# Loop Quantization of a 3D AbelianBF Model with σ-Model Matter

Authors: Diego C. M. Mendonça; Olivier Piguet
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

The main goal of this work is to explore the symmetries and develop the dynamics associated with a 3D AbelianBF model coupled to scalar fields submitted to a sigma model like constraint, at the classical and quantum levels. Background independence, on which the model is founded, strongly constrains its nature. We adapt to the present model the techniques of Loop Quantum Gravity in order to construct its physical Hilbert space and its observables.


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## Body

## 1. Introduction

The now quasi-hundred-year-old General Relativity as a theory of gravitation, despite its tremendous successes in accounting for predicting phenomena, still lacks a quantum version. Previous perturbative attempts have shown the nonrenormalizability of the theory [1], whereas the pioneering nonperturbative approach of Wheeler and DeWitt [2–5] had its successes concentrated in reduced “minisuperspace” models dedicated to cosmology. However, very important progresses have been made in the last decades, especially in the framework of Loop Quantum Gravity (LQG) [6–9], based on the canonical Hamiltonian approach of Dirac [10, 11] applied to the Ashtekar-Barbero [12–14] parametrization of the theory. General Relativity, as a background independent theory—in the sense that no background geometry is givena priori, geometry being dynamical—is a fully constrained theory, its Hamiltonian being merely a sum of constraints generating the gauge invariances of the theory. The LQG program entails the difficult task of implementing the constraints of the theory as quantum operators in some predefined kinematic Hilbert space and to solve them, thus leaving as a subspace the physical Hilbert space in which act the self-adjoint operators representing the observables of the theory. Some of the constraints have been resolved, but the last one, the so-called scalar constraint. The latter has resisted up to now a complete solution, the most popular approach being that of “spin foams” [15, 16].By contrast, the lower-dimensional gravitation theories are much more easy to handle, since they can be described as topological gauge theories, when not coupled to matter [17–23]. Coupling them to matter however lets them lose their topological character, excepted in some special cases, where a complete and rather simple loop quantization can be achieved [24, 25].The purpose of this paper is to present the loop quantization of a background independent theory of theBF type [26, 27] with the Abelian group U(1) as a gauge group. Background independence means that no metric is introduced in the manifold on which the theory is defined, the sole requirement being the invariance under diffeomorphisms, beyond the gauge invariance. The BF fields are coupled to a complex scalar “matter” field subject to a σ-model type of constraint. Background independence severely restricts the form of the action. It turns out that the topological nature of the BF theory persists in the sense that no local degrees of freedom are present. However, provided the topology of 2-dimensional space is nontrivial, global degrees of freedom are present in the theory. We consider spaces with point-like singularities, in which cases a nontrivial physical Hilbert space and global observables are explicitly constructed. A non-Abelian version is presently under study [25].The model and its gauge invariances are presented in Section2, its classical analysis is done in Section 3 together with the separation of the first and second class constraints and the definition of the Dirac brackets, and the quantization is presented in Section 4. Brief conclusions are given at last.


## 2. Formulation of the Model

### 2.1. The Gauge Invariances and the Action

The field content of the model is aU(1) connection form A=Aμ(x)dxμ, a “B” form B=Bμ(x)dxμ, a complex scalar field ϕ(x), and a 3-form field e=(1/3!)eμνρdxμdxνdxρ, transforming as (Wedge symbols ∧ are not written explicitly; space-time indices μ,ν,… take the values 0, 1, and 2; later on, space indices will be denoted by the letters a,b,… taking the values 1, 2; Aμ and θ are taken as imaginary)
(1)Aμ′=Aμ′+g-1∂μg=Aμ+∂μθ,B′=B,ϕ′=gϕ,ϕ-′=g-1ϕ-,e′=e,
under U(1) gauge transformations (Aμ and θ are taken as purely imaginary; Bμ is real) g(x)=exp⁡⁡θ(x).The most general action, invariant under the gauge transformations (1) and background independent, reads
(2)Sgeneral=∫M3K(ϕ-ϕ)BF+λ(ϕ-ϕ)BDϕ-Dϕ+eμϕ-ϕ-R,
where K, λ, and μ are arbitrary real functions of ϕ-ϕ, assumed to be analytic in their argument and to fulfill the conditions
(3)K(0)≠0λ(0)≠0μ(0)=0;F=dA, and D denotes the covariant derivative (covariant with respect to the gauge transformation (1)):
(4)Dϕ=(d-A)ϕ,Dϕ-=(d+A)ϕ-.
The integration is performed over some 3-dimensional differential manifold M3. The action is obviously invariant under the diffeomorphisms of M3. Having no metric at our disposal, it is clear that no other term, such as,for example, a potential term, can be added.The parameterR can be taken equal to 1 through a renormalization of the field e, and one easily shows that one can reduce the function μ(ϕ-ϕ) to the form μ=ϕ-ϕ through a field redefinition ϕ→ϕ′(ϕ,ϕ-)=ϕf(ϕ-ϕ), with f(ϕ-ϕ)=μ(ϕ-ϕ)/ϕ-ϕ, compatible with the gauge invariance (1). Having done this, we have the field equation ϕ-ϕ=1 which implies that the functions K and λ can be replaced by constants. One of them, let us say λ, can be given the value 1 through a renormalization of the field B. We are thus left with only the constant K as independent parameter. We will take the valued K=1 without loss of generality. The final action is then
(5)S=∫M3BF+BDϕ-Dϕ+e(ϕ-ϕ-1).
One recognizes in (5) a BF action coupled with scalar fields and a Lagrange multiplier field e assuring the σ-model type constraint ϕ-ϕ=1.It turns out that this action (5) has two more invariances under the gauge transformations
(6)A′=A,B′=B+dψ,ϕ′=ϕ,e′=e-dψF,(7)A′=A+dη,B′=B,ϕ′=ϕ,e′=e+dηdB,
where the scalars ψ(x) and η(x) are the transformation parameters. The transformation (6) coincides with the usual topological type transformations of the BF model, in the absence of the fields ϕ and e. Invariance under (7) is specific for the model.In order to check the invariances of the action (up to boundary terms), as well as for all the manipulations involving partial integrations, it is useful to remember that the covariant derivativeD, defined by DX=dX-qAX, where q is the U(1) charge of the field X, obeys the Leibniz rule. The respective U(1) charges of the basic fields A, B, ϕ, ϕ-, and e are 0,0,1,-1, and 0. Let us also note the useful identity
(8)Dϕ-Dϕ=dϕ-dϕ+Ad(ϕ-ϕ).
The field equations read
(9)δSδB=F+Dϕ-Dϕ=*0,δSδA=dB-Bd(ϕ-ϕ)=*0,δSδϕ-=BFϕ-dBDϕ+eϕ=*0,δSδϕ=BFϕ--dBDϕ-+eϕ-=*0,δSδe=ϕ-ϕ-1=*0,
where the symbol =* means “on shell” equality, that is, “equations of motion being fulfilled.” The last equation is equivalent to
(10)ϕx=*eiφx,ϕ-x=*e-iφx,φarealphase.
This system of equations is equivalent to the simpler one:
(11)F=*0,dB=*0,e=*0,ϕ-ϕ-1=*0.


