on the phase structure of quantum gravity€¦ · a.c. longhitano, phys.rev.d22:1166,1980. the...

The gravitational Higgs phenomenon GraviGUT The main question NCG

On the phase structure of quantum gravity

Roberto Percacci1

1SISSA, Trieste, Italy

Quantum gravity in ParisMarch 23, 2017


Outline

1 The gravitational Higgs phenomenon

2 GraviGUT

3 The main question

4 NCG


Basic question

Why is the gravitational connection not dynamical?


Higgs mechanism I

Certain gauge fields are experimentally seen to be massive.How to reconcile a mass with gauge invariance?Higgs field φ with values in some vectorspace VW a G-invariant potential with minima in G/H ⊂ VAdapted coordinates in V : φ = (ρ, σ)Dynamics gives 〈φ2〉 = ρ2

0Can choose “unitary gauge” φ = (ρ, σ0)


Higgs mechanism II

DecomposeA = A|L(H) + A|P

where L(G) = L(H)⊕ P

Dφ = ∂φ+ Aφ = (Dρ,Dσ) where

Dρ = ∂ρ and Dσ = ∂σ + AiKi(σ)

in particular Dσ0 = Ai |PKi(σ0)

in unitary gauge

(Dφ)2 7→ (∂ρ)2 + ρ20(A|P)2


Higgs mechanism III

only Goldstone bosons are necessary for HiggsmechanismHiggs particle “only” necessary for perturbativerenormalizability


Higgsless Higgs mechanism

Higgs field h = ρ− ρ0 also has mass ≈ λρ0W = λ

4 (φ2 − ρ20)2

limλ→∞W with ρ0=const

ρ decouples in the limit, leaves gauged nonlinear sigma model

T. Appelquist, C.W. Bernard, Phys.Rev.D22:200,1980.A.C. Longhitano, Phys.Rev.D22:1166,1980.


Low Energy EFT

For p2 � m2h,

ρ = ρ0

For p2 � m2A,

A|P = 0

orDσ = 0

Example: Meissner effect in superconductivity


Gravity with more Variables

Spacetime manifold M, dimM = 4

frame field (a.k.a. soldering form) θaµ, detθ 6= 0

pseudo-fiber metric, γab signature +,+,+,−linear connection, Aµa

b (structure group GL(4))

First two carry nonlinear realizations of GL(4)


Induced structures in TM

If we think of θ : TM → EE real vectorbundle with fiber dimension 4local bases {∂µ} in TM and {ea} in E

thengµν = θa

µ θbν γab

Γλµν = θ−1

aµAλa

bθbν + θ−1

aµ∂λθ

aν


Torsion and Nonmetricity

Θµaν = ∂µθ

aν − ∂νθa

µ + Aµab θ

bν − Aνa

b θbµ

∆λab = −∂λγab + Aλca γcb + Aλc

b γac


Gauge invariance

G = AutGL(4)E

θaµ(x) 7→ θ′

aµ(x ′) = Λ−1a

b(x) θbν(x)

∂xν

∂x ′µ

γab(x) 7→ γ′ab(x ′) = Λca(x) Λd

b(x) γcd (x)

Aµab(x) 7→ A′µ

ab(x ′) =

∂xν

∂x ′µ(Λ−1a

c(x)Aνcd (x)Λd

b(x)

+Λ−1ac(x)∂νΛc

b(x))

0→ AutGL(4)M E → AutGL(4)E → DiffM → 0

is split: θ∗ : DiffM → AutGL(4)Eθ∗(f ) = θ ◦ Tf ◦ θ−1


Goldstone Bosons

AutGL(4)E acts transitively on metric and soldering form

γ(x) ∈ GL(4)/SO(3,1)

γ ∈ {fiber metrics} ≈ AutGL(4)E/AutSO(3,1)E

θ ∈ {isomorphisms TM → E} ≈ AutGL(4)E/DiffM


Metric gauge

θaµ = δa

µ

unbroken group DiffMgµν = γµν , Γλ

µν = Aλµν

Θµaν = Γµ

aν − Γν

aµ


Vierbein gauge

γab = ηab

unbroken group AutSO(3,1)Mgµν = θa

µ θbν ηab

∆λab = Aλab + Aλba


Metric and vierbein gauge

Not enough freedom to fix both simultaneously

In general gauge: connection and two “Goldstone bosons”In either of the two “unitary” gauges: connection and one“Goldstone boson”


Low energy action

S(A, γ, θ) = SG(A, γ, θ) + Sm(A, γ, θ)

where

SG =

∫d4x

√|g|[M2

P θaµθb

νFµνab + . . .]

