A possible interpretation of theBarbero–Immirzi parameter and its

consequencesSimone Mercuri

IGC - The Pennsylvania State University

International Loop Quantum Gravity seminar

March 3, 2009

ILQG – p. 1/32

Motivations: The Barbero–Immirzi (BI) parameter is a free constantthat appears in the spectra of area and volume operators of LQG.Many authors have contributed to shed light on the origin of thisambiguity.a Here we present an extension of the idea originallyproposed by Gambini, Obregon and Pullin.

Idea: The BI parameter is a quantization ambiguity analogous to theθ-angle of Yang–Mills gauge theories.

Further extension: We present a motivation to promote the BIparameter to be a field rather than a constant, digressing on thedynamics determined by the coupling of the BI field with theNieh–Yan topological density. By the way, an extension of the Holstaction to spacetimes with torsion is also discussed.

aRovelli & Thiemann, (1998), R. Gambini et al., (1999), Perez & Rovelli (2005), Freidel et al. (2005),Chou et al. (2005), Randono (2005).

ILQG – p. 2/32

Outline

General remarks: brief review of the role of the BI parameter inclassical and quantum gravity.

Einstein-Cartan theory: description of the interaction between spinormatter fields and gravity. Torsion and its geometrical features.

Generalization of the Holst action: brief description of the Nieh–Yantopological density and its relation to the Pontryagin densities.Generalization of the Holst theory to spacetimes with torsion.Comparison between the BI parameter and the vacuum angle ofYang–Mills gauge theories.

ILQG – p. 3/32

Outline

Promoting the BI parameter to be a field: motivation to consider thehypothesis that the BI parameter is actually a field rather than aconstant. Proposal on the nature of the BI parameter. Dynamics andcomparison with previously proposed models.

Interaction with fermions and Peccei–Quinn mechanism: introductionof fermion fields and dynamical consequences related to thepresence of the BI field. Description of the PQ mechanism fordetermining the value of the BI parameter.

Canonical theory: constraints of the new theory with the BI field.Proposal for the Ashtekar–Barbero formulation.

Concluding remarks and discussion: summary and discussion of thedescribed results, proposals for future research.

ILQG – p. 4/32

General Remarks: the BI parameter

The BI parameter is a free constant appearing in the spectra of length,area and volume operators of LQG, e.g., the area spectrum is:

Aγ = 8πγℓ2Pl

∑

k

√

jk (jk + 1) . (1)

Rovelli and Thiemann in 1998 noted that the canonical transformationU(γ) : (A,E) → (A′, E′), where A′i

α = γAiα + (1 − γ) Γi

α and E′βk = 1

γEβ

k ,does not correspond to a unitary transformation in the quantum theory.

A rescaling of the canonical variables usually generates different, butunitary equivalent quantum representations.

So, what happens here?

ILQG – p. 5/32

General Remarks: the BI parameter

One possibility is that the BI parameter has a topological origin.

If, in fact, the BI parameter has a topological origin, then different choicesof γ leads to inequivalent representations of the quantum canonicalcommutations relations.

Gambini, Obregon and Pullin suggested that it exists an interestinganalogy between BI parameter and the θ-ambiguity appearing inYang–Mills gauge theories. Both do not affect the classical theory, butproduce striking effects in the quantum regime.

But in pure gravity this analogy is not completely convincing.

The BI parameter appears as a multiplicative constant in front of the Holstmodification, which cannot be ascribable among topological densities.

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General Remarks: the Holst action

The Holst action is:

SHol [e, ω] = SHP [e, ω] + SMod [e, ω] = − 1

16πG

∫

ea ∧ eb ∧(

⋆Rab + βRab)

,

(2)

where for later convenience we defined β = − 1γ

.

The signature is +,−,−,− and we set ~ = c = 1; [G] = M−2 = L2.The Holst modification vanishes on (half-)shell, namely on the solution ofthe homogeneous second Cartan structure equation. Specifically, wehave that:

dea + ωab ∧ eb = 0 =⇒ Ra

b ∧ eb = 0 , (3)

namely if the connection 1-form ωab satisfies the structure equation, itsassociated curvature 2-form satisfies the cyclic Bianchi identity. So that

The term SMod does not affect the classical equations of motion of GR.

