SO*(2N) coherent states for loop quantum gravity

1.1 MB·

by  Florian Girelli

SO∗(2N) coherent states for loop quantum gravity
Florian Girelli1,∗ and Giuseppe Sellaroli1,†
1Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada
(Dated: January 27, 2017)
ASU(2)intertwinerwithN legscanbeinterpretedasthequantumstateofaconvexpolyhedronwith
N faces(whenworkingin3d). WeshowthattheintertwinerHilbertspacecarriesarepresentationof
thenon-compactgroupSO∗(2N). Thisgroupcanbeviewedasthesubgroupofthesymplecticgroup
Sp(4N,R) which preserves the SU(2) invariance. We construct the associated Perelomov coherent
states and discuss the notion of semi-classical limit, which is more subtle that we could expect.
Our work completes the work by Freidel and Livine [1, 2] which focused on the U(N) subgroup of
SO∗(2N).
7
1
0
CONTENTS
2

n
Introduction 2
a
J
I. Polyhedron parametrization 3
5
2
II. Coherent states for the polyhedron with fixed area: a review 4

] A. Harmonic oscillators and intertwiner 4
h
B. Intertwiner as U(N) representation 6
p
C. U(N) coherent states 6
-
h 1. U(N) coherent states `a la Perelomov 6
t 2. Matrix elements and semi-classical limit 8
a
m
III. A new coherent state for the SU(2) intertwiner 9
[ A. The Lie group SO∗(2N) and its Lie algebra so∗(2N) 9
1 1. The Lie group SO∗(2N) 10
v 2. The Lie algebra so∗(2N) 10
9 B. SO∗(2N) Perelomov coherent states for the intertwiner 11
1 1. SO∗(2N) Perelomov coherent states 11
5
2. Expectation values of observables 13
7
3. Relating SO∗(2N) and some Bogoliubov transformations 13
0
. C. Semi-classical limit 15
1
1. Recovering the spinor variables 15
0
2. Geometric interpretation of the semi-classical states 17
7
1
: Discussion 20
v
i
X Acknowledgements 20
r
a A. Perelomov coherent states 20
B. Proof of Proposition 1 21
C. Proof of the Propositions of Section IIIB2 22
1. Proof of Proposition 2 23
2. Proof of Proposition 3 24
3. Proof of Proposition 4 26
D. Proof of proposition 6 27
∗ [email protected]
† [email protected]
2
Case 1: all λ are distinct 29
α
Case 2: some λ are identical 30
α
References 31
INTRODUCTION
The spinorial formalism for loop quantum gravity (LQG) [3] provides a different way to parameterize the LQG
Hilbertspaceandassuchprovidespromisingavenuestoaddresssomeproblemsencounteredinthefield,suchas1: how
to construct the intertwiner observables when dealing with a quantum group (to introduce a non-zero cosmological
constant) [5], how to implement the simplicity constraints in a natural way [6], how to calculate various types of
entropies [1, 7].
One of the key results of this formalism is that it provides a closed algebra, spanned by E ,F ,F˜ , to express
ab ab ab
any intertwiner observables2. This algebra, in fact a Lie algebra, contains u(N) as a subalgebra (for a N legged
intertwiner) which is generated by the E . As a consequence, Freidel and Livine have shown that the space of
ab
N-legged intertwiners with fixed total area carries a specific representation of U(N) [1]. They showed furthermore
that a U(N) coherent states (`a la Perelomov) could be interpreted as a semi-classical polyhedron with N faces and
fixed area [2]. The rest of the algebra has not been fully studied yet and this is what we intend to do here.
Provided we redefine the observables E with respect to the usual convention, the full algebra of observables is
ab
isomorphicasacomplexalgebratoso(2N,C). Welookthenfortherealalgebrawhichwouldhaveu(N)asitscompact
sub-algebraandsuchthattheF ,F˜ areantisymmetricunderthepermutationa↔b. Thereisanuniquechoice[8],
ab ab
given by so∗(2N) which spans a non-compact group SO∗(2N). This group has not been studied much, in particular
its representation theory is not completely known (for N > 1). However, applications of SO∗(2N) in physics have
been already considered in the past. For example, it has been suggested to use SO∗(2N) as a generalized space-time
symmetry or as a dynamical algebra containing SO(3,1) [9]. In our work, we show that the intertwiner Hilbert space
provides an infinite-dimensional representation of the so∗(2N) Lie algebra, parametrized in terms of the total area.