Sm =

∫d4x

√|g|[M2(Θ ·Θ + ∆ ·∆ + Θ ·∆) + . . .

]


The Higgs Mechanism v.I

flat background: A = 0, θ = 1, γ = η

Θµaν = Aµa

ν − Aνaµ

∆µab = Aµab + Aµba

Fµνab = ∂µAνa

b − ∂νAµab + Aµa

c Aνcb − Aνa

c Aµcb

S contains

12

∫d4x

√|det g|M2 Q(A,A)

generically Q is non-degenerate.There is a more general point of view.


Levi–Civita Connection

given θ, γ, there is a unique A s.t. Θ = 0, ∆ = 0

Aabc =12(Eabc + Ecab − Ebac

)+

12(Cabc + Cbac − Ccab

)where

Eabc = θ−1aλ ∂λκbc

Cabc = γad θdλ

(θ−1

bµ ∂µθ

−1cλ − θ−1

cµ ∂µθ

−1bλ)

Any connection A can be split uniquely in A = A + Φ

then S(A, γ, θ) = S(A(θ, γ) + Φ, θ, γ) = S′(Φ, θ, γ)


The Higgs Mechanism v.II

Θµaν = Φµ

aν − Φν

aµ

∆µab = Φµab + Φµba

Fµνab = Fµνa

b + ∇µφνab − ∇νφµa

b + φµa

c φνc

b − φνac φµ

cb

therefore

SP(A, γ, θ) = SP(A + Φ, γ, θ) = SH(γ, θ) + SQ(Φ, γ, θ)

where

SQ(φ, γ, θ) =12

∫d4x

√|det g| M2

P QP(Φ,Φ)

andSm(φ, γ, θ) =

12

∫d4x

√|det g| M2 Qm(Φ,Φ)


Gravitational Higgs Phenomenon

Generically all components of φ are massive.(Not true for the Palatini action, since QP has nontrivial kernel)At energy scales p2 � M2

P

φ = 0⇐⇒ {Θ = 0 and ∆ = 0}


Gravitational Higgs Phenomenon

gravity is a gauge theory of GL(4) with two Goldstonebosonsthere are two unitary gaugesHiggsless Higgs phenomenon occurs at Planck scale,giving mass to Φ (equivalently A)at low energy A = A(θ, γ)

Θ = 0 and ∆ = 0 means that the theory is in a “Higgs”phase


Questions

why is the metric nondegenerate?what is the dynamical origin of the Planck scale?does the connection propagate at ultra-Planckian scales?


Grand Unification

use Higgs phenomenon with G1 ×G2 ⊂ Gto do list:

identify GUT group Gfit particles in irreps of Gwrite G-invariant actionexplain symmetry breaking (select order parameter, orbit,potential)check that new particles not seen at low energy have highmass


Grand unification: SO(10)

(eL , νL ,eR , νR ,ur ,g,bL ,d r ,g,b

L ,ur ,g,bR ,d r ,g,b

R )

16 complex 2 component Weyl spinors of Lorentz4 doublets and 8 singlets of SU(2)LRepeat three times. (nν = 2.984± 0.008 measured at LEP)

Fit exactly in the 16 of SO(10)!explains hypercharge assignments


A symmetry breaking chain

SO(10)↓

SO(4)× SO(6) ≈ SU(2)R × SU(2)L × SU(4)↓

SU(2)R×SU(2)L×SU(3)C×U(1)B−L↓

U(1)EM × SU(2)L × SU(3)C↓

U(1)EM × SU(3)C

requires at least 45, 16 and 10


GraviGUT I

use gravitational Higgs phenomenon to constructunified theory of gravity and all other interactions.to do list:

identify GraviGUT group Gfit particles in irreps of Gwrite G-invariant actionexplain symmetry breaking (select order parameter, orbit,potential)check that new particles not seen at low energy have highmass