### 2.2. Diffeomorphism Invariance

In the present theory, like in the topological theories of the Chern-Simons orBF type, the invariance under the diffeomorphisms is a consequence of the invariance under the gauge transformations (1), (6), and (7), up to field equations. Indeed, with the diffeomorphisms being generated by the Lie derivative Lξ=iξd+diξ along an infinitesimal vector field ξ when acting on forms (iξ is the interior derivative, with iξdxμ=ξμ), one checks that
(12)LξA=*d(iξA),LξB=d(iξB),Lξϕ=*iiξdφϕ,Lξϕ-=*-iiξdφϕ-,Lξe=*0,
where φ is the phase of the field ϕ defined in (10). One sees that these infinitesimal diffeomorphisms are given, on-shell, by a combination of the three gauge invariances, with the respective field dependent infinitesimal parameters given by
(13)θ=iiξdφ,ψ=iξB,η=iξ(A-idφ).


## 2.1. The Gauge Invariances and the Action

The field content of the model is aU(1) connection form A=Aμ(x)dxμ, a “B” form B=Bμ(x)dxμ, a complex scalar field ϕ(x), and a 3-form field e=(1/3!)eμνρdxμdxνdxρ, transforming as (Wedge symbols ∧ are not written explicitly; space-time indices μ,ν,… take the values 0, 1, and 2; later on, space indices will be denoted by the letters a,b,… taking the values 1, 2; Aμ and θ are taken as imaginary)
(1)Aμ′=Aμ′+g-1∂μg=Aμ+∂μθ,B′=B,ϕ′=gϕ,ϕ-′=g-1ϕ-,e′=e,
under U(1) gauge transformations (Aμ and θ are taken as purely imaginary; Bμ is real) g(x)=exp⁡⁡θ(x).The most general action, invariant under the gauge transformations (1) and background independent, reads
(2)Sgeneral=∫M3K(ϕ-ϕ)BF+λ(ϕ-ϕ)BDϕ-Dϕ+eμϕ-ϕ-R,
where K, λ, and μ are arbitrary real functions of ϕ-ϕ, assumed to be analytic in their argument and to fulfill the conditions
(3)K(0)≠0λ(0)≠0μ(0)=0;F=dA, and D denotes the covariant derivative (covariant with respect to the gauge transformation (1)):
(4)Dϕ=(d-A)ϕ,Dϕ-=(d+A)ϕ-.
The integration is performed over some 3-dimensional differential manifold M3. The action is obviously invariant under the diffeomorphisms of M3. Having no metric at our disposal, it is clear that no other term, such as,for example, a potential term, can be added.The parameterR can be taken equal to 1 through a renormalization of the field e, and one easily shows that one can reduce the function μ(ϕ-ϕ) to the form μ=ϕ-ϕ through a field redefinition ϕ→ϕ′(ϕ,ϕ-)=ϕf(ϕ-ϕ), with f(ϕ-ϕ)=μ(ϕ-ϕ)/ϕ-ϕ, compatible with the gauge invariance (1). Having done this, we have the field equation ϕ-ϕ=1 which implies that the functions K and λ can be replaced by constants. One of them, let us say λ, can be given the value 1 through a renormalization of the field B. We are thus left with only the constant K as independent parameter. We will take the valued K=1 without loss of generality. The final action is then
(5)S=∫M3BF+BDϕ-Dϕ+e(ϕ-ϕ-1).
One recognizes in (5) a BF action coupled with scalar fields and a Lagrange multiplier field e assuring the σ-model type constraint ϕ-ϕ=1.It turns out that this action (5) has two more invariances under the gauge transformations
(6)A′=A,B′=B+dψ,ϕ′=ϕ,e′=e-dψF,(7)A′=A+dη,B′=B,ϕ′=ϕ,e′=e+dηdB,
where the scalars ψ(x) and η(x) are the transformation parameters. The transformation (6) coincides with the usual topological type transformations of the BF model, in the absence of the fields ϕ and e. Invariance under (7) is specific for the model.In order to check the invariances of the action (up to boundary terms), as well as for all the manipulations involving partial integrations, it is useful to remember that the covariant derivativeD, defined by DX=dX-qAX, where q is the U(1) charge of the field X, obeys the Leibniz rule. The respective U(1) charges of the basic fields A, B, ϕ, ϕ-, and e are 0,0,1,-1, and 0. Let us also note the useful identity
(8)Dϕ-Dϕ=dϕ-dϕ+Ad(ϕ-ϕ).
The field equations read
(9)δSδB=F+Dϕ-Dϕ=*0,δSδA=dB-Bd(ϕ-ϕ)=*0,δSδϕ-=BFϕ-dBDϕ+eϕ=*0,δSδϕ=BFϕ--dBDϕ-+eϕ-=*0,δSδe=ϕ-ϕ-1=*0,
where the symbol =* means “on shell” equality, that is, “equations of motion being fulfilled.” The last equation is equivalent to
(10)ϕx=*eiφx,ϕ-x=*e-iφx,φarealphase.
This system of equations is equivalent to the simpler one:
(11)F=*0,dB=*0,e=*0,ϕ-ϕ-1=*0.