Indeed, if the u(N) observables can be understood as transformations between intertwiners with fixed areas, the left-
over of the algebra, spanned by F˜ ,F , can be interpreted as maps between intertwiners which create or annihilate
ab ab
quanta of area. As such given a N legged intertwiner with a given total area, any other N legged intertwiner can
be obtained from it by a suitable SO∗(2N) transformation. Said otherwise, if we think of a N-legged intertwiner as
parametrizedintermsofthestatesof2N harmonicoscillators,invariantunderaglobalSU(2)transformation,thenany
otherN leggedintertwinercanbeobtainedbyasymplectomorphism(orBogoliubovtransformations)whichpreserves
the SU(2) invariance. Hence, we will show that SO∗(2N) can be seen as the subgroup of Sp(4N,R) preserving the
SU(2) invariance.
Once we have identified the Lie algebra/group, we can construct a new intertwiner coherent state (for a thorough
reviewonintertwinercoherentstatessee[10]). Notethattherearedifferentoptionstogeneralizethestandardconcept
ofcoherentstateforanharmonicoscillator. Indeedtheharmonicoscillatorcoherentstatesatisfiestwokeyproperties:
the creation operator acts diagonally on the coherent state and the Heisenberg group acts coherently on the state. It
istypicallyonlywhendealingwithHeisenberggrouplikestructuresthatwecanhavebothofthesepropertiesatonce.
To generalize the notion of coherent state to the SO∗(2N) case, we therefore have the choice: we retain any of these
properties to construct the state. The construction of coherent states which diagonalize the creation operators F˜
ab
hasbeenperformedin[6]. Thesestatesareactuallytailoredtosolvetheso-calledholomorphicsimplicityconstraints.
The other option, to keep a coherent action of the group, falls into the Gilmore-Perelomov program to construct
coherent states [11, 12]. The group SO∗(2N) being non-compact makes things a bit easier and in fact these coherent
stateswereverysuccinctlystudiedinPerelomov’sbook[12],albeitnotfortheintertwinerrepresentation. Weprovide
here the full details of their construction in a different representation than [12]. We determine the matrix elements of
the generators E ,F ,F˜ and their expectations values with respect to these states.
ab ab ab
The construction of a coherent state allows for the study of the semi-classical limit. We expect to recover a convex
polyhedronwithN faces[13]. Weshowthatthiscanbethecase, withsomeextrasubtletiesdependingonthematrix
ζ parametrizing the coherent state. This N×N matrix being antisymmetric, has a rank which is even, rank(ζ)=2k,
and clearly bounded by N. We will note λ2, α = 1,..,k the eigenvalues of ζ∗ζ. If all these eigenvalues are distinct,
α
1 Formorereferencessee[4].
2 TheoperatorsE ,F ,F˜ areinvariantundertheglobalSU(2)transformationsbutarenotself-adjointoperators. Sostrictlyspeaking
ab ab ab
therearenotobservables. Howeverwecanconstructpolynomialfunctionsoftheseoperatorswhichwillbeself-adjoint.
3
we obtain a (discrete) family of k polyhedra with N faces. In particular, if rank(ζ) = 2, we recover one polyhedron
withN facesaswecouldexpect. λ (ormoreexactlyafunctionofit)definesthetotalareaofeachofthepolyhedron
α
α. However if some λ are identical, we actually get some continuous families of polyhedra, each of the polyhedron
α
having a total area specified by λ .
α
Itisinterestingthatthecoherentstateswehaveconstructedalreadyappearedintheliterature[14–18]duetotheir
nice features to perform calculations. Note however that they were always defined in terms of a matrix ζ of rank 2,
sothatthereisnoissuewiththesemi-classicallimit. Finally, manyoftheresultspresentedhere, especiallyregarding
the construction of the coherent state, were also presented as part of the PhD thesis [19].
