Department of Mathematics and Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX79409, USAE-mail: [email protected], [email protected], [email protected], [email protected]
Abstract.
The authors previously introduced a diffeomorphism-invariant definition of a homogeneous andisotropic sector of loop quantum gravity, along with a program to embed loop quantum cosmology into it.The present paper works out that program in detail for the simpler, but still physically non-trivial, case wherethe target of the embedding is the homogeneous, but not isotropic, Bianchi I model. The diffeomorphism-invariant conditions imposing homogeneity and isotropy in the full theory reduce to conditions imposingisotropy on an already homogeneous Bianchi I spacetime. The reduced conditions are invariant under theresidual diffeomorphisms still allowed after gauge fixing the Bianchi I model. We show that there is aunique embedding of the quantum isotropic model into the homogeneous quantum Bianchi I model that(a) is covariant with respect to the actions of such residual diffeomorphisms, and (b) intertwines both the(signed) volume operator and at least one directional Hubble rate. That embedding also intertwines allother operators of interest in the respective loop quantum cosmological models, including their Hamiltonianconstraints. It thus establishes a precise equivalence between dynamics in the isotropic sector of the BianchiI model and the quantized isotropic model, and not just their kinematics. We also discuss the adjointrelationship between the embedding map defined here and a projection map previously defined by Ashtekarand Wilson-Ewing. Finally, we highlight certain features that simplify this reduced embedding problem,but which may not have direct analogues in the embedding of homogeneous and isotropic loop quantumcosmology into the full theory of general relativity.
1. Introduction
Quantum gravity is a domain of physics in which contact with observation remains a challenge, due tothe extreme nature of the Planck scale where effects of the corresponding theory are expected to becomerelevant. That being said, due to cosmic expansion, the entire visible universe was once Planck sized. Indeed,cosmology has emerged as a promising domain in which to observe potential effects of quantum gravity [1–4],and perhaps such effects have even already been observed [5, 6].Loop quantum gravity (LQG) is a minimalist approach to a theory of quantum gravity guided foremostby Einstein’s general principle of relativity, which in modern times is reformulated as diffeomorphismcovariance, or background independence. Loop quantum cosmology (LQC) is a quantization of thehomogeneous isotropic sector of gravity using the same techniques as loop quantum gravity. To derive theeffects of LQG on cosmology, the nearly exact homogeneity and isotropy of the early universe is exploitedby using LQC for calculations. The relative simplicity of LQC allows for exact solutions to dynamics as wellas the construction of a complete set of Dirac observable operators.One can ask whether LQC, a quantization of a symmetry reduced sector of gravity, accurately reflectsthe physics of full loop quantum gravity. When the choices made in the quantizations of a field theory andits corresponding symmetry-reduced model are chosen to be appropriately compatible, symmetry reductionand quantization can indeed commute [7]. In order to ask whether LQC reflects the appropriate sector ofLQG, one must first specify what this sector is. It should be the quantum analogue of the homogeneousisotropic sector of classical gravity - that is, it should be the space of states in LQG which are homogeneousand isotropic in some sense which is compatible with the diffeomorphism invariance of the theory. A proposalfor such a sector has been defined in the prior work [8, 9] by finding diffeomorphism covariant phase space uantum isotropy and the reduction of dynamics in Bianchi I some maximal symmetry group on the spatial slice — the symmetry conditions .These phase space functions are furthermore readily quantizable on the loop quantum gravity Hilbert space,so that the simultaneous kernel of the corresponding operators defines the desired sector in question. Thesecond step is to find some embedding of LQC states into the states of this sector. The work [8, 9] did thisfor a non-interacting toy example and sketched how to embed LQC into full LQG.The value of constructing an embedding of LQC into full LQG is not simply to both clarify the meaningof homogeneous isotropic in LQG as well as to understand how well LQC represents the physics of this sector.The value, more importantly, lies in its potential to associate each quantization choice in the full theory witha corresponding choice in the reduced theory. With such an association in hand, contact between LQC andobservation can provide not only a test of LQG, but can also guide choices made in quantizing the full theory.There are a number of programs which have been introduced to establish such an association [10–16]. Theadvantages of the present program are that (1) it is compatible with the dynamics in the full theory, inthe sense that diffeomorphism covariance is left intact without gauge fixing, and (2) it is compatible withthe full space of states in LQG, in the sense that one does not need to restrict to states with support in alattice. Since the so-called ‘ µ -scheme’ in LQC arises from requirements of diffeomorphism covariance [16–18],it is reasonable to hope that the above two properties of the present strategy will enable a derivation of the µ -scheme from full LQG without inserting it by hand, in contrast to other approaches up until now. Still,we expect there to be a relation between the approach followed here and at least the approaches of [10, 14]:The map from LQC states to (gauge-fixed, lattice-truncated) LQG states implicit in these latter approachesare based on coherent states, and the range of this implicit embedding is the span of all coherent stateswith homogeneous isotropic labels. This space is precisely the simultaneous kernel of quantum operatorscorresponding to the ‘holomorphic part’ of the appropriate symmetry conditions [7, 19], which are complexin a way exactly analogous to the complex symmetry conditions considered in the strategy of the presentwork.The goal of the present paper is to complete the program of [8, 9], but in the simpler case of embeddingLQC into Bianchi I
LQC, in which homogeneity, but not isotropy, holds a priori . The goal of doing thisis to see how the program can be carried out to completion in this simpler, but still realistic case, therebysolidifying confidence in the program as well as providing an opportunity to gain intuition that will aid inapplying it to embed into full LQG. The results turn out to be cleaner, more satisfactory, and more revealingthan we had expected.In the Bianchi I model, the fully diffeomorphism-invariant condition imposing homogeneity and isotropyintroduced in [8, 9] reduces to a residual diffeomorphism-invariant condition imposing only isotropy , whichcan be easily quantized in a manner similar to that suggested in [8,9] for the full theory. We furthermore findthat there exists a unique embedding from isotropic to Bianchi I LQC states that is covariant with respectto (canonical) residual diffeomorphisms, and also intertwines the operators in the two theories correspondingto the signed volume and a single directional Hubble rate. This uniquely determined embedding has imagecontained in the kernel of the quantum isotropy conditions. It furthermore intertwines the Hamiltonianconstraints in the two theories, as well as all physically meaningful operators. Interestingly, it is preciselythe adjoint of the projection from Bianchi I to isotropic LQC proposed by Ashtekar and Wilson-Ewing in [20].The rest of this paper is organized as follows. In section 2 we review the Bianchi I model as defined byAshtekar and Wilson-Ewing in [20]. We then derive in section 3 the restriction, to the Bianchi I phase space,of the constraints proposed in [8] imposing diffeomorphism invariant homogeneity and isotropy. The Poissonbrackets of these symmetry conditions among themselves are calculated with an eye toward quantum theory.The general quantization strategy presented in [20] is then used to provide symmetry constraint operators onthe Bianchi I Hilbert space, whose simultaneous kernel defines the ‘quantum isotropic sector’ of Bianchi I.Section 3 ends with a review of the isotropic model. In section 4, we derive the embedding of this modelinto the quantum isotropic sector of Bianchi I, and exhibit its properties. The successes of the results aresufficiently surprising that we devote section 5 to clarifying the classical origins of these successes. Lastlywe close with a discussion. uantum isotropy and the reduction of dynamics in Bianchi I