## 2.2. Diffeomorphism Invariance

In the present theory, like in the topological theories of the Chern-Simons orBF type, the invariance under the diffeomorphisms is a consequence of the invariance under the gauge transformations (1), (6), and (7), up to field equations. Indeed, with the diffeomorphisms being generated by the Lie derivative Lξ=iξd+diξ along an infinitesimal vector field ξ when acting on forms (iξ is the interior derivative, with iξdxμ=ξμ), one checks that
(12)LξA=*d(iξA),LξB=d(iξB),Lξϕ=*iiξdφϕ,Lξϕ-=*-iiξdφϕ-,Lξe=*0,
where φ is the phase of the field ϕ defined in (10). One sees that these infinitesimal diffeomorphisms are given, on-shell, by a combination of the three gauge invariances, with the respective field dependent infinitesimal parameters given by
(13)θ=iiξdφ,ψ=iξB,η=iξ(A-idφ).


## 3. Hamiltonian Analysis and Constraints

We apply here the canonical formalism of Dirac [10, 11] for systems with constraints. Supposing that the space-time manifold admits a “time” × “space” foliation M3=R×Σ, where the space slice Σ is some two-dimensional manifold, we first rewrite the action as the time integral
(14)S=∫dtL(A,A˙,B,B˙,ϕ,ϕ˙,ϕ-,ϕ-˙,e,e˙)
of a Lagrangian function
(15)L(A,A˙,B,B˙,ϕ,ϕ˙,ϕ-,ϕ-˙,e,e˙)=∫Σd2xB~a∂tAa-B~aDaϕ-∂tϕ+B~aDaϕ∂tϕ-+AtC1+BtC2+e~C5,
where
(16)C1=∂aB~a+B~a∂aϕ-ϕ,C2=F~+εabDaϕ-Dbϕ,C5=ϕ-ϕ-1,F~=12εabFab,B~a=εabBb,e~=13!εμνρeμνρ.
Following the canonical procedure, we identify the conjugate momenta of each field X, ΠX=δL/δX˙:
(17)ΠAt=0,ΠBt=0,Πe~=0,Πϕ=-B~aDaϕ-,Πϕ-=B~aDaϕ,ΠAa=B~a,ΠBa=0,
satisfying together with the X’s the equal time Poisson bracket relations
(18)Xαx,ΠXβy=δβαδ2x,y,Xαx,Xβy=ΠXαx,ΠXβy=0,
where the indices α, β run over all components of all fields. The Legendre transform Hc=-L+∑α‍∫d2xΠXαX˙α yields the canonical Hamiltonian
(19)Hc=-∫d2xAtC1+BtC2+e~C5,
with the C’s given in (16).Noting that the velocities do not appear in any of (17) for the momenta, we conclude that all of these equations are (primary) constraints [10, 11]. The equality sign must be replaced by the “weak equality” sign ≈, meaning that the constraints are solved at the end, after all calculations involving Poisson brackets are done. We remark that the last two constraints in (17) are second class, their brackets being nonzero: {ΠAa(x)-εabBb(x),ΠBc(y)}=εabδ2(x,y). These constraints can be solved as strong equalities
(20)ΠAa=εabBb=B~a,ΠBa=0,
provided the Poisson brackets are replaced by the corresponding Dirac brackets, which read
(21)Aax,B~by=δabδ2(x,y),
the other brackets being left unchanged. We use the same notation {·,·} for these Dirac brackets.We are left with the five constraints(22)ΠAt≈0,ΠBt≈0,Πe~≈0,(23)C3x=Πϕ+B~aDaϕ-≈0,C4(x)=Πϕ--B~aDaϕ≈0.
The stability of the three constraints (22) under the Hamiltonian evolution requires the three secondary constraints
(24)C1(x)≈0,C2(x)≈0,C5(x)≈0,
