Renormalisation with SU(1, 1) coherent states on the LQC Hilbert space

We describe how SU(1,1) Perelomov coherent states can be constructed and used on the standard LQC Hilbert space. At a technical level, this requires us to find a factor ordering for the operators representing the so called CVH algebra that preserves its classical SU(1,1) structure. We present such a (rather involved) ordering choice. This allows us to transfer recently established results on coarse graining cosmological states from direct quantisations of the CVH algebra to the standard loop quantum cosmology (LQC) Hilbert space and full theory embeddings thereof. We explicitly discuss how the SU(1,1) representation spaces used in this latter approach are embedded into the LQC Hilbert space and how the SU(1,1) representation label sets a lower cut-off for the loop quantum gravity spins (= U(1) representation labels in LQC). Our results provide an explicit example of a non-trivial renormalisation group flow with a scale set by the SU(1, 1) representation label and interpreted as the minimally resolved geometric scale.


Introduction
Loop quantum gravity [1,2] is a non-perturbative approach to a quantum theory of the gravitational field. At its core are quantisation techniques similar to those of lattice gauge theory, but augmented to apply to background-independent theories. The key step in this procedure is to perform a quantisation of the gravitational field in terms of connection variables such that the gravitational degrees of freedom are represented as (non-regular) lattices on which the quantum dynamics acts. The representation labels of the involved gauge groups, in the standard formulation SU(2) spins j SU (2) , turn out to specify (some of the) the geometric properties of such lattices, in particular their physical proper size. This leads to the notion of a quantum geometry, a priori referring to spatial slices in canonical quantisations and space-times in path-integral formulations. For simplicity, let us consider a linear scaling of "size" with j SU (2) , as will be the case in this paper, e.g. via the volume operator in the loop quantum cosmology setting.
For computation of dynamical processes and an eventual comparison to experiment, one is, as usual, faced with the problem of different possible states describing the same (quantum) geometry. If one is far away from the Planck scale, such states may be represented equally well by many small spins on fine lattices, or few large ones on coarse lattices. These descriptions should be connected via a renormalisation group flow that renormalises the operators involved in the description. The study of such flows has received increasing attention in recent years in the loop quantum gravity literature, see e.g. [3,4,5,6,7,8,9,10,11,12,13,14,15]. As a result of the complexity of the involved analytical problem, explicit results are however scarce. Rather, for dynamical computations, one usually works in the limit of large j SU (2) for which convenient asymptotic formula exist and defers the renormalisation problem to a later stage. While this typically leads to the correct semiclassical limit for curvatures much lower (and discretisation much larger) than the Planck scale, it is unclear to what kind of a low spin formulation, if any, this would correspond. As low spins are expected to be relevant in the high curvature regime where quantum gravity effects should be important, see e.g. [16] for a recent treatment, this question is rather pressing.
In this paper, we are going to tackle this problem in a simplified setting that considers quantum states representing spatially flat, homogeneous and isotropic cosmology. Recent proposals [17,18,14] approximate such states as product states on N identical (fiducial) cells, each corresponding to a copy of a Hilbert space that represents a single quantum cosmology restricted to one cell. The main technical simplification for coarse graining is that in this approximation, interactions between different cells can be neglected, leading to an effectively 0+1 dimensional problem when considering the possible interactions that may arise along the renormalisation group flow.
A key ingredient in our analysis is a recent study [15] of coarse graining in quantum cosmological models based on an SU(1, 1) structure [19,20,21,22] that allows to exactly compute a coarse graining flow under the assumption that the involved operators, the so-called CVH algebra (see below), are isomorphic to the generators of su(1, 1). While [15] started from a classical Poisson algebra with this property, we are going to construct such operators directly on the loop quantum cosmology (LQC) Hilbert space in this paper. The SU(1, 1) structure will then immediately yield an explicit and non-trivial renormalisation group flow under a change of scale, i.e. the transition from many small to few large spins. In this process, the SU(1, 1) representation label j functions as a lower cutoff for the involved U(1)-analogues of the SU(2) spins j SU (2) in loop quantum cosmology. This paper is organised as follows: Basic concepts of loop quantum cosmology are recalled in section 2. Section 3 reviews [15] and the associated main idea of using the SU(1, 1) structure of the CVH algebra for coarse graining. Our main result, an explicit realisation of the CVH algebra on the LQC Hilbert space and the implied renormalisation group flow, is presented in section 4. We conclude in section 5 and briefly survey the representation theory of SU(1, 1) in the appendix.

