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Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary General Relativity as an Effective Field Theory Sven Faller Theoretical Physics 1 University of Siegen Theory Seminar 18.12.2006 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary table of contents 1 Introduction 2 Quantum Gravity 3 Effective Field Theory of Gravity 4 Leading Quantum Corrections 5 Evaluation of the Vertex Corrections 6 Gravitational Potential 7 Potential Definitions 8 Summary Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation all known field theories: quantum field theories gravity quantization - Feynman´s Gedankenexperiment gravity must be a quantum field theory problem: consistent quantization method unknown Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltman present energies, quantum gravity non-renormalizable low-energy predictions independent of high-energy influence Donoghue: possible solution Effective Field Theory of Gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation of Quantization Feynman´s Gedankenexperiment: two-slit diffraction experiment with gravity detector characteristic for a quantum field ⇒ should be described by an amplitude rather than a probability Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Motivation of Quantization Feynman´s Gedankenexperiment: two-slit diffraction experiment with gravity detector characteristic for a quantum field ⇒ should be described by an amplitude rather than a probability Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Laws (1687) law of inertia no external force : ddt~r = ~v = const. ⇒ inertial frame of reference (IS) second law force ∝ inertia mass mi ⇒ ~ = mi · ~a. F third law actio est reactio Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Laws (1687) law of inertia no external force : ddt~r = ~v = const. ⇒ inertial frame of reference (IS) second law force ∝ inertia mass mi ⇒ ~ = mi · ~a. F third law actio est reactio Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Laws (1687) law of inertia no external force : ddt~r = ~v = const. ⇒ inertial frame of reference (IS) second law force ∝ inertia mass mi ⇒ ~ = mi · ~a. F third law actio est reactio Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Relativity three different masses: inertia mass mi , passive gravitational mass mG and active gravitational mass MG third law: passive and active gravitational mass equal force of gravity ~ 12 (~r ) = −G m1 m2 ~r1 − ~r2 F |~r1 − ~r2 |3 problem: equality of inertia and passive masses experimental measurements: verification of equality, bases for Einstein’s Principle of Equivalence Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Relativity three different masses: inertia mass mi , passive gravitational mass mG and active gravitational mass MG third law: passive and active gravitational mass equal force of gravity ~ 12 (~r ) = −G m1 m2 ~r1 − ~r2 F |~r1 − ~r2 |3 problem: equality of inertia and passive masses experimental measurements: verification of equality, bases for Einstein’s Principle of Equivalence Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Relativity three different masses: inertia mass mi , passive gravitational mass mG and active gravitational mass MG third law: passive and active gravitational mass equal force of gravity ~ 12 (~r ) = −G m1 m2 ~r1 − ~r2 F |~r1 − ~r2 |3 problem: equality of inertia and passive masses experimental measurements: verification of equality, bases for Einstein’s Principle of Equivalence Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Relativity three different masses: inertia mass mi , passive gravitational mass mG and active gravitational mass MG third law: passive and active gravitational mass equal force of gravity ~ 12 (~r ) = −G m1 m2 ~r1 − ~r2 F |~r1 − ~r2 |3 problem: equality of inertia and passive masses experimental measurements: verification of equality, bases for Einstein’s Principle of Equivalence Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Newton’s Relativity three different masses: inertia mass mi , passive gravitational mass mG and active gravitational mass MG third law: passive and active gravitational mass equal force of gravity ~ 12 (~r ) = −G m1 m2 ~r1 − ~r2 F |~r1 − ~r2 |3 problem: equality of inertia and passive masses experimental measurements: verification of equality, bases for Einstein’s Principle of Equivalence Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Einstein’s Special Relativity Newton: Galilei transformations between IS Einstein 1905: Newton’s Theory must be specialized by universality of the velocity of light in all frames x 7−→ x 0 = Λ x + a (Lorentz transformation) Postulate general transformation for the line element must satisfy ds2 = ηαβ dx µ dx ν = c2 dt 2 − d ~x 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Einstein’s Special Relativity Newton: Galilei transformations between IS Einstein 1905: Newton’s Theory must be specialized by universality of the velocity of light in all frames x 7−→ x 0 = Λ x + a (Lorentz transformation) Postulate general transformation for the line element must satisfy ds2 = ηαβ dx µ dx ν = c2 dt 2 − d ~x 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Einstein’s Special Relativity Newton: Galilei transformations between IS Einstein 1905: Newton’s Theory must be specialized by universality of the velocity of light in all frames x 7−→ x 0 = Λ x + a (Lorentz transformation) Postulate general transformation for the line element must satisfy ds2 = ηαβ dx µ dx ν = c2 dt 2 − d ~x 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity General Relativity Einstein (1916): Die Grundlagen der allgemeinen Relativitätstheorie. Ann. d. Physik, 322(10):891-921 Newton: space R3 and parameter time Rt Einstein : new relations between space-time and mass ⇒ curved space-time mannifold curvature of space = measure for mass: „matter tells space how to curve, and space tells matter how to move“ (Miesner, 1973) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity General Relativity Einstein (1916): Die Grundlagen der allgemeinen Relativitätstheorie. Ann. d. Physik, 322(10):891-921 Newton: space R3 and parameter time Rt Einstein : new relations between space-time and mass ⇒ curved space-time mannifold curvature of space = measure for mass: „matter tells space how to curve, and space tells matter how to move“ (Miesner, 1973) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity General Relativity Einstein (1916): Die Grundlagen der allgemeinen Relativitätstheorie. Ann. d. Physik, 322(10):891-921 Newton: space R3 and parameter time Rt Einstein : new relations between space-time and mass ⇒ curved space-time mannifold curvature of space = measure for mass: „matter tells space how to curve, and space tells matter how to move“ (Miesner, 1973) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity General Relativity Einstein (1916): Die Grundlagen der allgemeinen Relativitätstheorie. Ann. d. Physik, 322(10):891-921 Newton: space R3 and parameter time Rt Einstein : new relations between space-time and mass ⇒ curved space-time mannifold curvature of space = measure for mass: „matter tells space how to curve, and space tells matter how to move“ (Miesner, 1973) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity General Relativity Einstein (1916): Die Grundlagen der allgemeinen Relativitätstheorie. Ann. d. Physik, 322(10):891-921 Newton: space R3 and parameter time Rt Einstein : new relations between space-time and mass ⇒ curved space-time mannifold curvature of space = measure for mass: „matter tells space how to curve, and space tells matter how to move“ (Miesner, 1973) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Principle of Equivalence „At every space-time point in an arbitrary gravitational field it is possible to choose a „locally inertial coordinate system“ such that, within sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in absence of gravitation.