with C1, C2, and C5 as given in (16). It will turn out convenient to replace C1 by the equivalent constraint:
(25)C1′(x)≈0,withC1′x=C1-ϕC3+ϕ-C4=∂aB~a-ϕΠϕ+ϕ-Πϕ-.
The constraints (22) can be put strongly to zero, the corresponding fields At, Bt, and e~ playing now the roles of Lagrange multipliers λ1, λ2, and λ5. Introducing also Lagrange multipliers fields for the primary constraints (23), we define the total Hamiltonian as
(26)HT=∑m=15‍Cmλm,
where we have defined the functionals
(27)Cmλm=∫‍d2xλm(x)Cm(x),
considering the Lagrangian multiplier fields λm as smooth test functions.Since this Hamiltonian is entirely made of constraints—a characteristic of theories with general covariance—the stability of our five constraintsCα, α=1,…,5, boils down to examine the matrix Mmn(x,y)≈{Cm(x),Cn(y)} of their Poisson brackets, written up to constraints, hence the ≈ sign. Indeed, their stability condition reads (summation convention is assumed)
(28)C˙m=Cm,HT=Mmnλn=0.
This provides a system of equations for the λ’s; which can be solved for some of the λ’s in terms of the remaining ones. The matrix M reads
(29)M=00000000ϕC20000-(ϕC1′+C3)00-ϕC2(ϕC1′+C3)0-ϕ000ϕ0×δ2(x-y),
where we have substituted the constraint C3 with the equivalent one
(30)C3′=ϕC3-ϕ-C4.
One sees that the first three constraints, C1, C2, and C3′, are first class;that is, their Poisson brackets with any other constraint are constraints: they generate three gauge invariances of the theory. The last two, namely, C4 and C5, however are second class. Indeed, denoting them by χp (p=1,2), their Poisson brackets form the matrix Cpq of nonvanishing determinant on the constraint surface:
(31)C=0-ϕϕ0.
These second class constraints may be written as strong equalities, provided the Poisson brackets are substituted by the Dirac brackets [10, 11]
(32)X,YD=X,Y-∑p,q‍X,χpC-1pqχq,Y.
The second class constraints χp can be solved for ϕ- and Πϕ-  in terms of the now independent fields Aa, B~a, ϕ, and Πϕ:
(33)ϕ-=1ϕ,Πϕ-=B~aDaϕ.
The independent fields obey the Dirac bracket relations
(34)Aax,B~byD=δabδ2(x-y),ϕx,ΠϕyD=δ2(x-y),Aax,ΠϕyD=Da1ϕδ2(x-y),B~ax,ΠϕyD=-B~a1ϕδ2(x-y),(otherbracketsvanishing).
This system can be diagonalized through the redefinition
(35)Π=Πϕ-B~aDa1ϕ,
with the result
(36)Aax,B~byD=δabδ2(x-y),ϕx,ΠyD=δ2(x-y),(otherbracketsvanishing).
Finally, the remaining three constraints read, taking (35) into account,
(37)C1=∂aB~a,C2=F~,C3=ΦΠ.
They are first class (their Dirac brackets are indeed zero) and generate the three gauge invariances defined by δiX={X,Ci[ϵi]} (i=1,2,3) using the functional notation (27):
(38)δ1Aa=-∂aϵ1,δ2Aa=0,δ3Aa=0,δ1B~a=0,δ2B~a=-ϵab∂bϵ2,δ3B~a=0,δ1ϕ=0,δ2ϕ=0,δ3ϕ=ϵ3ϕ,δ1Π=0,δ2Π=0,δ3Π=-ϵ3Π.
We see that the U(1) gauge invariance is split in two invariances generated by C1 and C3, corresponding to the invariances (1) and (7) of the Lagrangian formalism. The invariance generated by C2 corresponds to the topological type invariance (6).