Basics of the LQC Hilbert space
In this section, we briefly review the Hilbert space structure of loop quantum cosmology to the extend necessary for this paper. Seminal papers on the subject include [23,24,25], see [26,27] for reviews. We consider spatially flat, homogeneous and isotropic cosmology where the gravitational sector is described by the canonical pair {b, v} = 1, where v is the signed spatial volume and b is proportional to the mean curvature. We work in units where = 12πG = c = 1.
The aim of LQC is to quantise a cosmological model while mimicking key steps from full loop quantum gravity. The result should be considered as an "inspired model" unless one refers to a precise embedding of such a model into a full theory context, see e.g. [28,29,30,18]. To avoid unnecessary technicalities and references to full loop quantum gravity, we introduce LQC as the synthesis of a spatially flat, homogeneous and isotropic quantum cosmological model in the presence of a spatial volume quantised in integer multiples of a fundamental scale λ > 0. It follows that wave functions in the volume representation have support only on λZ and the natural scalar product reads Hence, there cannot be an operator corresponding to b as it would act as a derivative on a discontinuous function. Rather, the shift operators have a well defined action on basis states |v as and are self-adjoint.
One is therefore forced to regularise operators corresponding to b or its powers via such exponentials, the simplest choice being b → sin(λb)/λ. Such a replacement is referred to as a polymerisation and is analogous to using holonomies around closed loops instead of field strengths in lattice gauge theory. We note that there is, at least so far and in this simple model, no continuum limit implied that would remove the correction terms O(λ 2 b 3 ). Rather, such terms should be interpreted as higher derivative quantum corrections to the effective action that are suppressed by the scale λ. In the spatially flat homogeneous and isotropic setting, this means that corrections become relevant once the matter energy density becomes close to 1/λ 2 . In LQC, one argues that λ ≈ 1 is a natural choice [25], leading to corrections close to the Planck curvature. One generically finds that cosmological singularities are resolved by such corrections, although exotic counterexamples can be constructed [31].
The above Hilbert space that we will denote as H LQC can be identified with the square integrable functions on U(1), where every integer value of v/λ corresponds to a representation. Expansion in the v-basis can thus be understood as a Peter-Weyl decomposition of a function on U(1), where the range of b is compactified to [0, 2π/λ). An extension to the Bohr compactification of the real line is possible where the main difference is that n may be any real number, see e.g. [24]. We will not consider this possibility here as the simplest choices for the dynamics preserve an evenly spaced lattice of v-values.

Quantising the CVH algebra and coarse graining
In this section, we will review recent results on using the CVH algebra in the context of LQC. Seminal papers about using an SU(1, 1) structure in LQC include [19,20]. The algebra was directly quantised in [21,22] using a LQC inspired regularisation preserving the SU(1, 1) structure, following the seminal ideas of [32]. The usefulness of the associated SU(1, 1) coherent states in the context of coarse graining was pointed out in [15].