2. Review of Bianchi I
The spacetime metric in the Bianchi I model has the formd s = − N ( t ) d t + a x ( t ) d x + a y ( t ) d y + a z ( t ) d z . (1)The natural (co-)triad field on a homogeneous slice of constant t is e ia ( t ) := a i ( t )˚ e ia , where ˚ e ia := d x ia (2)is the fiducial (co-)triad. Note that there is no sum over the index i in this definition of e ia . We will writeall such sums explicitly. Meanwhile, the extrinsic curvature of a homogeneous slice is K ab ( t ) = 12 L u q ab ( t ) = X i a i ( t ) ˙ a i ( t ) N ( t ) ˚ e ia ˚ e ib , (3)where u = N ( t ) ∂∂t is the future-directed, unit normal to the homogeneous slice. We will omit any explicit t -dependence below.Geometrically, the spatial coordinates x i in (1) can be defined as affine parameters along three mutuallyorthogonal congruences of parallel geodesics in the Euclidean spatial geometry of the Bianchi I model.Moreover, the directions of those congruences are fixed in (3) to coincide with the principal axes of theextrinsic curvature tensor K ab . Given appropriate Cauchy data for the Bianchi I model, consisting of aEuclidean spatial metric q ab and a homogeneous extrinsic curvature K ab , the spatial coordinates so definedare unique up to (a) affine reparameterizations ϕ ( m , b ) : x i ˜ x i := m i x i + b i of each congruence, with each m i = 0, and (b) permutations ϕ π : x i ˜ x i := x π ( i ) of the coordinate axes, with π ∈ S . Any choice ofsuch coordinates defines a canonical diffeomorphism from the spatial slice to R . The present coordinateambiguity therefore reflects the restricted diffeomorphism group Diff ≈ ( S ⋉ R × ) ⋉ R mapping R toitself, i.e. , the group of spatial diffeomorphisms that preserve the partial gauge-fixing conditions implicit in(1) and (3).The loop quantization of general relativity originates in the Ashtekar formulation of the classical theory.The basic variables of that formulation are the densitized triad E ai := | det e | e ai = | a x a y a z | a i ˚ E ai , (4)and the Ashtekar connection with Barbero–Immirzi parameter γ . The latter is given by γ A ia := Γ ia + γ K ab e bi = γ ˙ a i N ˚ e ia , (5)where Γ ia is the spin connection form for e ai , relative to a flat reference connection. Spatial geometry is alreadyflat in the Bianchi I model, so it is simplest to choose the reference connection to be the spin connection,whence Γ ia = 0. The symplectic structure in Ashtekar gravity generally has the formΩ( δ , δ ) := 2 κγ Z V δ [1 γ A ia δ E ai , (6)where κ = 8 πG Newton . The integral in (6) diverges when the field perturbations involved are homogeneousand the spatial slice V is not compact. But, precisely due to that homogeneity, it then makes sense torestrict the integral to a compact fiducial cell V , i.e. , to a finite, rectangular volume with edges parallel tothe coordinate axes [20]. The symplectic structure then reduces toΩ( δ , δ ) = 2 κγ X i δ [1 c i δ p i or Ω = 1 κγ X i d c i ∧ d p i , (7)where we have introduced the reduced phase space coordinates ( c i , p i ) such that γ A ia =: c i ˚ e ia L i and E ai =: L i L x L y L z p i ˚ E ai . (8) uantum isotropy and the reduction of dynamics in Bianchi I L i := | ∆ x i | of the edges of the fiducial cell V enter these definitions to render thecanonical coordinates independent of the initial choice of adapted coordinates x i in (1). It will be convenientto exclude those points of the phase space corresponding to degenerate spatial geometries, i.e. , having oneor more of the p i equal to zero. Our Bianchi I phase space is therefore Γ ∼ = R × R × topologically, wherethe second factor excludes the three coordinate planes in R where at least one p i vanishes.Now we consider the transformations of the Bianchi I phase space induced by the restricted spatialdiffeomorphisms described above. The phase-space transformation associated with a diffeomorphism ϕ ( m , b ) mapping each coordinate axis to itself follows immediately from the pull-backs ϕ ∗ ( m , b ) ˚ e ia = m i ˚ e ia ❀ ϕ ( m , b ) : ( c i , p i ) (˜ c i , ˜ p i ) := (cid:18) m i c i , | m x m y m z | m i p i (cid:19) . (9)The translation parameter b in ϕ ( m , b ) has no effect in phase space, as one would expect for a homogeneousmodel. The situation is slightly more complicated for the diffeomorphisms ϕ π : R → R that interchangethe coordinate axes because ϕ ∗ π ˚ e i = ˚ e π ( i ) , which generally differs from ˚ e i . Thus, whereas the definition (8)of the coordinates ( c i , p i ) presumes that A x ∝ d x , A y ∝ d y , and so forth, the pullback ϕ ∗ π A ia no longernecessarily satisfies this parallelism condition. This difficulty is easy to fix, however, by incorporating anappropriate, internal gauge rotation R ∈ SO (3) such that ϕ ( π,R ) : γ A ia γ ˜ A ia := X j R ij ϕ ∗ π γ A ja = X j R ij c j ˚ e π ( j ) a L j is again proportional to ˚ e ia , and similarly for the physical (co-)triad e ia . The rotation here must be chosensuch that R ij = 0 unless π ( j ) = i . The set of rotations mapping the coordinate axes into one another likethis form the (chiral) octahedral group O ⊂ SO (3), i.e. , the subgroup of rotations preserving the unit cube.For any fixed π ∈ S , there are exactly four rotations satisfying the above condition, differing from oneanother by half-rotations about one of the coordinate axes. Choosing any one of them leads to ϕ ( π,R ) : ( c i , p i ) (˜ c i , ˜ p i ) := (cid:18) m i c π − ( i ) , m i p π − ( i ) (cid:19) with m i := R iπ − ( i ) L i L π − ( i ) . (Note that R iπ − i ( i ) = ± | m x m y m z | = 1 by definition.) Composing with an appropriate scalingtransformation from (9) thus leads to a transformation that simply permutes the ( c i , p i ) coordinates inpairs.The residual automorphism group Aut R is the set of distinct phase-space transformations inducedby the restricted diffeomorphisms described above. In detail, Aut R ∼ = [Diff × SO (3)] q /K Γ is isomorphicto the group of restricted diffeomorphisms, extended to include (homogeneous) internal gauge rotations,then restricted to preserve the parallelism of (8), and finally quotiented by the (normal) subgroup K Γ ofsuch transformations that act as the identity in phase space. The resulting group is naturally a semi-directproduct Aut R = (cid:0) Dil R × Par R ) ⋊ Rot R of three distinct factors, consisting of(i) anisotropic dilatations ϕ t ∈ Dil R ∼ = R , labeled by t ∈ R and having the form ϕ t ( c i , p i ) := (cid:0) e − t i c i , e t i − T p i (cid:1) with T := P i t i ; (10)(ii) partial reflections ϕ ζ ∈ Par R ∼ = S , labeled by ζ ∈ {± } and having the form ϕ ζ ( c i , p i ) := (cid:0) ζ i c i , ζ i p i (cid:1) ; (11)(iii) and residual rotations ϕ π ∈ Rot R ∼ = S , labeled by π ∈ S and having the form ϕ π ( c i , p i ) := (cid:0) c π ( i ) , p π ( i ) (cid:1) . (12)Note that the partial reflections and residual rotations together define a natural action of the (achiral)octahedral group Par R ⋊ Rot R ∼ = O h ⊂ O (3), which is the full isometry group of the unit cube, includingreflections. The residual automorphism group has a non-trivial center Z (Aut R ), consisting of uantum isotropy and the reduction of dynamics in Bianchi I S ) isotropic dilations ϕ T ∈ Dil S ∼ = R + , labeled by T ∈ R and having the form ϕ T ( c i , p i ) := (cid:0) e − T/ c i , e − T/ p i (cid:1) ; (13)(ii S ) and isotropic reflections ϕ Z ∈ Par S ∼ = S , labeled by