## 4. Quantization

### 4.1. Kinematical Hilbert Space

The constraintsC1 and C3 will be solved at the quantum level in this section, whereas the last one, C2, is left for the next section. Following the lines of Loop Quantum Gravity [6–9], we will construct a kinematical Hilbert space Hkin whose vectors Ψ are subjected to the constraints C1 and C3 in the forms C^1Ψ=0 and C^3Ψ=0, where C^i are operators representing the classical Ci. Choosing the fields Aa and ϕ as configuration space coordinates, our task will be to define wave functionals (we use the “bra” and “ket” Dirac notation, with 〈A,ϕ∣Ψ〉=Ψ[A,ϕ]) Ψ[A,ϕ] and the scalar product 〈Ψ∣Ψ′〉. The fields are now promoted to operators A^a, ϕ^, B^a, and Π^ obeying the canonical commutation relations corresponding to the classical Dirac brackets (36):
(39)A^ax,B^by=iħδabδ2x-y,ϕ^x,Π^y=iħδ2x-y,(otherbracketsvanishing).A^ and ϕ^ act multiplicatively and B^ and Π^ as functional derivatives:
(40)B^axΨA,ϕ=-iħδΨA,ϕδAax,Π^xΨA,ϕ=-iħδΨA,ϕδϕx.
Everything up to now is purely formal since we have still no proper Hilbert space. But we can already solve the constraint C^3(x)Ψ[ϕ,A]=-iħϕ^δΨ[ϕ,A]/δϕ(x)=0: the wave functional only depends on A, Ψ=Ψ[A].In order to construct a scalar product defined by an appropriate integration measure in configuration space, we first restrict the space of wave functionals to the set of functions of finite numbers of holonomies of the connectionA—the “cylindrical functions.” If γ is an orientated curve in Σ (a “link”), the holonomy of A on γ is defined as the exponentiated line integral
(41)hγA=exp⁡⁡∫γ‍A.
Given a “graph,”that is, a finite set Γ={γ1,…,γN} of links, a “cylindrical function”  ΨΓ,ψ[A] is function ψ of the holonomies of Γ:
(42)ΨΓ,ψA=ψhγ1A,…,hγNA.
The cylindrical functions associated with all graphs on Σ form the vectorial space Cyl, in which we can define a sesquilinear scalar product using the Haar measure dμ(g) of the gauge group. For U(1), the (normalized) measure is given by (1/2π)∫dθf(g(θ)) for g parameterized as g(θ)=exp⁡⁡(iθ). First, for two cylindrical functions defined on the same graph,
(43)Γ,ψ∣Γ,ψ′=∫G⊗N∫dμg1⋯dμ(gN)ψg1,…,gN*×ψ′(g1,…,gN).
Next, for two cylindrical functions corresponding to two different graphs Γ and Γ′, one defines
(44)Γ,ψ∣Γ′,ψ′=∫G⊗N∫dμg1⋯dμ(gN^)ψg1,…,gN*×ψ′g1,…,gN′,
where Γ^ is the union graph Γ∪Γ′ consisting of N^≤(N+N′) links.With this scalar product in hands we dispose of a norm so one can define a Hilbert spaceHCyl through the Cauchy completion of Cyl.An orthonormal basis ofHCyl may be defined using the Peter-Weyl theorem—which in the Abelian U(1) case is nothing but the Fourier series theorem. Basis elements are the cylindrical functions
(45)ΨΓ,n→A=χn1hγ1A⋯χnNhγNA,wheren→=n1,…,nN,nk∈Z,nk≠0,
and χn(g) is the character of the irreducible unitary representation of “charge” n∈Z. In the parametrization g=exp⁡⁡(iθ), χn(g)=exp⁡⁡(inθ). The orthonormality condition
(46)Γ,n→∣Γ′,n→′
is an obvious consequence of the theory of Fourier series. The prescription of nonvanishing charges nk avoids an overcounting of the basis vectors which would otherwise occur since a graph with a zero charge link would give the same function as the graph with this link omitted. Therefore, the basis must be completed with the zero charge function Ψ∅ corresponding to the empty set ∅. These basis vectors Γ,n→ will be called “charge networks” in analogy with the spin networks of Loop Quantum Gravity [6–9]. A particular consequence of these definitions is that vectors corresponding to different graph are orthogonal, and thus the Hilbert space HCyl is the infinite direct sum of spaces HCyl,Γ, each of them being associated with a single graph Γ. With this sum being performed over the noncountable set of all graphs, HCyl is a nonseparable Hilbert space.Let us now turn to the constraintC1 in (37), which corresponds to the invariance under the U(1) gauge transformations δ1 of (38). It will be fulfilled by demanding the gauge invariance of the basis cylindrical functions (45). Under a gauge transformation Aa′=Aa′+∂aω, the holonomy (41) transforms as
(47)hγA′=hγAexp⁡⁡(ω(xf)-ω(xi)),
where xi and xf are the coordinates of the initial and end points of the link γ, respectively. Thus gauge invariance of a charge network functional ΨΓ,n→ follows from the requirement of a “charge conservation law”;that is, the sum of charges entering a vertex of Γ (point of intersection of links) must be zero, with the convention that the charge entering a vertex is positive if the vertex lies at the end of the link and negative if it lies at the beginning. This requires in particular that the graphs must be closed since no zero charge links are allowed. An example is depicted in Figure 1.Figure 1
Closed graphΓ with three links and two vertices. Link γk carrying a charge nk, charge conservation at vertex v1 (or v2) amounts to -n1-n2+n3=0.The vectors ofHCyl obeying the condition of gauge invariance span the nonseparable “kinematical” Hilbert space Hkin⊂HCyl.