CVH algebra and regularisation
For classical spatially flat homogeneous and isotropic cosmology, the CVH algebra is formed by the so called complexifier C = vb (owing its name to [33]), the spatial volume v, and the gravitational part of the Hamiltonian constraint H g = − 1 2 vb 2 , where b is proportional to the mean curvature. The main feature of this set of phase space functions is that it forms a Poisson algebra isomorphic to su(1, 1) using the bracket {b, v} = 1. Via the identification or equivalently With the definition k ± := k x ± ik y , the algebra reads We will use this latter form throughout the paper. A generalisation to the complete Hamiltonian constraint including a massless scalar field is possible, see e.g. [22], although we will not consider it in this paper due to the value that the Casimir operator takes (see section 3.2 for more details). It should also be noted that other choices for the su(1, 1) generators are possible. Further details on the representation theory of SU(1, 1) can be found in the appendix.
It will be relevant for later to also spell out the Poisson brackets of CVH algebra in terms of C, v, and H g :

Group quantisation
A direct implementation of the CVH algebra on the LQC Hilbert space is not possible due to the appearance of b outside of exponentials. One therefore needs to find a regularised version of this algebra in terms of operators that are well defined on H LQC . Suitable regularisations were obtained in [21,22], e.g.
from which (3.4) can be easily verified. Here, v m is a free constant that can be identified with a minimal volume and will be relevant later in assigning suitable SU(1, 1) representations. To see this, we compute the classical value of the Casimir operator as This expression should be matched, at least to leading order, with the value of the Casimir operator as determined by the representation choice.
It is straight forward to quantise our system by promoting j, k ± to the generators of su(1, 1), see the appendix for an overview of the relevant representation theory. The representation problem is thereby already solved, it only remains to pick a suitable subclass of SU(1, 1) representations. In order to be able to transfer the ideas of [15] and focus only on the gravitational sector, we choose representations from the discrete class with representation label j ∈ N/2 and positive eigenvalues forĵ z . For such a representation the Casimir operator takes the value j(j − 1). This suggests to identify The subleading correction in j can be attributed to ordering choices for this specific operator and as we will see in the following, the identification (3.8) is precise for another large class of operators that we will be interested in for coarse graining, i.e. 2λj is the minimal eigenvalue of the volume operator.
An interesting choice of quantum states is given by the normalised SU(1, 1) Perelomov coherent states [34], which are characterised by the representation label j ∈ N/2 and a spinor z ∈ C 2 , where we abbreviated L = 1 2 (|z 0 | 2 − |z 1 | 2 ). They have the property that the SU(1, 1) action in any representation with label j transfers directly to the spinor, which is in the defining representation: This property will later allow to relate the dynamics between finer and coarser scales labelled by j.

Coarse graining
It was shown in [15] that the coherent states 3.9 allow for a natural coarse graining operation. One first notes that the expectation values ofĝ =ĵ z ,k ± factor into where f g are three functions depending only on z. This suggests to interpret j as an extensive scale of the system, while z sets the intensive state, i.e. ratios of extensive quantities. We note that this is consistent with the classical interpretation of j z , k ± if v m also scales extensively, which is precisely the case for the identification (3.8) along with the additional observation that j is the minimal eigenvalue of j z = v/(2λ). The coarse graining operation now looks as follows: We consider N independent (= non-interacting) copies of our system, where the quantum state in each copy is given by (3.9) with the same j 0 , z 1 . The interpretation of j as a scale in turn suggests that one may obtain the same physics if one instead considers a single copy labeled by j, z, where j = N j 0 . This proposition turns out to be exactly correct for the expectation values of any power ofĵ α =ĵ z ,k ± if one compares the coarse grained operators to a sum of the corresponding operators at the non-coarse grained level as suggested by their extensive nature 2 : ĵ n α j = ĵ α,1 + . . . +ĵ α,N n j 0 ,1,...,N (3.12) = n r 1 ,...,r j =0: In the first line on the right hand side, the additional subscript onĵ α refers to one of the N copies of the system, and the expectation value is taken in the product state of N states with labels j 0 , z. The second line uses the multinomial theorem to split the expression into products of expectation values of powers ofĵ α . In the third line, another convenient form is given where terms with the same powers are collected.
Furthermore, the eigenvalues and their probability distributions are exactly reproduced. If the dynamics is generated by a linear combination of j z , k ± (as will be the case for the gravitational part of the Hamiltonian constraint), it is also correctly reproduced due to (3.10). We note that these results have been derived in [15] using only the representation theory of SU(1, 1) and the choice of states (3.9). Hence, they are independent of the application to quantum cosmology that we focus on in this paper and only require a classical Poisson algebra of extensive quantities that is isomorphic to su(1, 1).

