Properties of the Rovelli-Smolin-Depietri Volume

Home , Jerzy Lewandowski

arXiv:2104.11010v2 [gr-qc] 27 Apr 2021 prtri h pcso monochromatic of intertwiners spaces the in volume operator Rovelli-Smolin-DePietri the of Properties pcso U2 nain esr itrwnr)wt l pn equ spins all with (intertwiners) tensors invariant SU(2) of spaces h omlgclsco fteter.I h aeo pn1 spin of case the In b can theory. and the isotropic of are sector that cosmological states invariant the gauge SU(2) of study ecl hmspin them call We ersnainspaces representation yisato ntesae ftno products tensor of spaces the eigenstates. the in of action degeneracy its of p by pattern diagonalization the the on light simplify some signiﬁcantly shed which gravity, quantum loop in Abstract. Warsa 02-093 7, Pateura Research, Nuclear for Centre National Kisielowski Marcin rbe opeey eso httevlm prtri proportion is factor. operator proportionality volume the the calculate that show we completely: problem esuysm rpriso h oel-mlnDPer oueoper volume Rovelli-Smolin-DePietri the of properties some study We j oohoai neties uhsae r motn nthe in important are spaces Such intertwiners. monochromatic H j i i , 1 = N , . . . , aeldwt spins with labelled , H j 1 ⊗ . . . H ⊗ j / N esleteeigenvalue the solve we 2 j fteirdcbeSU(2) irreducible the of i h prtri deﬁned is operator The ,Poland w, ∈ al ple oextract to applied e 1 2 j N 1 lt dniyand identity to al ersrc to restrict We . = . . . olmand roblem = j N = ator j . Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 2

1. Introduction

The idea that isotropic states in loop quantum gravity should be described by monochromatic intertwiners appeared in the literature a couple of times. In spin-foam cosmology the conditions of homogeneity and isotropy are imposed on the coherent states [1]. The coherent states are linear combinations of spin-network states for different spins but the main contribution comes from states with all spins equal. As a result, the leading order amplitudes involve the monochromatic intertwiners only. In group field theory the states are restricted to isotropic states by considering monochromatic intertwiners maximizing the volume [2, 3]. In our recent approach [4] we study homogeneous- isotropic invariant subspaces of the scalar constraint operators proposed in [5, 6, 7, 8]. The component of the homogeneous-isotropic invariant space corresponding to spin network with 0 loops is defined using spin j monochromatic intertwiners. The scalar constraint operator adds and subtracts loops labelled with a given spin l. If j = l the entire homogeneous-isotropic invariant space is described by monochromatic spin j intertwiners only. The last example is our main motivation for this work. While in the previous two examples the focus is on 4-valent intertwiners, in our approach it is necessary to study the spaces of monochromatic intertwiners Inv ⊗N with arbitrary Hj valence N (or at least in some range when the cut-off on the number of loops is used). We believe that spin 1/2 spaces are of particular interest because they seem to describe the smallest homogeneous-isotropic grains of space, from which the quantum geometry of space is built (see [9] for the discussion of the quantum grains of space). In loop quantum gravity [10, 11, 9, 12, 13, 14, 15, 16] the volume operator plays an important role. It appears in the gravitational as well as in the matter part of the quantum scalar constraint operators. In loop quantum cosmology [17, 18, 19, 20] the volume operator is used to construct one of the two fundamental observables. An evolution of the operator in certain coherent states is calculated to show that the theory predicts Big Bounce instead of Big Bang – the quantum universe contracts until it reaches a minimal volume at which it bounces and starts to expand. It is therefore unavoidable to study the volume operator when analyzing cosmological sector of loop quantum gravity. There are two proposals for the volume operator: we will call them Rovelli-Smolin- DePietri volume operator [21, 22] and Ashtekar-Lewandowski volume operator [23]. The quantum operators correspond to the same classical object. In [23], this fact is underlined in the nomenclature: they are called the volume operator in extrinsic and intrinsic regularization, respectively. The operators are defined in the spaces of intertwiners of valence at least 3 and vanish in the spaces of intertwiners of valence 3 [24]. In the case of 4-valent intertwiners the Rovelli-Smolin-DePietri volume operator and Ashtekar-Lewandowski volume operator coincide. In this simplest, non-trivial case some spectral properties of the volume operator were studied analytically [25, 26, 27, 28]. In the case of Rovelli-Smolin-DePietri volume operator and higher valent monochromatic intertwiners, bounds were found for the eigenvalues [29] but more detailed spectral Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 3 properties in the higher valent cases are usually studied numerically. Our numerical study revealed a property not reported previously in the literature – the volume operator is proportional to identity matrix in the spaces of spin 1/2 monochromatic intertwiners. In this paper we prove it analytically and calculate the proportionality factor. The idea is that the Rovelli-Smolin-DePietri volume operator commutes with the permutations of the indices of the invariant tensors. It turns out that the space of ⊗N spin 1/2 monochromatic intertwiners Inv 1 is an irreducible representation space H 2 of the group of permutations of N elements, denoted here by SN . The property of the volume operator follows from Schur’s lemma. In this paper we will give even more general result – we will find a decomposition of the space of spin j monochromatic intertwiners into subspaces invariant under the action of the volume operator. Our main tool will be techniques used in the context of Schur-Weyl duality as presented in [30]. Our results explain to some extent the pattern of degeneracy of eigenstates of the Rovelli-Smolin-DePietri volume operator in the spaces on monochromatic spin j intertwiners. Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 4

2. The volume operator and the space of monochromatic intertwiners

The volume operator corresponds to classical observable which is a volume of a region R of the space manifold Σ. There are two proposals for the quantum operator, which we will call Rovelli-Smolin-DePietri volume operator [21, 25, 22] and Ashtekar-Lewandowski volume operator [23]. The difference between the operators is just a quantization ambiguity – they correspond to the same classical object. In [23] they are called, respectively, the volume operator in the extrinsic and intrinsic regularization. Our result will apply only to the former proposal. In this paper we will call it simply the volume operator. We start this section by recalling the definition of the Loop Quantum Gravity Hilbert space and both versions of the volume operator (see section 2.1). In section 2.2 we recall the definition of the spin networks and discuss the action of the volume operator on the spin-network states. This action is described by a family of operators, each acting in the space of SU(2) invariant tensors (intertwiners) Inv ( ... ). In Hj1 ⊗ ⊗ HjN this paper we restrict to monochromatic intertwiners, i.e. to elements of Inv ⊗N . Hj In section 2.3 we discuss the action of the permutation group S on Inv ⊗N and N Hj show that the volume operator commutes with this action. The results presented in this paper are based on this observation.

2.1. Deﬁnition of the volume operator The Hilbert space of LQG is a space of cylindrical functions on the space of SU(2) connections on Σ equipped with the Ashtekar-Lewandowski measure [10, 11]. In order to deﬁne the Hilbert space one considers oriented graphs γ embedded in the space manifold Σ. A graph γ is a collection of oriented compact 1-dimensional submanifolds called links Links(γ) meeting at the nodes Nodes(γ). A complex function Ψ on the ‡ space of SU(2) connections on Σ is called cylindrical if there is a graph γ and a complex function ψ : SU(2)N C such that →

Ψ(A)= ψ(Al1 ,...,AlN ), (1)

where l ,..., l are all links of the graph γ and Al SU(2) is the parallel transport of 1 N ∈ the connection A along the link l. We will say that the function Ψ is cylindrical with respect to the graph γ. Given two graphs γ1,γ2 and functions Ψ1, Ψ2 cylindrical with

respect to γ1 and γ2, respectively, it is always possible to ﬁnd a graph γ such that Ψ1 and Ψ2 are cylindrical with respect to γ. The scalar constraint between Ψ1 and Ψ2 is deﬁned by

Ψ1 Ψ2 = dg1 ...dgN Ψ1(g1,...,gN )Ψ2(g1,...,gN ), (2) h | i N ZSU(2) where N is the number of links in γ. The kinematical Hilbert space of LQG is the Cauchy completion of the space of cylindrical functions with respect to the scalar product (2).