### 4.2. Physical Hilbert Space

The last constraint to be imposed is the curvature constraintC2 in (37), whose quantum expression is F^Ψ=0. Its general solution is given by a wave functional Ψ[A] whose argument A is a connection with null curvature. It is sufficient to impose this condition on the basis vectors of Hkin (charge networks), which will select the basis of the physical Hilbert space Hphys⊂Hkin.The condition of null curvature means that,locally, there exists a scalar function φ such that
(48)Aa=∂aφ.
The rest of the discussion depends on the topology of the space sheet Σ.Let us begin with the case where the topology ofΣ is that of R2. Then (48) holds globally, with the result that the holonomy associated with any link γ with initial and final end points xi and xf takes the form
(49)hγA=exp⁡⁡(φ(xf)-φ(xi)).
Together with the fact that the graph associated with any charge network Γ,n→ is closed and that the charge conservation condition must hold at each vertex, one easily sees that its wave functional ΨΓ,n→ is equal to 1. In other words, the graph Γ shrinks to a single point, and we are left with the sole vector ∅. The physical Hilbert space is reduced to a trivial 1-dimensional space.The next case is that with the topology ofR2∖{O}, the 2-dimensional plane with one point O suppressed. There are now two classes of closed graphs, those with O inside and those with O outside. Two examples of the former class are shown in Figure 2.Figure 2
Two charge network graphs with the singular pointO “inside.”Applying the charge conservation condition as in the previous case shows that any charge network graph with the pointO “outside” reduces to a point with the resulting wave functional equal to 1, defining the empty state described by the vector ∅. On the other hand, any charge network graph with the point O “inside” is equivalent to a single loop γ with O inside, with the resulting wave functional equal to a unimodular complex number
(50)A∣n=ΨnA=exp⁡⁡(inQ),
with n∈Z the charge of the loop. The value of the “flux” Q, given by
(51)exp⁡⁡(iQ)=hCA,
where C is a closed positively oriented loop around the singular point O, is independent of the form and size of the loop, and the value of n is computed using the charge conservation condition. Figure 2 shows an example of two such equivalent graphs. The basis of the physical Hilbert space Hphys then consists of the vectors n, n∈Z, with 〈n∣n′〉=δnn′. For n=0, one has 0 = ∅, corresponding to the former class of graphs. One notes that the integer number n can be interpreted also as a winding number of the loop: to wind n times around the singular point with charge 1 or to wind 1 time the singular point with charge n yields the same wave functional.The generalization to a plane withN singular points, R2∖{O1,…,ON}, is straightforward. The basis vectors of Hphys read n→=n1,…,nN where nk is the charge (or winding number) of a loop encircling the kth singular point, all the other singular points remaining outside of it. The corresponding wave functional is explicitly given by
(52)A∣n→=Ψn→A=exp⁡⁡i∑k=1N‍nkQk,
where Qk is the flux associated with the kth singular point, defined by
(53)exp⁡⁡(iQk)=hCkA,
where
(54)Ck=closedloopencirclingpositivelyonetimethesingularpointOkandleavingasidealltheotherones.
The orthonormality relations are
(55)n1,…,nN∣n1′,…,nN′′=δNN′∏k=1N‍δnknk′.Hphys is separable.One remarks that diffeomorphism invariance, which in the classical theory is a consequence of its gauge invariances, is explicit in the quantum theory constructed here, once all constraints are fulfilled. Note that the states of the (nonseparable) kinematical Hilbert space, which still do not obey the curvature constraintC2, are not diffeomorphism invariant since they depend on the location and form of the associated graphs.


### 4.3. Observables

It follows from the above discussion that no nontrivial observables do exist in the case of a trivial topology such as that ofR2. On the other side, with a nontrivial topology such as that of R2 with N singular points Ok, there is a set of N observables L^k, k=1,…,N, simultaneously diagonalized in the basis (52) of Hphys:
(56)L^kn→=nkn→,k=1,…,N.
They are explicitly given by
(57)L^k=∫Σd2xXa(k)(x)B^a(x),
where Xa(k) is a closed 1-form (dX(k)=0), such that its integral on a loop Ck as defined by (54) takes the value i/ħ, whereas its integral on a loop Cl around another singular point Ol vanishes. Explicitly:
(58)∫CkX(l)=iħδkl,
the result depending only on the homotopy class of Ck. In a polar coordinate frame (r,θ) centred in Ok, a particular solution (a “physical” interpretation may be to view -iX as the analogue of a 2-dimensional magnetic field whose source is a point current of magnitude 1/ħ located in Ok) for the 1-form X(k) is given by Ar=0 and Aθ=i/(2πħ). The result (56) follows from the expression (52) for the basis vector functionals, together with (53) and the differentiation formula (taking into account the support property of X(k))
(59)∫Σd2xXakxδδAaxhCkA=∫CkXhCkA.
The operators L^k thus defined are obviously self-adjoint in Hphys and form a complete commutative set of observables.