In Section I, we review the different parametrization of a classical convex polyhedron, introducing the classical
spinorial formalism. In Section II, we introduce the quantum version of the spinorial formalism, ie the harmonic
oscillatorsrepresentation. WereviewtheconstructionoftheU(N)coherentstates`a la Perelomov unlikewhatFreidel
andLivinedidin[2]. Wediscussinparticularthesemi-classicallimittoidentifytheclassicalspinorswhichparametrize
the semi-classical polyhedron. We will use the same approach to deal with the SO∗(2N) coherent states which we
defineinSectionIII.Wedeterminetheexpectationvaluesofthebasicobservablesandthevarianceofthe(total)area
with respect to these states. We also explain how these coherent states can be viewed as a specific class of squeezed
states. Finally, we discuss how in the semi-classical limit, we can recover a discrete family of polyhedra and/or a
continuous one, depending on the nature of coherent state.
I. POLYHEDRON PARAMETRIZATION
A polyhedron with N faces in R3 can be reconstructed from the N normals V(cid:126) ∈R3 of its faces [20] which satisfy
a
what is called the closure condition,
N
C =(cid:88)V(cid:126) =(cid:126)0. (1)
a
a=1
Kapovich and Milson [21] introduced a phase space structure on the space of polyhedra for fixed areas given by
|V(cid:126) | = V . The closure condition (1) can then be seen as a momentum map implementing global rotations. Their
a a
phase space is given by the symplectic reduction
PKM =(S2×..×S2)//SO(3), V(cid:126) =V vˆ (2)
N a a a
with Poisson bracket on S2
{Vi,Vj}=δ (cid:15)ij Vk, {V ,Vj}=0, ∀a,b. (3)
a b ab k a a b
PKM is a space with dimension 2N−6. From the loop quantum gravity perspective, it is important to also have the
N
area as a variable. One of the strengths of the so-called spinor approach is to provide such parametrization. To have
a phase space structure, one usually extends the Kapovich-Milson phase space by replacing S2 by C2 ∼R4 (cid:51)(V(cid:126),φ).
One of the extra degrees of freedom is the area (ie the norm of the vector) whereas the other3 one can be seen as
(cid:18) (cid:19)
x
a phase φ. If we note the pair of complex numbers4 which we call the spinors5, |z(cid:105) = ∈ C2, then the maps
y
between the spinors and the vector/phase variables are the following:
1
V(cid:126) = (cid:104)z|(cid:126)σ|z(cid:105), (cid:126)σ being the Pauli matrices and |V(cid:126)|=V,
2
√
eiθ (cid:18) V +V (cid:19) V +iV
|z(cid:105)= √ √ z , eiφ = z y . (4)
2 eiφ V −Vz (cid:112)V2−V2
z
Hence we see that given V(cid:126) we can reconstruct the spinor up to a phase θ. In the spinorial approach, the polyhedron
phase space [10] is now given by
Pspin =C2N//SU(2), with {z ,z }=−iδ , the other brackets being 0. (5)
N a b ab
3 Weconsideraspaceofevendimensionasotherwisewecannothaveaproperphasespace.
4 Wechangenotationwithrespecttotheusualnotationinordertoavoidtoomanyindiceslater.
(cid:18) (cid:19)
y
5 Wehavethat(cid:104)z|=(x,y)andwewillalsouse|z]= aswellas[z|=(y,−x).
−x
4
ThesymplecticreductionbySU(2)isgivenbytheclosureconstraintmomentummapexpressedinthespinorvariables
N N
(cid:88) 1(cid:88)
|z (cid:105)(cid:104)z |= (cid:104)z |z (cid:105)1. (6)
a a 2 a a
a a
One of the key advantages of the spinor formalism is that it allows to construct a closed algebra of observables [3].
We introduce the SU(2) invariant quantities
• e =(cid:104)z |z (cid:105) which changes the area of the faces a and b while keeping the total area fixed. If a=b it provides
ab a b
the value of the area of the face a.
• f˜ =[z |z (cid:105) which changes the area of the faces a and b while adding one unit to the total area.
ab a b
• f =(cid:104)z |z ] which changes the area of the faces a and b while subtracting one unit to the total area.
ab a b
Any observable built in terms of the normals V(cid:126) such as the norm |V(cid:126) | or the relative angle V(cid:126) ·V(cid:126) can be defined in
a a a b
terms of these observables.