ILQG – p. 7/32

General Remarks: the Holst action

Since the Holst modification is not a topological term, the sameconclusion cannot be drawn in spacetimes with torsion.

The Bianchi cyclic identity is modified by the presence of torsion, i.e.

dea + ωab ∧ eb = T a =⇒ Ra

b ∧ eb = d(ω)T a , (4)

The case of fermion fields coupled to gravity is particularly instructive. It iswell known that the presence of fermions coupled to first order Palatinigravity generates torsion in spacetime, thus revealing some interestingfeatures about the Holst theory and suggesting a possible generalization.

ILQG – p. 8/32

Einstein–Cartan theory: generalities

We can describe a system of spin-1/2 fields coupled to gravity via theEinstein–Cartan action:

SEC

[

e, ω, ψ, ψ]

= − 1

16πG

∫

ea ∧ eb ∧ ⋆Rab

+i

2

∫

⋆ ea ∧(

ψγaDψ −Dψγaψ +i

2meaψψ

)

, (5)

where the covariant derivatives are defined as:

Dψ = dψ − i

4ωabΣabψ and Dψ = dψ +

i

4ψΣabω

ab. (6)

Spinor fields generate torsion: the variation with respect to ωab gives

d(ω)ea = T a = −4πG ⋆(

ea ∧ ebJb(A)

)

where Jd(A) = ψγdγ5ψ . (7)

The unique solution is: ωab(

e, ψ, ψ)

= ω◦

ab (e) + 2πG ǫabcde

cJd(A).

ILQG – p. 9/32

Einstein–Cartan theory: generalitiesIt is worth noting that the Cartan structure equation does not contain anydynamical information, it is an algebraic relation that makes it possible touniquely express the Lorentz valued connection ωab as a function of theother fields, namely the gravitational and the spinor fields.

By pulling back the EC action on the solution of the structure equation, weobtain the following effective action:

Seff

[

e, ψ, ψ]

= − 1

16πG

∫

1

2ǫabcd e

a ∧ eb ∧R◦

cd

+i

2

∫

⋆ ea ∧(

ψγaD◦ψ −D

◦ψγaψ +

i

2meaψψ

)

+3

2πG

∫

dV ηabJa(A)J

b(A) ,

known as EC effective action

ILQG – p. 10/32

Einstein–Cartan theory: Holst case

For pure gravity the Holst action is dynamically equivalent to HP action:in the presence of minimally coupled fermions the situation changes:

SECH

[

e, ω, ψ, ψ]

= − 1

16πG

∫

ea ∧ eb ∧(

⋆Rab + βRab)

+i

2

∫

⋆ ea ∧(

ψγaDψ −Dψγaψ +i

2meaψψ

)

, (8)

The structure equation becomes:

d(ω)ea = τa = − 4πG

1 + β2⋆(

ea ∧ ebJb(A)

)

−4πGβ

1 + β2ea ∧ ebJ

b(A) . (9)

The new torsion tensor depends explicitly on the BI parameter. It differsfrom the one calculated before because of the presence of an additionaltrace component which vanishes as soon as β = 0. Strangely enough thetrace component is an internal axial vector rather than a vector.

ILQG – p. 11/32

Einstein–Cartan theory: Holst case

The solution of the new Cartan structure equation can be easilycalculated:

ωab(

e, ψ, ψ)

= ω◦

ab (e) +2πG

1 + β2ǫab

cdecJd

(A) − 8πGβ

1 + β2e[aJ

b](A) . (10)

By pulling the action back on the solution above, we obtain:

Seff

[

e, ψ, ψ]

= − 1

16πG

∫

1

2ǫabcd e

a ∧ eb ∧R◦

cd

+i

2

∫

⋆ ea ∧(

ψγaD◦ψ −D

◦ψγaψ +

i

2meaψψ

)

+3

2πG

1

1 + β2

∫

dV ηabJa(A)J

b(A) , (11)

The effective action deviates from the EC one and the BI parameteracquires a classical physical meaning.