GraviGUT II

G1 = SO(1,3), G2 = SO(10),

=⇒ G = SO(1,13) or G = SO(3,11)

keep dimM=4,enlarge fibers of E to have dimension N > 4order parameter is soldering form

γ =

[η 00 1N−4

], θ is 4× N matrix, e.g. 〈θ〉 =

[140

]


Fermions I

F. Nesti, R.P., Phys. Rev. D 81, 025010 (2010) arXiv:0909.4537 [hep-th]

Assume γab = ηab, G = SO(3,11)

In SO(10) GUT one family isη ∈ 2C × 16C of SO(3,1)× SO(10)


Fermions II

Let BΣ∗ij = ΣijB and ψc = Bψ∗. Define ψ± by (ψ±)c = ±ψ±(Majorana spinors)Define ψL/R by γψL/R = ∓ψL/R (Weyl spinors).In signature (3,11) [γ,B] = 0 so we can define Majorana-Weylspinors ψL/R±. These have 64 real components.

Decomposing ψL+ under SO(3,1)× SO(10) ⊂ SO(3,11) wefind it is equivalent to η.:

64R = 2C × 16C

Remark: SO(1,13) has Weyl 64C

64C = 2C × 16C + 2C × 16C


Fermions III

DµψL+ =

(∂µ +

12

AijµΣ

(3,11)L ij

)ψL+

let Σ†ijA = −A Σij

then ψ†L+(Aγ i)LDψL+ is one-form in 14 of SO(3,11)

S =

∫ψ†L+(Aγ i)LDψL+ ∧ θj ∧ θk ∧ θ` φijk` .


Fermions IV

Assuming the following VEVs:{φmnrs = εmnrsφijk` = 0 otherwise

{θmµ = Mem

µ

θaµ = 0 otherwise

one gets

S =

∫d4x√

g η†σµ∇µη ,

where now

∇µ = D(4+10)µ = ∂µ +

12

Γabµ Σ

(3,1)ab +

12

Aabµ (10)Σ

(10)ab


GraviGUT III

Gravitational Higgs phenomenon:

A =

[A(4) HHT A(10)

]kinetic term of θ gives mass to A(4), H,SO(10) remains unbroken

R.P. Phys. Lett. B 144, 37 (1984), Nucl. Phys. B 353, 271, (1991).


Status of GraviGUT

kinematics well understoodfermionic content and dynamics okbosonic action for broken phase can be writtenhard to write action that works in both phases


The main question for a QT of spacetime

What makes the VEV of the soldering form/metric nonzero?

Related questions:- How does an extended spacetime arise?- What is the origin of the Planck scale?


This conference

Causal dynamical triangulations (Jurkiewicz)String theory (West)LQG (Geiller)NCG (Steinacker, Barrett)Asymptotic safety (Wetterich)


What you see is what you get

Theory is subject to constraints also at high energy.A.H. Chamseddine, V.Mukhanov, “On Unification of Gravity andGauge Interactions” JHEP 1603 (2016) 020, arXiv:1602.02295

Recently extended to GUTsG.K.Karananas, M. Shaposhnikov, “Gauge coupling unificationwithout leptoquarks” arXiv:1703.02964 [hep-ph]


Pick the right solution

Dynamics admits solutions with θ = 0 and also with θ 6= 0.Just pick the one having θ of maximal rank.