1 1 1
|V(cid:126) |2 = e2 , V(cid:126) ·V(cid:126) = e e − e e . (7)
a 4 aa a b 2 ab ba 4 aa bb
Hence the spinor variables provide a finer parametrization of the polyhedron phase space, a parametrization which
furthermore closes in terms of the Poisson bracket, unlike the observables expressed in terms of the normals such as
V(cid:126) ·V(cid:126) .
a b
{e ,e } =−i(δ e −δ e ), {e ,f }=−i(δ f −δ f ), {f ,f }={f˜ ,f˜ }=0
ab cd cb ad ad cb ab cd ad bc ac bd ab cd ab cd
(cid:16) (cid:17)
{e ,f˜ }=−i δ f˜ −δ f˜ , {f ,f˜ }=−i(δ e +δ e −δ e −δ e ). (8)
ab cd bc ad bd ac ab cd db ca ca db cb da da cb
The observables eab form the classical version of the u(N) algebra, whereas the eab together with the fab and the f(cid:101)ab
form a so∗(2N) algebra. We will discuss in more details these structures in Section III.
II. COHERENT STATES FOR THE POLYHEDRON WITH FIXED AREA: A REVIEW
A. Harmonic oscillators and intertwiner
We consider 2N quantum harmonic oscillators (A ,B ), with the only non-zero commutators
a a
[A ,A†]=[B ,B†]=1δ , (9)
a b a b ab
which act on the Fock basis
|n ,n (cid:105) ≡|n (cid:105) ⊗|n (cid:105) , n ,n ∈N. (10)
A B HO A HO B HO A b
These harmonic oscillators are the quantum version of the spinors of Section I. The observable generators are then
obtained by quantizing directly their classical definition. We choose the symmetric ordering so that zz→A†A+ 1
2
which leads to the following quantum observables6.
Eab =A†aAb+Ba†Bb+δab1, Fab =BaAb−AaBb, F(cid:101)ab =Ba†A†b−A†aBb†. (11)
We emphasize the presence of the δ 1 term in the definition of E which is not usually present in the spinorial
ab ab
formalism where a different ordering is used. Using the harmonic oscillator commutation relations (9) allows to
recover
[Eab,Ecd]=δcbEad−δadEcb, [Eab,F(cid:101)cd]=δbcF(cid:101)ad−δbdF(cid:101)ac, [Eab,Fcd]=δadFbc−δacFbd, (12a)
[Fab,F(cid:101)cd]=δdbEca+δcaEdb−δcbEda−δdaEcb, [Fab,Fcd]=[F(cid:101)ab,F(cid:101)cd]=0, (12b)
6 Thisorderingwasalsonoticedinthefirstfootnotesof[2].
5
It is essential to use this quantization scheme in order to recover this Lie algebra structure which we will identify to
be the so∗(2N) Lie algebra.
It will prove useful to also introduce the notation7
Eα :=αabEab, F(cid:101)ζ :=ζabF(cid:101)ab, Fζ :=ζabFab, α,ζ ∈MN(C), (13)
These elements satisfy the commutation relations
[Eα,Eβ]=E[α,β], [Eα,F(cid:101)ζ]=F(cid:101)αζ+ζαt, [Eα,Fζ]=−Fα∗ζ+ζα, [Fw,F(cid:101)ζ]=E(ζ−ζt)(w−wt)∗. (14)
The action of observable generators on the intertwiner follows from the Schwinger-Jordan representation of su(2)
representations. Explicitly, we realize an intertwiner in terms of the harmonic oscillator representations, which allows
in turns to have an action of observable generators on the intertwiner space.
The su(2) generators are realized in terms of harmonic oscillators as
1
J = (A†A−B†B), J =A†B, J =B†A, (15)
z 2 + −
while the su(2) irreps are
|j,m(cid:105)=|j+m,j−m(cid:105) =|n ,n (cid:105) , m∈{−j,..,j}. (16)
HO A B HO
One can easily check that the Casimir can be expressed in terms of the E operator.
J2 = 1(E−1)(E+1), E :=A†A+B†B+1, (17)
4
with
E|j,m(cid:105)=(2j+1)|j,m(cid:105), (18)
thatis,insomesense,E provides(almost)asquarerootoftheCasimir. Weextendthisconstructiontotheintertwiner
space as follows. We denote by Inv (H ⊗···⊗H ) the set of SU(2) invariant vectors in the tensor product of
SU(2) j1 jN
N SU(2) irreducible unitary representations, that is those that are annihilated by the total angular momentum
N
J(cid:126):=(cid:88)J(cid:126)(a), (19)
a=1
which we can identify with N-legged intertwiners. We then introduce the Jordan-Schwinger representation for each
leg, i.e., we use 2N harmonic oscillators8
J(a) = 1(cid:0)A†A −B†B (cid:1), J(a) =A†B , J(a) =B†B . (20)
z 2 a a a a + a a − a a
These vector operators can be seen as the quantization of the polyhedron normals V(cid:126) .