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Einstein–Cartan theory: extension

To construct an action which reduces to the Ashtekar–Romano–Tate onefor β = ± i, we have to modify also the fermionic sector:a

S[

e, ω, ψ, ψ]

= − 1

16πG

∫

ea ∧ eb ∧(

⋆Rab + βRab)

+i

2

∫

⋆ ea ∧[

ψγaS5(β)Dψ −DψS5

(β)γaψ]

,

where S5(β) = 1 + iβγ5.

THIS ACTION IS DYNAMICALLY EQUIVALENT TO THAT OF THEEINSTEIN-CARTAN THEORY

Note that for β = i only the left handed fermions interact with the self dualAshtekar connections. The above action can be considered a naturalgeneralization of the ART one for arbitrary values of the BI parameter.

aSM, Phys. Rev. D73, 084016, (2006) [gr-qc/0601013] (see also SM, gr-qc/0610026).

ILQG – p. 13/32

Einstein–Cartan theory: extension

To understand the reason why the introduced modifications do not affectthe classical theory, consider:

I =

∫[

β

16πGRab ∧ ea ∧ eb +

β

2⋆ ea ∧

(

ψγaγ5Dψ −Dψγ5γaψ)

]

. (12)

By using the solution of the Cartan structure equation obtained by varyingthe non-minimal action, we can rewrite the fermionic term, so that weobtain

I =β

16πG

∫

[

Rab ∧ ea ∧ eb − T a ∧ Ta

]

= − β

16πG

∫

d (ea ∧ T a) = −SNY [e, ω] , (13)

The integrand above is known as Nieh–Yan topological density.It represents a generalization of the Holst term to spacetimes with torsion.

ILQG – p. 14/32

Holst theory: generalization

Consequently: we propose the following generalized action for spacetimeswith torsion

Snew

[

e, ω, ψ, ψ]

= − 1

16πG

∫

ea ∧ eb ∧ ⋆Rab + Smatter

[

e, ω, ψ, ψ]

− β

16πG

∫

(

ea ∧ eb ∧Rab − T a ∧ Ta

)

, (14)

REMARKS:

the proposed action describes the same classical dynamics of theEinstein(-Cartan) action;

the BI parameter disappears from the effective classical action;

it generalizes the Holst approach to Riemann–Cartan spacetimes;

the new action suggests a possible physical interpretation of the BIparameter in analogy with the θ-angle of Yang-Mills gauge theories.

ILQG – p. 15/32

From EC to AB formulation of GravityThe action now contains a true topological term, which provides insightinto the nature of the BI parameter, completing the picture proposed byGambini, Obregon and Pullin. Specifically. . .

. . . the Ashtekar–Barbero constraints of GR can be obtained by rescalingthe heuristic state functional of the Einstein–Cartan theory by the

exponential of the Nieh–Yan functional.a

WORK IN PROGRESS. . . More rigorously, one should work with the welldefined states of LQG, we expect that the rescaling of the states by theexponential of the Nieh–Yan functional multiplied by the constant β̃generates a shift of the BI parameter β → β + β̃.

The rescaling is motivated by a particular large gauge transformationrelated to the temporal gauge fixing, specifically consisting in modifyingthe Wigner boost parameter which fixes the gauge.

aSM, PRD 77, (2008), 024036

ILQG – p. 16/32

End of first part

General remarks: brief review of the role of the BI parameter inclassical and quantum gravity.

Einstein-Cartan theory: description of the interaction between spinormatter fields and gravity. Torsion and its geometrical features.

Generalization of the Holst action: brief description of the Nieh–Yantopological density and its relation to the Pontryagin densities.Generalization of the Holst theory to spacetimes with torsion.Comparison between the BI parameter and the vacuum angle ofYang–Mills gauge theories.

ILQG – p. 17/32

Second part

Promoting the BI parameter to be a field: motivation to consider thehypothesis that the BI parameter is actually a field rather than aconstant. Proposal on the nature of the BI parameter. Dynamics andcomparison with previously proposed models.

Interaction with fermions and Peccei–Quinn mechanism: introductionof fermion fields and dynamical consequences related to thepresence of the BI field. Description of the PQ mechanism fordetermining the value of the BI parameter.