## 4.1. Kinematical Hilbert Space

The constraintsC1 and C3 will be solved at the quantum level in this section, whereas the last one, C2, is left for the next section. Following the lines of Loop Quantum Gravity [6–9], we will construct a kinematical Hilbert space Hkin whose vectors Ψ are subjected to the constraints C1 and C3 in the forms C^1Ψ=0 and C^3Ψ=0, where C^i are operators representing the classical Ci. Choosing the fields Aa and ϕ as configuration space coordinates, our task will be to define wave functionals (we use the “bra” and “ket” Dirac notation, with 〈A,ϕ∣Ψ〉=Ψ[A,ϕ]) Ψ[A,ϕ] and the scalar product 〈Ψ∣Ψ′〉. The fields are now promoted to operators A^a, ϕ^, B^a, and Π^ obeying the canonical commutation relations corresponding to the classical Dirac brackets (36):
(39)A^ax,B^by=iħδabδ2x-y,ϕ^x,Π^y=iħδ2x-y,(otherbracketsvanishing).A^ and ϕ^ act multiplicatively and B^ and Π^ as functional derivatives:
(40)B^axΨA,ϕ=-iħδΨA,ϕδAax,Π^xΨA,ϕ=-iħδΨA,ϕδϕx.
Everything up to now is purely formal since we have still no proper Hilbert space. But we can already solve the constraint C^3(x)Ψ[ϕ,A]=-iħϕ^δΨ[ϕ,A]/δϕ(x)=0: the wave functional only depends on A, Ψ=Ψ[A].In order to construct a scalar product defined by an appropriate integration measure in configuration space, we first restrict the space of wave functionals to the set of functions of finite numbers of holonomies of the connectionA—the “cylindrical functions.” If γ is an orientated curve in Σ (a “link”), the holonomy of A on γ is defined as the exponentiated line integral
(41)hγA=exp⁡⁡∫γ‍A.
Given a “graph,”that is, a finite set Γ={γ1,…,γN} of links, a “cylindrical function”  ΨΓ,ψ[A] is function ψ of the holonomies of Γ:
(42)ΨΓ,ψA=ψhγ1A,…,hγNA.
The cylindrical functions associated with all graphs on Σ form the vectorial space Cyl, in which we can define a sesquilinear scalar product using the Haar measure dμ(g) of the gauge group. For U(1), the (normalized) measure is given by (1/2π)∫dθf(g(θ)) for g parameterized as g(θ)=exp⁡⁡(iθ). First, for two cylindrical functions defined on the same graph,
(43)Γ,ψ∣Γ,ψ′=∫G⊗N∫dμg1⋯dμ(gN)ψg1,…,gN*×ψ′(g1,…,gN).
Next, for two cylindrical functions corresponding to two different graphs Γ and Γ′, one defines
(44)Γ,ψ∣Γ′,ψ′=∫G⊗N∫dμg1⋯dμ(gN^)ψg1,…,gN*×ψ′g1,…,gN′,
where Γ^ is the union graph Γ∪Γ′ consisting of N^≤(N+N′) links.With this scalar product in hands we dispose of a norm so one can define a Hilbert spaceHCyl through the Cauchy completion of Cyl.An orthonormal basis ofHCyl may be defined using the Peter-Weyl theorem—which in the Abelian U(1) case is nothing but the Fourier series theorem. Basis elements are the cylindrical functions
(45)ΨΓ,n→A=χn1hγ1A⋯χnNhγNA,wheren→=n1,…,nN,nk∈Z,nk≠0,
and χn(g) is the character of the irreducible unitary representation of “charge” n∈Z. In the parametrization g=exp⁡⁡(iθ), χn(g)=exp⁡⁡(inθ). The orthonormality condition
(46)Γ,n→∣Γ′,n→′
is an obvious consequence of the theory of Fourier series. The prescription of nonvanishing charges nk avoids an overcounting of the basis vectors which would otherwise occur since a graph with a zero charge link would give the same function as the graph with this link omitted. Therefore, the basis must be completed with the zero charge function Ψ∅ corresponding to the empty set ∅. These basis vectors Γ,n→ will be called “charge networks” in analogy with the spin networks of Loop Quantum Gravity [6–9]. A particular consequence of these definitions is that vectors corresponding to different graph are orthogonal, and thus the Hilbert space HCyl is the infinite direct sum of spaces HCyl,Γ, each of them being associated with a single graph Γ. With this sum being performed over the noncountable set of all graphs, HCyl is a nonseparable Hilbert space.Let us now turn to the constraintC1 in (37), which corresponds to the invariance under the U(1) gauge transformations δ1 of (38). It will be fulfilled by demanding the gauge invariance of the basis cylindrical functions (45). Under a gauge transformation Aa′=Aa′+∂aω, the holonomy (41) transforms as
(47)hγA′=hγAexp⁡⁡(ω(xf)-ω(xi)),
where xi and xf are the coordinates of the initial and end points of the link γ, respectively. Thus gauge invariance of a charge network functional ΨΓ,n→ follows from the requirement of a “charge conservation law”;that is, the sum of charges entering a vertex of Γ (point of intersection of links) must be zero, with the convention that the charge entering a vertex is positive if the vertex lies at the end of the link and negative if it lies at the beginning. This requires in particular that the graphs must be closed since no zero charge links are allowed. An example is depicted in Figure 1.Figure 1
Closed graphΓ with three links and two vertices. Link γk carrying a charge nk, charge conservation at vertex v1 (or v2) amounts to -n1-n2+n3=0.The vectors ofHCyl obeying the condition of gauge invariance span the nonseparable “kinematical” Hilbert space Hkin⊂HCyl.