“ relation between accelerated local IS x α and static frame of reference x̄ µ described by metric tensor, which leaves line element ds2 invariant: gµν = ηαβ Sven Faller ∂x α ∂x β ∂ x̄ µ ∂ x̄ ν General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Principle of Equivalence „At every space-time point in an arbitrary gravitational field it is possible to choose a „locally inertial coordinate system“ such that, within sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in absence of gravitation.“ relation between accelerated local IS x α and static frame of reference x̄ µ described by metric tensor, which leaves line element ds2 invariant: gµν = ηαβ Sven Faller ∂x α ∂x β ∂ x̄ µ ∂ x̄ ν General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Principle of Equivalence „At every space-time point in an arbitrary gravitational field it is possible to choose a „locally inertial coordinate system“ such that, within sufficiently small region of the point in question, the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in absence of gravitation.“ relation between accelerated local IS x α and static frame of reference x̄ µ described by metric tensor, which leaves line element ds2 invariant: gµν = ηαβ Sven Faller ∂x α ∂x β ∂ x̄ µ ∂ x̄ ν General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Principle of General Covariance General laws of nature should be expressed in terms of equations which are true in all frames of reference and transform covariantly by arbitrary substitutions. general coordinate transformation: x 7−→ x 0 = f (x) Principle of General Covariance is not an invariance principle like Principle of Galilean or Special Relativity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Principle of General Covariance General laws of nature should be expressed in terms of equations which are true in all frames of reference and transform covariantly by arbitrary substitutions. general coordinate transformation: x 7−→ x 0 = f (x) Principle of General Covariance is not an invariance principle like Principle of Galilean or Special Relativity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Principle of General Covariance General laws of nature should be expressed in terms of equations which are true in all frames of reference and transform covariantly by arbitrary substitutions. general coordinate transformation: x 7−→ x 0 = f (x) Principle of General Covariance is not an invariance principle like Principle of Galilean or Special Relativity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Recall: Lorentz Invariance global coordinate change: x µ 7−→ x 0µ = Λµν x ν Minkowski metric ηµν invariant fields transform as scalars, vectors, etc. φ(x) 7−→ φ0 (x 0 ) = φ(x) Aµ (x) 7−→ A0µ (x) = Λµν (x) Aν (x) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Recall: Lorentz Invariance global coordinate change: x µ 7−→ x 0µ = Λµν x ν Minkowski metric ηµν invariant fields transform as scalars, vectors, etc. φ(x) 7−→ φ0 (x 0 ) = φ(x) Aµ (x) 7−→ A0µ (x) = Λµν (x) Aν (x) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Recall: Lorentz Invariance global coordinate change: x µ 7−→ x 0µ = Λµν x ν Minkowski metric ηµν invariant fields transform as scalars, vectors, etc. φ(x) 7−→ φ0 (x 0 ) = φ(x) Aµ (x) 7−→ A0µ (x) = Λµν (x) Aν (x) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Covariant Derivative local coordinate changes require covariant derivative: Dµ Aν = ∂µ Aν + Γνµλ Aλ = Aν ,µ + Γνµλ Aλ ≡ Aν ;µ affine connection Γλµν (geometric interpretation) for scalar fields: Φ;µ ≡ Φ,µ = ∂µ Φ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Covariant Derivative local coordinate changes require covariant derivative: Dµ Aν = ∂µ Aν + Γνµλ Aλ = Aν ,µ + Γνµλ Aλ ≡ Aν ;µ affine connection Γλµν (geometric interpretation) for scalar fields: Φ;µ ≡ Φ,µ = ∂µ Φ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Covariant Derivative local coordinate changes require covariant derivative: Dµ Aν = ∂µ Aν + Γνµλ Aλ = Aν ,µ + Γνµλ Aλ ≡ Aν ;µ affine connection Γλµν (geometric interpretation) for scalar fields: Φ;µ ≡ Φ,µ = ∂µ Φ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Riemann Space (R4 ) metric definition: ds2 = gµν (x) dx µ dx ν affine connection: Γλµν = 12 g λσ ∂µ gνσ + ∂ν gνσ − ∂σ gµν λ Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rµσν Riemann curvature tensor: R λµσν = Γλµν,σ − Γλµσ,ν + Γλτσ Γτµν − Γλτν Γτµσ Ricci tensor: Rµν = R λµλν ≡ Rνµ Ricci scalar: R = g µν Rµν Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Riemann Space (R4 ) metric definition: ds2 = gµν (x) dx µ dx ν affine connection: Γλµν = 12 g λσ ∂µ gνσ + ∂ν gνσ − ∂σ gµν λ Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rµσν Riemann curvature tensor: R λµσν = Γλµν,σ − Γλµσ,ν + Γλτσ Γτµν − Γλτν Γτµσ Ricci tensor: Rµν = R λµλν ≡ Rνµ Ricci scalar: R = g µν Rµν Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Riemann Space (R4 ) metric definition: ds2 = gµν (x) dx µ dx ν affine connection: Γλµν = 12 g λσ ∂µ gνσ + ∂ν gνσ − ∂σ gµν λ Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rµσν Riemann curvature tensor: R λµσν = Γλµν,σ − Γλµσ,ν + Γλτσ Γτµν − Γλτν Γτµσ Ricci tensor: Rµν = R λµλν ≡ Rνµ Ricci scalar: R = g µν Rµν Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Riemann Space (R4 ) metric definition: ds2 = gµν (x) dx µ dx ν affine connection: Γλµν = 12 g λσ ∂µ gνσ + ∂ν gνσ − ∂σ gµν λ Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rµσν Riemann curvature tensor: R λµσν = Γλµν,σ − Γλµσ,ν + Γλτσ Γτµν − Γλτν Γτµσ Ricci tensor: Rµν = R λµλν ≡ Rνµ Ricci scalar: R = g µν Rµν Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Riemann Space (R4 ) metric definition: ds2 = gµν (x) dx µ dx ν affine connection: Γλµν = 12 g λσ ∂µ gνσ + ∂ν gνσ − ∂σ gµν λ Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rµσν Riemann curvature tensor: R λµσν = Γλµν,σ − Γλµσ,ν + Γλτσ Γτµν − Γλτν Γτµσ Ricci tensor: Rµν = R λµλν ≡ Rνµ Ricci scalar: R = g µν Rµν Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Fundamentals General Relativity Riemann Space (R4 ) metric definition: ds2 = gµν (x) dx µ dx ν affine connection: Γλµν = 12 g λσ ∂µ gνσ + ∂ν gνσ − ∂σ gµν λ Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rµσν Riemann curvature tensor: R λµσν = Γλµν,σ − Γλµσ,ν + Γλτσ Γτµν − Γλτν Γτµσ Ricci tensor: Rµν = R λµλν ≡ Rνµ Ricci scalar: R = g µν Rµν Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization General relativity as a gauge theory (Sabbata 1985) Poincaré group is non abelian cf. Yang-Mills theory 1 1 a a µν F = − trF 2 Lgauge = − Fµν 4 2 gravity: introduction of vierbein- or tetrad fields eµλ̄ Lgauge = − e µ ν 2 √ ω ) ≡ 2 −g R e e R λ̄σ̄ (ω 2g λ̄ σ̄ µν κ with g = det[gµν ] and κ2 = 32πG. Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization General relativity as a gauge theory (Sabbata 1985) Poincaré group is non abelian cf. Yang-Mills theory 1 1 a a µν F = − trF 2 Lgauge = − Fµν 4 2 gravity: introduction of vierbein- or tetrad fields eµλ̄ Lgauge = − e µ ν 2 √ ω ) ≡ 2 −g R e e R λ̄σ̄ (ω 2g λ̄ σ̄ µν κ with g = det[gµν ] and κ2 = 32πG. Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization General relativity as a gauge theory (Sabbata 1985) Poincaré group is non abelian cf. Yang-Mills theory 1 1 a a µν F = − trF 2 Lgauge = − Fµν 4 2 gravity: introduction of vierbein- or tetrad fields eµλ̄ Lgauge = − e µ ν 2 √ ω ) ≡ 2 −g R e e R λ̄σ̄ (ω 2g λ̄ σ̄ µν κ with g = det[gµν ] and κ2 = 32πG. Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Background Field Method introduced by ’t Hooft and Veltmann (1974) gravitational field expanded about smooth background metric ḡµν gµν = ḡµν + κhµν g µν = ḡ µν − κhµν + κ2 hλµ hλν + . . . classical equations of motion: ḡµν quantum field hµν : all dynamical information Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Background Field Method introduced by ’t Hooft and Veltmann (1974) gravitational field expanded about smooth background metric ḡµν gµν = ḡµν + κhµν g µν = ḡ µν − κhµν + κ2 hλµ hλν + . . . classical equations of motion: ḡµν quantum field hµν : all dynamical information Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Background Field Method introduced by ’t Hooft and Veltmann (1974) gravitational field expanded about smooth background metric ḡµν gµν = ḡµν + κhµν g µν = ḡ µν − κhµν + κ2 hλµ hλν + . . . classical equations of motion: ḡµν quantum field hµν : all dynamical information Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Background Field Method introduced by ’t Hooft and Veltmann (1974) gravitational field expanded about smooth background metric ḡµν gµν = ḡµν + κhµν g µν = ḡ µν − κhµν + κ2 hλµ hλν + . . . classical equations of motion: ḡµν quantum field hµν : all dynamical information Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Gravitational Action Einstein-Hilbert action Svac = R matter action Sm = d 4 x Lm Sgr = Svac + Sm = Z R 4 d 4x d x √ √ −g 2 κ2 R 2 −g 2 R + Lm κ further gauge invariant terms √ 2 2 µν L = −g λ + 2 R + c1 R + c2 Rµν R + . . . + Lm κ upper bound: constants c1 , c2 < 1074 (Stelle 1978) λ ≡ −8πGΛ, cosmological constant Λ ≡ 0 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Gravitational Action Einstein-Hilbert action Svac = R matter action Sm = d 4 x Lm Sgr = Svac + Sm = Z R 4 d 4x d x √ √ −g 2 κ2 R 2 −g 2 R + Lm κ further gauge invariant terms √ 2 2 µν L = −g λ + 2 R + c1 R + c2 Rµν R + . . . + Lm κ upper bound: constants c1 , c2 < 1074 (Stelle 1978) λ ≡ −8πGΛ, cosmological constant Λ ≡ 0 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Gravitational Action Einstein-Hilbert action Svac = R matter action Sm = d 4 x Lm Sgr = Svac + Sm = Z R 4 d 4x d x √ √ −g 2 κ2 R 2 −g 2 R + Lm κ further gauge invariant terms √ 2 2 µν L = −g λ + 2 R + c1 R + c2 Rµν R + . . . + Lm κ upper bound: constants c1 , c2 < 1074 (Stelle 1978) λ ≡ −8πGΛ, cosmological constant Λ ≡ 0 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Gravitational Action Einstein-Hilbert action Svac = R matter action Sm = d 4 x Lm Sgr = Svac + Sm = Z R 4 d 4x d x √ √ −g 2 κ2 R 2 −g 2 R + Lm κ further gauge invariant terms √ 2 2 µν L = −g λ + 2 R + c1 R + c2 Rµν R + . . . + Lm κ upper bound: constants c1 , c2 < 1074 (Stelle 1978) λ ≡ −8πGΛ, cosmological constant Λ ≡ 0 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Gravitational Action Einstein-Hilbert action Svac = R matter action Sm = d 4 x Lm Sgr = Svac + Sm = Z R 4 d 4x d x √ √ −g 2 κ2 R 2 −g 2 R + Lm κ further gauge invariant terms √ 2 2 µν L = −g λ + 2 R + c1 R + c2 Rµν R + . . . + Lm κ upper bound: constants c1 , c2 < 1074 (Stelle 1978) λ ≡ −8πGΛ, cosmological constant Λ ≡ 0 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Gravitational Action Einstein-Hilbert action Svac = R matter action Sm = d 4 x Lm Sgr = Svac + Sm = Z R 4 d 4x d x √ √ −g 2 κ2 R 2 −g 2 R + Lm κ further gauge invariant terms √ 2 2 µν L = −g λ + 2 R + c1 R + c2 Rµν R + . . . + Lm κ upper bound: constants c1 , c2 < 1074 (Stelle 1978) λ ≡ −8πGΛ, cosmological constant Λ ≡ 0 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Expansion: Vacuum Lagrangian metric expansion: ff p p κ α κ2 α β κ2 ` α ´2 −g = −ḡ 1 − hα − hβ hα + hα + O(h3 ) 2 4 8 Lagrangian expansion » – p 2 p 2 (1) (2) −gR = −ḡ 2 R̄ + Lgr + Lgr + . . . , 2 κ κ ˆ µν ˜ 1 (1) Lgr = hµν ḡ R̄ − 2R̄ µν , κ 1 1 (2) Lgr = Dα hµν Dα hµν − Dα h Dα h + Dα h Dβ hαβ − Dα hµβ Dβ hµα 2 2 „ « ´ ` λ 1 2 1 hνλ − h hµν . + R̄ h − hµν hµν + R̄ µν 2hµ 4 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Expansion: Vacuum Lagrangian metric expansion: ff p p κ α κ2 α β κ2 ` α ´2 −g = −ḡ 1 − hα − hβ hα + hα + O(h3 ) 2 4 8 Lagrangian expansion » – p 2 p 2 (1) (2) −gR = −ḡ 2 R̄ + Lgr + Lgr + . . . , 2 κ κ ˆ µν ˜ 1 (1) Lgr = hµν ḡ R̄ − 2R̄ µν , κ 1 1 (2) Lgr = Dα hµν Dα hµν − Dα h Dα h + Dα h Dβ hαβ − Dα hµβ Dβ hµα 2 2 „ « ´ ` λ 1 2 1 hνλ − h hµν . + R̄ h − hµν hµν + R̄ µν 2hµ 4 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Expansion: Matter Lagrangian e.g. scalar particle: Lm = Lagrangian expansion: √ −g 1 2g µν ∂ φ∂ φ µ ν − 12 m2 φ2 p ¯ ˘ (0) (1) (2) −ḡ Lm + Lm + Lm + . . . ´ 1` = ∂µ φ∂ µ φ − m2 φ2 2 κ = − hµν T µν 2 ` ´ 1 ≡ ∂µ φ∂ µ φ − ḡµν ∂λ φ∂ λ φ − m2 φ2 (energy-momentum-tensor) 2 „ « „ « 2 ˆ ˜ 1 µλ ν 1 κ 1 = κ2 h hλ − hhµν ∂µ φ∂ν φ − hλσ hλσ − hh ∂µ φ ∂ µ φ − m2 φ2 2 4 8 2 Lm = (0) Lm (1) Lm T µν (2) Lm Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Expansion: Matter Lagrangian e.g. scalar particle: Lm = Lagrangian expansion: √ −g 1 2g µν ∂ φ∂ φ µ ν − 12 m2 φ2 p ¯ ˘ (0) (1) (2) −ḡ Lm + Lm + Lm + . . . ´ 1` = ∂µ φ∂ µ φ − m2 φ2 2 κ = − hµν T µν 2 ` ´ 1 ≡ ∂µ φ∂ µ φ − ḡµν ∂λ φ∂ λ φ − m2 φ2 (energy-momentum-tensor) 2 „ « „ « 2 ˆ ˜ 1 µλ ν 1 κ 1 = κ2 h hλ − hhµν ∂µ φ∂ν φ − hλσ hλσ − hh ∂µ φ ∂ µ φ − m2 φ2 2 4 8 2 Lm = (0) Lm (1) Lm T µν (2) Lm Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Einstein Equation Rµν − 1 κ2 gµν R = Tµν 2 4 ḡµν satisfies Einstein equation Lagrangian terms linear in quantum field hµν vanish one-loop order: 2 R̄ (0) + Lm κ2 ff L0 = p −ḡ Lquad = p ˘ (2) (2) ¯ −ḡ Lg + Lgf + Lghost + Lm Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Einstein Equation Rµν − 1 κ2 gµν R = Tµν 2 4 ḡµν satisfies Einstein equation Lagrangian terms linear in quantum field hµν vanish one-loop order: 2 R̄ (0) + Lm κ2 ff L0 = p −ḡ Lquad = p ˘ (2) (2) ¯ −ḡ Lg + Lgf + Lghost + Lm Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Einstein Equation Rµν − 1 κ2 gµν R = Tµν 2 4 ḡµν satisfies Einstein equation Lagrangian terms linear in quantum field hµν vanish one-loop order: 2 R̄ (0) + Lm κ2 ff L0 = p −ḡ Lquad = p ˘ (2) (2) ¯ −ḡ Lg + Lgf + Lghost + Lm Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Einstein Equation Rµν − 1 κ2 gµν R = Tµν 2 4 ḡµν satisfies Einstein equation Lagrangian terms linear in quantum field hµν vanish one-loop order: 2 R̄ (0) + Lm κ2 ff L0 = p −ḡ Lquad = p ˘ (2) (2) ¯ −ḡ Lg + Lgf + Lghost + Lm Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Quantization Problems field equations non linear coupling constant κ has mass dimension coupling grows with energy possible solution: Effective Field Theory separate high enery fluctuations from small quantum fluctuations at ordinary energies Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Quantization Problems field equations non linear coupling constant κ has mass dimension coupling grows with energy possible solution: Effective Field Theory separate high enery fluctuations from small quantum fluctuations at ordinary energies Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Quantization Problems field equations non linear coupling constant κ has mass dimension coupling grows with energy possible solution: Effective Field Theory separate high enery fluctuations from small quantum fluctuations at ordinary energies Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Quantization Quantization Problems field equations non linear coupling constant κ has mass dimension coupling grows with energy possible solution: Effective Field Theory separate high enery fluctuations from small quantum fluctuations at ordinary energies Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Introduction Effective Lagrangian low-energy d.o.f.: hµν + ghost fields + matter fields Z Z[J] = [dφ][dhµν ]eiSeff (φ,ḡ,h,J) p R Seff = d 4 x −ḡ Leff , Leff = Lgr + Lm effective Lagrangian = expansion in powers of hµν (0) (2) (0) (2) (4) Lgr = Lgr + Lgr + Lgr + . . . Lm = Lm + Lm + . . . Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Introduction Effective Lagrangian low-energy d.o.f.: hµν + ghost fields + matter fields Z Z[J] = [dφ][dhµν ]eiSeff (φ,ḡ,h,J) p R Seff = d 4 x −ḡ Leff , Leff = Lgr + Lm effective Lagrangian = expansion in powers of hµν (0) (2) (0) (2) (4) Lgr = Lgr + Lgr + Lgr + . . . Lm = Lm + Lm + . . . Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Introduction Effective Lagrangian low-energy d.o.f.: hµν + ghost fields + matter fields Z Z[J] = [dφ][dhµν ]eiSeff (φ,ḡ,h,J) p R Seff = d 4 x −ḡ Leff , Leff = Lgr + Lm effective Lagrangian = expansion in powers of hµν (0) (2) (0) (2) (4) Lgr = Lgr + Lgr + Lgr + . . . Lm = Lm + Lm + . . . Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Graviton Progpagator second order Lagrangian Lgr harmonic gauge → gauge fixing Lagrangian Lgf γδ 1 αβ quantum field hµν bilinear Lagrangian Lfree ∆−1 gr = − 2 h αβγδ h graviton propagator in harmonic gauge αβ q µν = 1 i η αµ η βν + η αν η βµ − η αβ η µν 2 2 q + i Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Graviton Progpagator second order Lagrangian Lgr harmonic gauge → gauge fixing Lagrangian Lgf γδ 1 αβ quantum field hµν bilinear Lagrangian Lfree ∆−1 gr = − 2 h αβγδ h graviton propagator in harmonic gauge αβ q µν = 1 i η αµ η βν + η αν η βµ − η αβ η µν 2 2 q + i Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Graviton Progpagator second order Lagrangian Lgr harmonic gauge → gauge fixing Lagrangian Lgf γδ 1 αβ quantum field hµν bilinear Lagrangian Lfree ∆−1 gr = − 2 h αβγδ h graviton propagator in harmonic gauge αβ q µν = 1 i η αµ η βν + η αν η βµ − η αβ η µν 2 2 q + i Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Graviton Progpagator second order Lagrangian Lgr harmonic gauge → gauge fixing Lagrangian Lgf γδ 1 αβ quantum field hµν bilinear Lagrangian Lfree ∆−1 gr = − 2 h αβγδ h graviton propagator in harmonic gauge αβ q µν = 1 i η αµ η βν + η αν η βµ − η αβ η µν 2 2 q + i Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Vertex Factors vertex factors at one-loop order q −→ p0 `- p0 p `0 % p Sven Faller k −→ %` &q General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Scalar-Graviton-Vertex vertexZfactor τµν = i 0 ∂ ∂ ∂ d 4 x d 4 x1 d 4 x2 d 4 x3 ei(px1 −p x2 +qx3 ) · ∂φ(x1 ) ∂φ(x2 ) ∂hµν (x3 ) » –ff ` κ αβ 1 γ 2 2´ · − h · ∂α φ(x)∂β φ(x) − ηαβ ∂γ φ(x)∂ φ(x) − m φ(x) 2 scalar-graviton-vertex q p0 iκ 0 0 0 2 µν = − pµ pν + pν pµ − ηµν p · p − m 2 −→ p Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Scalar-Graviton-Vertex vertexZfactor τµν = i 0 ∂ ∂ ∂ d 4 x d 4 x1 d 4 x2 d 4 x3 ei(px1 −p x2 +qx3 ) · ∂φ(x1 ) ∂φ(x2 ) ∂hµν (x3 ) » –ff ` κ αβ 1 γ 2 2´ · − h · ∂α φ(x)∂β φ(x) − ηαβ ∂γ φ(x)∂ φ(x) − m φ(x) 2 scalar-graviton-vertex q p0 iκ 0 0 0 2 µν = − pµ pν + pν pµ − ηµν p · p − m 2 −→ p Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections S-Matrix Feynman diagrams → invariant matrix element iM long range interaction: Mfull = Aq 2 (1 + ακ2 q 2 + βκ2 q 2 ln(−q 2 ) + γκ2 q 2 √m −q 2 +. . . ) R-matrix: R = S − 1 p, pT incoming, pT0 , p0 outgoing momentum: hp0 |R|pi = (2π)4 δ 4 (p0 + pT0 − p − pT ) iM Born approximation: nonrelativistic limit position-space potential V (~r ) = − 1 1 2m1 2m2 Sven Faller Z d 3~q i ~q ·~r e M (2π)3 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections S-Matrix Feynman diagrams → invariant matrix element iM long range interaction: Mfull = Aq 2 (1 + ακ2 q 2 + βκ2 q 2 ln(−q 2 ) + γκ2 q 2 √m −q 2 +. . . ) R-matrix: R = S − 1 p, pT incoming, pT0 , p0 outgoing momentum: hp0 |R|pi = (2π)4 δ 4 (p0 + pT0 − p − pT ) iM Born approximation: nonrelativistic limit position-space potential V (~r ) = − 1 1 2m1 2m2 Sven Faller Z d 3~q i ~q ·~r e M (2π)3 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections S-Matrix Feynman diagrams → invariant matrix element iM long range interaction: Mfull = Aq 2 (1 + ακ2 q 2 + βκ2 q 2 ln(−q 2 ) + γκ2 q 2 √m −q 2 +. . . ) R-matrix: R = S − 1 p, pT incoming, pT0 , p0 outgoing momentum: hp0 |R|pi = (2π)4 δ 4 (p0 + pT0 − p − pT ) iM Born approximation: nonrelativistic limit position-space potential V (~r ) = − 1 1 2m1 2m2 Sven Faller Z d 3~q i ~q ·~r e M (2π)3 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections S-Matrix Feynman diagrams → invariant matrix element iM long range interaction: Mfull = Aq 2 (1 + ακ2 q 2 + βκ2 q 2 ln(−q 2 ) + γκ2 q 2 √m −q 2 +. . . ) R-matrix: R = S − 1 p, pT incoming, pT0 , p0 outgoing momentum: hp0 |R|pi = (2π)4 δ 4 (p0 + pT0 − p − pT ) iM Born approximation: nonrelativistic limit position-space potential V (~r ) = − 1 1 2m1 2m2 Sven Faller Z d 3~q i ~q ·~r e M (2π)3 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections S-Matrix Feynman diagrams → invariant matrix element iM long range interaction: Mfull = Aq 2 (1 + ακ2 q 2 + βκ2 q 2 ln(−q 2 ) + γκ2 q 2 √m −q 2 +. . . ) R-matrix: R = S − 1 p, pT incoming, pT0 , p0 outgoing momentum: hp0 |R|pi = (2π)4 δ 4 (p0 + pT0 − p − pT ) iM Born approximation: nonrelativistic limit position-space potential V (~r ) = − 1 1 2m1 2m2 Sven Faller Z d 3~q i ~q ·~r e M (2π)3 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Expansion: Gravitational Potential lowest order: V (r ) = −G m1r·m2 (Newton) higher order effects: O(v 2 /c 2 ), O(Gm/rc 2 ) general form: G(m1 + m2 ) G m 1 m2 V (r ) = − 1+a· . . . r r c2 dimensional analysis: loop diagrams → extra power of κ2 ∼ G, factor ~ gravitational potential: general form Gm1 m2 G(m1 + m2 ) G~ V (r ) = − 1+α + β 2 3 + ... r rc 2 r c Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Expansion: Gravitational Potential lowest order: V (r ) = −G m1r·m2 (Newton) higher order effects: O(v 2 /c 2 ), O(Gm/rc 2 ) general form: G(m1 + m2 ) G m 1 m2 V (r ) = − 1+a· . . . r r c2 dimensional analysis: loop diagrams → extra power of κ2 ∼ G, factor ~ gravitational potential: general form Gm1 m2 G(m1 + m2 ) G~ V (r ) = − 1+α + β 2 3 + ... r rc 2 r c Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Expansion: Gravitational Potential lowest order: V (r ) = −G m1r·m2 (Newton) higher order effects: O(v 2 /c 2 ), O(Gm/rc 2 ) general form: G(m1 + m2 ) G m 1 m2 V (r ) = − 1+a· . . . r r c2 dimensional analysis: loop diagrams → extra power of κ2 ∼ G, factor ~ gravitational potential: general form Gm1 m2 G(m1 + m2 ) G~ V (r ) = − 1+α + β 2 3 + ... r rc 2 r c Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Expansion: Gravitational Potential lowest order: V (r ) = −G m1r·m2 (Newton) higher order effects: O(v 2 /c 2 ), O(Gm/rc 2 ) general form: G(m1 + m2 ) G m 1 m2 V (r ) = − 1+a· . . . r r c2 dimensional analysis: loop diagrams → extra power of κ2 ∼ G, factor ~ gravitational potential: general form Gm1 m2 G(m1 + m2 ) G~ V (r ) = − 1+α + β 2 3 + ... r rc 2 r c Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Expansion: Gravitational Potential lowest order: V (r ) = −G m1r·m2 (Newton) higher order effects: O(v 2 /c 2 ), O(Gm/rc 2 ) general form: G(m1 + m2 ) G m 1 m2 V (r ) = − 1+a· . . . r r c2 dimensional analysis: loop diagrams → extra power of κ2 ∼ G, factor ~ gravitational potential: general form Gm1 m2 G(m1 + m2 ) G~ V (r ) = − 1+α + β 2 3 + ... r rc 2 r c Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Tree Level p k0 q iM = m1 m2 k = ταβ (k , k 0 ) · iP αβγδ q 2 + i · τγδ (p, p0 ) p0 nonrelativitstic position space potential Z κ2 d 3~q i ~q ·~r 1 κ2 1 V (~r ) = − m1 m2 e = − m1 m2 ~q 2 8 8 4πr (2π)3 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Tree Level p k0 q iM = m1 m2 k = ταβ (k , k 0 ) · iP αβγδ q 2 + i · τγδ (p, p0 ) p0 nonrelativitstic position space potential Z κ2 d 3~q i ~q ·~r 1 κ2 1 V (~r ) = − m1 m2 e = − m1 m2 ~q 2 8 8 4πr (2π)3 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Vertex Corrections - Overview (a) (b) = (c) + + + (d) + (e) Sven Faller + (f) General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections General Form QED: Ward identiy ⇔ vertex: energy conservation ∂µ T µν = 0 momentum conservation: qµ V µν ≡ 0 general vertex form k2 q k1 V µν = h k2 | T µν | k1 i 1 2 µν µ ν 2 ν µ = F1 (q ) k1 k2 + k1 k2 + q g 2 µ ν 2 µν 2 + F2 (q ) q q − g q . Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections General Form QED: Ward identiy ⇔ vertex: energy conservation ∂µ T µν = 0 momentum conservation: qµ V µν ≡ 0 general vertex form k2 q k1 V µν = h k2 | T µν | k1 i 1 2 µν µ ν 2 ν µ = F1 (q ) k1 k2 + k1 k2 + q g 2 µ ν 2 µν 2 + F2 (q ) q q − g q . Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections General Form QED: Ward identiy ⇔ vertex: energy conservation ∂µ T µν = 0 momentum conservation: qµ V µν ≡ 0 general vertex form k2 q k1 V µν = h k2 | T µν | k1 i 1 2 µν µ ν 2 ν µ = F1 (q ) k1 k2 + k1 k2 + q g 2 µ ν 2 µν 2 + F2 (q ) q q − g q . Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Form Factors tree-level limit −−−−−−−−→ normalization condition: F1 ≡ 0 F2 no normalization condition form factors dimensionless Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Form Factors tree-level limit −−−−−−−−→ normalization condition: F1 ≡ 0 F2 no normalization condition form factors dimensionless Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Form Factors tree-level limit −−−−−−−−→ normalization condition: F1 ≡ 0 F2 no normalization condition form factors dimensionless Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Feynman Rules Scattering Potential Vertex Corrections Form Factors tree-level limit −−−−−−−−→ normalization condition: F1 ≡ 0 F2 no normalization condition form factors dimensionless Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary First Loop Diagram k2 q ←− x ` µν m k1 d 4` 1 (2π)4 `2 (` − q)2 [(` − k2 )2 − m2 ] µν · τρσ (k2 − `, k2 , m) τλκ (k1 , k2 − `, m) ταβγδ (`, q) . = iP σραβ iP γδλκ Z i Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Results form factor F1 (q 2 ) result from our Donoghue Akhundov et al. ln(−q 2 ) - 3/4 - 3/4 - 5/4 2 π √ m −q 2 1/16 1/16 - 1/16 form factor F2 (q 2 ) result from our Donoghue Akhundov et al. Sven Faller ln(−q 2 ) 7/3 3 -7/3 2 π √ m −q 2 7/8 7/8 - 7/8 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Results form factor F1 (q 2 ) result from our Donoghue Akhundov et al. ln(−q 2 ) - 3/4 - 3/4 - 5/4 2 π √ m −q 2 1/16 1/16 - 1/16 form factor F2 (q 2 ) result from our Donoghue Akhundov et al. Sven Faller ln(−q 2 ) 7/3 3 -7/3 2 π √ m −q 2 7/8 7/8 - 7/8 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Second Loop Diagram x µν ` k2 m k1 = iP αβλκ iP γδρσ Z Vαβγδ Sven Faller µν d 4 ` τλκρσ (`, q) (2π)4 `2 (` − q)2 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Results form factor F1 (q 2 ) result from our Donoghue Akhundov et al. ln(−q 2 ) 0 0 0 2 π √ m −q 2 0 0 0 form factor F2 (q 2 ) result from our Donoghue Akhundov et al. Sven Faller ln(−q 2 ) -13/3 -13/3 7/3 2 π √ m −q 2 0 0 0 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Results form factor F1 (q 2 ) result from our Donoghue Akhundov et al. ln(−q 2 ) 0 0 0 2 π √ m −q 2 0 0 0 form factor F2 (q 2 ) result from our Donoghue Akhundov et al. Sven Faller ln(−q 2 ) -13/3 -13/3 7/3 2 π √ m −q 2 0 0 0 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Result: Vertex Correction form factors » – 3 1 π2 m κ2 q 2 2 p − ln(−q ) + , 32π 2 4 16 −q 2 – » κ2 m2 7 π2 m 2 p . F2 (q 2 ) = −2 ln(−q ) + 32π 2 8 −q 2 F1 (q 2 ) = 1 + tree-level normalized → factor κ/2i V µν = − «–„ » „ « κ2 q 2 3 iκ 1 π2 m 1 2 µν µ ν 2 ν µ p 1+ − ln(−q ) + k k + k k + q η 1 2 1 2 2 32π 2 4 16 2 −q 2 » – „ « 2m 7 π 1 κ3 m2 − −2 ln(−q 2 ) + p q µ q ν − q 2 η µν . 64π 2 8 2 −q 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Result: Vertex Correction form factors » – 3 1 π2 m κ2 q 2 2 p − ln(−q ) + , 32π 2 4 16 −q 2 – » κ2 m2 7 π2 m 2 p . F2 (q 2 ) = −2 ln(−q ) + 32π 2 8 −q 2 F1 (q 2 ) = 1 + tree-level normalized → factor κ/2i V µν = − «–„ » „ « κ2 q 2 3 iκ 1 π2 m 1 2 µν µ ν 2 ν µ p 1+ − ln(−q ) + k k + k k + q η 1 2 1 2 2 32π 2 4 16 2 −q 2 » – „ « 2m 7 π 1 κ3 m2 − −2 ln(−q 2 ) + p q µ q ν − q 2 η µν . 64π 2 8 2 −q 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Vacuum Polarisation - Diagrams = + (a) + (b) (c) vacuum polarisation tensor Παβγδ → graviton propagator correction ∆αβγδ + ∆αβµν iΠµνρσ ∆ρσγδ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Vacuum Polarization counter-term, graviton self energy and ghost → Veltman and ’t Hooft (1974) matter loop » ´` ´ ´ i κ2 −1 ` α β 1 ` α γ q q − q 2 η αβ q γ q δ − q 2 η γδ − q q − q 2 η αγ 32π 2 4 5 30 – ` ´ ´` ´ 1 ` α δ · q β q δ − q 2 η βδ − q q − q 2 η αδ q β q γ − q 2 η βγ − 2m34 η αβ η γδ 30 ff ´ 2 ` ´ 2 2 αβ ` γδ 2 − m3 η η q − q γ q δ − m32 η γδ η αβ q 2 − q α q β ln(−q 2 ) 3 3 Π̃αβγδ = Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Vacuum Polarization counter-term, graviton self energy and ghost → Veltman and ’t Hooft (1974) matter loop » ´` ´ ´ i κ2 −1 ` α β 1 ` α γ q q − q 2 η αβ q γ q δ − q 2 η γδ − q q − q 2 η αγ 32π 2 4 5 30 – ` ´ ´` ´ 1 ` α δ · q β q δ − q 2 η βδ − q q − q 2 η αδ q β q γ − q 2 η βγ − 2m34 η αβ η γδ 30 ff ´ 2 ` ´ 2 2 αβ ` γδ 2 − m3 η η q − q γ q δ − m32 η γδ η αβ q 2 − q α q β ln(−q 2 ) 3 3 Π̃αβγδ = Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary One Particle Irreduzible Diagrams k2 q k10 = k1 + k20 q + Sven Faller + General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Gravitational Potential 1PI-diagrams → S-matrix contributions ˆ ˜ 4µναβ iΠαβγδ 4 γδρσ V2ρσ (k10 , k20 , −q, m) iM = V1µν (k1 , k2 , q, m1 ) 4 µνρσ +4 position space gravitational potential G m1 m2 G(m1 + m2 ) 167 G ~ V (r ) = − 1− − r 30π r 2 c3 r c2 include massless Neutrino-loop gravitational potential G m1 m2 G(m1 + m2 ) 167 Nν G~ V (r ) = − 1− − + r 30π 40π r 2 c3 r c2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Gravitational Potential 1PI-diagrams → S-matrix contributions ˆ ˜ 4µναβ iΠαβγδ 4 γδρσ V2ρσ (k10 , k20 , −q, m) iM = V1µν (k1 , k2 , q, m1 ) 4 µνρσ +4 position space gravitational potential G m1 m2 G(m1 + m2 ) 167 G ~ V (r ) = − 1− − r 30π r 2 c3 r c2 include massless Neutrino-loop gravitational potential G m1 m2 G(m1 + m2 ) 167 Nν G~ V (r ) = − 1− − + r 30π 40π r 2 c3 r c2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Gravitational Potential 1PI-diagrams → S-matrix contributions ˆ ˜ 4µναβ iΠαβγδ 4 γδρσ V2ρσ (k10 , k20 , −q, m) iM = V1µν (k1 , k2 , q, m1 ) 4 µνρσ +4 position space gravitational potential G m1 m2 G(m1 + m2 ) 167 G ~ V (r ) = − 1− − r 30π r 2 c3 r c2 include massless Neutrino-loop gravitational potential G m1 m2 G(m1 + m2 ) 167 Nν G~ V (r ) = − 1− − + r 30π 40π r 2 c3 r c2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Gravitational Potential (so far) our result: V (r ) = − 167 G m1 m2 G(m1 + m2 ) Nν G~ − 1− + r 30π 40π r 2 c3 r c2 Donoghue G m1 m2 G(m1 + m2 ) (135 + 2Nν ) G ~ V (r ) = − 1− − . r r c2 30π 2 r c3 Akhundov et al. G m1 m2 G(m1 + m2 ) 107 G ~ V (r ) = − 1+ − r r c2 30π 2 r 2 c2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Further Potential Definition Hamber and Liu (1995) m1 q m2 m1 Sven Faller m2 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Further Contributions double-seagull diagramm: V4 G m1 m2 2G(m1 + m2 ) 14 G ~ =− − r π r 2 c3 r c2 triangle diagrams: Vi◦h Sven Faller G m1 m2 11 G ~ =− r 2π r 2 c3 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Further Contributions double-seagull diagramm: V4 G m1 m2 2G(m1 + m2 ) 14 G ~ =− − r π r 2 c3 r c2 triangle diagrams: Vi◦h Sven Faller G m1 m2 11 G ~ =− r 2π r 2 c3 General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Further Contributions cross-boxed diagramms (Bjerrum-Bohr 2003) + V (r ) = − 47 m1 m2 G2 . 3 π r3 Gravitational Potential G m1 m2 G(m1 + m2 ) 64 + Nν G ~ V (r ) = − 1+ − r 40π r 2 c3 r c2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Further Contributions cross-boxed diagramms (Bjerrum-Bohr 2003) + V (r ) = − 47 m1 m2 G2 . 3 π r3 Gravitational Potential G m1 m2 G(m1 + m2 ) 64 + Nν G ~ V (r ) = − 1+ − r 40π r 2 c3 r c2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Summary full theory of quantum gravity unknown effective field theory of gravity low energy effects separated from high-energy effects one-loop order quantum predictions evaluate leading quantum corrections → effective potential and Schwarzschild-metric understanding quantum nature of gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Summary full theory of quantum gravity unknown effective field theory of gravity low energy effects separated from high-energy effects one-loop order quantum predictions evaluate leading quantum corrections → effective potential and Schwarzschild-metric understanding quantum nature of gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Summary full theory of quantum gravity unknown effective field theory of gravity low energy effects separated from high-energy effects one-loop order quantum predictions evaluate leading quantum corrections → effective potential and Schwarzschild-metric understanding quantum nature of gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Summary full theory of quantum gravity unknown effective field theory of gravity low energy effects separated from high-energy effects one-loop order quantum predictions evaluate leading quantum corrections → effective potential and Schwarzschild-metric understanding quantum nature of gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Summary full theory of quantum gravity unknown effective field theory of gravity low energy effects separated from high-energy effects one-loop order quantum predictions evaluate leading quantum corrections → effective potential and Schwarzschild-metric understanding quantum nature of gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Summary full theory of quantum gravity unknown effective field theory of gravity low energy effects separated from high-energy effects one-loop order quantum predictions evaluate leading quantum corrections → effective potential and Schwarzschild-metric understanding quantum nature of gravity Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Thanks supervisor: Prof. Dr. T. Mannel second supervisor: Dr. A. Khodjamirian for usefull tips and discussions Dr. Th. Feldmann Dr. E. Bjerrum-Bohr Prof. Dr. F. Donoghue Dipl.