The CVH algebra on the LQC Hilbert space
In the previous section, we have recalled how group quantisation can be used to directly quantise a classical algebra that has been identified with a Lie algebra. The straight forward coarse graining properties of the Perelomov coherent states make such a quantisation particularly attractive and one would like to import it somehow to the standard LQC Hilbert space. There however, one does not start with quantising a classical algebra that has been identified with su(1, 1), but with operatorsv, e inλb that correspond to terms which were used in the classical definition (3.6) of the Lie algebra generators. One therefore needs to consider derived operators in a certain ordering that should be (partially) fixed by requiring that the operators reproduce the su(1, 1) commutation relations. In the following, we will present such an ordering choice and discuss how the corresponding SU(1, 1) representation spaces can be identified as subspaces of the LQC Hilbert space. Related results for the light-like representation with label j = 0 were recently reported in [36].

Warmup: no minimal volume
Due to the rather cumbersome computation necessary to derive the general result, we first consider the simplifying choice v m = 0 in this subsection and highlight why it is necessary to go beyond it.
We start by choosing j z =v 2λ motivated by the wish of having at least one simple operator and the identification of a minimal volume by 2λj. Other choices are possible as e.g. in (3.2). Our strategy is then to pick a simple regularisation ofĤ g , deriveĈ via the commutation relations following from (3.5), and check the resulting operator for consistency with the other commutation relations.
Our trial ansatz forĤ g is the symmetric orderinĝ  Using this definition, we compute To verify that the algebra really is closed, we still need to calculate the commutator ofĤ g witĥ C. Using we find With these results, we can define the operatorŝ which fulfill the su(1,1) algebra We should now study the properties of the representation we found, i.e. find the representation label j and study how the representation space is embedded in the LQC Hilbert space. Let us first note that due the actionĵ z |j, m = m |j, m , the accessible volume eigenstates |ρ correspond to eigenvalues 2λm, m = j, j + 1, j + 2, . . ..
Next, we would like to fix j by requiringk − |j, j = 0. For this, we computê It follows that the minimal eigenvalue of the volume operator is λ, corresponding to j = 1 2 . As a cross-check, we can explicitly compute the action of the Casimir operator aŝ which is consistent due to j(j − 1) = −1/4 for j = 1/2.
We conclude that while we found a factor ordering for the CVH algebra that reproduces the correct commutation relations, we are restricted to the j = 1/2 representation. For applications of coarse graining as discussed in section 3.3, it is interesting to also find explicit realisations of the CVH algebra on H LQC for all j ∈ N/2. The observation of section 3.2, cited from [22], that there is a one-parameter family of classical Poisson algebras labelled by v m ∼ j suggests that a similar one-parameter family may yield the representations for j > 1/2. As we will show in the next subsection, this expectations turns out to be correct.

Regularisation with minimal volume
Inspired by (3.6), we again choose j z =v 2λ . For H g , we make the ansatẑ whereṽ m is a free constant. As we will see later, consistency of the algebra determines it to bẽ v m = v m − λ and it thus includes a subtle quantum correction to the naive classical expectation.
Proceeding as before, we calculateĈ via the commutator ofĤ g andv aŝ The next step consists in computing Ĉ ,v as Comparison with (4.5) shows that thev term obtains corrections for non-zeroṽ m .
As before, we are left with the commutator Ĥ g ,Ĉ . To simplify the calculation, we will introduce the shorthand notation c = cos (λb),ŝ = sin (λb),ĉ(n) = cos (nλb),ŝ(n) = sin (nλb), (4.17) Straight forward but cumbersome computations yield To calculate the commutator ofĈ and (4.16), one first shows that (4.20) Putting this result together with (4.19) yields As a last step, we compute v, 4λ 2Ĥ g + Analogously to (4.8), we are now in a position to define the su(1, 1) generators aŝ Our previous calculations imply that their algebra reproduces that of su(1, 1) as Sinceĵ z did not change as compared to the previous subsection, the accessible volume eigenstates |ρ still have the eigenvalues 2λm, m = j, j + 1, j + 2, . . .. To fix the representation, we again need to calculate the action ofk ± on volume eigenstates. Using the intermediate resultŝ The action ofk − vanishes forṽ m = ±λ(2m−1). As m ! = j in this case, we find using v m :=ṽ m +λ for the choice v m >ṽ m > 0 and j > 0. Again, we can confirm this result via the Casimir operator asĈ For j = 1/2 ⇔ v m = λ, we obtain the results from the previous subsection as a cross-check.