The submanifolds need to satisfy certain properties, see [10, 5]. ‡ Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 5

i The volume operators can be deﬁned using ’angular momentum like’ operators Jn,I:

d i l n i t=0ψ(Al1 ,...,AlI exp(tτ ),...,AlN ), if I is outgoing from , i dt| (Jn,IΨ)(A)=  (3) d i  i ψ(Al ,..., exp( tτ )Al ,...,Al ), if l is incoming to n.  dt|t=0 1 − I N I i i 1 i Here, τ are su(2) matrices related to the Pauli matrices by τ = 2i σ . Given a region R in Σ the action of the volume operator VˆR on function cylindrical with respect to a graph γ is a sum over operators, each depending only on small neighbourhood of the corresponding node of the graph: 3 8πG~γ 2 Vˆ Ψ= κ qˆn Ψ. (4) R 0 c3 | | n∈Nodes(γ)∩R X p Both proposals for the volume operator share the general form but diﬀer in the choice of operatorsq ˆn. (i) In Ashtekar-Lewandowski volume operator the operator is:

Nn Nn Nn 1 i j k qˆnΨ= ǫ(l˙ , l˙ , l˙ )ǫ Jn J Jn , (5) 8 I J K ijk ,I n,J ,K I J I K J X=1 X= X= where ǫ(l˙I , l˙J , l˙K) = sgn(ω(l˙I, l˙J , l˙K)),ω is the orientation 3-form on Σ and l˙I , l˙J , l˙K

are vectors tangent to the links lI , lJ , lK at the node n. (ii) In the Rovelli-Smolin-DePietri volume operator the operator is:

Nn Nn Nn 1 i j k qˆnΨ= ǫ Jn Jn Jn . (6) 8 | ijk ,I ,J ,K| I=1 J=I K=J X X X Let us notice that the operator does not depend on the embedding of the graph in Σ.

2.2. Spin networks The kinematical Hilbert space of Loop Quantum Gravity is spanned by the gauge-variant spin-network states . A gauge-variant spin network s is a triple (γ, ρ, ι): § an oriented graph γ embedded in the space manifold Σ, • a labelling ρ of the links of γ with unitary irreducible representations of the SU(2) • group:

Links(γ) l ρl GL( l), (7) ∋ 7→ ∈ H where GL(V ) is the space of linear automorphisms of V , l is the Hilbert space in H which the SU(2) representation acts,

Typically a spin-network state is gauge invariant (see for example [31, 32]). On the other hand in the § standard text-books [10, 11] the spin networks that are not gauge invariant are considered. We follow the nomenclature from [11] and use a notion of gauge-variant spin networks. Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 6

a labelling ι of the nodes of γ with tensors: • ∗ ∗ Nodes(γ) n ιn l ... l l ... l , (8) ∋ 7→ ∈ H 1 ⊗ H M ⊗ H M+1 ⊗ ⊗ H N where the links l1,..., lM are incoming to the node n and the links lM+1,..., lN are ∗ outgoing from the node, l denotes the Hilbert space dual to l. H H A gauge-variant spin network such that each tensor ιn is invariant with respect to the action of the SU(2) group will be simply called a spin network. In this case we will write ∗ ∗ ιn Inv l ... l l ... l . A subspace spanned by all spin networks ∈ H 1 ⊗ H M ⊗ H M+1 ⊗ ⊗ H N is the space of solutions of the Gauss constraint. Each gauge-variant spin network s deﬁnes a complex function ψ : SU(2)N C by s → a unique contraction [32]:

y ψs(ul1 ,...,ulN )= ρl(ul) ιn , uli SU(2). (9) l  n  ∈ ∈Links(O γ) ∈Nodes(O γ) The corresponding gauge-variant spin-network state  is 

Ψs(A)= ψs(Al1 ,...,AlN ). (10)

The action of the operatorq ˆn on a gauge-variant spin-network state Ψs is deﬁned by its action on the space of tensors associated to the node n. Let us focus on a single node n and assume for simplicity that all links are outgoing from n. By applying the deﬁnition i of the operator Jn,I (3) to the spin-network contraction (9) we obtain:

i ′ (Jn,IΨs)(A)=Ψs (A), (11) ′ ′ ′ ′ ′ where s =(γ , ρ , ι ) is the same as s =(γ, ρ, ι) except for the tensor ιn which is related to ιn by the following relation:

′ ′ i

½ ½ ½ ιn =(½ ... ρ (τ ) ... ) ιn. (12) ⊗ ⊗ ⊗ I ⊗ ⊗ ⊗ · l n In the formula above ρI equals ρln,I for the unique link n,I deﬁned by the pair ( ,I) and

′ d su(2) τ ρ (τ)= ρ(exp(tτ)) End( l ) (13) ∋ 7→ dt|t=0 ∈ H n,I is the representation of the su(2) Lie algebra induced by the representation ρ of SU(2). We will denote by J i a linear operator in the space ... deﬁned by: I Hρ1 ⊗ ⊗ HρN

i ′ i

½ ½ ½ J = ½ ... ρ (τ ) ... . (14) I ⊗ ⊗ ⊗ I ⊗ ⊗ ⊗ In the following, for each spin j we will fix a representation of the SU(2) group (a common choice is to take the Wigner D-matrices): ρ : SU(2) GL( ). (15) j → Hj ~ The Rovelli-Smolin-DePietri volume operator is defined by a family of operators V~j, j = j ,...,j each defined on the space ... : { 1 N } Hj1 ⊗ ⊗ HjN 3 8πG~γ 2 V~ = κ qˆ~ , (16) j 0 c3 | j| q Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 7

where N N N 1 i j k qˆ~ = ǫ J J J . (17) j 8 | ijk I J K | I=1 J=I K=J X X X It is straightforward to check that V~j commutes with the left action of the SU(2) group:

V~ ρ (u) ... ρ (u)= ρ (u) ... ρ (u) V~ u SU(2). (18) j j1 ⊗ ⊗ jN j1 ⊗ ⊗ jN j ∀ ∈ As a result, the space Inv ( ... ) is an invariant space of the operator and Hj1 ⊗ ⊗ HjN V~ descends to an operator on Inv ( ... ). j Hj1 ⊗ ⊗ HjN

2.3. The space of monochromatic intertwiners In this paper we will study the volume operator in the spaces of monochromatic intertwiners. Following [29] we will say that a space of intertwiners Inv( ... ) Hj1 ⊗ ⊗ HjN is monochromatic if all spins are equal j1 = j2 = ... = jN = j. There is a natural right action of the permutation group SN in the space of monochromatic intertwiners. Consider the right action of S in the space ⊗N : N Hj (v ... v ) σ = v ... v . (19) 1 ⊗ ⊗ N · σ(1) ⊗ ⊗ σ(N) This action commutes with the left action of the GL( ) group. In particular for any Hj u SU(2) and ι ⊗N we have: ∈ ∈ Hj ρ (u)⊗N (ι σ)= ρ (u)⊗N ι σ. (20) j · · j · · ⊗N ⊗N For ι Inv we have ρj(u) ι = ι and therefore: ∈ Hj · ⊗N ρj(u) (ι σ)= ι σ. (21) · · · This means that if ι Inv ⊗N then ι σ Inv ⊗N . Let us notice that the resulting ∈ Hj · ∈ Hj representation of the group S in Inv ⊗N is unitary (in the canonical scalar product N Hj pulled-back with the canonical embedding Inv ⊗N ⊗N ). Hj ⊂ Hj The operators J i are covariant under the action of the group S : I N J i ((v ... v ) σ)= J i (v ... v ) σ. (22) I 1 ⊗ ⊗ N · σ(I) 1 ⊗ ⊗ N · It is straightforward to verify that the operator V~j is invariant under this action:

V~ ((v ... v ) σ)= V~ (v ... v ) σ. (23) j 1 ⊗ ⊗ N · j 1 ⊗ ⊗ N · Indeed, if we writeq ˆ~j in an equivalent form:

1 i j k qˆ~ = ǫ J J J , (24) j 48 | ijk I J K| I J,I K,J K 6= X6= 6= we obtain:

1 i j k qˆ~ ((v ... v ) σ)= ǫ J J J (v ... v ) σ = j 1 ⊗ ⊗ N · 48 | ijk σ(I) σ(J) σ(K)|· 1 ⊗ ⊗ N · I J,I K,J K ! 6= X6= 6= = qˆ~ (v ... v ) σ. (25) j 1 ⊗ ⊗ N · Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 8

In the last equality we used the fact the σ is a bijection and we changed the sum to be ′ ′ ′ over I = σ(I), J = σ(J), K = σ(K). Let us notice that the invariance of V~j under the action of the permutation group holds only for the Rovelli-Smolin-DePietri volume operator.

