Intro to LQG Simone Speziale Centre de Physique Theorique de - PowerPoint PPT Presentation

Intro to LQG Simone Speziale Centre de Physique Theorique de Luminy, Marseille, France LAPP-LAPTH 25-02-2012

Outline Motivations SU(2) singlets and polyhedra Applications Conclusions Speziale — Introduction to Loop quantum gravity 2/28

Outline Motivations SU(2) singlets and polyhedra Applications Conclusions Speziale — Introduction to Loop quantum gravity Motivations 3/28

Motivations Einstein’s equations: R µν ( g ) − 1 2 g µν R ( g ) = 8 πG c 4 T µν The source of spacetime curvature is the energy-momentum tensor of matter What is the response of spacetime in situations where the quantum nature of matter is dominant? Speziale — Introduction to Loop quantum gravity Motivations 4/28

Motivations Einstein’s equations: R µν ( g ) − 1 2 g µν R ( g ) = 8 πG c 4 T µν − Λ g µν The source of spacetime curvature is the energy-momentum tensor of matter What is the response of spacetime in situations where the quantum nature of matter is dominant? Speziale — Introduction to Loop quantum gravity Motivations 4/28

Motivations Einstein’s equations: R µν ( g ) − 1 2 g µν R ( g ) = 8 πG c 4 T µν − Λ g µν The source of spacetime curvature is the energy-momentum tensor of matter What is the response of spacetime in situations where the quantum nature of matter is dominant? Structure of equations A ( g ) ∂ 2 g + B ( g )( ∂g ) 2 + C ( g ) = 8 πG c 4 T e-m analogy • gauge part: diffeos A µ �→ A µ + ∂ µ λ • constrained part: Newton’s law ∇ · E = ρ • degrees of freedom: Two Two spin-1 polarizations Speziale — Introduction to Loop quantum gravity Motivations 4/28

Perturbative expansion Perturbative approach: g µν = η µν + h µν = ⇒ Two spin-2 polarizations, gravitational waves Speziale — Introduction to Loop quantum gravity Motivations 5/28

Perturbative expansion Perturbative approach: g µν = η µν + h µν = ⇒ Two spin-2 polarizations, gravitational waves Key to quantization: the splitting introduces a a background spacetime, and a quadratic term in the action. = ⇒ tools of quantum field theory become available However! • Goroff and Sagnotti (’86), Van de Ven (’91): As long since suspected, general relativity is not a perturbatively renormalizable quantum field theory = ⇒ only valid as an effective field theory Speziale — Introduction to Loop quantum gravity Motivations 5/28

Perturbative expansion Perturbative approach: g µν = η µν + h µν = ⇒ Two spin-2 polarizations, gravitational waves Key to quantization: the splitting introduces a a background spacetime, and a quadratic term in the action. = ⇒ tools of quantum field theory become available However! • Goroff and Sagnotti (’86), Van de Ven (’91): As long since suspected, general relativity is not a perturbatively renormalizable quantum field theory = ⇒ only valid as an effective field theory • Could the problem be in the perturbative treatment, rather than in the theory itself? • Maybe the problem lies in this splitting: can one quantize the full gravitational field at once? Speziale — Introduction to Loop quantum gravity Motivations 5/28

Perturbative expansion Perturbative approach: g µν = η µν + h µν = ⇒ Two spin-2 polarizations, gravitational waves Key to quantization: the splitting introduces a a background spacetime, and a quadratic term in the action. = ⇒ tools of quantum field theory become available However! • Goroff and Sagnotti (’86), Van de Ven (’91): As long since suspected, general relativity is not a perturbatively renormalizable quantum field theory = ⇒ only valid as an effective field theory • Could the problem be in the perturbative treatment, rather than in the theory itself? • Maybe the problem lies in this splitting: can one quantize the full gravitational field at once? Case for background-independence Speziale — Introduction to Loop quantum gravity Motivations 5/28

A paradigm shift kinematics dynamics QFT: | p i , h i � quanta: momenta, helicities, etc. Feynman diagrams Speziale — Introduction to Loop quantum gravity Motivations 6/28

A paradigm shift kinematics dynamics QFT: | p i , h i � quanta: momenta, helicities, etc. Feynman diagrams At the Planck scale: Speziale — Introduction to Loop quantum gravity Motivations 6/28