Embedding of the SU(1, 1) representation spaces and dynamics
Let us now collect our results. As shown before, by regularising the CVH algebra as (4.23) on H LQC , we can correctly reproduce the su(1, 1) algebra. A representation with label j is obtained by choosing the minimal eigenvalue of the volume operator (as obtained fromĵ z ) to be v m = 2λj. It follows from the representation theory of SU(1, 1) that by acting with operators from the CVH algebra, in particulark + , we obtain states with higher volume, but never go below the minimal volume 2λj. Therefore, in a representation j, the support of wave functions in the SU(1, 1) representation spaces embedded into H LQC is restricted to ρ ∈ 2λ(j + N 0 ).
In particular, it follows that the dynamics generated by any linear combination ofĵ z ,k ± preserves this subspace. This observation suggests an improved regularisation of H g aŝ For eigenvalues ofv much larger than the minimal eigenvalue 2λj, the last line approaches zero and one reobtains (4.14). In turn, the square root in that expression is well approximated bŷ v in this limit and one obtains the more common expression (4.3). The differences to (4.3) for j = 1/2 should thus be seen as a convenient choice of ordering that allows for a very simple coarse graining operation as discussed in the next subsection. In other words, j can be used as a scale so that the dependence ofĤ improved g on j can be seen as a renormalisation group flow.

Coarse graining
The results obtained in this section, combined with [15], allow to define a coarse graining operation as follows. Using (3.12), we know that N independent but identical copies of our quantum cosmological system described by a product state with labels j 0 , z in each cell can be equivalently described by a single copy with coherent state labelled by j, z with j = N j 0 . The only requirement for this is that the coarse grainedĵ z ,k ± satisfy the su(1, 1) algebra and are represented in the representation with label j. We established this before.
It is of interest to compare this coarse graining flow with that of section 3.3. We first note that in section (3), we used a different quantisation procedure that quantises a classical Poisson algebra isomorphic to su (1,1). Therefore, the su(1, 1) generators are immediately available on the Hilbert space and no assembling of them from other more fundamental operators was necessary as in section 4. This means thatĵ z ,k ± are unambiguously defined, while their supposed constituents, such asv, sin (λb), and v m are not. The point of view taken in [15] was that v m in (3.6) (that is analogous toṽ m in section 4), also scales extensively with the system size j. This may be achieved by defining v m as the lowest eigenvalue ofv, while, e.g., a definition over the Casimir operator eigenvalue j(j −1) suggests a quantum correction to the extensive scaling when comparing with (3.7). Such discrepancies are expected because there is no unique definition of operators that are not contained in the sub-algebra of observables that one (unambiguously) represents.
Let us now turn to the coarse graining flow of this section. Our quantisation procedure represented the operatorsv and e iλnb unambiguously. We then assembled su(1, 1) generators from them in (4.23), leading to the consistency requirementṽ m = 2λ(j − 1/2). In contrast to the results from section 3.3, this implies a quantum correction to the extensive nature of k ± (andĤ improved g ) in the naive classical limit where operators are replaced by their classical counterparts. The extensive nature ofĵ z =v/(2λ) remains unaffected, asṽ m is not present there. This observation still allows us to coarse grain the system exactly, the difference to section (3.3) is however that coarse grained operators are not simply given by the operators that would naively correspond to the sum of the fundamental operators, but to those derived before, e.g. (4.30), which exhibit a correction to the naive extensive scaling in j in the form j → j − 1/2 andv →v ± λ in some places.
As mentioned before, the dependence of (4.30) on j can thus be interpreted as a renormalisation group flow with scale j. Fundamental physics takes place at j = 1/2. Coarse graining to a scale j > 1/2, analogous to a block-spin transformation that joins N = 2j spins into one, introduces a non-trivial dependence of the operators on the scale, as e.g. in the generator of the gravitational dynamics (4.30). Let us note again that other (inequivalent) constructions of the su(1, 1) operators may be possible where the coarse graining flow could look different. We merely present a concrete example that can be studied analytically.