## 4.2. Physical Hilbert Space

The last constraint to be imposed is the curvature constraintC2 in (37), whose quantum expression is F^Ψ=0. Its general solution is given by a wave functional Ψ[A] whose argument A is a connection with null curvature. It is sufficient to impose this condition on the basis vectors of Hkin (charge networks), which will select the basis of the physical Hilbert space Hphys⊂Hkin.The condition of null curvature means that,locally, there exists a scalar function φ such that
(48)Aa=∂aφ.
The rest of the discussion depends on the topology of the space sheet Σ.Let us begin with the case where the topology ofΣ is that of R2. Then (48) holds globally, with the result that the holonomy associated with any link γ with initial and final end points xi and xf takes the form
(49)hγA=exp⁡⁡(φ(xf)-φ(xi)).
Together with the fact that the graph associated with any charge network Γ,n→ is closed and that the charge conservation condition must hold at each vertex, one easily sees that its wave functional ΨΓ,n→ is equal to 1. In other words, the graph Γ shrinks to a single point, and we are left with the sole vector ∅. The physical Hilbert space is reduced to a trivial 1-dimensional space.The next case is that with the topology ofR2∖{O}, the 2-dimensional plane with one point O suppressed. There are now two classes of closed graphs, those with O inside and those with O outside. Two examples of the former class are shown in Figure 2.Figure 2
Two charge network graphs with the singular pointO “inside.”Applying the charge conservation condition as in the previous case shows that any charge network graph with the pointO “outside” reduces to a point with the resulting wave functional equal to 1, defining the empty state described by the vector ∅. On the other hand, any charge network graph with the point O “inside” is equivalent to a single loop γ with O inside, with the resulting wave functional equal to a unimodular complex number
(50)A∣n=ΨnA=exp⁡⁡(inQ),
with n∈Z the charge of the loop. The value of the “flux” Q, given by
(51)exp⁡⁡(iQ)=hCA,
where C is a closed positively oriented loop around the singular point O, is independent of the form and size of the loop, and the value of n is computed using the charge conservation condition. Figure 2 shows an example of two such equivalent graphs. The basis of the physical Hilbert space Hphys then consists of the vectors n, n∈Z, with 〈n∣n′〉=δnn′. For n=0, one has 0 = ∅, corresponding to the former class of graphs. One notes that the integer number n can be interpreted also as a winding number of the loop: to wind n times around the singular point with charge 1 or to wind 1 time the singular point with charge n yields the same wave functional.The generalization to a plane withN singular points, R2∖{O1,…,ON}, is straightforward. The basis vectors of Hphys read n→=n1,…,nN where nk is the charge (or winding number) of a loop encircling the kth singular point, all the other singular points remaining outside of it. The corresponding wave functional is explicitly given by
(52)A∣n→=Ψn→A=exp⁡⁡i∑k=1N‍nkQk,
where Qk is the flux associated with the kth singular point, defined by
(53)exp⁡⁡(iQk)=hCkA,
where
(54)Ck=closedloopencirclingpositivelyonetimethesingularpointOkandleavingasidealltheotherones.
The orthonormality relations are
(55)n1,…,nN∣n1′,…,nN′′=δNN′∏k=1N‍δnknk′.Hphys is separable.One remarks that diffeomorphism invariance, which in the classical theory is a consequence of its gauge invariances, is explicit in the quantum theory constructed here, once all constraints are fulfilled. Note that the states of the (nonseparable) kinematical Hilbert space, which still do not obey the curvature constraintC2, are not diffeomorphism invariant since they depend on the location and form of the associated graphs.


## 4.3. Observables

It follows from the above discussion that no nontrivial observables do exist in the case of a trivial topology such as that ofR2. On the other side, with a nontrivial topology such as that of R2 with N singular points Ok, there is a set of N observables L^k, k=1,…,N, simultaneously diagonalized in the basis (52) of Hphys:
(56)L^kn→=nkn→,k=1,…,N.
They are explicitly given by
(57)L^k=∫Σd2xXa(k)(x)B^a(x),
where Xa(k) is a closed 1-form (dX(k)=0), such that its integral on a loop Ck as defined by (54) takes the value i/ħ, whereas its integral on a loop Cl around another singular point Ol vanishes. Explicitly:
(58)∫CkX(l)=iħδkl,
the result depending only on the homotopy class of Ck. In a polar coordinate frame (r,θ) centred in Ok, a particular solution (a “physical” interpretation may be to view -iX as the analogue of a 2-dimensional magnetic field whose source is a point current of magnitude 1/ħ located in Ok) for the 1-form X(k) is given by Ar=0 and Aθ=i/(2πħ). The result (56) follows from the expression (52) for the basis vector functionals, together with (53) and the differentiation formula (taking into account the support property of X(k))
(59)∫Σd2xXakxδδAaxhCkA=∫CkXhCkA.
The operators L^k thus defined are obviously self-adjoint in Hphys and form a complete commutative set of observables.


## 5. Conclusions

What we have shown, using the Dirac canonical scheme together with the LQG quantization procedure, is that the three-dimensional AbelianBF model, minimally coupled to a scalar field obeying a σ-model type of constraint, has the same degrees of freedom as the pure BF model. These degrees of freedom are nonlocal, of purely topological nature, characterized by the topological nature of space. They are represented by a complete set of N commuting observables L^k in the case of the space topology being that of  R2 with N points omitted (N “punctures”).Two main conclusions can be drawn. First, the model we have presented is a simple example of how restrictive is the assumption of background invariance. It eliminates from the action an infinity of terms which otherwise would be present if one only postulatesU(1) gauge invariance, such as potential terms, for instance. It is also interesting to note that invariance under both gauge transformations (6) and (7) is a consequence of background invariance and U(1) gauge invariance.The second main conclusion is that we have succeeded to implement the loop quantization scheme up to the construction of the physical Hilbert space and its observables. The implementation has turned out to be rather simple, in contrast to the difficulties which one encounters in 4-dimensional gravity [6–9]. This simplicity, in our case, originates from the topological character of the theory, where the diffeomorphism constraints reduce to a very simple curvature constraint. Moreover, we have been able to show explicitly how different topologies of space lead to different Hilbert spaces and sets of observables, which is a very hard problem in genuine quantum gravity.Similar achievements for 3-dimensional theories, non-Abelian but without coupling with matter, may be found in [28, 29]. The generalization to a non-Abelian BF theory coupled with matter is not straightforward [25].



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