a
The E satisfy the commutation relations
ab
[E ,E ]=δ E −δ E , (21)
ab cd cb ad ad cb
which are those of a u(n)C algebra. These operators can be used to construct all the usual LQG observables, namely
J(cid:126)(a)·J(cid:126)(b) ≡2A A −A A −(1−2δ )A , (22)
ab ba a b ab a
where
1
A := (E −δ 1), A :=A . (23)
ab 2 ab ab a aa
We are going to interpret the eigenvalues of the operator A
a
A |j ,m (cid:105)=j |j ,m (cid:105) (24)
a a a a a a
(cid:80)
as the area associated to the leg a, hence we will refer to the A ’s as area operators. The operator A:= A gives
a a a
the total area of the intertwiner.
7 Weusethecomplexconjugateofζ inFζ toensurethat(Fζ)† =F(cid:101)ζ,whichwillhappenwhenthe(SO∗(2N))representationisunitary
asweshallseelater.
8 ItisimplicitlyassumedthattheoperatorswithsubscriptaonlyactonHja.
6
B. Intertwiner as U(N) representation
It was shown in [1] that the space of intertwiners with a fixed total area9 J ∈N
(cid:77)
HJ = Inv (V ⊗···⊗V ) (25)
N SU(2) j1 jN
(cid:80)aja=J
has the structure of an irreducible unitary representation of U(N), whose infinitesimal action is given by the E
ab
operators we defined10. Explicitly,
HJ ≡[J +1,J +1,1,...,1], (26)
N
where the [λ ,λ ,...,λ ], with
1 2 N
λ ≥λ ≥···≥λ ≥0, (27)
1 2 N
denotes the U(N) representation with highest weight vector |λ(cid:105), for which
E |λ(cid:105)=λ |λ(cid:105) and E |λ(cid:105)=0, ∀a<b. (28)
aa a ab
This particular choice of λ’s is required for the SU(2) invariance. The dimension of U(N) representations can be
computed with the hook-length formula [22]
dim[λ ,...,λ ]= (cid:89) λa−λb+b−a, (29)
1 N b−a
a<b
which in our specific case gives
(cid:18) (cid:19)(cid:18) (cid:19)
λ −λ +1 λ +N −2 λ +N −3
dim[λ ,λ ,1,...,1]= 1 2 1 2 , (30)
1 2 λ λ −1 λ −1
1 1 2
so that
(cid:18) (cid:19)(cid:18) (cid:19)
1 J +N −1 J +N −2 (N +J −1)!(N +J −2)!
dimHJ = = , (31)
N J +1 J J J!(J +1)!(N −1)!(N −2)!
which is indeed the dimension of the space of N-legged intertwiners with fixed total area.
C. U(N) coherent states
WewillnowrevisittheconstructionofU(N)coherentstatesfortheintertwinerrepresentation,originallypresented
in [2].
1. U(N) coherent states `a la Perelomov
Working with the representation HJ , we will use the highest weight vector (the N legged intertwiner where only 2
N
legs have a non-zero area)
1
|ψJ(cid:105):= (cid:112) F(cid:101)1J2|0(cid:105) (32)
J!(J +1)!
as our fixed state. One can easily check that this is indeed the highest weight, i.e.,
E |ψ (cid:105)=0, ∀a<b. (33)
ab J
9 Thefactthatthetotalareamustbeanintegerfollowsfromtheselectionrulesoftheadditionofangularmomenta.
10 We recall that our definition differs from that of [1], namely our E have an additional δ term, which as we will see is essential to
ab ab
constructtheSO∗(2N)representation.
