On the phase structure of quantum gravity Roberto Percacci 1 1 SISSA, - - PowerPoint PPT Presentation

The gravitational Higgs phenomenon GraviGUT The main question NCG

On the phase structure of quantum gravity

Roberto Percacci1

1SISSA, Trieste, Italy

Quantum gravity in Paris March 23, 2017

The gravitational Higgs phenomenon GraviGUT The main question NCG

Outline

1

The gravitational Higgs phenomenon

2

GraviGUT

3

The main question

4

NCG

The gravitational Higgs phenomenon GraviGUT The main question NCG

Basic question Why is the gravitational connection not dynamical?

The gravitational Higgs phenomenon GraviGUT The main question NCG

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 V W a G-invariant potential with minima in G/H ⊂ V Adapted coordinates in V: φ = (ρ, σ) Dynamics gives φ2 = ρ2 Can choose “unitary gauge” φ = (ρ, σ0)

The gravitational Higgs phenomenon GraviGUT The main question NCG

Higgs mechanism II Decompose A = 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 → (∂ρ)2 + ρ2

0(A|P)2

The gravitational Higgs phenomenon GraviGUT The main question NCG

Higgs mechanism III

• nly Goldstone bosons are necessary for Higgs

mechanism Higgs particle “only” necessary for perturbative renormalizability

The gravitational Higgs phenomenon GraviGUT The main question NCG

Higgsless Higgs mechanism Higgs field h = ρ − ρ0 also has mass ≈ λρ0 W = λ

4(φ2 − ρ2 0)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.

The gravitational Higgs phenomenon GraviGUT The main question NCG

Low Energy EFT For p2 ≪ m2

h,

ρ = ρ0 For p2 ≪ m2

A,

A|P = 0

• r

Dσ = 0 Example: Meissner effect in superconductivity

The gravitational Higgs phenomenon GraviGUT The main question NCG

Gravity with more Variables Spacetime manifold M, dimM = 4 frame field (a.k.a. soldering form) θaµ, detθ = 0 pseudo-fiber metric, γab signature +, +, +, − linear connection, Aµab (structure group GL(4)) First two carry nonlinear realizations of GL(4)

The gravitational Higgs phenomenon GraviGUT The main question NCG

Induced structures in TM If we think of θ : TM → E E real vectorbundle with fiber dimension 4 local bases {∂µ} in TM and {ea} in E then gµν = θaµ θbν γab Γλµν = θ−1aµAλabθbν + θ−1

a µ∂λθaν

The gravitational Higgs phenomenon GraviGUT The main question NCG

Torsion and Nonmetricity Θµaν = ∂µθaν − ∂νθaµ + Aµab θbν − Aνab θbµ ∆λab = −∂λγab + Aλca γcb + Aλcb γac

The gravitational Higgs phenomenon GraviGUT The main question NCG

Gauge invariance G = AutGL(4)E θaµ(x) → θ′a

µ(x′) = Λ−1a b(x) θbν(x) ∂xν

∂x′µ γab(x) → γ′

ab(x′) = Λca(x) Λd b(x) γcd(x)

Aµa

b(x)

→ A′

µ a b(x′) = ∂xν

∂x′µ

• Λ−1ac(x)Aνc

d(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)E θ∗(f) = θ ◦ Tf ◦ θ−1

The gravitational Higgs phenomenon GraviGUT The main question NCG

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

The gravitational Higgs phenomenon GraviGUT The main question NCG

Metric gauge θaµ = δa

µ

unbroken group DiffM gµν = γµν, Γλµν = Aλµν Θµaν = Γµaν − Γνaµ

The gravitational Higgs phenomenon GraviGUT The main question NCG

Vierbein gauge γab = ηab unbroken group AutSO(3,1)M gµν = θaµ θbν ηab ∆λab = Aλab + Aλba

The gravitational Higgs phenomenon GraviGUT The main question NCG

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”

The gravitational Higgs phenomenon GraviGUT The main question NCG

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 gravitational Higgs phenomenon GraviGUT The main question NCG

The Higgs Mechanism v.I flat background: A = 0, θ = 1, γ = η Θµaν = Aµaν − Aνaµ ∆µab = Aµab + Aµba Fµνa

b

= ∂µAνa

b − ∂νAµa b + Aµac Aνc b − Aνac Aµc b

S contains 1 2

• d4x
• | det g| M2 Q(A, A)

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

The gravitational Higgs phenomenon GraviGUT The main question NCG

Levi–Civita Connection given θ, γ, there is a unique ¯ A s.t. ¯ Θ = 0, ¯ ∆ = 0 ¯ Aabc = 1 2