Z ∈ {± } and having the form ϕ Z ( c i , p i ) := (cid:0) Z c i , Z p i (cid:1) . (14)We refer to Z (Aut R ) ⊳ Aut R as the isotropic automorphism group not only because its elements “actisotropically” in the Bianchi I phase space, but also because it is naturally isomorphic to the group ofresidual automorphisms analogous to Aut R for the fully reduced, isotropic model to be discussed in thenext section. The quotient group Aut R /Z (Aut R ) plays a pivotal role in relating the Bianchi I model to itsisotropic reduction. This quotient can be identified with the (normal) subgroup Aut R ⊳ Aut R having T = 0in (10) and ζ x ζ y ζ z = 1 in (11). We refer to this as the proper residual automorphism group becauseits elements preserve both the symplectic structure (7) and the orientation of the physical triad (2).Turning now to the dynamics of the Bianchi I model, recall that the gravitational part of the classicalHamiltonian constraint involves the curvature of the homogeneous Ashtekar connection (5): γ F abi := d γ A iab + X jk ǫ ijk γ A ja γ A kb = L i L x L y L z c x c y c z c i X jk ǫ ijk ˚ e ja ˚ e kb . (15)The coordinate scales L i enter because the result is expressed in terms of the fiducial triad. One can relateit instead to the physical triad by solving for the original scale factors a i : p i = L x L y L z L i | a x a y a z | a i ❀ a i = 1 L i vol( p ) p i with vol( p ) := | p x p y p z | / . (16)Geometrically, vol( p ) is the proper volume of the fiducial cell. Substituting into (15) then gives γ F abi = c x c y c z c i sgn( p x p y p z ) p i Σ abi with Σ abi := X jk ǫ ijk e ja e kb . (17)This result can be expressed compactly in terms of the directional Hubble rates θ i := L u a i a i = ˙ a i N a i = c i p i γ vol( p ) ❀ γ A ia = γ K ia = γ θ i e ia . (18)Each Hubble rate is invariant under anisotropic dilatations (10) and partial reflections (11), and they permutecovariantly under the residual rotations of (12). Meanwhile, the curvature of (17) is given by γ F abi = γ θ x θ y θ z θ i Σ abi = γ X jk ǫ ijk θ j θ k e ja e kb . (19)This yields a compact expression for (the gravitational part of) the Hamiltonian constraint: H g [ N ] := 12 κ Z V X ij N E ai E bj | det E | / (cid:18)X k ǫ ij k γ F abk − γ ) K [ ai K b ] j (cid:19)! = − sgn(det e ) κγ Z V N X k γ F k ∧ e k = − vol[ N ]( p ) κ X i
0) monotonically with p i . Thus, a Schr¨odinger representationbased on p is closely related to a Schr¨odinger representation based on λ , though the two have different naturalinner products since d p = | p | / vol( p ) d λ . More importantly, however, the vector field in question is aconstant multiple of ∂∂λ i on each of its integral curves, and therefore generates a rigid translation in λ i . Itfollows that the natural (Schr¨odinger) quantization of ∆ i ( s ) is such that (cid:10) λ (cid:12)(cid:12) ˆ∆ i ( s ) := (cid:28) λ + ~ κγs p | p | / | λ i || λ x λ y λ z | ss e i (cid:12)(cid:12)(cid:12)(cid:12) , (31)where e i is the canonical basis vector in R and s is a length scale to be fixed below. As usual, the dualbasis vectors h λ | here map a state | ψ i to its value ψ ( λ ) at a particular point λ ∈ R . Note that, if we were toreplace | p i | with p i on the right side of (28), then the flow of this vector field would reverse in the half-space λ i <
0, and therefore would not be globally integrable [21]. (For s <
0, for example, the flow would convergeon the plane λ i = 0 from both sides in finite affine parameter “time,” and one cannot continue to integratethrough that plane.)Although the Schr¨odinger representation based on p motivates the quantization (31) of ∆ i ( s ), theresulting operator needs to act in the “polymer” Hilbert space [20] of loop quantum cosmology. The innerproduct on this space is the sum h φ, ψ i = X p ¯ φ ( p ) ψ ( p ) = X λ ¯ φ ( λ ) ψ ( λ ) . (32)The distinction is important. The ordering | p i | / c i
7→ | ˆ p i | / ˆ c i chosen in (29) is the unique one thatleaves a constant wave function ψ ( p ) = ψ invariant under the action of the resulting translation operator.But this ordering is not Hermitian in the Schr¨odinger representation based on p , and its exponential ˆ∆ i ( s )is not unitary: this is because the Lebesgue measure d p is not invariant under a rigid translation in λ .The polymer representations based on p and λ are the same, however, so (31) is unitary in loop quantumcosmology.Typically one would take the limit s → after quantization to remove the regulator in (31), but thatlimit does not exist in loop quantum cosmology. Instead, one fixes a certain finite value of s to define the curvature and Hamiltonian constraint operators by setting s := ~ κ | γ | p j ( j + 1) , (33) uantum isotropy and the reduction of dynamics in Bianchi I j = so that s is the minimal quantum of area in full loop quantum gravity.This fixes the length scale introduced in (31). Then one chooses the area scale p from (30) such that theratio of dimensional factors in (31) is one half: p = ~ κγ p j ( j + 1) . (34)With these choices, the basic operators of loop quantum cosmology act according to (cid:10) λ (cid:12)(cid:12) ˆ∆ i ( s ) := (cid:28) λ + | λ i || λ x λ y λ z | s s e i (cid:12)(cid:12)(cid:12)(cid:12) and (cid:10) λ (cid:12)(cid:12) ˆ p i := p P ( λ i ) (cid:10) λ (cid:12)(cid:12) , (35)where again P ( λ i ) := sgn( λ i ) λ i . The regularization scheme for the Hamiltonian constraint in [20,21], whichwe extend here to other operators that are needed to enforce the quantum symmetry conditions, simply setsall s i = s . Accordingly, we introduce the shorthandsˆ∆ i := ˆ∆ i ( s ) ❀ (cid:10) λ (cid:12)(cid:12) ˆ∆ i := (cid:28) λ + | λ i | | Λ | e i (cid:12)(cid:12)(cid:12)(cid:12) with Λ := λ x λ y λ z . (36)For purposes of comparison, the basic holonomy operators ˆ E ± i from [21] ‡ correspond to ˆ∆ ± i in the notationwe use here. Note that the effect of ˆ∆ ± i is to shift only the λ i component of the argument of the given wavefunction ψ ( λ ) := h λ | ψ i such that Λ Λ ± := Λ ± sgn(Λ λ i ).The standard approach in loop quantum cosmology is to reduce the regularized Hamiltonian constraint(26) to the scalar form (27) prior to quantization. To do this, write (27) in the form H g [ N ]( s ) = − vol[ N ]( p ) κ X i 3. Reduction to the Isotropic Model The companion paper [8] selects the homogeneous and isotropic section of general relativity by setting S [ f, g ] := B [ f ] vol[ g ] − vol[ f ] B [ g ] ≈ , (41)for arbitrary smearing fields f ij and g ij , where B [ f ] := sgn(det e ) X ij Z V F i ∧ e j f ij and vol[ f ] := Z V tr f | det e | . (42)The curvature appearing in the definition of B [ f ] is that of the complexified connection A ia := A ia + i α e ia , (43)where α is an arbitrary, but fixed, real constant with units of inverse length. The conditions (41) imposinghomogeneity and isotropy in this approach are diffeomorphism covariant in the sense that replacing both thefundamental fields ( A, E ) and the smearing fields ( f, g ) with their images under a spatial diffeomorphismleaves S [ f, g ] unchanged. Requiring (41) for all choices of the smearing fields therefore selects those points( A, E ) of the phase space that are invariant under some action, as opposed to under a fixed action, of thesymmetry group for the appropriate class of isotropic and homogeneous cosmologies.The symmetry conditions (41) simplify considerably when restricted to the phase space of Bianchi Icosmologies described in the previous section. Specifically, (43) becomes A i = i ˚ e i L i := (cid:18) c i + i α vol( p ) p i (cid:19) ˚ e i L i , and the functionals from (42) become B [ f ] = X i p x p y p z x y z p i i vol ( p ) Z V f ii | det e | and vol[ f ] = X i Z V f ii | det e | , respectively. These both depend on the smearing field f ij only through the average values f i := 1vol( p ) Z V f ii | det e | of its diagonal components over the fiducial cell. Note that such an average is independent of the(homogeneous) triad field in a Bianchi I geometry. Using these averages, together with the definition (18)of the directional Hubble rates as functions on phase space, then gives B [ f ] = X i f i B i := vol( p ) X i f i Y j = i ( γθ j + i α ) ! and vol[ f ] = vol( p ) X i f i . (44)Finally, substituting these expressions into (41) gives the symmetry conditions S [ f, g ] = vol( p ) X ij f i g j ( B i − B j ) ≈ uantum isotropy and the reduction of dynamics in Bianchi I B i − B j = B i ς j − ς i B j in the sum, where ς i denotes the vector with all components equal to one. Doing so shows that the symmetryconditions hold for all smearing fields if and only if B ∧ ς = 0, meaning that the two vectors are proportional,and thus that B x = B y = B z . Furthermore, we have that( γθ y + i α )( γθ z + i α ) = ( γθ x + i α )( γθ z + i α ) = ( γθ y + i α )( γθ z + i α ) ⇔ θ x = θ y = θ z . (46)Thus, the full content of the symmetry conditions (41) in the Bianchi I model is just that all three directionalHubble rates are the same.In order to impose the symmetry conditions (45) simultaneously in the canonical formalism, one mustcheck that their Poisson algebra closes. To do so, first define S [ f, g ] = X ijk ǫ ijk f i g j S k with Sk := vol( p ) X lm ǫ klm B l . (47)The symmetry conditions, for all smearing fields f ij and g ij , is equivalent to S x = S y = S z = 0 due to thehomogeneity of the Bianch I model. Furthermore, { S x , S y } = { (cid:0) c x p x + i α vol( p ) (cid:1)(cid:0) c y p y − c z p z (cid:1) , (cid:0) c y p y + i α vol( p ) (cid:1)(cid:0) c z p z − c x p x (cid:1) } = (cid:0) c x p x + i α vol( p ) (cid:1)(cid:0) c z p z − c x p x (cid:1) { c y p y − c z p z , i α vol( p ) } + (cid:0) c y p y + i α vol( p ) (cid:1)(cid:0) c y p y − c z p z (cid:1) { i α vol( p ) , c z p z − c x p x } + (cid:0) c y p y − c z p z (cid:1)(cid:0) c z p z − c x p x (cid:1)(cid:16) { c x p x , i α vol( p ) } + { i α vol( p ) , c y p y } (cid:17) = 0 , (48)and cyclic permutations. We have used { c i p i , c j p j } = 0 in passing to the second line here, as well as { c i p i , vol( p ) } = κγ vol( p ) in the final step. This is a stronger result than in the full theory [8], where thePoisson algebra of the symmetry conditions is closed ( i.e. , the Poisson bracket of two S ′ s is a sum of termsproportional to S ’s) but not trivial. A similar calculation shows that { S x , ¯ S y } = − i ακγ (cid:0) c y p y − c z p z (cid:1)(cid:0) c z p z − c x p x (cid:1) vol( p ) . (49)Although this Poisson bracket does not vanish everywhere in phase space, it does vanish when the symmetryconditions hold. Again, this is a stronger result than in the full theory [8], where the Poisson brackets ofthe symmetry conditions and their complex conjugates generally do not vanish even weakly, i.e. , on thesubmanifold where the symmetry conditions hold. This result is attributable to the proportionality betweeneach symmetry condition and its complex conjugate in the homogeneous Bianchi I model with coefficientnon-zero and smooth throughout Γ,¯ S y = (cid:0) c y p y − i α vol( p ) (cid:1)(cid:0) c z p z − c x p x (cid:1) = c y p y − i α vol( p ) c y p y + i α vol( p ) S y =: η y S y , (50)as it implies immediately { S x , ¯ S y } = { S x , η y } S y + { S x , S y } η y ≈ . Let Γ ∼ = R × R × denote the classical phase space of the Bianchi I model constructed in the previoussection. Let ¯Γ ⊂ Γ denote the classical isotropic sector on which the symmetry conditions (41) hold,or equivalently, on which θ x = θ y = θ z . There are only two independent conditions here, so ¯Γ is (locally)a 4-dimensional submanifold of the 6-dimensional phase space Γ. We can pull the symplectic structure (7)back to ¯Γ by first writingΩ = 1 κγ X i d c i ∧ d p i = 1 κ X i d (cid:18) θ i vol( p ) p i (cid:19) ∧ d p i = 1 κ d vol( p ) X i θ i d ln | p i | ! uantum isotropy and the reduction of dynamics in Bianchi I θ x = θ y = θ z =: θ , thend ln vol( p ) = 12 X i d ln | p i | ❀ Ω = 2 κ d θ ∧ d vol( p ) . (51)This is clearly degenerate, with a kernel consisting of vectors tangent to ¯Γ ⊂ Γ that change neither thecommon value θ of the directional Hubble rates, nor the proper volume vol( p ) of the fiducial cell.The appropriate, non-degenerate isotropic phase space is the quotient manifold Γ S ∼ = R × R × ,consisting of equivalence classes of points ( c i , p i ) ∈ ¯Γ on which the geometric means c := √ c x c y c z and p := √ p x p y p z both take constant values. It is sometimes convenient to use the signed volume v := sgn p | p | / instead of p itself as a phase-space coordinate for the isotropic model. The volume and Hubble rate(s) arevol( p ) = | p | / = | v | and θ = p θ x θ y θ z = c sgn pγ | p | / = cγv / (52)respectively, on each equivalence class in Γ S . They therefore descend to well-defined functions on the reducedphase space. Any function of these quantities likewise descends to Γ S , including in particular the gravitationalpart (20) of the Hamiltonian constraint H g = − κγ | p | (3 n +1) / c = − κγ | v | n +1 / c , (53)where we have fixed the lapse N = vol( p ) n , as well as the regularized Hubble rates (25)¯ θ i ( s ) = sgn pγs sin cs | p | / = 1 γs sin csv / , and the regularized Hamiltonian constraint (27) derived from them. The symplectic structure (51) becomesΩ = 3 κγ d c ∧ d p = 2 κγ | v | / d c ∧ d v = 2 κ d( θ sgn p ) ∧ d v. This clearly descends to Γ S as well, where it is equivalent to the standard Poisson bracket { c, p } = κγ ofisotropic loop quantum cosmology.As mentioned in the previous section, the residual automorphism group for the isotropic model isnaturally isomorphic to the center Z (Aut R ) of the residual automorphism group for the Bianchi I model. Indetail, the isotropic dilatations (13) and isotropic reflections (14) act on Γ S via ϕ T : ( c, p ) (cid:0) e − T/ c, e − T/ p (cid:1) and ϕ Z : ( c, p ) (cid:0) Zc, Zp (cid:1) , (54)respectively. More importantly, however, the complementary subgroup Aut R ∼ = Aut R /Z (Aut R ) of properresidual automorphisms acts transitively on the equivalence class of points ( c i , p i ) ∈ ¯Γ corresponding to anygiven point ( c, p ) ∈ Γ S of the isotropic phase space. To see this, first observe that( c i , p i ) = (cid:18) cpθ θ i p i , p i p p (cid:19) = ϕ t ◦ ϕ ζ (cid:18) cθ θ i , ( p, p, p ) (cid:19) with ( t i := ln | p i | − ln | p | ζ i := sgn( pp i )for any point ( c i , p i ) ∈ Γ, where θ := p θ x θ y θ z denotes the geometric mean of the directional Hubble rates.The residual automorphism on the right here is proper because P i t i = 0 and ζ x ζ y ζ z = 1 by construction.Inverting it shows that every ( c i , p i ) ∈ Γ can be put in a “partly diagonal” form with p x = p y = p z = p by anappropriate proper residual automorphism. Furthermore, the resulting phase-space point is “fully diagonal”in the sense that c x = c y = c z = c as well if and only if ( c i , p i ) ∈ Γ lies in the classical isotropic sector ¯Γ ⊂ Γwhere θ x = θ y = θ z = θ . This fact characterizes the classical isotropic sector ¯Γ ⊂ Γ purely in terms of theaction of the residual automorphism group: uantum isotropy and the reduction of dynamics in Bianchi I Theorem 1. A point ( c i , p i ) ∈ Γ of the Bianchi I phase space lies in the classical isotropic sector ¯Γ ⊂ Γ ifand only if there exists a residual automorphism ϕ ∈ Aut R such that ϕ ◦ ϕ π ◦ ϕ − ( c i , p i ) = ( c i , p i ) (55) for all residual rotations ϕ π ∈ Rot R . One may choose ϕ ∈ Aut R to be proper without loss of generality.3.3. Quantum Isotropy and the Isotropic Model Working in the Hilbert space H of the Bianchi I model, we define the (regularized) operator analogues ofthe functions S i from (47) that define the classical isotropic sector as follows:ˆ S x ( s ) = | ˆ v | (cid:0) γ ˆ θ x ( s ) + i α (cid:1) | ˆ v | (cid:0) γ ˆ θ z ( s ) − γ ˆ θ y ( s ) (cid:1) + (cid:0) γ ˆ θ z ( s ) − γ ˆ θ y ( s ) (cid:1) | ˆ v | (cid:0) γ ˆ θ x ( s ) + i α (cid:1) | ˆ v | , (56)and cyclic permutations. The regularized Hubble rate operators ˆ θ i ( s ) are defined in (38), and the orderingprescription adopted here at the quantum level mimics that of the Hamiltonian constraint from (40). The quantum isotropic sector is the subspace V symm ⊂ H of Bianchi I states that are annihilated by all threeoperators ˆ S i . It is not obvious at the moment that any such states exist. But we will