7
The isotropy subgroup of |ψ (cid:105) is given by U(2)×U(N −2), so that, following Perelomov, the coherent states are
J
going to labelled by elements of the quotient space
U(N)
, (34)
U(2)×U(N −2)
which is isomorphic to the Grassmannian
Gr (Cn)={ξ ∈so(N,C)|rank(ξ)=2}/∼ where ξ ∼χ ⇔ ξ =λχ, 0(cid:54)=λ∈C. (35)
2
U(N) acts on the equivalence classes [ξ]∈Gr (CN) as
2
g(cid:46)[ξ]=(cid:2)gξgt(cid:3). (36)
The equivalence class with representative
(cid:18) (cid:19) (cid:18) (cid:19)
σ 0 0 −1
ξ = , with σ = (37)
0 0 0 1 0
satisfies
g(cid:46)[ξ ]=[ξ ] ⇔ g ∈U(2)×U(N −2). (38)
0 0
For every ξ there is a (non-unique) unitary matrix, which we will denote by g , such that
[ξ]
(cid:18) (cid:19)
σ 0
ξ =λg gt (39)
[ξ] 0 0 [ξ]
forsomeλ. Thenotationg isconsistentsinceforeachχ∈[ξ]wecanusethesameunitarymatrixinthefactorisation.
[ξ]
We then have
(cid:104) (cid:105)
[ξ]= g ξ gt =g (cid:46)[ξ ]. (40)
[ξ] 0 [ξ] [ξ] 0
We are now in a position to define the coherent states. For each [ξ]∈Gr (CN) we define the state
2
(cid:18)1 (cid:19)J (cid:0)1tr(ξ∗ξ)(cid:1)−J2
|J,ξ(cid:105)=NJ(ξ) 2F(cid:101)ξ |0(cid:105), NJ(ξ)= (cid:112)2 J!(J +1)! , (41)
which one can check to be normalised to 1, see the end of Appendix C3. Note that the state does not depend on the
representative ξ, as
λJ
|J,λξ(cid:105)= |J,ξ(cid:105)=eiθ(λ)|J,ξ(cid:105), ∀λ(cid:54)=0. (42)
|λ|J
Moreover, we have
|J,ξ (cid:105)≡|ψ (cid:105). (43)
0 J
To show that these states are indeed Perelomov coherent states, we have to show that they arise from the action of
the group on the state |ψ (cid:105). To do so, we are going to show a more general result: instead of showing the coherence
J
under the group U(N), we are going to show the coherence under GL(N,C) which contains U(N) as a subgroup.
Proposition 1. [2] The action of GL(N,C)∼=U(N)C on the highest weight vector |ψJ(cid:105) is
det(g) (cid:18)1 (cid:19)J
g|ψJ(cid:105)= (cid:112)J!(J +1)! 2F(cid:101)gξ0gt |0(cid:105).
For the proof see Appendix B. It follows in particular that
det(g ) (cid:18)1 (cid:19)J det(g ) (cid:18)1 (cid:19)J
g[ξ]|ψJ(cid:105)= (cid:112)J!(J[+ξ]1!) 2F(cid:101)g[ξ]ξ0g[tξ] |0(cid:105)= (cid:112)J!(J[+ξ]1!)λ−J 2F(cid:101)ξ |0(cid:105) (44)
where
1
|λ|2 = tr(ζ∗ζ), (45)
2
that is
|J,ξ(cid:105)=eiθ(ξ)g |ψ (cid:105). (46)
[ξ] J
The coherence under U(N), up to a phase, follows then naturally.