Canonical theory: constraints of the new theory with the BI field.Proposal for the Ashtekar–Barbero formulation.

ILQG – p. 18/32

Motivation from particle physicsLet us consider the following action:

Snew

[

e, ω, ψ, ψ, q, q]

= SHP [e, ω] + Smatter

[

e, ω, ψ, ψ, q, q]

− β

16πG

∫

(

ea ∧ eb ∧Rab − T a ∧ Ta

)

, (15)

where the matter term describes leptons as well as quarks.

It is well known that the quarks mass matrices M resulting from thespontaneous breaking of the SU(2) × U(1) symmetry are neither diagonalnor Hermitian. We can diagonalize M by chiral rotating the quark fields:

qR → q′R = ei

2nfArg det M

qR , (16a)

qL → q′L = e− i

2nfArg det M

qL . (16b)

Such a chiral transformation introduces a divergent termin the effective action.

ILQG – p. 19/32

Motivation from particle physicsWe recall that, in spacetime with torsion, the jacobian of a chiral rotation ofthe fermionic measure in the Euclidean path-integral contains a divergentterm,a i.e.

δqδqU(1)A−→ δqδqe

i

8π2

Rα[Rab∧Rab+2Λ2(Ta∧T a−ea∧eb∧Rab)] ,

Λ being the regulator and α the parameter of the transformation.

The resulting action is

S[

e, ω, ψ, ψ, q, q]

= SHP [e, ω] + SD

[

e, ω, ψ, ψ, q, q]

+1

8π2α

∫

Rab ∧Rab

+1

16πG

(

β +4G

παΛ2

)∫

(

T a ∧ Ta − ea ∧ eb ∧Rab)

.

As soon as we try to remove the regulator we obtain a divergence.

aChandía & Zanelli, PRD 55, (1997), 7580. ILQG – p. 20/32

Motivation from particle physicsLet us introduce in the fundamental action the field β(x), which interactswith the gravitational field through the NY term, i.e.

STot

[

e, ω, ψ, ψ, q, q, β]

= SHP [e, ω] + SD

[

e, ω, ψ, ψ, q, q]

+1

16πG

∫

β(x)(

T a ∧ Ta − ea ∧ eb ∧Rab)

. (17)

Dynamically, the new action (17) is no longer equivalent to the EC one.

The new action above is invariant under a rescaling of β(x), so that,without affecting the dynamical content of the theory, we can incorporatethe divergence in the definition of a new field β′ (x) = β (x) + 4G

παΛ2.

In this new framework the BI parameter, β0, can be naturally associated tothe expectation value of the field β(x), namely β0 = 〈β(x)〉.

ILQG – p. 21/32

The BI field: effective dynamicsLet us study the effective dynamics of the BI field in the pure gravitationalcase:

Sgrav = − 1

16πG

∫

ea ∧ eb ∧ ⋆Rab − 1

16πG

∫

β(

ea ∧ eb ∧Rab − T a ∧ Ta

)

.

By varying the action with respect to the connection 1-form ωab andmanipulating it, we obtain the following structure equation:

dea + ωab ∧ eb = T a =

1

2ǫab

cd∂bβ ec ∧ ed .

Pulling back the action on the solution of the structure equation, wehave:a

Seff = − 1

16πG

∫

ea ∧ eb ∧ ⋆Rab +1

2

∫

⋆dβ̃ ∧ dβ̃ , (18)

where we defined β̃(x) = 14

√

3πGβ(x).

[

β̃(x)]

= M = L−1.

aSM, arXiv:0902.2764. Calcagni & SM, arXiv:0902.0957

ILQG – p. 22/32

The BI field: remarks

When the parameter β is promoted to be a field, the NY term in theaction ceases to be topological, generating a modification withrespect to the EC theory.

To preserve the usual transformation properties of the irreduciblecomponents of torsion, the β̃(x) field has to be a pseudo-scalar.Interestingly enough, the pseudo-scalar nature of β̃(x) is notassumed a priori, but is a geometrical consequence of the theory.

The BI field decouples from gravity and behaves exactly as a(pseudo) scalar field. In the previous models a based on the usualHolst action, φ = sinhβ plays the role of a scalar field rather than βitself.

aTaveras & Yunes, PRD 78, (2008). Torres-Gomez & Krasnov, arXiv:0811.1998. Castellani et al., (1991).