see in the next sectionthat indeed they do by showing that all three operators annihilate every state in a particular embedding ofthe Hilbert space of the fully isotropic theory into H .To compare the isotropic sector of the quantum Bianchi I model to the quantum isotropic model —wherein isotropy is imposed at the classical level, prior to quantization — we must of course review thequantization of the fully reduced model itself. It proceeds [16] similarly to that of the Bianchi I modelpresented in detail above. We introduce the the exponentials∆( s ) := exp − ic s | p | / , and motivate their quantization by recalling that c becomes a differential operator in a Schr¨odingerquantization based on p , and ˆ∆( s ) := exp s ~ κγ | p | / ddp = exp s ~ κγ ddv . a shift operator acting on ˆ v eigenstates as h v | ˆ∆( s ) = D v + s ~ κγ (cid:12)(cid:12)(cid:12) = D v + 12 ss v (cid:12)(cid:12)(cid:12) . The Hubble rate (52) can again be expressed as limits of combinations of ∆( s ) and v : θ = lim s → θ ( s ) := lim s → sgn v (∆( − s ) − ∆(2 s ))2 iγs . Weyl ordering yields the regulated operatorˆ θ ( s ) = sgn ˆ v ( ˆ∆( − s ) − ˆ∆(2 s )) + ( ˆ∆( − s ) − ˆ∆(2 s )) sgn ˆ v iγs = (cid:16) ˆ∆( − s ) Θ (cid:0) | ˆ v | + ( v s/s ) sgn ˆ v (cid:1) − ˆ∆(2 s ) Θ (cid:0) | ˆ v | − ( v s/s ) sgn ˆ v (cid:1)(cid:17) sgn ˆ v γs . (57)Following [16], we again take the limits to s = s , so that ˆ θ = ˆ θ ( s ). As in the Bianchi I case, the isotropicHamiltonian constraint (53) can be expressed in terms of v and θ and quantized using a symmetric ordering,yielding the constraint operator of [16],ˆ H g = − κ | ˆ v | n ˆ θ | ˆ v | n ˆ θ | ˆ v | n . (58) uantum isotropy and the reduction of dynamics in Bianchi I 4. Embedding The definition of a gauge and diffeomorphism-invariant homogeneous isotropic sector in full loop quantumgravity is only the first part of the strategy outlined in [8]. The second part is to define an embedding ι Full of the isotropic model into this sector, and use this embedding to compare operators and dynamics in thetwo models. The strategy presented there is to define ι Full by stipulating the following conditions:(i) ι Full should map states into the quantum homogeneous isotropic sector. That is, it’s image should beannihilated by the symmetry constraint operators ˆ S [ f, g ] for all f and g .(ii) ι Full should intertwine two pairs of operators ( ˆ O i Full , ˆ O iS ), i = 1 , O i Full ◦ ι Full = ι Full ◦ ˆ O iS , (59)corresponding to the two dimensions of the homogeneous isotropic phase space.The first condition fixes the image of ι Full , while the second condition fixes how states in this image areidentified with states in the homogeneous isotropic model. If we use ι Full to identify homogeneous isotropicstates with full theory states, the second condition (59) simply states that ˆ O i Full should have the same actionon homogeneous isotropic states as ˆ O iS .For the present paper, the task is to find an embedding ι of the isotropic quantum model into theBianchi I quantum model. The analogue of the above conditions is then(i) ˆ S i ◦ ι = 0 for all i .(ii) ˆ O i ◦ ι = ι ◦ ˆ O iS for two pairs of operators ( ˆ O i , ˆ O iS ) in the Bianchi I and isotropic models, i = 1 , even once the Gauss and diffeomorphism constraints areimposed . As a consequence, in the Bianchi I case, there is an additional covariance condition which can andmust be stipulated:(iii) ι should be covariant under all residual automorphisms well defined in the quantum theory.As we shall argue below, conditions (i) and (iii) are expected to have the same content from classical analysis,and, in the quantum theory, we will see that (iii) implies (i). For this reason, we impose (iii), and let (i)follow as a consequence. ¶ In fact, the classical analysis will lead us to expect not only the equivalence of (i)and (iii), but also the equivalence of(i) ˆ S † i ◦ ι = 0with both of these, and we will see explicitly in the quantum theory that (iii) implies not only (i), but (i) aswell.In this section, our imposition of (iii) — basically equivalent to (i) — and (ii) will uniquely determine ι .This is consistent with the results found for the toy model in appendix B of [8]. Once uniquely determined, ι can be used to compare other operators ( ˆ O, ˆ O s ) in the two models, again via the intertwining conditionˆ O ◦ ι Full = ι Full ◦ ˆ O S . (60)Note that if ˆ O S is not known, the above equation will also uniquely determine it. Hence, the above equationcan also be thought of as defining a map from Bianchi I operators preserving the isotropic sector, to operatorsin LQC. Remarkably, in the end, we will find that ι maps all of the physically relevant operators in Bianchi Iintroduced in section 2.3 exactly to the corresponding operators in the isotropic theory introduced in section3.3. This includes the Hamiltonian constraint operators in the two models, so that the embedding ι willestablish that the isotropic model captures both the kinematics and dynamics of the isotropic sector of thequantum Bianchi I exactly . ¶ Once a single superselection sector is picked in the Bianchi I model [20], the implication also goes in the opposite direction.However, the argument for superselection comes from a specific dynamics. Part of the purpose of this work is to test compatibilityof the dynamics in the isotropic and Bianchi I models, so that we preferred our presentation to be independent of any one choiceof dynamics, and hence independent of any superselection. uantum isotropy and the reduction of dynamics in Bianchi I Let Dil oR denote the proper anisotropic dilitations , that is, the dilitations preserving the volume ofthe fiducial cell. The subgroup of the residual automorphisms introduced in subsection 2.1 that arecanonical transformations, hence with unitary action on quantum states, we call the canonical residualautomorphisms Aut CR . Explicitly, it is generated by the proper anisotropic dilitations, the partialreflections, and the residual rotations, Aut CR = (Dil oR × Par R ) ⋊ Rot R . From equations (10), (11), and(12), for each t x , t y ∈ R , ζ ∈ {± } , and π ∈ S , the actions of these three types of transformations in thequantum theory is given by ˆ ϕ ( t x ,t y , − t x − t y ) | p x , p y , p z i = | e t x p x , e t y p y , e − t x − t y p z i , ˆ ϕ ζ | ( p i ) i = | ( ζ i p i ) i , (61)ˆ ϕ π | ( p i ) i = | ( p π ( i ) ) i . As discussed in section 2, the residual automorphisms, when acting on the isotropic phase space, reduce tothe group of isotropic automorphisms Aut S . For canonical residual automorphisms T := P i =1 t i = 0, sothat Aut CR reduces to the even smaller group of isotropic reflections Par S . That is, the actions of the properanisotropic dilitations and residual rotations on the isotropic phase space are trivial, while the action of thepartial reflections is given by (54), so that the quantum action is given byˆ ϕ ( t x ,t y , − t x − t y ) | p i = | p i , ˆ ϕ ζ | p i = | ζ ζ ζ p i , (62)ˆ ϕ π | p i = | p i . For the purpose of actuallyderiving the embedding, it is convenient to label the momentum basis in Bianchi I using λ x , λ y , and Λ (30,36), and to label the momentum basis in the isotropic theory also by Λ. In terms of these labels, the actionof the most general canonical residual automorphism ϕ := ϕ ( t x ,t y , − t x − t y ) ◦ ϕ ζ ◦ ϕ π is given byˆ ϕ | λ x , λ y , Λ i = (cid:12)(cid:12)(cid:12) ζ x e t x / λ π ( x ) , ζ y e t y / λ π ( y ) , ζ x ζ y ζ z Λ E ˆ ϕ | Λ i = | ζ x ζ y ζ z Λ i . Imposing covariance of ι under all such transformations, ˆ ϕ ◦ ι = ι ◦ ˆ ϕ , leads to the following condition on thematrix elements of ι : h λ x , λ y , Λ | ι | Λ ′ i = h ζ x e − t x / λ π ( x ) , ζ y e − t y / λ π ( y ) , ζ x ζ y ζ z Λ | ι | ζ x ζ y ζ z Λ ′ i (63)for all t x , t y , ζ , π . First setting ζ z = ζ x ζ y and π = id, imposing this condition for all t x , t y , ζ x , ζ y leads to h λ x , λ y , Λ | ι | Λ ′ i = h β x λ x , β y λ y , Λ | ι | Λ ′ i for all β x , β y ∈ R × . Since λ x = 0 and λ y = 0, setting β x = λ − x and β y = λ − y , we have h λ x , λ y , Λ | ι | Λ ′ i = h , , Λ | ι | Λ ′ i =: C (Λ; Λ ′ )for all ( λ x , λ y , Λ) ∈ R × . These matrix elements then furthermore satisfy (63) for all canonical residualautomorphisms if and only C (Λ; Λ ′ ) additionally satisfies C ( − Λ; − Λ ′ ) = C (Λ; Λ ′ ). Explicitly, the resultingembedding then takes the form ι | Λ ′ i = X λ x ,λ y , Λ =0 C (Λ; Λ ′ ) | λ x , λ y , Λ i = X Λ =0 C (Λ; Λ ′ ) X λ x ,λ y =0 | λ x , λ y , Λ i =: X Λ =0 C (Λ; Λ ′ ) ι | Λ i . (64) uantum isotropy and the reduction of dynamics in Bianchi I ι is necessarily non-normalizable in the polymer inner product (32). Define Cyl BI to be the space of finite linear combinations of momentumeigenstates in the Bianchi I Hilbert space. Then, what we are saying is that covariance under canonicalresidual automorphisms forces the image of ι to be represented in the algebraic dual Cyl ∗ BI , which includespossibly non-normalizable linear combinations of momentum eigenstates. This is similar to what happenswhen solving the diffeomorphism constraint in the full theory [24]. ι then Automatically Satisfies Quantum Isotropy Independent of Ordering Ambiguity. Lemma 1. ι intertwines both ˆ v and ˆ θ i ( s ) for all s . Proof. That ι intertwines ˆ v is immediate: ι ◦ ˆ v | Λ i = v Λ ι | Λ i = v Λ X λ x ,λ y =0 | λ x , λ y , Λ i = ˆ v X λ x ,λ y =0 | λ x , λ y , Λ i = ˆ v ◦ ι | Λ i . For the ˆ θ i ( s ), it is sufficient to consider ˆ θ z ( s ). Starting from equations (38) and (35), we have for all | Λ i ,ˆ θ z ( s ) ◦ ι | Λ i = (cid:20) ∆ i ( − s ) Θ (cid:16)(cid:12)(cid:12)(cid:12) ˆΛ (cid:12)(cid:12)(cid:12) + ( s/s ) sgn ˆ λ z (cid:17) − ∆ i (2 s ) Θ (cid:16)(cid:12)(cid:12)(cid:12) ˆΛ (cid:12)(cid:12)(cid:12) − ( s/s ) sgn ˆ λ z (cid:17) (cid:21) sgn ˆ λ z iγs X λ x ,λ y =0 | λ x , λ y , Λ i = 12 iγs X λ x ,λ y =0 (cid:20) sgn (cid:18) Λ λ x λ y (cid:19) Θ (cid:18) | Λ | + ( s/s ) sgn (cid:18) Λ λ x λ y (cid:19)(cid:19) (cid:12)(cid:12)(cid:12) λ x , λ y , Λ + ( s/s ) sgn( λ x λ y ) E − sgn (cid:18) Λ λ x λ y (cid:19) Θ (cid:18) | Λ | − ( s/s ) sgn (cid:18) Λ λ x λ y (cid:19)(cid:19) (cid:12)(cid:12)(cid:12) λ x , λ y , Λ − ( s/s ) sgn( λ x λ y ) E(cid:21) = sgn Λ2 iγs X λ x ,λ y =0 (cid:20) { sgn( λ x λ y } Θ ( | Λ | + ( s/s ) { sgn( λ x λ y ) } sgn Λ) (cid:12)(cid:12)(cid:12) λ x , λ y , Λ + ( s/s ) { sgn( λ x λ y ) } E + {− sgn( λ x λ y ) } Θ ( | Λ | + ( s/s ) {− sgn( λ x λ y ) } sgn Λ) (cid:12)(cid:12)(cid:12) λ x λ y , Λ + ( s/s ) {− sgn( λ x λ y ) } E(cid:21) = sgn Λ2 iγs X λ x ,λ y =0 (cid:20) Θ ( | Λ | + ( s/s ) sgn Λ) (cid:12)(cid:12)(cid:12) λ x , λ y , Λ + s/s E − Θ ( | Λ | − ( s/s ) sgn Λ) (cid:12)(cid:12)(cid:12) λ x , λ y , Λ − s/s E(cid:21) = ι sgn Λ2 iγs (cid:20) Θ ( | Λ | + ( s/s ) sgn Λ) (cid:12)(cid:12)(cid:12) Λ + s/s E − Θ ( | Λ | − ( s/s ) sgn Λ) (cid:12)(cid:12)(cid:12) Λ − s/s E(cid:21) = ι θ ( s ) (cid:12)(cid:12)(cid:12) Λ E . In going from line 3 to line 4, we have used the fact that the first and second terms are identical except thatthe signs in braces in the first term are all sgn( λ x λ y ), whereas all those in the second term are − sgn( λ x λ y ),so that in exactly one of the two terms these signs are all +1 and in the other they are − (cid:4) Theorem 2. ι as given in (64) satisfies α ′ ˆ S i ( s ) ◦ ι = 0 for all choices of regularization parameter s and all choices of complexification parameter α ′ , and independentof the choice of coefficients C (Λ; Λ ′ ) . Proof. uantum isotropy and the reduction of dynamics in Bianchi I | Λ i , we have α ′ ˆ S x ( s ) ◦ ι | Λ i = γ | ˆ v | / (cid:16) ( γ ˆ θ x ( s ) + iα ′ ) | ˆ v | (ˆ θ y ( s ) − ˆ θ z ( s )) + (ˆ θ y ( s ) − ˆ θ z ( s )) | ˆ v | ( γ ˆ θ x ( s ) + iα ′ ) (cid:17) | ˆ v | / ·· X Λ ′ ∈ R × C (Λ ′ ; Λ) ι | Λ i = γ X Λ ′ ∈ R × C (Λ ′ ; Λ) | ˆ v | / (cid:16) ( γ ˆ θ x ( s ) + iα ′ ) | ˆ v | (ˆ θ y ( s ) − ˆ θ z ( s )) + (ˆ θ y ( s ) − ˆ θ z ( s )) | ˆ v | ( γ ˆ θ x ( s ) + iα ′ ) (cid:17) | ˆ v | / ◦ ι | Λ i = γ X Λ ′ ∈ R × C (Λ ′ ; Λ) ι ◦ | ˆ v | / (cid:16) ( γ ˆ θ ( s ) + iα ′ ) | ˆ v | (ˆ θ ( s ) − ˆ θ ( s )) + (ˆ θ ( s ) − ˆ θ ( s )) | ˆ v | ( γ ˆ θ ( s ) + iα ′ ) (cid:17) | ˆ v | / | Λ i = 0Whence α ′ ˆ S z ( s ) ◦ ι = 0 for all s and α ′ . Similarly, α ′ ˆ S y ( s ) ◦ ι = α ′ ˆ S z ( s ) ◦ ι = 0 for all s and α ′ . (cid:4) Note that if any other ordering of ˆ S i had been chosen in (56), this theorem would still hold. Furthermore,for the case α ′ = − α , this theorem implies that not only ˆ S i ≡ α ˆ S i ( s ) annihilates ι , but also its adjoint ˆ S † i ≡ − α ˆ S i ( s ). This contrasts with the full theory analysis in [8], where one only expects it to be possiblefor an embedding to be annihilated by one of \ S [ f, g ], \ S [ f, g ], not both. Thus, the condition satisfied by ι inthe Bianchi I case is much stronger. The possibility of this was expected due to equation (46) in the classicaltheory, and this will be discussed in section 5. Consistency with Classical Theory It may seem puzzling that canonical residual automorphism covarianceof ι implies that its image satisfies our quantum isotropy condition: Is not the former simply a condition ofconsistency with gauge symmetry, whereas the latter is an actual physical restriction on states? It may seemequally puzzling that it simultaneously implies that the adjoint of our isotropy condition is satisfied on theimage of ι .These puzzles are resolved if one carefully translates these logical relations to the classical theory, wherewe will see that it holds as well. The canonical residual automorphism covariance of ι implies that the imageof ι is invariant under the identity component of this group, the proper anisotropic dilitations. The classicalanalogue of imposing invariance under a unitary flow in quantum theory, e t ˆ X | Ψ i = | Ψ i , is to impose thatthe corresponding generators be zero: ˆ X | Ψ i = 0 ❀ X ≈ 0. The proper anisotropic dilitations are the flowson space generated by vector fields of the form X t x ,t y := t x x ∂∂x + t y y ∂∂y − ( t x + t y ) z ∂∂z . The correspondingcanonical generators on the phase space are thus X t x ,t y = 1 κγ Z V A ia L X tx,ty ˜ E ai d x = 1 κγ ( − t x ( pc ) x − t y ( pc ) y − ( − t x − t y )( pc ) z )= 1 κγ (( pc ) z − ( pc ) x ) t x + 1 κγ (( pc ) z − ( pc ) y ) t y = − S y κγ ( p ) y t x + S x κγ ( p ) x t y = − S y κγ (( pc ) y + iα vol( p )) t x + S x κγ (( pc ) x + iα vol( p )) t y . = − S y κγ (( pc ) y − iα vol( p )) t x + S x κγ (( pc ) x − iα vol( p )) t y . (65)The key point is that these generators are not constraints — anisotropic dilitations do not approach theidentity at infinity, so that they are not generated by the diffeomorphism constraint. Thus, their vanishingimposes a non-trivial restriction on the physical degrees of freedom. In fact, it is immediate from the aboveform that the vanishing of the above generators for all t x , t y is equivalent to S x ≈ S y ≈ 0, which is equivalentto S i ≈ i — our classical isotropy condition. At the same time, it is equivalent to S i ≈ 0. Finally,that S i is proportional to S i with coefficient everywhere smooth and non-vanishing — equation (50) — isconsistent with the expectation that ˆ S i ◦ ι = 0 and ˆ S † i ◦ ι = 0 be equivalent in quantum theory. uantum isotropy and the reduction of dynamics in Bianchi I Imposition of canonical residualautomorphism covariance and quantum isotropy has not yet uniquely determined the embedding ι . But thiswas expected: These conditions have only restricted the image of ι . As noted in [8], in order to achieveuniqueness of ι , one expects to impose two more conditions, such as the intertwining of two operators.The basic variables in the isotropic theory are p and c , so it is natural to try to impose intertwining ofcorresponding operators with appropriate operators in the Bianchi I theory. One can indeed require that ι intertwine ˆ p with ˆ v / in Bianchi I, which is equivalent to requiring that ι intertwine the signed volume ˆ v inboth theories. However, c has no operator analogue in the quantum theory, but rather only exponentials of c have operator analogues. Because of this, it is natural to instead require intertwining of an appropriate oneparameter family of exponentials of c , or operators contructed therefrom. We choose