8
2. Matrix elements and semi-classical limit
We will compute the matrix elements of the u(n)C generators in the coherent state basis following the procedure
used in [2, 10]. Let |J,ξ) be the unnormalised coherent state
1
|J,ξ)= |J,ξ(cid:105). (47)
N (ξ)
J
We know from the proof of (1) that, for any α∈M (C),
n
(cid:20)1 (cid:16) (cid:17)(cid:21)J
(J,η|eEα|J,ξ)=etr(α)(J,η|J,eαξeαt)=J!(J +1)!etr(α) tr η∗eαξeαt , (48)
2
which we can use to find
(J,η|E |J,ξ)= d (cid:8)(J,η|eθEα|J,ξ)(cid:9) =J!(J +1)! d (cid:40)eθtr(α)(cid:20)1tr(cid:16)η∗eθαξeθαt(cid:17)(cid:21)J(cid:41) . (49)
α dθ θ=0 dθ 2
θ=0
Computing the derivative we find that
tr(η∗αξ)
(cid:104)J,η|E |J,ξ(cid:105)=N (η)N (ξ)(J,η|E |J,ξ)=(cid:104)J,η|J,ξ(cid:105)tr(α)+2J(cid:104)J −1,η|J −1,ξ(cid:105) . (50)
α J J α (cid:112)
tr(η∗η)tr(ξ∗ξ)
In particular, choosing α=∆ we get
ab
(ξ∗ξ)
(cid:104)E (cid:105)=(cid:104)J,ξ|E |J,ξ(cid:105)=δ +2J ab. (51)
ab ab ab tr(ξ∗ξ)
This expression can be simplified using the fact that any rank-2 complex anti-symmetric matrix ξ can be written as
(cid:114) (cid:18) (cid:19)
1 0 −1
ξ =λUMUt, with M =λσ⊕0 , λ= tr(ξ∗ξ), σ = (52)
N−2 2 1 0
and where U ∈U(N) is a unitary matrix. We can then introduce the 2N spinors
√ (cid:18)U (cid:19)
|z (cid:105)= J a1 (53)
a U
a2
satisfying by construction
(cid:88) (cid:88)1 (cid:88)
|z (cid:105)(cid:104)z |= (cid:104)z |z (cid:105)1 , (cid:104)z |z (cid:105)=2J. (54)
a a 2 a a 2 a a
a a a
Hence from their definition, the closure constraint is satisfied. Note that as the matrix ξ has rank 2 and is antisym-
metric, the equivalence class [ξ] can be parametrized in terms of N spinors in many different ways (all related to each
other by GL(2,C) transformations). These spinors will however not necessarily satisfy the closure constraint. This
parametrization of ξ was used a lot in [2] for doing calculations; in particular, it was discussed how some SL(2C)
transformationcanbeusedtogetthemtoclose. Weemphasizethatthesespinorshavenothingtodowiththespinors
thatweusedtodefinethesemi-classicallimit. Thesemi-classicalspinorsweobtaineddosatisfytheclosureconstraint,
which is expected since after all we are dealing with an intertwiner or a polyhedron, hence an object invariant under
the global SU(2) transformations.
In terms of the spinors we have
(cid:104)J,ξ|E |J,ξ(cid:105)=δ +(cid:104)z |z (cid:105). (55)
ab ab a b
In a similar fashion, we can compute
(cid:104)J,ξ|EαEβ|J,ξ(cid:105)= ddθ ddϕ (cid:8)(cid:104)J,ξ|eθEαeϕEβ|J,ξ(cid:105)(cid:9)θ=0,ϕ=0 (56)
9
to find variances and covariances. We will concentrate on the area operators
1
A = (E −1), (57)
a 2 aa
for which we find
δ 1 1
Cov(A ,A )=(cid:104)A A (cid:105)−(cid:104)A (cid:105)(cid:104)A (cid:105)= ab(cid:104)z |z (cid:105)+ (cid:104)z |z ][z |z (cid:105)− (cid:104)z |z (cid:105)(cid:104)z |z (cid:105) (58)
a b a b a b 4 a a 4J b a a b 4J a a b b
and
1 1
Var(A )=Cov(A ,A )= (cid:104)z |z (cid:105)− (cid:104)z |z (cid:105)2. (59)
a a a 4 a a 4J a a
√
Note that both (cid:104)A (cid:105) and Var(A ) are of order 1 in J, so that the coefficient of variation Var(Aa) approaches 0 when
a a (cid:104)Aa(cid:105)
the total area J is large. We can thus think of the coherent state |J,ζ(cid:105) as being peaked, in the large J limit, on the
classical geometry obtained by introducing the vectors
1
V(cid:126)a = (cid:104)z |(cid:126)σ|z (cid:105); (60)
2 a a
these satisfy
(cid:88)V(cid:126)a =0, |V(cid:126)a|=(cid:104)A (cid:105), (61)
a
a
so we can think of them as the normal vectors to a polyhedron with N faces f , with area(f ) = (cid:104)A (cid:105) and total
a a a
surface area J =(cid:104)A(cid:105). Note that our spinors are not unique: the unitary matrix appearing in (52) is defined up to a
transformation
(cid:18) (cid:19)
X 0
U →UV, V = , X ∈SU(2), Y ∈U(N −2); (62)
0 Y
under the same transformation, the spinors change as
|z (cid:105)→Xt|z (cid:105), (63)
a a
while the vectors undergo a global SO(3) rotation. These are the natural symmetries of the polyhedron, which is
defined only up to a global rotation.