-Phys. M. Jung Dipl.-Phys. N. Offen Dipl.-Phys. K. Grybel Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion - Affine Connection background field method: metric expansion gµν = ḡµν + κhµν g µν = ḡ µν − κhµν + κ2 hλµ hλν + . . . affine connection: Γλµν = Γ̄λµν + −Γλµν + =Γλµν with 1 Γ̄λµν = ḡ λσ ∂µ ḡσν + ∂ν ḡσµ − ∂σ ḡµν 2 κ λσ λ ḡ D h + D h − D h Γ = µ σν ν σµ σ µν µν − 2 κ2 λγ λ Γ = − h Dµ hγν + Dν hµγ − Dγ hµν µν = 2 Sven Faller (O(h0 )) , (O(h1 )) , (O(h2 )) . General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion - Affine Connection background field method: metric expansion gµν = ḡµν + κhµν g µν = ḡ µν − κhµν + κ2 hλµ hλν + . . . affine connection: Γλµν = Γ̄λµν + −Γλµν + =Γλµν with 1 Γ̄λµν = ḡ λσ ∂µ ḡσν + ∂ν ḡσµ − ∂σ ḡµν 2 κ λσ λ ḡ D h + D h − D h Γ = µ σν ν σµ σ µν µν − 2 κ2 λγ λ Γ = − h Dµ hγν + Dν hµγ − Dγ hµν µν = 2 Sven Faller (O(h0 )) , (O(h1 )) , (O(h2 )) . General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion - Affine Connection background field method: metric expansion gµν = ḡµν + κhµν g µν = ḡ µν − κhµν + κ2 hλµ hλν + . . . affine connection: Γλµν = Γ̄λµν + −Γλµν + =Γλµν with 1 Γ̄λµν = ḡ λσ ∂µ ḡσν + ∂ν ḡσµ − ∂σ ḡµν 2 κ λσ λ ḡ D h + D h − D h Γ = µ σν ν σµ σ µν µν − 2 κ2 λγ λ Γ = − h Dµ hγν + Dν hµγ − Dγ hµν µν = 2 Sven Faller (O(h0 )) , (O(h1 )) , (O(h2 )) . General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion: Curvature Riemann curvature tensor: β β β β β λ β λ R βαµν = Dµ Γβ αν − Dν Γαµ + Γαν Γλµ − Γαµ Γλν ≡ R̄ αµν + R αµν + R = αµν − Ricci scalar: R = ḡ αµ R − κhαµ Rαµ + κ2 hγα hγµ R̄αµ = αµ − ´˜ 1 ` β µ γ´ 1 ˆ β` µ 2 = κ − Dµ hγ D hβ + Dβ hν 2Dµ hνµ − Dν hµ 2 2 ´` ´ 1` ν ν + Dµ hβ + Dβ hµ − Dν hµβ Dµ hνβ + Dν hβµ − Dβ hνµ 4 ´ 1` 1 β β µ − 2Dµ hνµ − Dν hµ Dν hβ − hαµ Dµ Dα hβ 4 2 ff ´ 1 µ ` α β α β α µ + hα Dβ D hµ + Dµ hβα − Dβ hµ + κ2 hµ hβ R̄α 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion: Curvature Riemann curvature tensor: β β β β β λ β λ R βαµν = Dµ Γβ αν − Dν Γαµ + Γαν Γλµ − Γαµ Γλν ≡ R̄ αµν + R αµν + R = αµν − Ricci scalar: R = ḡ αµ R − κhαµ Rαµ + κ2 hγα hγµ R̄αµ = αµ − ´˜ 1 ` β µ γ´ 1 ˆ β` µ 2 = κ − Dµ hγ D hβ + Dβ hν 2Dµ hνµ − Dν hµ 2 2 ´` ´ 1` ν ν + Dµ hβ + Dβ hµ − Dν hµβ Dµ hνβ + Dν hβµ − Dβ hνµ 4 ´ 1` 1 β β µ − 2Dµ hνµ − Dν hµ Dν hβ − hαµ Dµ Dα hβ 4 2 ff ´ 1 µ ` α β α β α µ + hα Dβ D hµ + Dµ hβα − Dβ hµ + κ2 hµ hβ R̄α 2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Gauge Fixing and Ghost Field Yang-Mills field theory gauge fixing introduce Fadeev-Popov ghost fields gauge fixing Lagrangian (’t Hoof and Veltman 1974) p 1 1 µ ν µλ Lgf = −ḡ D hµν − Dµ h Dλ h − D h 2 2 ghost field Lagrangian (ebd.) p Lghost = −ḡ η ?µ Dλ Dλ ḡµν − R̄µν η ν complex ghostfield η: only contribution from vacuum polarization to the graviton propagator Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Gauge Fixing and Ghost Field Yang-Mills field theory gauge fixing introduce Fadeev-Popov ghost fields gauge fixing Lagrangian (’t Hoof and Veltman 1974) p 1 1 µ ν µλ Lgf = −ḡ D hµν − Dµ h Dλ h − D h 2 2 ghost field Lagrangian (ebd.) p Lghost = −ḡ η ?µ Dλ Dλ ḡµν − R̄µν η ν complex ghostfield η: only contribution from vacuum polarization to the graviton propagator Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Gauge Fixing and Ghost Field Yang-Mills field theory gauge fixing introduce Fadeev-Popov ghost fields gauge fixing Lagrangian (’t Hoof and Veltman 1974) p 1 1 µ ν µλ Lgf = −ḡ D hµν − Dµ h Dλ h − D h 2 2 ghost field Lagrangian (ebd.) p Lghost = −ḡ η ?µ Dλ Dλ ḡµν − R̄µν η ν complex ghostfield η: only contribution from vacuum polarization to the graviton propagator Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Gauge Fixing and Ghost Field Yang-Mills field theory gauge fixing introduce Fadeev-Popov ghost fields gauge fixing Lagrangian (’t Hoof and Veltman 1974) p 1 1 µ ν µλ Lgf = −ḡ D hµν − Dµ h Dλ h − D h 2 2 ghost field Lagrangian (ebd.) p Lghost = −ḡ η ?µ Dλ Dλ ḡµν − R̄µν η ν complex ghostfield η: only contribution from vacuum polarization to the graviton propagator Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Gauge Fixing and Ghost Field Yang-Mills field theory gauge fixing introduce Fadeev-Popov ghost fields gauge fixing Lagrangian (’t Hoof and Veltman 1974) p 1 1 µ ν µλ Lgf = −ḡ D hµν − Dµ h Dλ h − D h 2 2 ghost field Lagrangian (ebd.) p Lghost = −ḡ η ?µ Dλ Dλ ḡµν − R̄µν η ν complex ghostfield η: only contribution from vacuum polarization to the graviton propagator Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Gauge Fixing and Ghost Field Yang-Mills field theory gauge fixing introduce Fadeev-Popov ghost fields gauge fixing Lagrangian (’t Hoof and Veltman 1974) p 1 1 µ ν µλ Lgf = −ḡ D hµν − Dµ h Dλ h − D h 2 2 ghost field Lagrangian (ebd.) p Lghost = −ḡ η ?µ Dλ Dλ ḡµν − R̄µν η ν complex ghostfield η: only contribution from vacuum polarization to the graviton propagator Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Field Theories - Overview renormalizabel and non-renormalizable field theories general form of the Lagrangian L = L(c1 , c2 , . . . , cn ) low energy structure determined by finite parameters c1 , c2 , . . . cn two different typs of quantum field theories asympotically free theories - ultraviolet stable theories ultraviolet unstable theories ultraviolet unstable theories low energy limit of fundamental theory - no difference between renormalizable and effective non-renormalizable theory Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Field Theories - Overview renormalizabel and non-renormalizable field theories general form of the Lagrangian L = L(c1 , c2 , . . . , cn ) low energy structure determined by finite parameters c1 , c2 , . . . cn two different typs of quantum field theories asympotically free theories - ultraviolet stable theories ultraviolet unstable theories ultraviolet unstable theories low energy limit of fundamental theory - no difference between renormalizable and effective non-renormalizable theory Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Field Theories - Overview renormalizabel and non-renormalizable field theories general form of the Lagrangian L = L(c1 , c2 , . . . , cn ) low energy structure determined by finite parameters c1 , c2 , . . . cn two different typs of quantum field theories asympotically free theories - ultraviolet stable theories ultraviolet unstable theories ultraviolet unstable theories low energy limit of fundamental theory - no difference between renormalizable and effective non-renormalizable theory Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Field Theories - Overview renormalizabel and non-renormalizable field theories general form of the Lagrangian L = L(c1 , c2 , . . . , cn ) low energy structure determined by finite parameters c1 , c2 , . . . cn two different typs of quantum field theories asympotically free theories - ultraviolet stable theories ultraviolet unstable theories ultraviolet unstable theories low energy limit of fundamental theory - no difference between renormalizable and effective non-renormalizable theory Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Field Theories - Overview renormalizabel and non-renormalizable field theories general form of the Lagrangian L = L(c1 , c2 , . . . , cn ) low energy structure determined by finite parameters c1 , c2 , . . . cn two different typs of quantum field theories asympotically free theories - ultraviolet stable theories ultraviolet unstable theories ultraviolet unstable theories low energy limit of fundamental theory - no difference between renormalizable and effective non-renormalizable theory Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Field Theories - Overview renormalizabel and non-renormalizable field theories general form of the Lagrangian L = L(c1 , c2 , . . . , cn ) low energy structure determined by finite parameters c1 , c2 , . . . cn two different typs of quantum field theories asympotically free theories - ultraviolet stable theories ultraviolet unstable theories ultraviolet unstable theories low energy limit of fundamental theory - no difference between renormalizable and effective non-renormalizable theory Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Field Theories - Overview renormalizabel and non-renormalizable field theories general form of the Lagrangian L = L(c1 , c2 , . . . , cn ) low energy structure determined by finite parameters c1 , c2 , . . . cn two different typs of quantum field theories asympotically free theories - ultraviolet stable theories ultraviolet unstable theories ultraviolet unstable theories low energy limit of fundamental theory - no difference between renormalizable and effective non-renormalizable theory Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Field Theory two different types of effecive field theories decoupling effective field theories heavy degrees of freedom integrated out effective level no light particles Lagrangian general form X 1 X giD OiD Leff = LD≤4 + ΛD−4 D>4 iD non-decoupling effective field theories fundamental to effective level by phase transistion spontaneously broken symmetry → light pseudo-Goldstone bosons Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective General Relativity quantization process: first order curvature renormalization higher order needed one-loop Feynman rules → higher order curvature no contribution gravitational action S = Svac + Sm + Sgf + Sghost Z 2R 4 √ = d x −g + Lm + Lgf + Lghost κ2 quantum degrees of freedom: gravitation field hµν , ghostfields ηµ , ηµ∗ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective General Relativity quantization process: first order curvature renormalization higher order needed one-loop Feynman rules → higher order curvature no contribution gravitational action S = Svac + Sm + Sgf + Sghost Z 2R 4 √ = d x −g + Lm + Lgf + Lghost κ2 quantum degrees of freedom: gravitation field hµν , ghostfields ηµ , ηµ∗ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective General Relativity quantization process: first order curvature renormalization higher order needed one-loop Feynman rules → higher order curvature no contribution gravitational action S = Svac + Sm + Sgf + Sghost Z 2R 4 √ = d x −g + Lm + Lgf + Lghost κ2 quantum degrees of freedom: gravitation field hµν , ghostfields ηµ , ηµ∗ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective General Relativity quantization process: first order curvature renormalization higher order needed one-loop Feynman rules → higher order curvature no contribution gravitational action S = Svac + Sm + Sgf + Sghost Z 2R 4 √ = d x −g + Lm + Lgf + Lghost κ2 quantum degrees of freedom: gravitation field hµν , ghostfields ηµ , ηµ∗ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective General Relativity quantization process: first order curvature renormalization higher order needed one-loop Feynman rules → higher order curvature no contribution gravitational action S = Svac + Sm + Sgf + Sghost Z 2R 4 √ = d x −g + Lm + Lgf + Lghost κ2 quantum degrees of freedom: gravitation field hµν , ghostfields ηµ , ηµ∗ Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Effective Quantum Gravity R √ Sgrav = d 4 x −g κ22 R + c1 R 2 + c2 Rµν R µν + · · · + Lm parameter ci finite number values unknown free parameters low-energy limit: effective quantum gravity Z 2 4 √ 2 µν Sgrav = d x −g 2 R + c1 R + c2 Rµν R + Lm κ higher energy: theory renormalization value of c1 , c2 shifted → renormalization effects absorbed by parameters Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Contributions gravitational contributions (0) (2) Lgr (Λ) , Lgr (R) , (4) Lgr (R 2 ) matter field scalar matter field: (0) Lm (φ, m) , (2) Lm (φ, m, R) massless matter field: (0) L̄m = 0 , (2) L̄m (φ, R) , Sven Faller (0) L̄m (φ, R, R 2 ) General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Contributions gravitational contributions (0) (2) Lgr (Λ) , Lgr (R) , (4) Lgr (R 2 ) matter field scalar matter field: (0) Lm (φ, m) , (2) Lm (φ, m, R) massless matter field: (0) L̄m = 0 , (2) L̄m (φ, R) , Sven Faller (0) L̄m (φ, R, R 2 ) General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Contributions gravitational contributions (0) (2) Lgr (Λ) , Lgr (R) , (4) Lgr (R 2 ) matter field scalar matter field: (0) Lm (φ, m) , (2) Lm (φ, m, R) massless matter field: (0) L̄m = 0 , (2) L̄m (φ, R) , Sven Faller (0) L̄m (φ, R, R 2 ) General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Contributions gravitational contributions (0) (2) Lgr (Λ) , Lgr (R) , (4) Lgr (R 2 ) matter field scalar matter field: (0) Lm (φ, m) , (2) Lm (φ, m, R) massless matter field: (0) L̄m = 0 , (2) L̄m (φ, R) , Sven Faller (0) L̄m (φ, R, R 2 ) General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Counter Terms loop-diagrams → UV-divergencies UV-divergencies separated non-divergent parts by constant ci and di regulation one-loop order: Veltman and ’t Hoof (1974) (1) LM p ff −ḡ 1 7 2 µν = R̄ + R̄µν R̄ 8π 2 120 20 with =4−D MS-scheme (r ) c1 = c1 + 1 960π 2 and (r ) c2 = c2 + 7 160π 2 two-loop order (2) LM = 209κ 1p −ḡ R αβγδ R γδρσ R ρσαβ 2 2 2880(16π ) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Counter Terms loop-diagrams → UV-divergencies UV-divergencies separated non-divergent parts by constant ci and di regulation one-loop order: Veltman and ’t Hoof (1974) (1) LM p ff −ḡ 1 7 2 µν = R̄ + R̄µν R̄ 8π 2 120 20 with =4−D MS-scheme (r ) c1 = c1 + 1 960π 2 and (r ) c2 = c2 + 7 160π 2 two-loop order (2) LM = 209κ 1p −ḡ R αβγδ R γδρσ R ρσαβ 2 2 2880(16π ) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Counter Terms loop-diagrams → UV-divergencies UV-divergencies separated non-divergent parts by constant ci and di regulation one-loop order: Veltman and ’t Hoof (1974) (1) LM p ff −ḡ 1 7 2 µν = R̄ + R̄µν R̄ 8π 2 120 20 with =4−D MS-scheme (r ) c1 = c1 + 1 960π 2 and (r ) c2 = c2 + 7 160π 2 two-loop order (2) LM = 209κ 1p −ḡ R αβγδ R γδρσ R ρσαβ 2 2 2880(16π ) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Counter Terms loop-diagrams → UV-divergencies UV-divergencies separated non-divergent parts by constant ci and di regulation one-loop order: Veltman and ’t Hoof (1974) (1) LM p ff −ḡ 1 7 2 µν = R̄ + R̄µν R̄ 8π 2 120 20 with =4−D MS-scheme (r ) c1 = c1 + 1 960π 2 and (r ) c2 = c2 + 7 160π 2 two-loop order (2) LM = 209κ 1p −ḡ R αβγδ R γδρσ R ρσαβ 2 2 2880(16π ) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Counter Terms loop-diagrams → UV-divergencies UV-divergencies separated non-divergent parts by constant ci and di regulation one-loop order: Veltman and ’t Hoof (1974) (1) LM p ff −ḡ 1 7 2 µν = R̄ + R̄µν R̄ 8π 2 120 20 with =4−D MS-scheme (r ) c1 = c1 + 1 960π 2 and (r ) c2 = c2 + 7 160π 2 two-loop order (2) LM = 209κ 1p −ḡ R αβγδ R γδρσ R ρσαβ 2 2 2880(16π ) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Evaluation Vertex Factors momentum space vertex factors Z Vµ1 ν1 ,...,µm νn = +i d 4 x d 4 x1 . . . d 4 xn d 4 y1 . . . d 4 ym ei(p1 x1 +···+pn xn +q1 y1 +···+qm ym ) δ δ δ δ · ... · · ... · · µm νm δJ1 (x1 ) δJn (xn ) δH1ν1 µ1 (y1 ) δHm (ym ) ´ ` · Lint φ1 , . . . , φn , H1 , . . . Hm (x) · sources of gravity: J1 , . . . , Jn µm νm external and internal gravity field: H1µ1 ν1 , . . . , Hm incoming p1 , . . . , pn , outgoing q1 , . . . qm momentum Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Evaluation Vertex Factors momentum space vertex factors Z Vµ1 ν1 ,...,µm νn = +i d 4 x d 4 x1 . . . d 4 xn d 4 