to require intertwiningof the regularized isotropic Hubble rate ˆ θ ( s ) (57) with one of the regularized directional Hubble rates (38)— specifically, we arbitrarily choose ˆ θ z ( s ) for this purpose. With this condition imposed, we shall see that ι is uniquely determined up to an overall constant, and will then automatically intertwine ˆ θ ( s ) with the otherdirectional Hubble rates as well. Indeed, we shall see that the resulting unique ι will satisfy basically everyproperty that could be desired from such an embedding.Let Cyl S denote the space of finite linear combinations of volume eigenstates in the isotropic theory,so that its algebraic dual, Cyl ∗ S , may be identified with distributional states which include possibly non-normalizable linear combinations of volume eigenstates. Theorem 3. There exists an embedding ι from isotropic LQC states, Cyl ∗ S , to Bianchi I quantum states, Cyl ∗ BI , that (1.) is covariant under all canonical residual automorphisms, (2.) intertwines ˆ v in the twotheories, and (3.) intertwines ˆ θ ( s ) with ˆ θ z ( s ) for all s . This embedding is furthermore unique up to a(physically irrelevant) overall constant, given by ι = Cι for some C ∈ C . Proof. By the argument of section 4.3.1, condition (1.) imposes that ι be of the form ι | Λ i = X Λ ′ =0 C (Λ; Λ ′ ) ι | Λ ′ i for some C (Λ; Λ ′ ). Condition (2.) then forces C (Λ; Λ ′ ) = C (Λ) δ Λ , Λ ′ for some C (Λ), and, finally, condition(3.) forces C (Λ) to be a constant C . (cid:4) The overall constant C is not a physical ambiguity, because quantum states have meaning only up torescaling. Hence the embedding is physically unique, as was expected from the analysis of [8, 9]. From nowon we set ι to be equal to the embedding so selected, choosing C = 1, so that ι | Λ i = ι | Λ i = X λ x ,λ y =0 | λ x , λ y , Λ i . (66)This is the Bianchi I analogue of what we have called the volume embedding in the full theory [8]. Remark In selecting the unique embedding ι above, we have required that it intertwine ˆ θ ( s ) with ˆ θ z ( s ) forall s . One can alternatively require that, for all s , the more basic shift operators ˆ∆( s ) intertwine with a slightmodification of ˆ∆ z ( s ), namely ˆ∆ ′ i ( s ) := d exp (cid:0) − ip i c i s v (cid:1) , also unitary, for, e.g. i = z . The resulting selectedembedding is again the same. For this reason, these alternative shift operators are arguably more naturalbuilding blocks for the Bianchi I theory. Indeed, one could construct a Hamiltonian constraint operator fromthese alternative shift operators, and the result would be equivalent to the one used here and in [20] whenacting on states | λ i with sufficiently large volume. We have not used this alternative simply in order to beconsistent with [20]. In the full theory paper [8], we gave two arguments against the use of the volume embedding in the generalcase. Here, we address each of them, and show they don’t apply in the simpler case of embedding intoBianchi I. First, we noted that the superposition which defines the volume embedding is in no way peaked uantum isotropy and the reduction of dynamics in Bianchi I | Λ i , ι | Λ i is a superposition of states | λ x , λ y , Λ i for which the condition λ x = λ y = λ z is not satisfied. However, this condition merely describes the dimensions of the fiducial cell; it has nothing todo with the isotropy of the phase space variables ( q ab , K ab ). Rather, the correct isotropy condition is the onethat been the subject of this paper: That states should be annihilated by the operators (56). From theorem2 we know that ι in fact does map all isotropic states into the isotropic sector of Bianchi I.The second objection was that the definition of the volume embedding depends critically on the choiceof basis used to define it. However, in the present Bianchi I context, there is no ambiguity at all in theembedding. As already shown above, ι is (up to an overall constant) the unique embedding which iscovariant under canonical residual automorphisms and which intertwines the signed volume and any oneof the directional Hubble rates. s , ι Intertwines All of the Directional Hubble Rates ˆ θ i ( s ) with ˆ θ ( s ) . This follows from the factthat ι = ι , the volume embedding, and Lemma 1. ι Intertwines the Hamiltonian Constraint Operators of the Isotropic and Bianchi I Models. This isimmediate from the expressions (40) and (58) for these Hamiltonian constraint operators, together withthe properties ˆ v ◦ ι = ι ◦ ˆ v and ˆ θ i ( s ) ◦ ι = ι ◦ ˆ θ ( s ) noted above. ι is the Adjoint of the Projector of Ashtekar and Wilson-Ewing. In [20], Ashtekar and Wilson-Ewing definea projector from Bianchi I states to isotropic LQC states given by h Λ | ˆ P Ψ i = ( ˆ P Ψ)(Λ) = X λ x ,λ y Ψ( λ x , λ y , Λ) = X λ x ,λ y h λ x , λ y , Λ | Ψ i for all Ψ, so that h Λ | ˆ P = X λ x ,λ y h λ x , λ y , Λ | and hence ˆ P † | Λ i = X λ x ,λ y | λ x , λ y , Λ i = ι | Λ i whence ˆ P † = ι . Technical remark: Though ˆ P maps normalizable states in the Bianchi I Hilbert space H to normalizablestates in the isotropic Hilbert space H S , it is unbounded and hence only densely defined. As a consequence,its adjoint in the sense of a densely defined map H S → H need not, and in fact does not, exist. However,the adjoint in the algebraic dual sense always exists. The domain of ˆ P can be taken to be, for example,Cyl BI ; with this choice, its range is Cyl S . The adjoint in the algebraic dual sense, ˆ P † : Cyl ∗ S → Cyl ∗ BI canthen be restricted to a map ˆ P † : H S → Cyl ∗ BI . This is the map which equals our selected embedding ι upto constant rescaling, mapping all non-zero states in H S into non-normalizable states in Cyl ∗ BI . 5. Origins of the Embedding Properties In contrast to what is expected in the full theory [8], we have seen above that, for the embedding intoBianchi I, the following holds:(i) Not only is it possible to impose the quantization of the symmetry conditions (47) consistently inthe quantum theory, but also possible to simultaneously impose their adjoint. Furthermore, a naturalembedding of the quantum isotropic model into the common kernel of the quantum conditions and theiradjoint exists. uantum isotropy and the reduction of dynamics in Bianchi I H S which turns out to be exactly the corresponding operator in the isotropic model.These are surprisingly strong results. The first result implies that the quantization of the real and imaginaryparts of S i — d Re S i := ( S i + S † i ) and d Im S i := i ( S i − S † i ) — each annihilate the image of ι . As first arguedby Dirac [25], it is physically correct to impose a given system of real constraints strongly in quantum theoryonly if it forms a first class system . Do Reˆ S i and Imˆ S i form such a system? Indeed they do. This is equivalentto none other than the Poisson bracket (46).In the following, we will see that the second result is likewise foreshadowed by classical Poisson bracketsthat indicate that, in fact, one expects the second result to be true for every operator invariant under properanisotropic dilitations , and thus in particular for every operator invariant under residual automorphisms.Finally, we note that the Poisson brackets foreshadowing both of the above results hold thanks to the factthat S is proportional to S with coefficient smooth and non-vanishing everywhere, and trace the source ofthis to an observation about the physics of the Bianchi I phase space. ι will intertwine all proper-dilitation invariant operators We here prove that any function F on the Bianchi I phase space invariant under proper anisotropic dilitations— and hence in particular any F invariant under residual automorphisms — satisfies { F, S i } = X j λ ij S j (67)for some matrix of phase space functions λ ij . This leads to the expectation an operator corresponding toeach such quantity will preserve the quantum isotropic sector, hence preserve the image of the embedding,and therefore be intertwined with corresponding operators in the isotropic theory, a fact which we havealready seen is true for F equal to the volume of the fiducial cell, the directional Hubble rates, and