The property (23) can be expressed by saying that V~j is an element of the commutator algebra (as deﬁned in [30]): B = ϕ End(U): ϕ(v g)= ϕ(v) g, v U, g G , (26) { ∈ · · ∀ ∈ ∈ } where U = Inv ⊗N ,G = S . In order to understand this notion better, let us Hj N consider some examples. If U is an irreducible (right) representation space of G, d = dim U, then by Schur’s lemma it follows that

B = c½ : c C = C. (27) { d ∈ } ∼ ⊕ni In similar manner if U = Ui , where Ui is an irreducible (right) representation space of G, then (see [33] for more details)

B = m ½d : m Mn (C) = Mn (C), (28) { ⊗ i ∈ i } ∼ i where M (C) denotes the ring of n n complex matrices. In general if U = U ⊕ni ni i × i i i is the decomposition of U into irreducible subspaces, then L B M (C). (29) ∼= ni i M Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 9

3. Decomposition of the space of monochromatic intertwiners

The main tool in this paper is the decomposition of the space of monochromatic intertwiners into B-irreducible spaces, where B is the commutator algebra introduced

in section 2.3. In the basis adapted to this decomposition V~j has a block diagonal form. This property can be used to simplify the diagonalization problem. We will find the decomposition by applying some techniques described in [30] in the context of Schur-Weyl duality, which relates the finite dimensional representations of general linear and permutation groups. The representation theory of the permutation group is typically formulated using the language of group algebra and its representations rather than the group and its representations [30, 33]. Therefore we will introduce first the concept of the group algebra CG (see section 3.1). The irreducible representation spaces of the permutation group are labelled with the Young diagrams, which will be discussed in section 3.2. In section 3.3 we will discuss the representations of the permutation group. The decomposition will be presented in section 3.4 and in section 3.5 we will discuss its consequences for the eigenvalue problem.

3.1. The group algebra

Let us assume that the group G is finite, for example G = SN . A group algebra CG is a vector space with basis e ,g G: g ∈ CG = c e + ... + c e : c C,G = g ,...g (30) { 1 g1 n gn i ∈ { 1 n}} equipped with multiplication such that: e e = e . (31) g1 · g2 g1g2 A representationρ ˜ of a group algebra CG on a vector space V is an algebra homomorphism: ρ˜ : CG End(V ). (32) → A representation ρ : G GL(V ) of a group G extends by linearity to a representation → ρ˜ : CG End(V ) of the group algebra CG. Any statement about a representation of → a group G has an equivalent statement in the group algebra CG. A left action of CG on V makes it a left CG-module. For completeness we will recall the definition of a module. Let A be a ring. An Abelian group U is a left A-module if there is a mapping A U U, (a, u) au such that: × → 7→ a(u + u )= au + au , • 1 2 1 2 (a + a )u = a u + a u, • 1 2 1 2 a (a u)=(a a )u, • 1 2 1 2 1u = u. • Clearly CG is a ring (because any algebra is a ring), V is an Abelian group (with respect to the addition of vectors) and the action of CG on V satisfies the axioms above. Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 10

We can make V also a right CG-module using an anti-involution:

a = a e aˆ = a e −1 . (33) g g 7→ g g g G g G X∈ X∈ Left CG-module is turned into right CG-module by taking: v aˆ = a v. (34) · · A (left) regular representation of a ﬁnite group G is a group homomorphism ρ : G GL(CG) (35) → given by

ρ(g)egi = eggi . (36) In the language of group algebras: the regular representation is a left CG-module over CG. The regular representation R can be decomposed into its irreducible components

Wi. Each representation space Wi appears dim Wi times:

⊕ dim Wi R = Wi . (37) i M Each irreducible representation is isomorphic to a minimal left ideal in CG. Let us recall that a (left) ideal in a ring A is a subring I closed under (left) multiplication by elements of the ring A: AI = I. Each (left) ideal in CG is generated by an idempotent. This means in particular that for each space W there is e A such that e2 = e and i i ∈ i i Wi = Aei. (38)

We will present a construction of such idempotents for the permutation group Sd, called Young symmetrizers. We will introduce ﬁrst a convenient tool called Young diagrams and Young tableaux.

3.2. Young diagram and Young tableau A Young diagram is associated to a partition (λ ,...,λ ), λ λ ... λ 1,λ N (39) 1 k 1 ≥ 2 ≥ ≥ k ≥ i ∈ of a number d N: ∈ d = λ1 + ... + λk. (40)

A Young diagram has λi boxes in the i-th row. For example, the Young diagram corresponding to the partition 12 = 5 + 3 + 3 + 1 is:

′ ′ ′ The conjugate partition λ =(λ1,...,λr) is obtained by interchanging rows and columns in the Young diagram. For example, the Young diagram Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 11

corresponds to the partition conjugate to 12 = 5+3+3+1. A Young tableau is obtained from a Young diagram λ by inserting numbers 1,...,d into the boxes. Following [33] we will call a λ-tableau any Young tableau on the Young diagram corresponding to a partition λ. For example:

1 2 3 4 5 (41) 6 7 8 9 10 11 12 is a Young tableau on the Young diagram λ = (5, 3, 3, 1). Another Young tableau on this diagram could be:

1 2 6 7 5 (42) 10 11 8 3 4 9 12 Young tableau is called canonical if the boxes are labelled by consecutive numbers. For example the tableau (41) is the canonical Young tableau on the Young diagram λ = (5, 3, 3, 1). Young tableau is called standard if all rows and columns are increasing. Each canonical tableau is standard. Another example of standard tableau on this diagram is:

1 2 5 6 7 (43) 3 4 8 9 10 12 11 The tableau (42) is not standard.

3.3. Representations of the permutation group

The irreducible representations of the permutation group Sd can be constructed using the λ Young symmetrizers. For a given Young tableau t , two subgroups of Sd are constructed P (tλ)= g S : g preserves each row of tλ (44) { ∈ d } and Q(tλ)= g S : g preserves each column of tλ . (45) { ∈ d } Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 12

To these subgroups there correspond two elements in the group algebra CSd: λ λ a(t )= eg, b(t )= sgn(g)eg. (46) λ λ g∈XP (t ) g∈XQ(t ) A Young symmetrizer is a product of these elements: c(tλ)= a(tλ) b(tλ) CS . (47) · ∈ d λ Let tcan be the canonical λ-tableau (let us recall that in the canonical Young tableau the boxes of the Young diagram λ are labelled by consecutive numbers). The following notation is used: λ λ λ λ λ Pλ = P (tcan), Qλ = Q(tcan), aλ = a(tcan), bλ = b(tcan), cλ = c(tcan). (48) A central role is played by the following theorem (see Theorem 4.3 in [30]).

Theorem 1. The Young symmetrizer cλ is an idempotent up to a scalar factor: 2 cλ = nλcλ. (49)

The image of cλ by right multiplication on CSd is an irreducible representation Vλ of

Sd. Every irreducible representation can be obtained this way for a unique partition.

The number nλ in the theorem is given by d! nλ = . (50) dim Vλ

There is a convenient expression for the dimension of the space Vλ, called the hook length formula. The hook length of a box in a Young diagram is the number of boxes directly below or directly to the right from the box, including the box once. For example, in the following diagram each box is labelled by its hook length:

8 7 6 2 1 (51) 5 3 2 4 2 1 1

The dimension of the space Vλ is given by d! dim V = . (52) λ (Hook lengths) For the Young diagram Qλ = (5, 3, 3, 1) above the formula becomes: 12! dim V = . (53) λ 8 7 6 2 5 3 2 4 2 · · · · · · · · From the theorem follows that CSd has the following decomposition:

C ⊕ dim Vλ Sd ∼= Vλ , (54) λ M where the external direct sum is over all partitions of d and Vλ = Acλ, A = CSd. It turns out that the dimension of Vλ is also equal to the number of standard λ-tableaux (for Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 13

the deﬁnition of the standard tableau see the previous subsection). In fact, A = CSd is a direct sum of left ideals VTλ = AeTλ , where dim V e = λ c (55) Tλ d! Tλ and Tλ runs through all the standard λ-tableaux: C Sd = VTλ . (56) λ T M Mλ Each element a CS can be uniquely written as ∈ d

a = aTλ , (57) λ T X Xλ where a = ae V . Because of the property (57), we used an equality sign in the λ Tλ ∈ Tλ equation (56), while in (54) we used ∼= sign denoting an isomorphism. Let us notice

that for all standard Young tableaux on a diagram λ the spaces VTλ are isomorphic to V : V V . λ Tλ ∼= λ

3.4. Decomposition of the space of intertwiners Fundamental role in our approach is played by Lemma 6.22 from [30]. In [30] it is used in the context of Schur-Weyl duality. We will be interested in this lemma for its own sake. We will focus on the group of permutations but in the following lemma, G is any finite group. Let us recall that a map ϕ : U V is called a homomorphism of right → modules if it is a homomorphism of the Abelian groups U, V and satisfies ϕ(u a)= ϕ(u) a. (58) · · If U is a right-module over A = CG, the commutator algebra B is the algebra of automorphisms of U. Let us recall that in section 2.3 we used an equivalent definition: B = Hom (U, U)= ϕ : U U; ϕ(v g)= ϕ(v) g, v U, g G . (59) G { → · · ∀ ∈ ∈ } Let W be any left A-module. Consider a submodule N of a tensor product U C W ⊗ generated by va w v aw . The tensor product: { ⊗ − ⊗ } U W =(U C W ) /N (60) ⊗A ⊗ is a left B-module: b (v w)=(b v) w. (61) · ⊗ · ⊗ Let us notice that U A and U are isomorphic. The canonical isomorphism ⊗A π : U A U is ⊗A → π(v a)= v a. (62) ⊗ · Lemma 1. Let U be a finite-dimensional right A-module. (i) For any c A the restriction of the canonical map π to U Ac is an isomorphisms ∈ ⊗A of left B-modules U Ac and Uc. ⊗A Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 14

(ii) If W = Ac is an irreducible left A-module, then U W is an irreducible left ⊗A B-module.