A paradigm shift kinematics dynamics QFT: | p i , h i � quanta: momenta, helicities, etc. Feynman diagrams everything takes place in spacetime ⇒ quanta make up space and evolve into spacetime At the Planck scale: Speziale — Introduction to Loop quantum gravity Motivations 6/28

A paradigm shift kinematics dynamics QFT: | p i , h i � quanta: momenta, helicities, etc. Feynman diagrams everything takes place in spacetime ⇒ quanta make up space and evolve into spacetime kinematics dynamics LQG: | Γ , j e , i v � quanta: areas and volumes spin foams Speziale — Introduction to Loop quantum gravity Motivations 6/28

A paradigm shift kinematics dynamics QFT: | p i , h i � quanta: momenta, helicities, etc. Feynman diagrams everything takes place in spacetime ⇒ quanta make up space and evolve into spacetime kinematics dynamics LQG: | Γ , j e , i v � quanta: areas and volumes spin foams how do we recover semiclassical physics on a smooth spacetime? Speziale — Introduction to Loop quantum gravity Motivations 6/28

Stating the problem • LQG is a continuum theory with well-defined and interesting kinematics (spin networks, discrete spectra of geometric operators, etc.) • Models for the dynamics exist • Main open problem: how to test the theory and extract low-energy physics from it Why is it so hard? The quanta are exotic Speziale — Introduction to Loop quantum gravity Motivations 7/28

Stating the problem • LQG is a continuum theory with well-defined and interesting kinematics (spin networks, discrete spectra of geometric operators, etc.) • Models for the dynamics exist • Main open problem: how to test the theory and extract low-energy physics from it Why is it so hard? The quanta are exotic • photons − → electromagnetic waves • LQG quantum geometries − → smooth classical geometries = ⇒ Speziale — Introduction to Loop quantum gravity Motivations 7/28

Stating the problem • LQG is a continuum theory with well-defined and interesting kinematics (spin networks, discrete spectra of geometric operators, etc.) • Models for the dynamics exist • Main open problem: how to test the theory and extract low-energy physics from it Why is it so hard? The quanta are exotic • photons − → electromagnetic waves • LQG quantum geometries − → smooth classical geometries ◮ discrete ◮ non-commutative ◮ distributional (defined on graphs) = ⇒ Speziale — Introduction to Loop quantum gravity Motivations 7/28

Stating the problem • LQG is a continuum theory with well-defined and interesting kinematics (spin networks, discrete spectra of geometric operators, etc.) • Models for the dynamics exist • Main open problem: how to test the theory and extract low-energy physics from it Why is it so hard? The quanta are exotic • photons − → electromagnetic waves • LQG quantum geometries − → smooth classical geometries ◮ discrete ◮ non-commutative ◮ distributional (defined on graphs) = ⇒ Speziale — Introduction to Loop quantum gravity Motivations 7/28

Stating the problem • LQG is a continuum theory with well-defined and interesting kinematics (spin networks, discrete spectra of geometric operators, etc.) • Models for the dynamics exist • Main open problem: how to test the theory and extract low-energy physics from it Why is it so hard? The quanta are exotic • photons − → electromagnetic waves • LQG quantum geometries − → smooth classical geometries ◮ discrete ◮ non-commutative ◮ distributional (defined on graphs) = ⇒ Aim of the talk: showing the link between LQG on a fixed graph and a notion of discrete geometry Work in collaboration with L. Freidel, C. Rovelli, E. Bianchi and P. Don´ a Speziale — Introduction to Loop quantum gravity Motivations 7/28

Fundamental coupling constants Working hypothesis: the connection as a fundamental (and independent) variable 1 g µν �→ ( g µν , Γ ρ µν ) Lowest dimension operators and their coupling constants: √− g, √− gg µν R µν (Γ) , Λ 1 G G 1 more precisely: ( e I µ , ω IJ µ ) . Speziale — Introduction to Loop quantum gravity Motivations 8/28

Fundamental coupling constants Working hypothesis: the connection as a fundamental (and independent) variable 1 g µν �→ ( g µν , Γ ρ µν ) Lowest dimension operators and their coupling constants: √− g, √− gg µν R µν (Γ) , ǫ µνρσ R µνρσ (Γ) , Λ 1 1 G G γG • Classically irrelevant in the absence of torsion: Γ ρ � ρ ⇒ ǫ µνρσ R µνρσ (Γ( e )) = 0 � µν = = µν 1 more precisely: ( e I µ , ω IJ µ ) . Speziale — Introduction to Loop quantum gravity Motivations 8/28