theHamiltonian constraint operators. + Let us begin with a general argument that the analogue of (67) in the full theory almost holds. Thiswill allow us to see precisely the special property of the Bianchi I phase space that enables the argumentto be completed. Suppose we are given a function F on the full theory phase space Γ Full which is invariantunder all spatial diffeomorphisms and local gauge rotations – that is, invariant under all automorphisms ofthe SU (2) principal fiber bundle. Let Γ Full be the bundle-automorphism-covariant homogeneous isotropicsector, defined as the set of points η ∈ Γ Full such that S [ f, g ]( η ) = 0 for all f, g . From [8], η ∈ Γ Full if and onlyif there exists some action ρ of the Euclidean group E , via bundle automorphisms, such that ρ ( α ) η = η forall α ∈ E . Because F is automorphism invariant, its Hamiltonian flow cannot map one out of the symmetricsector Γ Full . Heuristically, one can see this because, in order for the flow of F to map a point in Γ Full outof itself, F would need to determine ‘where’ the inhomogeneity or ‘in which direction’ the anisotropy arises.But because F is invariant under diffeomorphisms and gauge rotations, this is not possible. More explicitly,let η ∈ Γ Full be given, and let ρ be the corresponding action of the Euclidean group. Let Φ tF : Γ Full → Γ Full denote the Hamiltonian flow generated by F on Γ Full . Because both F and the Poisson brackets on Γ Full are automorphism covariant, so is Φ tF for each t , so that ϕ ◦ Φ tF = Φ tF ◦ ϕ for all automorphisms ϕ and all t ∈ R . Thus, in particular, for all α ∈ E and all t ∈ R , we have ρ ( α )Φ tF ( η ) = Φ tF ( ρ ( α ) η ) = Φ tF ( η )so that Φ tF ( η ) ∈ Γ Full as well. Thus S [ f, g ](Φ tF ( η )) = 0 for all t . Taking the derivative with respect to t andsetting t to zero yields { F, S [ f, g ] } ( η ) = 0for all f, g , and all η ∈ Γ Full . As Γ Full is the zero set of S [ f, g ] for all f, g , and since the topology of Γ Full istrivial, it follows that { F, S [ f, g ] } = S [ h, k ] + S [˜ h, ˜ k ] (68) + The full theory paper [8] also includes a notion of average spatial curvature which is invariant under gauge anddiffeomorphisms. However, in the present Bianchi I framework average spatial curvature is identically zero and so is trivial. uantum isotropy and the reduction of dynamics in Bianchi I h, k, ˜ h, ˜ k depending on f and g and possibly the phase space point.The above argument goes through also for the Bianchi I case, with minimal modification. Let Γ denotethe Bianchi I phase space as in section 2. The only modifications required to adapt the above argument tothis case are the following:(i) The full group of bundle automorphisms is replaced by the canonical residual automorphisms, Aut CR .(ii) The Euclidean group is replaced with the residual rotation group, Rot R .That is, in the Bianchi I case one need only require that F , now a function on Γ, be invariant under Aut CR .Additionally, for all η ∈ Γ, by theorem 1, S i ( η ) = 0 (i.e., η ∈ Γ) if and only if there exists some proper,and hence canonical, residual automorphism ϕ such that η is invariant under the action ρ ( π ) := ϕ ◦ π ◦ ϕ − of all π ∈ Rot R . This, combined with the invariance of F and the Poisson brackets under Aut CR allows theabove argument in the Full theory to be repeated unchanged in the Bianchi I case. Thus (68) holds also inthe Bianchi I case, where it is more conveniently written as { F, S i } = X j (cid:0) h ij S j + k ij S j (cid:1) (69)for some possibly phase space dependent h ij , k ij . This is so far exactly analogous to the full theory. Whatis special in the Bianchi I case is equation (50), which allows (69) to be rewritten precisely in the form (67)claimed. Furthermore, equation (69) at any given phase space point η depends on F only in a neighborhoodof η . As a consequence, the invariance of F under the full group of residual canonical automorphisms is notrelevant for the validity of (69), but only invariance under the identity component of this group, namely thecanonical anisotropic dilitations. That is, it is actually sufficient for F to be invariant under the smallergroup of canonical anisotropic dilitations for (69) to hold. The volume of the fiducial cell, the directionalHubble rates, and the Hamiltonian constraint are all examples of such F ’s. Explicit Calculation in Cases of Interest We here explicitly calculate the matrix of phase space functions λ ij in (67) for the cases of F corresponding to the operators already shown to be intertwined by the embedding ι .We do this both because these cases are the most directly relevant for explaining these intertwining results,as well as to perform a check on the general arguments above. The volume of the fiducial cell. From the expressions (16), (44), (47) and { c i , p j } = κγδ ij , one calculates { S i , vol( p ) } = κγ γθ i + iα S i (70)so that, for F = vol( p ), λ ij = κγ γθ i + iα δ ji . The directional Hubble rates. Similarly, from the definition (18), { S i , θ k } = − κ p ) γθ k + iαγθ i + iα S i (71)so that for F = θ k , λ ij = − κ p ) γθ k + iαγθ i + iα δ ji . The Hamiltonian constraint. From equation (20), using N = vol( p ) n from equation (40), and using theabove two Poisson brackets, we have { S i , H g } = vol( p ) n γθ i + iα ) − nγ X j 6. Discussion In the work [8, 9], we introduced a gauge and diffeomorphism invariant — that is, principal-bundle-automorphism invariant — notion of homogeneous and isotropic states in full loop quantum gravity, togetherwith a strategy for constructing an embedding of loop quantum cosmology states into the space of such fulltheory states. We proposed that the resulting embedding be used to relate proposals for dynamics in fullLQG with choices of dynamics in LQC, where observational consequences can be more easily calculated.In the present paper, as a test, we have applied these ideas to the simpler case of Bianchi I, withsurprising success. In this simpler context, the automorphism-invariant conditions for homogeneity andisotropy reduce to residual-automorphism-invariant conditions S i ≈ S i . These operators are non-hermitian, and may be thought of as the “holomorphic part”of the symmetry conditions in the Gupta-Bleuler sense.Furthermore, we have shown that there exists a unique embedding, of isotropic LQC into Bianchi Istates, satisfying the following three conditions: uantum isotropy and the reduction of dynamics in Bianchi I θ z ( s ) in the Bianchi I model with the Hubble rateˆ θ ( s ) in the isotropic model for all s .The embedding ι so selected then automatically satisfies the following further properties: • It is annihilated by the quantum isotropy conditions ˆ S i — that is, it is an embedding into the sector ofquantum isotropy. • It intertwines all of the directional Hubble rates ˆ θ i with ˆ θ . • It intertwines the Hamiltonian constraint operators in the isotropic and Bianchi I models. • It is the adjoint of the projector from Bianchi I states to isotropic states introduced by Ashtekar andWilson-Ewing in [20].In particular, ι intertwines every operator of interest in the isotropic and Bianchi I models. From classicalanalysis, we in fact have seen that we expect all canonical residual automorphism invariant operators in theBianchi I and isotropic models, if appropriately quantized, to be intertwined by ι . Equally surprisingly, andperhaps at the root of this, we have seen that ι is not only annihilated by ˆ S i , but also by the adjoints ˆ S † i —by both the “holomorphic” and “anti-holomorphic” parts of the symmetry conditions. In section 5, we tracedthese last two surprising results to the fact that, in Bianchi I, Re S i is proportional to Im S i with coefficienteverywhere finite and smooth, a fact which does not hold in the full theory [8]. Though, as noted in [8],we expect the obvious interesting operators in the full theory to not preserve the quantum homogeneousisotropic sector, nevertheless, in this same work we have layed out a strategy to handle the expected resultingadded complication in the full theory case. 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