ILQG – p. 23/32

The BI field: coupling with fermionsLet me consider the following action:

STot = − 1

16πG

∫

ea ∧ eb ∧ ⋆Rab

− 1

16πG

∫

β(x)(

ea ∧ eb ∧Rab − T a ∧ Ta

)

+i

2

∫

⋆ea ∧(

ψγaDψ −Dψγaψ +i

2meaψψ

)

, (19)

Dynamically, the action (19) is not equivalent to the EC one, in fact thepresence of the new field β(x) modifies the expression of the torsion2-form, i.e.

dea + ωab ∧ eb = T a = ǫabcd

(

1

2ηbf∂fβ − 2πGJb

(A)

)

ec ∧ ed ,

where J(A) = eaJa(A) = ea

∑nf=1 ψfγ

aγ5ψf is the fermionic axial current.

ILQG – p. 24/32

The BI field: coupling with fermionsBy using the expression we found for torsion an interesting form of theeffective action can be easily calculated, namely

Seff = SHP

[

e, ω◦

]

+ SD

[

e, ω◦, ψ, ψ

]

+3

2πG

∫

dV ηabJa(A)J

b(A)

+1

2

∫

⋆dβ̃(x) ∧ dβ̃(x) −√

3πG

∫

⋆J(A) ∧ dβ̃(x) . (20)

An interaction between β̃(x) and the fermionic axial current appears.Remembering that the chiral anomaly prevents the axial current to beconserved, i.e. d ⋆ J(A) = − 1

192π2Rab ∧Rab, as noted previously, the β̃(x)

turns out to couple to the gravitational Pontryagin density, namely

Seff = SHP

[

e, ω◦

]

+ SD

[

e, ω◦, ψ, ψ

]

+3

2πG

∫

dV ηabJa(A)J

b(A)

+1

2

∫

⋆dβ̃(x) ∧ dβ̃(x) +

√3πG

192π2

∫

β̃(x)R◦

ab ∧R◦

ab . (21)

ILQG – p. 25/32

The BI field: coupling with fermionsThe effective action can also contain P and CP violating terms.

Possible sources of discrete symmetries violations are a θ term correlatedto the non-trivial global structure of the local gauge group. Moreover thepossible presence of quarks introduces another term proportional toα = Arg detM . We include these terms via the parameter θ̃ = θ + 1

8π2 α.So, finally:

Seff = SHP

[

e, ω◦

]

+ SD

[

e, ω◦, ψ, ψ

]

+3

2πG

∫

dV ηabJa(A)J

b(A)

+1

2

∫

⋆dβ̃(x) ∧ dβ̃(x) +1

8π2

∫

(

θ̃ +

√3πG

24β̃(x)

)

R◦

ab ∧R◦

ab . (22)

Focus the attention on the last term in the action. It is the only CPviolating term. Studying the effects connected with this term,a it could bepossible to put some restrictive limits on the expectation value of β̃(x).

asee, e.g., Alexander & Yunes (2008), Yunes & Spergel (2008), Alexander, Finn & Yunes (2007)

ILQG – p. 26/32

The BI field: Peccei–Quinn mechanismWe can relate the value of the BI parameter to that of the vacuum angle θ̃.In particular,

SCP =1

8π2

∫

(

θ̃ +

√3πG

24β̃(x)

)

R◦

ab ∧R◦

ab , (23)

being the only CP violating term, then the effective potential will be evenin θ̃ +

√3πG24 β̃(x), so it will have a stationary point in θ̃ +

√3πG24 β̃(x) = 0,

preserving the P and CP symmetries.a According to this mechanism, wehave β0 = −32θ̃.

The BI ambiguity is correlated to the vacuum angle of gravity.

Side remark: only the physical field β̃phys = β̃ −⟨

β̃⟩

interacts with gravity

via the Pontryagin density.

aPeccei & Quinn, PRL 38, 1440, (1977); PRD 16, 1791, (1977).