⊕mi (iii) If Wi = Aci are the irreducible left A-modules in the decomposition A ∼= i Wi , then L U = (U W )⊕mi = (Uc )⊕mi (63) ∼ ⊗A i ∼ i i i M M is the decomposition of U into irreducible left B-modules. The proof of this lemma can be found in [30]. We will not repeat it here. As we have shown in section 2.3 the space of invariant tensors U = Inv ⊗N is a Hj right module over A = CS . We have discussed above that A = CS has the following N N decomposition:

A = AeTλ , (64) λ T M Mλ where the direct sum is over all standard Young tableaux on the Young diagrams with

N boxes, eTλ are idempotents from (55) constructed from Young symmetrizers. This decomposition leads to a decomposition of U A ⊗A U A = U Ae . (65) ⊗A ⊗A Tλ λ T M Mλ By lemma 1 (ii) each component in this decomposition U Ae is an irreducible left ⊗A Tλ B-module. By using the canonical isomorphism π from (62) we have

U = UeTλ . (66) λ T M Mλ Each element u U has the following decomposition ∈

u = uTλ , (67) λ T X Xλ where uTλ = ueTλ is the component of u in the irreducible left B-module UeTλ . Let us summarize the construction by the following theorem. Theorem 2. Let Inv ⊗N be the space of tensors in ⊗N invariant under the action Hj Hj of the SU(2) group and S be the group of permutations of N elements. N (i) The subspace Inv ⊗N is invariant under the right action of the group S on Hj N ⊗N deﬁned by: Hj (v ... v ) σ = v ... v , v . (68) 1 ⊗ ⊗ N · σ(1) ⊗ ⊗ σ(N) i ∈ Hj (ii) Consider the commutator algebra: B = b : Inv ⊗N Inv ⊗N ; b(ι σ)= b(ι) σ, ι Inv ⊗N , σ S .(69) { Hj → Hj · · ∀ ∈ Hj ∈ N }

Let Tλ be a standard λ-tableau andeTλ be the idempotent corresponding to the Young

symmetrizer cTλ : dim V e = λ c . (70) Tλ N! Tλ Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 15

Let us denote by Inv ⊗N the image of e : Tλ Hj Tλ ⊗N ⊗N InvT = Inv eT . (71) λ Hj Hj λ The space Inv ⊗N has the following decomposition into irreducible left B- Hj modules: Inv ⊗N = Inv ⊗N , (72) Hj Tλ Hj λ Tλ M M where λ runs through all Young diagrams with N boxes, Tλ runs through all the standard λ-tableaux.

3.5. Consequences of the decomposition of the space of intertwiners for the eigenvalue problem The theorem has the following consequences. Consider a hermitian operator C : Inv ⊗N Inv ⊗N (73) Hj → Hj such that C(ι σ)= C(ι) σ ι Inv ⊗N , σ S . (74) · · ∀ ∈ Hj ∈ N An example of such operator is the loop quantum gravity volume operator. Clearly C is an element of the commutator algebra Hom (Inv ⊗N , Inv ⊗N ). Let us recall SN Hj Hj the deﬁnition of Inv ⊗N : Tλ Hj ⊗N ⊗N InvT = Inv eT . (75) λ Hj Hj λ From theorem 2 it follows that C (in the basis adapted to the decomposition) has block diagonal form. Let us denote by C : Inv ⊗N Inv ⊗N (76) Tλ Tλ Hj → Tλ Hj the corresponding blocks. The diagonalization problem of C reduces to the problem of diagonalizing the operators C , in particular if dim Inv ⊗N = 1 we obtain an Tλ Tλ Hj eigenvector of C . Tλ Let us notice that from the property (74) it follows that for a given λ all operators

CTλ are unitarily equivalent. Indeed, consider an action of the permutation group σ λ λ on λ-tableau t . Let us denote by tij the number in the box in the i-th row and j-th column in λ-tableau tλ. The λ-tableau σ tλ has a number σ(tλ ) in the box in the i-th · ij row and j-th column: (σ tλ) = σ(tλ ). (77) · ij ij λ λ Permuting the numbers in the boxes induces group isomorphism P (σt ) ∼= P (t ) and λ λ Q(σt ) ∼= Q(t ): P (σtλ)= σ P (tλ) σ−1, Q(σtλ)= σ Q(tλ) σ−1. (78) · · · · As a result, the Young symmetrizer transforms as: c(σ tλ)= σ c(tλ) σ−1. (79) · · · Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 16

It follows that the spaces Inv ⊗N and Inv ⊗N are unitarily equivalent. The Tλ Hj σ·Tλ Hj equivalence is given by an operator U : σ U : Inv ⊗N Inv ⊗N , U ι = ι σ−1. (80) σ Tλ Hj → σ·Tλ Hj σ · From the property (74) it follows that −1 Cσ·Tλ = UσCTλ Uσ . (81) Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 17

4. Monochromatic spin 1/2 intertwiners

In this section we will show that the volume operator is proportional to identity in ⊗N the spaces of spin 1/2 monochromatic intertwiners Inv 1 . We will calculate the H 2 proportionality factor. If the valence N of the intertwiner is odd, the intertwiner space is 0. If it is even, it will turn out to be the given by the n-th Catalan number (see section 4.1) which is equal to the number of standard λ-tableaux on a diagram λ with two rows of equal ⊗N 1 length: λ0 = (N/2,N/2). This means that dimension of each space InvTλ0 H 2 is not greater than 1. In section 4.2 we will show that it is non-trivial and therefore ⊗N ⊗N 1 1 dim InvTλ0 = 1. This means that the elements of InvTλ0 are eigenvectors H 2 H 2 of the volume operator and that the volume operator is proportion al to identity. In section 4.3 we will calculate the proportionality factor by calculating the trace of the operator.

4.1. Dimension of the space of invariant tensors

⊗N ⊗N Let us notice that the operator P : 1 1 : H 2 → H 2

P = dµ(g) ρ 1 (g) ρ 1 (g) ... ρ 1 (g) (82) 2 2 2 SU(2) ⊗ ⊗ ⊗ Z N

⊗N is a projector onto the subpace| of invariant{z tensors Inv} 1 . The dimension H 2 ⊗N DN = dim Inv 1 of the space of invariants is therefore H 2 N DN = dµ(g) Tr ρ 1 (g) . (83) 2 Z Let us notice, that D is non-zero if N is even: N = 2n, n N. This leads to the N ∈ following formula π 2n π 4 2 sin(2θ) 4n+1 2 4n+1 1 3 D = dθ sin2 θ = dθ(sin θ)2(cos θ)2n = B(n+ , ),(84) 2n π sin θ π 2π 2 2 Z0 Z0 where B(x, y) is the beta function. Since n is an integer, this expression simpliﬁes to 1 2n D = . (85) 2n n +1 n This is precisely the n-th Catalan number.