G. Lisi, L. Smolin, S. Speziale, J.Phys. A43 (2010) 445401arXiv:1004.4866 [gr-qc]


Conformal models

S =

∫d4x√

g[

12

(∇φ)2 − V (φ) + ξφ2R]

→ VEV of φ is the Planck mass


Ad-hoc models

Quantum graphityT. Konopka, F. Markopoulou, S. SeveriniPhys.Rev. D77 (2008) 104029 arXiv:0801.0861 [hep-th]

Self-organizing networksC. Trugenberger- Phys.Rev. D92 (2015) 084014 arXiv:1501.01408 [hep-th] |- Phys.Rev. E92 (2015) no.6, 062818 arXiv:1507.01820 [hep-th]- arXiv:1610.05934 [hep-th]


Self-consistent mean-field theory

Assuming 〈θ〉 = θ 6= 0, use θ in action to construct V (θ; θ).Check self-consistency a posteriori

R. Floreanini, R.P., E. Spallucci, Class. and Quantum Grav. 8, L193,(1991).R. Floreanini, R.P. Phys. Rev. D 46, 1566 (1992).


Bi-metric approach/asymptotic safety

Bi-metric actions appear in the functional RG approach toquantum gravity.Recent work by B. Knorr shows that at least in some cases

〈gµν − gµν〉 = 0

Otherwise, this approach is well-suited to discuss the Higgsphase.Unlikely to have access to the “unbroken” phase.


Lattice approaches

Lattice simulations of gravity see rich phase structure

Regge calculusEuclidean dynamical triangulationsCausal dynamical triangulations


New entry

A new class of multi-matrix models motivated by NCG

J. Barrett„ J.Math.Phys. 56 (2015) no.8, 082301 arXiv:1502.05383[math-ph]J. Barrett, L. Glaser, J.Phys. A49 (2016) 245001 arXiv:1510.01377[gr-qc]L. Glaser, arXiv:1612.00713 [gr-qc]


Fuzzy geometries

Subclass of matrix geometriesSpectral triple (A,H,D) with H = S ⊗M(n,C),(S spinor space of signature (p,q))with inner product(ψ1 ⊗m1, ψ2 ⊗m2) = (ψ1, ψ2) Trm†i m2andA = M(n,R),M(n,C),M(n/2,H)acts on H by ρ(a)(ψ ⊗m) = ψ ⊗ (am)Dirac operator

D(ψ ⊗m) =∑

i

ΓiLψ ⊗ [Li ,m] +

∑i

ΓiHψ ⊗ {Hi ,m}

where Γi ∈ C(p,q), ΓiL, Li antihermitian, Γi

H , Hi hermitian.


Path integral

Z =

∫(dD)e−S(D)

S(D) = g2 trD2 + g4 trD4


Observables

Measure the expectation value of the eigenvalues of D

〈λi〉 =

∫(dD)λi e−S(D)

the distribution of eigenvalues and the “order parameter”

F =

∑i(trHi)

2

n tr∑

i H2i

calculated by Monte Carlo methodP.Labus and R.P., to appear


TYPE (1,0)

D = {H, ·}

S = g4

(2NTrH4 + 8TrHTrH3 + 6(TrH)2

)+g2

(2NTrH2 + 2(TrH)2

)single-trace part of the action solvableE. Brezin, C. Itzykson, G. Parisi, J. B. Zuber, Commun. Math. Phys.59 (1978) 35.G. M. Cicuta, L. Molinari, E. Montaldi, Mod. Phys. Lett. A 1 (1986)125.

-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4

0.5

0.6

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8


TYPE (1,1)

Phase transition at g2 ≈ −2.4


TYPE (1,1)

-4 -2 2 4

0.1

0.2

0.3

0.4

0.5

g2 = −2.2,−2.4,−2.6


TYPE (1,1)

0 50 100 150 200

2

4

6

8

10

g2 = −2.2,−2.4,−2.6


TYPE (3,0)


Conclusions

A quantum theory of spacetime has to generate anextended geometry dynamically.Why is there a non-degenerate metric?Desirable to control phases of theory be a tunable“potential”. Models of fuzzy geometry seem to giveprecisely such a tool.There are hints from Monte Carlo simulations that suchmodels have a phase transition.Simplest models seem to give low-dimensional geometries.Hopefully some such model can produce d = 4Search just begun!

on the phase structure of quantum gravity€¦ · a.c. longhitano, phys.rev.d22:1166,1980. the...

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