• Eabc + Ecab − Ebac
• + 1

2

• Cabc + Cbac − Ccab
• where

Eabc = θ−1aλ ∂λκbc Cabc = γad θd λ

• θ−1

b µ ∂µθ−1cλ − θ−1cµ ∂µθ−1 b λ

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 gravitational Higgs phenomenon GraviGUT The main question NCG

The Higgs Mechanism v.II Θµaν = Φµaν − Φνaµ ∆µab = Φµab + Φµba Fµνa

b

= ¯ Fµνa

b + ¯

∇µφνa

b − ¯

∇νφµa

b + φµac φνc b − φνac φµc b

therefore SP(A, γ, θ) = SP(¯ A + Φ, γ, θ) = SH(γ, θ) + SQ(Φ, γ, θ) where SQ(φ, γ, θ) = 1 2

• d4x
• | det g| M2

P QP(Φ, Φ)

and Sm(φ, γ, θ) = 1 2

• d4x
• | det g| M2 Qm(Φ, Φ)

The gravitational Higgs phenomenon GraviGUT The main question NCG

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}

The gravitational Higgs phenomenon GraviGUT The main question NCG

Gravitational Higgs Phenomenon gravity is a gauge theory of GL(4) with two Goldstone bosons there are two unitary gauges Higgsless 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

The gravitational Higgs phenomenon GraviGUT The main question NCG

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

The gravitational Higgs phenomenon GraviGUT The main question NCG

Grand Unification use Higgs phenomenon with G1 × G2 ⊂ G to do list: identify GUT group G fit particles in irreps of G write G-invariant action explain symmetry breaking (select order parameter, orbit, potential) check that new particles not seen at low energy have high mass

The gravitational Higgs phenomenon GraviGUT The main question NCG

Grand unification: SO(10) (eL , νL , eR , νR , ur,g,b

L

, dr,g,b

L

, ur,g,b

R

, dr,g,b

R

) 16 complex 2 component Weyl spinors of Lorentz 4 doublets and 8 singlets of SU(2)L Repeat three times. (nν = 2.984 ± 0.008 measured at LEP) Fit exactly in the 16 of SO(10)! explains hypercharge assignments

The gravitational Higgs phenomenon GraviGUT The main question NCG

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

The gravitational Higgs phenomenon GraviGUT The main question NCG

GraviGUT I use gravitational Higgs phenomenon to construct unified theory of gravity and all other interactions. to do list: identify GraviGUT group G fit particles in irreps of G write G-invariant action explain symmetry breaking (select order parameter, orbit, potential) check that new particles not seen at low energy have high mass

The gravitational Higgs phenomenon GraviGUT The main question NCG

GraviGUT II G1 = SO(1, 3), G2 = SO(10), = ⇒ G = SO(1, 13)

• r

G = SO(3, 11) keep dimM=4, enlarge fibers of E to have dimension N > 4

• rder parameter is soldering form

γ = η 1N−4

• ,

θ is 4 × N matrix, e.g. θ = 14

The gravitational Higgs phenomenon GraviGUT The main question NCG

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 is η ∈ 2C × 16C of SO(3, 1) × SO(10)

The gravitational Higgs phenomenon GraviGUT The main question NCG

Fermions II Let BΣ∗

ij = ΣijB and ψc = Bψ∗. Define ψ± by (ψ±)c = ±ψ±

(Majorana spinors) Define ψL/R by ˆ γψL/R = ∓ψL/R (Weyl spinors). In signature (3, 11) [ˆ γ, B] = 0 so we can define Majorana-Weyl spinors ψL/R±. These have 64 real components. Decomposing ψL+ under SO(3, 1) × SO(10) ⊂ SO(3, 11) we find it is equivalent to η.: 64R = 2C × 16C Remark: SO(1, 13) has Weyl 64C 64C = 2C × 16C + 2C × 16C

The gravitational Higgs phenomenon GraviGUT The main question NCG

Fermions III DµψL+ =

• ∂µ + 1

2Aij

µΣ(3,11) L ij

• ψL+

let Σ†

ijA = −A Σij

then ψ†

L+(Aγi)LDψL+ is one-form in 14 of SO(3, 11)

S =

• ψ†

L+(Aγi)LDψL+ ∧ θj ∧ θk ∧ θℓ φijkℓ .