4.2. Decomposition of the space of invariant tensors In the following we will assume that N = 2n, n N. Let us calculate the dimension ∈ of the representation space Vλ of the permutation group SN corresponding to Young Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 18

diagram λ = (n, n). For example if N = 20, then λ = (10, 10) and the corresponding Young diagram is:

. (86)

The dimension of the space Vλ can be conveniently calculated using the hook length formula: (2n)! 1 2n dim V = = . (87) λ (n + 1)!n! n +1 n For example in the following diagram each box is labelled by its hook length:

11 10 9 8 7 6 5 4 3 2 (88) 10 9 8 7 6 5 4 3 2 1 Therefore for this diagram 20! dim V = . (89) λ 11!10! We will use this observation to prove the following lemma:

⊗N Lemma 2. For N odd the space Inv 1 is trivial and for N even it has the following H 2 decomposition into irreducible left B-modules: ⊗N ⊗N Inv 1 = InvTλ 1 , (90) H 2 H 2 T Mλ where λ = (N/2,N/2) and the direct sum is over all standard λ-tableaux. Each space ⊗N InvTλ 1 in this decomposition is 1-dimensional: H 2 ⊗N dim InvTλ 1 =1. (91) H 2 ⊗N Proof. Since dim Inv 1 = dim Vλ is the same as the number of standard λ- H 2 tableaux, it follows that ⊗N dim InvTλ 1 1, for λ = (N/2, N/2). (92) H 2 ≤ In fact it is non-trivial. Consider an invariant antisymmetric tensor ǫ Inv 1 1 : ∈ H 2 ⊗ H 2 A1 A2 B1B2 A1A2 ρ(u) B1 ρ(u) B2 ǫ = ǫ . (93) It is deﬁned uniquely up to a scale factor. Let us for simplicity choose the scale factor 1 1 in such a way that ǫ− 2 2 = 1. A non-trivial element is: − ιA1...AnAn+1...A2n = ǫ(A1|An+1|ǫA2|An+2| ...ǫAn)A2n , (94) where the brackets () denote symmetrization and the vertical lines distinguish the ﬁxed indices (those not used in the symmetrization). In other words, take ˜ιA1...AnAn+1...A2n = ǫA1An+1 ...ǫAnA2n . (95) Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 19

The invariant tensor ι is obtained from ˜ι by acting with the Young symmetrizer λ corresponding to the canonical Young tableau Tλ = tcan, where λ =(N/2,N/2): ι =˜ι c . (96) · λ Clearly ι is symmetric in the ﬁrst N/2 and the last N/2 indices, while it is antisymmetric under transposition of indices Ai and Ai+n. It is also invariant because ǫ is invariant. It is non-trivial, because: 1 ... 1 1 ... 1 n ι− 2 − 2 2 2 =( 1) . (97) − This shows that ⊗N λ dim Invtcan 1 = 1 for λ = (N/2, N/2). (98) H 2 ⊗N Since all the spaces InvTλ 1 corresponding to the same Young diagram λ are H 2 ⊗N λ isomorphic to Invtcan 1 , it follows that: H 2 ⊗N ⊗N Inv 1 = InvTλ 1 , (99) H 2 H 2 T Mλ where λ =(N/2,N/2) and the direct sum is over all standard λ-tableaux.

It follows that the volume operator V (and any element of the commutator algebra B) is diagonal in the basis adapted to the decomposition. From the discussion in section 3.5 it follows that the blocks VTλ are isospectral. As a result, all the diagonal ⊗N entries are equal. The volume operator restricted to the space Inv 1 is proportional H 2 to identity. We will ﬁnd the overall factor by calculating the trace of the operator.

4.3. The trace of the volume operator The calculation of the trace of the volume operator is based on the observation that the operators q : Inv ⊗N Inv ⊗N , IJK Hj → Hj i j k qIJK = ǫijkJ J J , I =J, I = K, J = K (100) | I J K| 6 6 6 are isospectral:

−1 qσ(I)σ(J)σ(K) = UσqIJKUσ , (101)

where Uσ is the unitary operator deﬁned by the right action of the permutation group on Inv ⊗N : Hj −1 Uσι = ι σ . (102) · As a result, the problem of calculating the trace of the operator q~j reduces to the problem of calculating the trace of q123: 1 N Tr q~ = Tr q . (103) j 8 3 123 Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 20

⊗N In order to calculate Tr q123 in the space Inv 1 , N = 2n, n N, we notice H 2 ∈ that there is an isomorphism 3 2 ⊗3 ⊗(N−3) ⊗N : Inv 1 k Inv k 1 Inv 1 , (104) E H 2 ⊗ H ⊗ H ⊗ H 2 → H 2 k 1 M= 2 deﬁned by ( (v w))A1...AN = vA1A2A3B1 ǫ wB2A4...AN , (105) E ⊗ B1B2 ⊗3 ⊗(N−3) 1 3 for all v Inv 1 k , w Inv k 1 ,k 2 , 2 . This isomorphism ∈ H 2 ⊗ H ∈ H ⊗ H 2 ∈ { } commutes with the operator q123: q (v w)= ((q v) w). (106) 123E ⊗ E 123 ⊗ ⊗3 In order to calculate q123 it is enough to know its action in the space Inv 1 k H 2 ⊗ H but this has been studied in [27, 28, 34]. In particular, from formulas (7.3) and (7.4) from [34] it is straightforward to calculate that

⊗3 q123v =0 v Inv 1 3 . (107) ∀ ∈ H 2 ⊗ H 2 ⊗4 The matrix of the operator q123 in Inv 1 is a 2 2 matrix, which turns out to be H 2 × diagonal:

√3 ⊗3 q123v = v v Inv 1 1 . (108) 4 ∀ ∈ H 2 ⊗ H 2 The trace of the operator q123 is therefore

√3 ⊗(N−2) √3 1 2n 2 Tr q123 = dim Inv 1 = − . (109) 2 · H 2 2 n n 1 − From equation (103) it follows that √3 1 2n 2n 2 Tr q~ = − . (110) j 16 n 3 n 1 − In order to calculate the factor λ,

q~j = λ½ (111) ⊗(N) we need to divide the trace by the dimension of the space Inv 1 : H 2 Tr q~ λ = j . (112) ⊗(N) dim Inv 1 H 2 We obtain: √3 λ = (N 2)N(N + 2). (113) 64 3! − · ~ 1 1 Therefore for j =( 2 ,..., 2 ) the volume operator is:

3 2 κ0 8πG~γ √3

V~ = (N 2)N(N + 2) ½. (114) j 8 c3 s 3! − · Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 21

Let us notice that the total spin is N 1 = N . In the limit of large total spin the expected · 2 2 asymptotics [29] is recovered: 3 V~ C(total spin) 2 . (115) || j|| ≈ Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 22

5. Higher spin monochromatic intertwiners

In the case of spin 1/2 monochromatic intertwiners we gave a full characterization of the volume operator. This was possible because the space of intertwiners turned out to be an irreducible space of the action of the permutation group. As a result, the volume operator is proportional to identity. In addition to this, the operator q123 takes a simple form, which allowed us to calculate the overall factor. In general the space of intertwiners splits into subspaces corresponding to more than one Young diagram and an irreducible B-module can be more than 1-dimensional. For the lowest spins and valences of the intertwiners our analysis predicts well the pattern of the degeneracy of the eigenvalues. In this section we will present some results of our numerical calculation of the dimensions of the irreducible spaces and the corresponding eigenvalues. In our numerical calculation we use the tree basis of the intertwiner space Inv ( ... ). We recall the deﬁnition in section 5.1 (see for example [35] for Hj1 ⊗ ⊗ HjN detailed presentation). In section 5.2 we calculate the matrices of the representation −1 of the permutation group σ Uσ, Uσι = ι σ . This allows us to compute the 7→ ⊗N · ⊗N dimensions of the spaces Inv = Inv λ (see section 5.3). We considered λ Hj tcan Hj N in the range 2,..., 6 and j in the range 1 , 1,..., 7. The results are presented in 2 table 1. Knowing dim Inv ⊗N , we can predict to some extend the degeneracy of λ Hj the eigenstates of the volume operator. In section 5.4 we discuss the methods we used to calculate the volume operator. In table 2 we presented the result of our numerical calculation of the eigenvalues of the volume operator. In section 5.5 we compare the degeneracy of the eigenstates for the volume matrices, summarized in table 2, with the the dimensions of the spaces Inv ⊗N , summarized in table 1. λ Hj 5.1. Tree basis A tree basis is built from invariant tensors C Inv ( ) and ǫj4 Inv ∗ ∗ . (116) j1j2j3 ∈ Hj1 ⊗ Hj2 ⊗ Hj3 ∈ Hj4 ⊗ Hj4 ∗ ∗ The spaces Inv ( j1 j2 j3 ) and Inv are 1-dimensional. Therefore the H ⊗ H ⊗ H Hj4 ⊗ Hj5 tensors C and ǫj4j5 are deﬁned uniquely up to scale and phase factors. A common j1j2j3 choice is:

A1A2A3 j1 j2 j3 j j−A1 Cj1j2j3 = , ǫA1A2 =( 1) δA1,−A2 . (117) A1 A2 A3 ! −

j1 j2 j3 where is the 3j-symbol. When the values of the spins j1, j2, j3, j are A1 A2 A3 ! clear from the context we will suppress them in the notation and write CA1A2A3 instead A1A2A3 j of Cj1j2j3 , ǫA1A2 instead of ǫA1A2 . A tree basis in Inv ( j1 ... jN ) is labelled by ~ H ⊗ ⊗ H a sequence of spins k =(k1,k2,...,kN−3) satisfying: j k k j + k , k + k + j N, (118) | I − I−2| ≤ I−1 ≤ I I−2 I−1 I−2 I ∈ Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 23

where I 2,...,N 1 , k = j , k = j . The basis element corresponding to the ∈{ − } 0 1 N−2 N sequence ~k is ′ ′ A1...A A A B B A B B A B − B − A − A ˜ι N = C 1 2 1 ǫ ′ C 1 3 2 ǫ ′ ...ǫ ′ C I−2 I I 1 ǫ ′ ...C N 3 N 1 N ,(119) ~k B1B1 B2B2 BI−2BI−2 BI−1BI−1 where the indices A ,...,A correspond to Hilbert spaces ,..., and the 1 N Hj1 HjN indices B ,...,B correspond to the Hilbert spaces ,..., − . The basis ˜ι~ 1 N−3 Hk1 HkN 3 k is orthogonal but not orthonormal. We will use an orthonormal basis N−2

ι~k = 2kI +1˜ι~k. (120) I=1 Y p ~ ~′ ′ We put the basis in the co-lexicographic order, i.e. we will say that k < k if kI

k where δ l is the Kronecker delta. Since any permutation is a product of the adjacent transpositions, we can calculate all matrices U , where σ S . σ ∈ 4 The calculation in the case of 4-valent intertwiners generalizes straightforwardly to N-valent intertwiners. The matrix elements of matrices corresponding to adjacent

transpositions U(II+1) are the following. (i) For I = 1, N−3 ~ U k =( 1)2j+k1 δki . (127) (1 2) ~l − li i=1 Y (ii) For any I 2, 3,...,N 2 the matrices U are expressed in terms of the ∈ { − } (II+1) 6j-symbol:

~ k j k U k = (2k + 1)(2l + 1)( 1)2j+kI−1+lI−1 I−2 I−1 δki ,(128) (II+1) ~l I−1 I−1 li − kI j lI−1 ( ) i6=I−1 p Y where we use the convention that k0 = j1,kN−2 = jN . (iii) For I = N 1, − N−3 ~k 2j+kN−3 ki U N N =( 1) δ . (129) ( −1 ) ~l − li i=1 Y Again, since any permutation can be expressed as a product of adjacent transpositions, any matrix U , σ S is a product of the matrices given above. σ ∈ N

5.3. Dimensions of the irreducible spaces According to lemma 1 (iii) the Hilbert space Inv ⊗N has the following decomposition: Hj ⊗N ⊗N ⊕mλ Inv = Invλ , (130) Hj ∼ Hj λ M where λ runs through all Young diagrams with N boxes and

⊗N ⊗N Inv = Inv λ . (131) λ Hj tcan Hj ⊗N The number of times Invλ appears in the decomposition is denoted by mλ. It is Hj equal to the dimension of the irreducible representation of S corresponding to λ and N can be calculated using the hook length formula (52). The dimensions of the spaces Inv ⊗N are equal to the trace of the idempotent e obtained by rescaling the Young λ Hj λ symmetrizer c (see theorem 2). Speciﬁcally, to the groups P and Q from section 3.3 λ λ λ correspond two operators in Inv ⊗N : Hj

Aλ = Uσ, Bλ = sgn(σ)Uσ. (132) σ P σ Q X∈ λ X∈ λ The space Inv ⊗N is the image of the operator P acting in Inv ⊗N : λ Hj λ Hj 1 P = B A . (133) λ (Hook lengths) λ λ Q Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 25

As a result the dimension of the space Inv ⊗N can be expressed in terms of the λ Hj trace of P : λ 1 dim Inv ⊗N = Tr B A . (134) λ Hj (Hook lengths) λ λ We calculated the dimensions of the spaces dim Inv ⊗N for N = 2, 3, 4, 5, 6, Q λ Hj j = 1 ,..., 7 together with their multiplicities m . The matrices U are calculated by 2 λ σ multiplying the adjacent transposition matrices (127), (128), (129). We calculate them in double accuracy using the library WIGXJPF for the 6j-symbol [36]. From equations (127), (128), (129) it is clear that the matrices are sparse. We use the Intel®MKL 2019.0 library to multiply and add the sparse matrices. The results are gathered in table 1. In the table we included only the Young diagrams λ such that dim Inv ⊗N =0 for at λ Hj 6 least one pair (N, j), N 2, 3, 4, 5, 6 , j 1 , 1,..., 7 . In our numerical calculations ∈{ } ∈{ 2 } the dimensions are double accuracy numbers but are only ǫ 10−16 away from integers. ≈ They are rounded to an integer. We will discuss the result in the next subsection.

5.4. Volume operator We calculated numerically the matrix of the volume operator in the tree basis and diagonalized it. In order to minimize the risk of making a mistake when entering the formulas or writing a computer code, we used two diﬀerent methods. First method is to construct the tree basis from the 3j-symbols and calculate the matrix elements of the volume operator using the formula for matrix elements of the angular momentum operators in the spin j representation (see for example [37]). Let us recall that the angular momentum operator J 3 is diagonal: J 3 jA >= A j A > . (135) | | 1 2 The operators J and J are conveniently expressed in term of operators J+ and J−: 1 1 1 1 J 1 = J + J , J 2 = J J , (136) 2 + 2 − 2i + − 2i − J jA>= (j A)(j + A + 1) j A +1 >, (137) +| − | J jA>= (j + A)(j A + 1) j A 1 > . (138) −| p − | − i The operators JI used inp the deﬁnition of the volume operator can be written in terms of the angular momentum operators:

i i

½ ½ ½ J = i ½ ... J ... . (139) I − ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ Second method is based on two observations, which we made already in section 1 4.3 when we calculated the overall factor of the volume operator in the case of spin 2 monochromatic intertwiners. We will repeat the formulas for completeness. (i) We recall that (see (101)):

−1 qσ(I)σ(J)σ(K) = UσqIJKUσ . (140)

Using such similarity transformations we can express each qIJK in terms of q123. Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 26 ❍ ❍ j ❍❍ 1 3 5 7 9 11 13 N λ ❍ 2 1 2 2 2 3 2 4 2 5 2 6 2 7 mλ ❍❍ (2) 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 2 (1,1) 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 (3) 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 3 (1,1,1) 1 0 1 0 0 0 1 0 0 0 1 0 0 0 1 (4) 1 0 1 1 1 1 2 1 2 2 2 2 3 2 3 4 (2,2) 2 1 1 1 2 2 2 3 3 3 4 4 4 5 5 (1,1,1,1) 1 0 0 1 0 1 1 1 1 2 1 2 2 2 2 (5) 1 0 0 0 1 0 0 0 2 0 0 0 3 0 0 (4,1) 4 0 0 0 1 0 1 0 2 0 3 0 4 0 5 (3,2) 5 0 0 0 1 0 1 0 3 0 2 0 6 0 5 5 (3,1,1) 6 0 1 0 0 0 3 0 1 0 6 0 3 0 10 (2,2,1) 5 0 0 0 1 0 0 0 3 0 2 0 5 0 4 (2,1,1,1) 4 0 0 0 0 0 1 0 1 0 2 0 3 0 4 (1,1,1,1,1) 1 0 0 0 1 0 0 0 1 0 0 0 2 0 0 (6) 1 0 1 0 2 0 3 0 4 0 6 0 8 0 10 (5,1) 5 0 0 1 0 2 1 4 2 7 4 11 7 16 11 (4,2) 9 0 1 0 3 0 6 1 11 3 17 6 26 11 37 (4,1,1) 10 0 0 1 0 3 1 6 3 11 6 18 11 27 18 (3,3) 5 1 0 2 0 4 0 7 0 11 1 16 2 23 4 6 (3,2,1) 16 0 0 0 1 2 3 5 7 10 14 18 23 29 36 (3,1,1,1) 10 0 0 0 1 0 3 1 6 3 11 6 18 11 27 (2,2,2) 5 0 1 0 2 0 4 0 7 0 11 1 16 2 23 (2,2,1,1) 9 0 0 1 0 2 0 5 1 9 2 15 5 23 9 (2,1,1,1,1) 5 0 0 0 0 0 1 0 2 1 4 2 7 4 11 (1,1,1,1,1,1) 1 0 0 0 0 1 0 1 0 2 0 3 0 4 0

Table 1. ⊗N ⊗N ⊕mλ We study the decomposition (130) Inv j ∼= λ Invλ j 1 H H for diﬀerent values of N = 2, 3, 4, 5, 6 and j = 2 , 1,..., 7. The table contains ⊗N L the dimensions of the spaces Invλ j and their multiplicities mλ. We included H ⊗N only the Young diagrams λ such that dim Invλ = 0 for at least one pair Hj 6 (N, j),N 2, 3, 4, 5, 6 , j 1 , 1,..., 7 . ∈{ } ∈{ 2 }