ILQG – p. 27/32

The BI field: Canonical TheoryThe gravitational theory with the BI field has been canonically formulatedin a recent paper.a Once fixed the temporal gauge and solved the secondclass constraints related to torsion, the evolution of the system in thephase space turns out to be limited by the following set of first classconstraints:

Ri := ǫ kij K

jαE

αk ≈ 0 , (24)

Hα := 2Eγi D[αK

iγ] + Π∂αβ̃ ≈ 0 , (25)

H := −8πG

2eEα

i Eγj

(

ǫijkRkαγ + 2Ki

[αKj

γ]

)

+1

2eΠ2 − 1

2e ∂αβ̃∂

αβ̃ ≈ 0 . (26)

where β̃ = 14

√

3πGβ, so that

[

β̃]

= L−1 = M and the symplectic structure

being:

{

Kiα(t, x), Eβ

j (t, x′)}

= δβαδ

ijδ(x, x

′) ,{

β̃(t, x),Π(t, x′)}

= δ(x, x′) , (27)

aCalcagni & SM, arXiv:0902.0957

ILQG – p. 28/32

Ashtekar–Barbero variablesOne is immediately tempted to define the new variables as:

Aiα(t, x) = − 1

β(t, x)Ki

α(t, x) + Γiα(t, x) , Pα

i (t, x) = −β(t, x)Eαi (t, x) ,

but this definition generates a complicated new canonical algebra.

The above definition breaks the rescaling symmetry of the theory

So, according to the interpretation of the BI parameter previouslyproposed, we suggest to define the new variables as:

Aiα(t, x) = − 1

β0Ki

α(t, x) + Γiα(t, x) . (28)

The expectation value β0 is completely arbitrary being associated to afree scalar field. But in the non-perturbative quantum theory the rescalingsymmetry cannot remain unbroken, generating a non-trivial effectivepotential which can dynamically fix the value of β0 = − 1

γ.

ILQG – p. 29/32

Ashtekar–Barbero constraintsBy introducing the new variables into the constraints, we can rewrite themin the Ashtekar–Barbero form, with the presence of the scalar field β̃, i.e.

Ri → Gi :=∂αEαi + ǫ k

ij AjαE

αk ≈ 0 , (29)

Hα → H′α :=Eγ

i Fiαγ + Π∂αβ̃ ≈ 0 , (30)

H → H′ := − 8πG

2eEα

i Eγj

(

ǫijkFkαγ + 2

(

1 + β20

β20

)

Ki[αK

j

γ]

)

+1

2eΠ2 − 1

2e ∂αβ̃∂

αβ̃ ≈ 0 , (31)

where F iαγ is the curvature of the SU(2) valued connection Ai

α.

We expect that the quantum theory breaks the U(1)A rescaling symmetryof the starting action.

The net result is the possible presence of an additional interaction throughthe Pontryagin class P =

∫

F i ∧ Fi (work in progress!).

ILQG – p. 30/32

ConclusionsThe BI parameter is a free constant appearing in the kinematicalobservables of LQG. Its value can be determined by studying theblack holes modes, but its nature is still debated.

Here, motivated by a previous suggestion of Gambini, Obregon andPullin, we initially proposed a topological interpretation of the BIparameter in analogy with the θ-angle of QCD.

To complete the analogy between the θ-angle of QCD and the BIparameter of LQG, we extended the Holst approach to torsionalspacetimes, where a true topological term containing the Holstmodification can be introduced.

We gave a general argument that the BI parameter has to bepromoted to be a field. The effective dynamics revealed that the BIfield behaves as a pseudo-scalar field.

ILQG – p. 31/32

ConclusionsMoreover, the introduction of fermions in this framework seems to beparticularly interesting and promising for new developments of thetheory.

We have showed how the analogous of the Peccei–Quinnmechanism can provide a dynamical determination of the BIparameter, which turns out to be related to the vacuum angle of thegravitational theory.

In order to study the outcomes in a rigorous way, we have tointroduce these new concepts in the well posed formalism of LQG.Through an analogous mechanism, the BI parameter could bedynamically related to other topological quantities. In particular, bystudying the interaction with fermions in the canonical framework,new interactions could appear, providing interesting effects.

ILQG – p. 32/32