Conclusion
In this paper, we have shown that it is possible to regularise operators corresponding to the CVH algebra on the LQC Hilbert space such that they satisfy the su(1, 1) Lie algebra. This result extends previous investigations [21,22,36] in which one started from a classical Poisson algebra with holonomy corrections. Due to extensive factor ordering problems, this result is non-trivial.
Based on the results of [15], it was then possible to define and explicitly perform a coarse graining operation in a system of N identical but independent copies of the same quantum cosmological system, each described by a Perelomov coherent state with labels j 0 , z. The result can be interpreted as a non-trivial renormalisation group flow from a volume scale j 0 to j = N j 0 . The observed difference to the coarse graining procedure in section 3.3, i.e. the corrections from the naive extensive scaling, are not of much concern as they do not affect the applicability of the results of [15]. Rather, they show that the coarse grained operators in section 4 are not naively given by the quantisation of the classically coarse grained objects, but contain quantum corrections. Clearly, such effects are expected and present in most systems. One should therefore interpret section 3.3 as an atypical example where the coarse grained operators can be obtained naively. When comparing the LQC Hilbert space to that of full LQG, e.g. via an embedding along the lines of [29,30], the magnetic quantum number m in the context of SU(1, 1) is analogous to a U(1) representation label, and thus to an SU(2) spin j. In contrast, the j labelling the SU(1, 1) representation functions as a lower cutoff for m, and thus the smallest resolved scale set by the SU(2) spins via the geometric operators. This strengthens the above interpretation of the SU(1, 1) j as a scale.
For future work, it is of obvious interest to check to which extend the computation performed in this paper can be generalised to more complicated systems. First, one may be interested in matter coupled to the gravitational sector. An easy choice is deparametrisation w.r.t. nonrotating dust, where the gravitational part of the Hamiltonian becomes the true Hamiltonian, see e.g. [38] and references therein. When quantising also the matter content, one can run into the problem that the classical value of Casimir operator is zero or negative, see e.g. [22], which selects different classes of SU(1, 1) representations, so that the analysis of [15] would have to be successfully repeated for these cases. Another route is to identify suitable su(1, 1) sub-algebras in increasingly complicated systems, such as spherical symmetry.
The Casimir operator is given bŷ This algebra is a non-compact form of su (2). There exist five possible groups of such representation. First, one can distinguish two classes, the continuous and the discrete one. We will not discuss the continuous ones here. The three discrete ones are given by m ∈ j, j + 1, j + 2, . . .
The last one can be obtained as the infinite dimensional Hilbert space generated by a bosonic harmonic oscillator. For our analysis, the first two cases are of interest. We restrict to positive m, which is case one, due to the interpretation of 2λm as the volume. As one can easily see, the action ofĈ ,ĵ z andk ± always results in new states belonging to this representation. The action ofk − on the lowest eigenstate |j, j vanishes.
Beside those infinite dimensional representations, also finite dimensional representations exit. Their dimension is, similar to the dimensions of the representations of su(2), given by 2j + 1. To find the generators, one we can simply take the su(2) generators and multiply two of them by i, which yields for the two-dimensional defining representation acting on the spinors (z 0 , z 1 ). The exponentiated action U = e iα i σ i preserves the pseudo-norm |z 0 | 2 − |z 1 | 2 .