III. A NEW COHERENT STATE FOR THE SU(2) INTERTWINER
As we have seen in Section I, the actual algebra of observables given in (8) is bigger than u(N). The usual
parametrization of the u(N) generators in the spinorial formalism does not contain the identity. By redefining these
generatorstoincludetheidentitywecanidentifythecommutationrelationsofso(2N,C). Wethenneedtoidentifythe
real form of this algebra which contain u(N). Thankfully, there is only one candidate given by so∗(2N) [8]. This Lie
algebra and its associated (non-compact) Lie group have not been studied much. For example the full representation
theory is not known to the best of our knowledge. As we are going to see in Section IIIB, the intertwiner space
providesaninfinite-dimensionalrepresentationofSO∗(2N),thankstotherealizationintermsofharmonicoscillators.
Afterhavingidentifiedthestructureofthealgebraofobservableswecanproceedinconstructingthecoherentstates
`a la Perelomov, study some of their properties and check their semi-classical limit. Note that we can also construct
different coherent states, not of the Perelomov type, by requiring not their coherence under the group action, but
insteadthe”creationoperators”F(cid:101)ab toactdiagonallyonthem[6]. Suchstatesallowtosolvethesimplicityconstraints
to build some 4d (Euclidian) holomorphic spin foam model [6].
A. The Lie group SO∗(2N) and its Lie algebra so∗(2N)
WesummarizesomeofthefeaturesoftheLiegroupSO∗(2N)andofitsLiealgebrathatwillbeusefultoconstruct
the Perelomov coherent states.
10
1. The Lie group SO∗(2N)
RecallthatSU(N,N)isthegroupofcomplexmatriceswithdeterminant1preservingtheindefiniteHermitianform
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
1 0 1 0
SU(N,N)= g ∈SL(2N,C), g∗ N g = N . (64)
0 −1 0 −1
N N
The non-compact Lie group G=SO∗(2N) is a subgroup of SU(N,N) such that
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19)(cid:27)
0 1 0 1
SO∗(2N)= g ∈SU(N,N), gt N g = N . (65)
1 0 1 0
N N
Elements of SO∗(2N) can be parametrised as 2×2 block matrices [12].
(cid:18) (cid:19)
A B
g = , A,B ∈M (C), with det(A)(cid:54)=0. (66)
−B A N
and
AA∗−BB∗ =1, A∗A−BtB =1, A∗B =−BtA, BAt =−ABt, (67)
and with inverse
(cid:18) A∗ Bt(cid:19)
g−1 = . (68)
−B∗ At
The maximal compact subgroup K ⊆SO∗(2N) is isomorphic to U(N), and is given by the elements of the form
(cid:18) (cid:19)
U 0
, U ∈U(N). (69)
0 U
The group is non-compact for all N ≥2, while SO∗(2)∼=U(1).
2. The Lie algebra so∗(2N)
The Lie algebra of SO∗(2N) is
(cid:26) (cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
0 1 0 1
so∗(2N)= V ∈su(N,N), Vt N =− N V . (70)
1 0 1 0
N N
Its elements are parametrised by 2×2 block matrices
(cid:18) (cid:19)
X Y
V = , X,Y ∈M (C), with X∗ =−X, Yt =−Y. (71)
−Y X N
Hence dimso∗(2N)=N(2N −1). A basis for so∗(2N)C ∼=so(2N,C) is given by the matrices
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
∆ 0 0 0 0 ∆ −∆
Eab = 0ab −∆ , Fab = ∆ −∆ 0 , F(cid:101)ab = 0 ab 0 ba , (72)
ba ab ba
where a,b=1,...,n and ∆ ∈M (C) is the matrix with entries
ab N
(∆ ) =δ δ . (73)
ab cd ac bd
The E matrices span the complexification of the subalgebra u(N). The commutation relations of the so∗(2N)
ab
complexified generators are (cf (8))
[Eab,Ecd]=δcbEad−δadEcb, [Eab,F(cid:101)cd]=δbcF(cid:101)ad−δbdF(cid:101)ac, [Eab,Fcd]=δadFbc−δacFbd, (74a)
[Fab,F(cid:101)cd]=δdbEca+δcaEdb−δcbEda−δdaEcb, [Fab,Fcd]=[F(cid:101)ab,F(cid:101)cd]=0, (74b)
and unitary representations are those for which
Ea†b =Eba, Fa†b =F(cid:101)ab. (75)