(ii) Let us also recall that q123 commutes with the isomorphism (see (105) and (106)): : Inv ⊗3 Inv ⊗(N−3) Inv ⊗N ,(141) E Hj ⊗ Hk ⊗ Hk ⊗ Hj → Hj k M deﬁned by: ( (v w))A1...AN = vA1A2A3B1 ǫ wB2A4...AN . (142) E ⊗ B1B2 As a result, in order to calculate q123 it is enough to know its action in the space of 4-valent intertwiners Inv ⊗3 . The second problem has an analytic solution Hj ⊗ Hk [27, 28, 34]. Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 27

1 3 5 We calculated the volume operator for valence N = 4, 5, 6 spins j = 2 , 1, 2 , 2, 2 . 1 3 The results are presented in table 2. Both methods were used for spins j = 2 , 1, 2 and we veriﬁed that with accuracy 10−6 they give the same results (typically they coincide with accuracy 10−15 but for zero eigenvalues the accuracy was lower). For higher spins the second method was used only, because it is considerably faster (around 800 times in our tests). Each entry of the table contains an eigenvalue of the volume operator and its degeneracy. The eigenvalue provided in the table is in the units of volume 3 8πG~γ 2 V0 = κ0 c3 . We assumed that two numerical eigenvalues correspond to the same true eigenvalue if the modulus of their diﬀerence does not exceed certain number ǫ . eigen We varied the parameter ǫ in the range 10−15, 10−14,..., 10−6 and obtained that the degeneracy pattern is stable in the range 10−14, 10−13,..., 10−6. According to analytic study [27], in the case of 4-valent intertwiners, the eigenvalues come in pairs except for the zero eigenvalue which is non-degenerate. The zero eigenvalue appears only in the spectrum of odd-dimensional matrices. Since the dimension of the space of 4-valent monochromatic spin j intertwiners is equal 2j + 1, the zero eigenvalue appears if j N. ∈ This is consistent with the numerical results that we obtained. The non-zero eigenvalues seem to have higher accuracy than zero eigenvalues. For example, our analytic result from section 4 coincides with the numerical calculation presented in the table with accuracy 10−15 but the zero eigenvalues predicted by the analytic study in the 4-valent case [27] are recovered with accuracy 10−7.

5.5. Degeneracy of the eigenvalues of the volume operator In most of the cases that we calculated, our analysis can be used to predict the pattern of the degeneracy of the eigenstates. For example, by looking at the entries for N =4, j =1 in table 1 we infer that the volume operator has 1 eigenvalue with degeneracy at least 1 corresponding to Inv ⊗4 , λ = (4), • λ H1 1 eigenvalue with degeneracy at least 2 corresponding to Inv ⊗4 , λ = (2, 2). • λ H1 It may very well turn out that there is only 1 eigenvalue with degener acy 3 but a quick glimpse at table 2 reveals that there are 2 eigenvalues with degeneracies precisely 1 and 2. In our examples this pattern repeats in most of the cases: the eigenvalues corresponding to different spaces Inv ⊗N are different. Exceptions are N =4, j = 3 and N =4, 5 . λ Hj 2 2 The numerical results seem to suggest that the eigenvalues corresponding to spaces for λ = (4) and λ = (1, 1, 1, 1) are the same. If Inv ⊗N has dimension greater than 1 then in principle we do not know if the λ Hj eigenvalues corresponding to Inv ⊗N are the same or different. In our calculation, λ Hj it turns out that such eigenvalues are different from each other. For example, in the 3 case N =6, j = 2 we could expect that the volume operator has ⊗6 1 eigenvalue with degeneracy 5 corresponding to Invλ 3 , λ = (5, 1), • H 2 ⊗6 1 eigenvalue with degeneracy 10 corresponding to Invλ 3 , λ = (4, 1, 1), • H 2 Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 28

N j=1/2 j=1 j=3/2 j=2 j=5/2 0.465302 2 0.000000 1 0.805927 2 0.000000 1 1.133249 2 4 0.930605 2 1.489473 2 1.403615 2 2.061107 2 2.137338 2 2.865738 2 1.201406 6 0.000000 1 2.293525 5 5 2.754087 5 2.911094 4 3.095988 1 0.930605 5 0.000000 1 1.810981 5 0.000000 1 2.707704 5 1.699044 9 2.527597 5 2.741033 9 3.662116 5 1.861210 5 2.783351 10 3.107533 5 4.126456 10 2.945708 9 3.761054 9 4.387806 9 3.001623 5 3.809809 1 4.392001 5 3.917444 16 4.871472 5 3.973291 9 5.118623 5 6 4.180309 10 5.144352 16 4.433841 5 5.344379 16 5.380997 10 5.533459 5 5.658935 9 5.734889 10 6.075997 1

Table 2. The table contains the result of our numerical computation of the eigenvalues of the volume operator and their degeneracies. We diagonalized the matrices of the ⊗N 1 3 5 volume operator in the spaces Inv j for N = 4, 5, 6 and j = 2 , 1, 2 , 2, 2 . Each H 3 2 8πG~γ entry is a pair: an eigenvalue in the units of volume V0 = κ0 c3 and its degeneracy. The matrix and its eigenvalues are calculated numerica lly. We consider two numerical eigenvalues to correspond to the same true eigenvalue if their diﬀerence −6 is not greater than ǫeigen = 10 .

2 eigenvalues with degeneracy 5 or 1 eigenvalue with degeneracy 10 corresponding • ⊗6 to Invλ 3 , λ = (3, 3), H 2 ⊗6 1 eigenvalue with degeneracy 9 corresponding to Invλ 3 , λ = (2, 2, 1, 1). • H 2 ⊗6 A quick glimpse at table 2 reveals that to the Hilbert space Invλ 3 , λ = (3, 3) H 2 correspond 2 eigenvalues with degeneracy 5. Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 29

6. Summary and outlook

The Rovelli-Smolin-DePietri volume operator can be defined by its action in the spaces of tensor products ... , where is the spin j representation space of the Hj1 ⊗ HjN Hj SU(2) group. We studied the sector of monochromatic spin j intertwiners Inv ⊗N . Hj There is a natural right action of the permutation group S on the space ⊗N : N Hj (v ... v ) σ = v ... v , σ S . (143) 1 ⊗ ⊗ N · σ(1) ⊗ ⊗ σ(N) ∀ ∈ N It descends to an action on Inv ⊗N . The Rovelli-Smolin-DePietri volume operator Hj commutes with this action. We studied consequences of this fact. We found a decomposition of the space of spin j monochromatic intertwiners Inv ⊗N : Hj Inv ⊗N = Inv ⊗N , (144) Hj Tλ Hj λ Tλ M M where λ runs through all Young diagrams with N boxes and Tλ runs through all standard λ-tableaux (we recall the definitions of Young diagram and standard λ-tableau in section 3.2). In a basis adapted to this decomposition the Rovelli-Smolin-DePietri volume operator takes a block diagonal form. For fixed λ the blocks corresponding to ⊗N different Tλ are isospectral. In the case of spin 1/2 intertwiners the space Inv 1 is H 2 non-trivial for N even. In this case, only one Young diagram contributes – it has two ⊗N rows with N/2 boxes each. Each space InvTλ 1 is 1-dimensional. This shows that H 2 the Rovelli-Smolin-DePietri volume operator is proportional to identity in the spaces of ⊗N spin 1/2 monochromatic intertwiners Inv 1 . H 2 This paper aims at simplifying the diagonalization problem of the Rovelli-Smolin- DePietri volume operator. In the spin 1/2 case we solved the diagonalization problem analytically. Since the volume operator is proportion to identity in the spaces of spin 1/2 intertwiners, it sufficed to calculate the overall factor. We obtained that in the ⊗N space of spin 1/2 monochromatic intertwiners Inv 1 the volume operator takes H 2 the following form: 3 2 κ0 8πG~γ √3 V 1 ,..., 1 = (N 2)N(N + 2) ½. (145) ( 2 2 ) 8 c3 s 3! − · For higher spins the simplification is not so drastic but still significant. The method presented in this paper allows to construct a basis adapted to the decomposition, in which the volume operator takes a block diagonal form. In the numerical study this may turn out to be useful, especially for large matrices. However, further work is needed to make this application practical. The calculation of the matrix elements of the Rovelli-Smolin-DePietri matrix becomes computationally intense when the valence N of the intertwiner N grows. One reason is that one should sum 3 matrices. Another reason is that the dimension of the matrices grows when N increases. We expect that in practical application one could overcome the second problem by calculating the blocks ⊗N corresponding to InvTλ 1 instead of trying to calculate the whole matrix at first H 2 Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 30