The gravitational Higgs phenomenon GraviGUT The main question NCG

Fermions IV Assuming the following VEVs: φmnrs = ǫmnrs φijkℓ = 0

• therwise

θm

µ = Memµ

θa

µ = 0

• therwise
• ne gets

S =

• d4x√g η†σµ∇µη ,

where now ∇µ = D(4+10)

µ

= ∂µ + 1 2Γab

µ Σ(3,1) ab

+ 1 2Aab

µ (10)Σ(10) ab

The gravitational Higgs phenomenon GraviGUT The main question NCG

GraviGUT III Gravitational Higgs phenomenon: A = A(4) H HT 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).

The gravitational Higgs phenomenon GraviGUT The main question NCG

Status of GraviGUT kinematics well understood fermionic content and dynamics ok bosonic action for broken phase can be written hard to write action that works in both phases

The gravitational Higgs phenomenon GraviGUT The main question NCG

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?

The gravitational Higgs phenomenon GraviGUT The main question NCG

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

The gravitational Higgs phenomenon GraviGUT The main question NCG

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 and Gauge Interactions” JHEP 1603 (2016) 020, arXiv:1602.02295

Recently extended to GUTs

G.K.Karananas, M. Shaposhnikov, “Gauge coupling unification without leptoquarks” arXiv:1703.02964 [hep-ph]

The gravitational Higgs phenomenon GraviGUT The main question NCG

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

• G. Lisi, L. Smolin, S. Speziale, J.Phys. A43 (2010) 445401

arXiv:1004.4866 [gr-qc]

The gravitational Higgs phenomenon GraviGUT The main question NCG

Conformal models S =

• d4x√g

1 2(∇φ)2 − V(φ) + ξφ2R

• →

VEV of φ is the Planck mass

The gravitational Higgs phenomenon GraviGUT The main question NCG

Ad-hoc models Quantum graphity

• T. Konopka, F

. Markopoulou, S. Severini Phys.Rev. D77 (2008) 104029 arXiv:0801.0861 [hep-th] Self-organizing networks

• C. 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]

The gravitational Higgs phenomenon GraviGUT The main question NCG

Self-consistent mean-field theory Assuming θ = ¯ θ = 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).

The gravitational Higgs phenomenon GraviGUT The main question NCG

Bi-metric approach/asymptotic safety Bi-metric actions appear in the functional RG approach to quantum 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 Higgs phase. Unlikely to have access to the “unbroken” phase.

The gravitational Higgs phenomenon GraviGUT The main question NCG

Lattice approaches Lattice simulations of gravity see rich phase structure Regge calculus Euclidean dynamical triangulations Causal dynamical triangulations

The gravitational Higgs phenomenon GraviGUT The main question NCG

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]

The gravitational Higgs phenomenon GraviGUT The main question NCG

Fuzzy geometries Subclass of matrix geometries Spectral 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 m2

and A = 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

Γi

Lψ ⊗ [Li, m] +

• i

Γi

Hψ ⊗ {Hi, m}

where Γi ∈ C(p, q), Γi

L, Li antihermitian, Γi H, Hi hermitian.

The gravitational Higgs phenomenon GraviGUT The main question NCG

Path integral Z =

• (dD)e−S(D)

S(D) = g2 trD2 + g4 trD4

The gravitational Higgs phenomenon GraviGUT The main question NCG

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 H2 i

calculated by Monte Carlo method

P .Labus and R.P ., to appear

The gravitational Higgs phenomenon GraviGUT The main question NCG

TYPE (1,0) D = {H, ·} S = g4

• 2NTrH4 + 8TrHTrH3 + 6(TrH)2

+g2

• 2NTrH2 + 2(TrH)2

single-trace part of the action solvable

• E. 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

The gravitational Higgs phenomenon GraviGUT The main question NCG

TYPE (1, 1) Phase transition at g2 ≈ −2.4

The gravitational Higgs phenomenon GraviGUT The main question NCG

TYPE (1, 1)

• 4
• 2

2 4 0.1 0.2 0.3 0.4 0.5

g2 = −2.2, −2.4, −2.6

The gravitational Higgs phenomenon GraviGUT The main question NCG

TYPE (1, 1)

50 100 150 200 2 4 6 8 10

g2 = −2.2, −2.4, −2.6

The gravitational Higgs phenomenon GraviGUT The main question NCG

TYPE (3, 0)

The gravitational Higgs phenomenon GraviGUT The main question NCG

TYPE (3, 0)

The gravitational Higgs phenomenon GraviGUT The main question NCG

Conclusions A quantum theory of spacetime has to generate an extended 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 give precisely such a tool. There are hints from Monte Carlo simulations that such models have a phase transition. Simplest models seem to give low-dimensional geometries. Hopefully some such model can produce d = 4 Search just begun!