place. There is a chance that the first problem can be simplified by using the properties of the basis such as symmetry or antisymmetry of the indices. The spin 1/2 case is interesting itself. It could be used as a toy model. It could be also a more fundamental assumption in the models studied – that the quantum space is built from a large number of small grains described by the spin 1/2 monochromatic intertwiners. The result concerning spin 1/2 intertwiners finds an immediate application to our approach to homogeneous-isotropic sector of loop quantum gravity [4]. Consider scalar constraint operator adding and subtracting loops labeled with spin 1/2 acting between the spaces of spin 1/2 monochromatic spin networks. In [4] we introduced an ad hoc cut-off in the number of loops. Adding a loop at a node n increases the valence N of the space of intertwiners associated to n by 2. From (145) we immediately see that the cut-off in the number of loops translates into a cut-off in the volume. Let us notice that the valence Nn of a node n can be written in the following form: Nn = Vn +2Ln, where Vn is the valence of the node excluding loops and Ln is the number of loops at the node n. As a result, the volume operator from (145) can be expressed by the number operator n introduced in [38, 39]. This observation can be used to study N the coherence properties of the new graph coherent states with respect to the volume operator. The analysis that we performed applies not only to the volume operator but to any element in the commutator algebra B = ϕ End(U): ϕ(v g)= ϕ(v) g, v U, g G , (146) { ∈ · · ∀ ∈ ∈ } where U = Inv ⊗N ,G = S . For example the Lorentzian part of the Hamiltonian Hj N operator proposed in [6] has the form: ˆ Lint l˙ l˙′ ˆ L Hn = ε( , )Hnl,l′ , (147) l,l′ X where the sum is over all pairs of links incident at the node n, ε(l˙, l˙′) = 0 if the vectors tangent to the links l, l′ at the node n are linearly dependent and 1 otherwise. If we had replaced the operator with

Lext L Hˆn = Hˆnl l′ , (148) | , | l,l′ X we would obtain an element of the commutator algebra, to which our techniques could be applied. It should be veriﬁed if both expressions (147) and (148) correspond to the same classical object. One could also make a similar change in the deﬁnition of the Euclidean part of the Hamiltonian operator. In this case the operator changes the graph and therefore maps one space of intertwiners into another space of intertwiners. We expect that the techniques from this paper can be extended to this case.

Acknowledgements

This work was supported by the National Science Centre, Poland grant No. 2018/28/C/ST9/00157. Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 31

References

[1] Eugenio Bianchi, Carlo Rovelli, and Francesca Vidotto. Towards Spinfoam Cosmology. Phys.Rev., D82:084035, 2010. [2] Daniele Oriti, Lorenzo Sindoni, and Edward Wilson-Ewing. Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates. Class. Quant. Grav., 33(22):224001, 2016. [3] Daniele Oriti, Lorenzo Sindoni, and Edward Wilson-Ewing. Bouncing cosmologies from quantum gravity condensates. Class. Quant. Grav., 34(4):04LT01, 2017. [4] Marcin Kisielowski. Homogeneous-isotropic sector of loop quantum gravity: new approach. Class. Quant. Grav., 37(18):185004, 2020. [5] Jerzy Lewandowski and Hanno Sahlmann. Symmetric scalar constraint for loop quantum gravity. Phys. Rev., D91(4):044022, 2015. [6] E. Alesci, M. Assanioussi, J. Lewandowski, and I. Mäkinen. Hamiltonian operator for loop quantum gravity coupled to a scalar field. Phys. Rev., D91(12):124067, 2015. [7] Mehdi Assanioussi, Jerzy Lewandowski, and Ilkka Mäkinen. New scalar constraint operator for loop quantum gravity. Phys. Rev., D92(4):044042, 2015. [8] Mehdi Assanioussi, Jerzy Lewandowski, and Ilkka Mäkinen. Time evolution in deparametrized models of loop quantum gravity. Phys. Rev., D96(2):024043, 2017. [9] Carlo Rovelli. Quantum Gravity. Cambridge University Press, 2004. [10] Abhay Ashtekar and Jerzy Lewandowski. Background independent quantum gravity: A Status report. Class.Quant.Grav., 21:R53, 2004. [11] Thomas Thiemann. Modern Canonical Quantum General Relativity. Cambridge University Press, 2007. [12] Carlo Rovelli. Zakopane lectures on loop gravity. PoS, QGQGS2011:003, 2011. [13] Carlo Rovelli. Loop quantum gravity: the first twenty five years. Class.Quant.Grav., 28:153002, 2011. [14] Muxin Han, Weiming Huang, and Yongge Ma. Fundamental structure of loop quantum gravity. Int.J.Mod.Phys., D16:1397–1474, 2007. [15] Abhay Ashtekar, Martin Reuter, and Carlo Rovelli. From General Relativity to Quantum Gravity. 2014. [16] Carlo Rovelli and Francesca Vidotto. Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory. Cambridge University Press, 2014. [17] Martin Bojowald. Loop quantum cosmology. Living Rev. Rel., 11:4, 2008. [18] Abhay Ashtekar and Parampreet Singh. Loop Quantum Cosmology: A Status Report. Class. Quant. Grav., 28:213001, 2011. [19] Ivan Agullo and Parampreet Singh. Loop Quantum Cosmology. In Abhay Ashtekar and Jorge Pullin, editors, Loop Quantum Gravity: The First 30 Years, pages 183–240. WSP, 2017. [20] Abhay Ashtekar, Tomasz Pawlowski, and Parampreet Singh. Quantum nature of the big bang. Phys. Rev. Lett., 96:141301, 2006. [21] Carlo Rovelli and Lee Smolin. Discreteness of area and volume in quantum gravity. Nucl. Phys. B, 442:593–622, 1995. [Erratum: Nucl.Phys.B 456, 753–754 (1995)]. [22] Roberto De Pietri and Carlo Rovelli. Geometry eigenvalues and scalar product from recoupling theory in loop quantum gravity. Phys. Rev. D, 54:2664–2690, 1996. [23] Abhay Ashtekar and Jerzy Lewandowski. Quantum theory of geometry. 2. Volume operators. Adv. Theor. Math. Phys., 1:388–429, 1998. [24] R. Loll. The Volume operator in discretized quantum gravity. Phys. Rev. Lett., 75:3048–3051, 1995. [25] R. Loll. Spectrum of the volume operator in quantum gravity. Nucl. Phys. B, 460:143–154, 1996. [26] Roberto De Pietri. Spin networks and recoupling in loop quantum gravity. Nucl. Phys. B Proc. Suppl., 57:251–254, 1997. [27] Johannes Brunnemann and Thomas Thiemann. Simplification of the spectral analysis of the Rovelli-Smolin-DePietri volume operator for monochromatic intertwiners 32

volume operator in loop quantum gravity. Class. Quant. Grav., 23:1289–1346, 2006. [28] Johannes Brunnemann and David Rideout. Properties of the volume operator in loop quantum gravity. I. Results. Class. Quant. Grav., 25:065001, 2008. [29] Seth A. Major and Michael D. Seifert. Modeling space with an atom of quantum geometry. Class. Quant. Grav., 19:2211–2228, 2002. [30] W. Fulton and J. Harris. Representation Theory: A First Course, volume 129 of Graduate texts in mathematics. Springer, New York, NY, 2004. [31] John C. Baez. Spin network states in gauge theory. Adv.Math., 117:253–272, 1996. [32] Wojciech Kaminski, Marcin Kisielowski, and Jerzy Lewandowski. Spin-Foams for All Loop Quantum Gravity. Class.Quant.Grav., 27:095006, 2010. [33] Bruce E Sagan. The symmetric group: representations, combinatorial algorithms, and symmetric functions, volume 203 of Graduate texts in mathematics. Springer, New York, NY, 2001. [34] Johannes Brunneman and David Rideout. Properties of the volume operator in loop quantum gravity. II. Detailed presentation. Class. Quant. Grav., 25:065002, 2008. [35] Ilkka Mäkinen. Introduction to SU(2) Recoupling Theory and Graphical Methods for Loop Quantum Gravity. 10 2019. [36] H. T. Johansson and C. Forssén. Fast and accurate evaluation of wigner 3j, 6j, and 9j symbols using prime factorization and multiword integer arithmetic. SIAM Journal on Scientific Computing, 38(1):A376–A384, 2016. [37] Alan Robert Edmonds. Angular momentum in quantum mechanics. Princeton University Press, 1957. [38] Mehdi Assanioussi. Polymer quantization of connection theories: Graph coherent states. Phys. Rev. D, 98(4):045016, 2018. [39] Mehdi Assanioussi. Graph coherent states for loop quantum gravity. Phys. Rev. D, 101(12):124022, 2020.