y1 . . . d 4 ym ei(p1 x1 +···+pn xn +q1 y1 +···+qm ym ) δ δ δ δ · ... · · ... · · µm νm δJ1 (x1 ) δJn (xn ) δH1ν1 µ1 (y1 ) δHm (ym ) ´ ` · Lint φ1 , . . . , φn , H1 , . . . Hm (x) · sources of gravity: J1 , . . . , Jn µm νm external and internal gravity field: H1µ1 ν1 , . . . , Hm incoming p1 , . . . , pn , outgoing q1 , . . . qm momentum Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Evaluation Vertex Factors momentum space vertex factors Z Vµ1 ν1 ,...,µm νn = +i d 4 x d 4 x1 . . . d 4 xn d 4 y1 . . . d 4 ym ei(p1 x1 +···+pn xn +q1 y1 +···+qm ym ) δ δ δ δ · ... · · ... · · µm νm δJ1 (x1 ) δJn (xn ) δH1ν1 µ1 (y1 ) δHm (ym ) ´ ` · Lint φ1 , . . . , φn , H1 , . . . Hm (x) · sources of gravity: J1 , . . . , Jn µm νm external and internal gravity field: H1µ1 ν1 , . . . , Hm incoming p1 , . . . , pn , outgoing q1 , . . . qm momentum Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Evaluation Vertex Factors momentum space vertex factors Z Vµ1 ν1 ,...,µm νn = +i d 4 x d 4 x1 . . . d 4 xn d 4 y1 . . . d 4 ym ei(p1 x1 +···+pn xn +q1 y1 +···+qm ym ) δ δ δ δ · ... · · ... · · µm νm δJ1 (x1 ) δJn (xn ) δH1ν1 µ1 (y1 ) δHm (ym ) ´ ` · Lint φ1 , . . . , φn , H1 , . . . Hm (x) · sources of gravity: J1 , . . . , Jn µm νm external and internal gravity field: H1µ1 ν1 , . . . , Hm incoming p1 , . . . , pn , outgoing q1 , . . . qm momentum Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Graviton-Graviton-Scalar-Vertex Lagrangian O(h2 ) (2) Lm = κ2 „ 1 µν ν 1 h h λ− hhµν 2 4 « ∂µ φ∂ν φ− vertex factor Z Vηλρσ = +i „ « ˆ ˜ 1 κ2 hλσ hλσ − hh ∂µ φ∂ µ φ−m2 φ2 8 2 d 4 x d 4 x1 d 4 x2 d 4 x3 d 4 x4 ei(px1 −p 0 x2 +kx3 −kx4 ) ∂ ∂ ∂ ∂ · · · ∂φ(x1 ) ∂φ(x2 ) ∂hηλ (x3 ) ∂hρσ (x4 ) » – ´ α κ2 ηλ 1` δ · h 1ηλαδ 1ρσβ − ηηλ 1ρσαβ + ηρσ 1ηλαβ ∂ φ(x)∂ β φ(x) 2 4 „ « ff ˜ ˆ 1 1 1 − 1ηλρσ − ηηλ − ηηλ ηρσ ∂ γ φ(x)∂γ φ(x) − m2 φ(x)2 hρσ 4 2 2 · Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Graviton-Graviton-Scalar-Vertex Lagrangian O(h2 ) (2) Lm = κ2 „ 1 µν ν 1 h h λ− hhµν 2 4 « ∂µ φ∂ν φ− vertex factor Z Vηλρσ = +i „ « ˆ ˜ 1 κ2 hλσ hλσ − hh ∂µ φ∂ µ φ−m2 φ2 8 2 d 4 x d 4 x1 d 4 x2 d 4 x3 d 4 x4 ei(px1 −p 0 x2 +kx3 −kx4 ) ∂ ∂ ∂ ∂ · · · ∂φ(x1 ) ∂φ(x2 ) ∂hηλ (x3 ) ∂hρσ (x4 ) » – ´ α κ2 ηλ 1` δ · h 1ηλαδ 1ρσβ − ηηλ 1ρσαβ + ηρσ 1ηλαβ ∂ φ(x)∂ β φ(x) 2 4 „ « ff ˜ ˆ 1 1 1 − 1ηλρσ − ηηλ − ηηλ ηρσ ∂ γ φ(x)∂γ φ(x) − m2 φ(x)2 hρσ 4 2 2 · Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Graviton-Graviton-Scalar-Vertex ηλ `0 - p0 `% p ρσ » – ´ ` α 0β ´ 1` iκ2 1ηλαδ 1ρσβ δ − ηηλ 1ρσαβ + ηρσ 1ηλαβ p p + pβ p0α 2 4 » – ff ` ´ 1 1 − 1ηλρσ − ηηλ ηρσ (p · p0 ) − m2 2 2 Vηλρσ = Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Three-Graviton-Vertex µν ταβγδ (k , q) = − iκ 2 – » 3 µν 2 µ ν µ ν µ ν Pαβγδ k k + (k − q) (k − q) + q q + η q 2 ˆ σλ µν σλ µν µσ νλ µσ νλ ˜ + 2qλ qσ 1αβ 1γδ + 1γδ 1αβ − 1αβ 1γδ − 1γδ 1αβ ˆ µλ µλ ´ νλ νλ ´ ν` µ` + qλ q ηαβ 1γδ + ηγδ 1αβ + ηγδ 1αβ + qλ q ηαβ 1γδ ` µν µν ´ σλ σλ ´˜ 2` µν + ηγδ 1αβ − q ηαβ 1γδ + ηγδ 1αβ − η qλ qσ ηαβ 1γδ ˆ ˘ λσ ν λσ µ µ ν + 2qλ 1αβ 1γδσ (k − q) + 1αβ 1γδσ (k − q) λσ ν µ λσ µ ν¯ 1αβσ k − 1γδ 1αβσ k − 1γδ ` λρ σ λρ σ ´˜ µ νσ νσ µ´ µν 2` + η qσ qλ 1αβ 1γδρ + 1γδ 1αβρ + 1αβ 1γδσ + q 1αβσ 1γδ » – 1 µν ` 2 ´ µσ ν νσ µ + k + (k − q) 1αβ 1γδσ + 1γδ 1αβσ − η Pαβγδ 2 ff ffff ` µν µν 2 2´ − 1γδ ηαβ k − 1αβ ηγδ (k − q) Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Born approximation p covariant normalized: hp0 |pi = 2Ep δ 3 (~p − ~p0 ), Ep = m2 + ~p2 Born approximation: hp0 |pi = −i Ṽ (~q )(2π)δ(E~p0 − E~p ) nonrelativistic limit: interaction potential Z 3 d ~pT 3 0 1 1 ~ −Ṽ (q ) = M δ (~p + ~p) 2m1 2m2 (2π)3 nonrelativistic limit: Ṽ (~q ) = − 2m11·2m2 M Fourier transformation to position-space: nonrelativistic limit: position-space potential Z 1 1 d 3~q i ~q ·~r V (~r ) = − e M 2m1 2m2 (2π)3 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Born approximation p covariant normalized: hp0 |pi = 2Ep δ 3 (~p − ~p0 ), Ep = m2 + ~p2 Born approximation: hp0 |pi = −i Ṽ (~q )(2π)δ(E~p0 − E~p ) nonrelativistic limit: interaction potential Z 3 d ~pT 3 0 1 1 ~ −Ṽ (q ) = M δ (~p + ~p) 2m1 2m2 (2π)3 nonrelativistic limit: Ṽ (~q ) = − 2m11·2m2 M Fourier transformation to position-space: nonrelativistic limit: position-space potential Z 1 1 d 3~q i ~q ·~r V (~r ) = − e M 2m1 2m2 (2π)3 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Born approximation p covariant normalized: hp0 |pi = 2Ep δ 3 (~p − ~p0 ), Ep = m2 + ~p2 Born approximation: hp0 |pi = −i Ṽ (~q )(2π)δ(E~p0 − E~p ) nonrelativistic limit: interaction potential Z 3 d ~pT 3 0 1 1 ~ −Ṽ (q ) = M δ (~p + ~p) 2m1 2m2 (2π)3 nonrelativistic limit: Ṽ (~q ) = − 2m11·2m2 M Fourier transformation to position-space: nonrelativistic limit: position-space potential Z 1 1 d 3~q i ~q ·~r V (~r ) = − e M 2m1 2m2 (2π)3 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Born approximation p covariant normalized: hp0 |pi = 2Ep δ 3 (~p − ~p0 ), Ep = m2 + ~p2 Born approximation: hp0 |pi = −i Ṽ (~q )(2π)δ(E~p0 − E~p ) nonrelativistic limit: interaction potential Z 3 d ~pT 3 0 1 1 ~ −Ṽ (q ) = M δ (~p + ~p) 2m1 2m2 (2π)3 nonrelativistic limit: Ṽ (~q ) = − 2m11·2m2 M Fourier transformation to position-space: nonrelativistic limit: position-space potential Z 1 1 d 3~q i ~q ·~r V (~r ) = − e M 2m1 2m2 (2π)3 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Born approximation p covariant normalized: hp0 |pi = 2Ep δ 3 (~p − ~p0 ), Ep = m2 + ~p2 Born approximation: hp0 |pi = −i Ṽ (~q )(2π)δ(E~p0 − E~p ) nonrelativistic limit: interaction potential Z 3 d ~pT 3 0 1 1 ~ −Ṽ (q ) = M δ (~p + ~p) 2m1 2m2 (2π)3 nonrelativistic limit: Ṽ (~q ) = − 2m11·2m2 M Fourier transformation to position-space: nonrelativistic limit: position-space potential Z 1 1 d 3~q i ~q ·~r V (~r ) = − e M 2m1 2m2 (2π)3 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion Form Factors dimensionless combinations: κ2 m2 , κ2 q 2 expansion: s « (−q 2 ) m2 F1 (q ) = 1 + d1 q + κ q l1 + l2 ln + l + . . . , 3 µ2 −q 2 s „ « (−q 2 ) m2 2 2 2 2 F2 (q ) = −4(d2 − d3 )m + κ m l4 + l5 ln + l6 + ... µ2 −q 2 2 2 2 2 „ (2) di : Lm contributions li : one-loop contributions l1 , l4 : divergent high enery contributions l2 , l3 , l5 , l6 : finite non-analytic low energy contributions Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion Form Factors dimensionless combinations: κ2 m2 , κ2 q 2 expansion: s « (−q 2 ) m2 F1 (q ) = 1 + d1 q + κ q l1 + l2 ln + l + . . . , 3 µ2 −q 2 s „ « (−q 2 ) m2 2 2 2 2 F2 (q ) = −4(d2 − d3 )m + κ m l4 + l5 ln + l6 + ... µ2 −q 2 2 2 2 2 „ (2) di : Lm contributions li : one-loop contributions l1 , l4 : divergent high enery contributions l2 , l3 , l5 , l6 : finite non-analytic low energy contributions Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion Form Factors dimensionless combinations: κ2 m2 , κ2 q 2 expansion: s « (−q 2 ) m2 F1 (q ) = 1 + d1 q + κ q l1 + l2 ln + l + . . . , 3 µ2 −q 2 s „ « (−q 2 ) m2 2 2 2 2 F2 (q ) = −4(d2 − d3 )m + κ m l4 + l5 ln + l6 + ... µ2 −q 2 2 2 2 2 „ (2) di : Lm contributions li : one-loop contributions l1 , l4 : divergent high enery contributions l2 , l3 , l5 , l6 : finite non-analytic low energy contributions Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion Form Factors dimensionless combinations: κ2 m2 , κ2 q 2 expansion: s « (−q 2 ) m2 F1 (q ) = 1 + d1 q + κ q l1 + l2 ln + l + . . . , 3 µ2 −q 2 s „ « (−q 2 ) m2 2 2 2 2 F2 (q ) = −4(d2 − d3 )m + κ m l4 + l5 ln + l6 + ... µ2 −q 2 2 2 2 2 „ (2) di : Lm contributions li : one-loop contributions l1 , l4 : divergent high enery contributions l2 , l3 , l5 , l6 : finite non-analytic low energy contributions Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion Form Factors dimensionless combinations: κ2 m2 , κ2 q 2 expansion: s « (−q 2 ) m2 F1 (q ) = 1 + d1 q + κ q l1 + l2 ln + l + . . . , 3 µ2 −q 2 s „ « (−q 2 ) m2 2 2 2 2 F2 (q ) = −4(d2 − d3 )m + κ m l4 + l5 ln + l6 + ... µ2 −q 2 2 2 2 2 „ (2) di : Lm contributions li : one-loop contributions l1 , l4 : divergent high enery contributions l2 , l3 , l5 , l6 : finite non-analytic low energy contributions Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Expansion Form Factors dimensionless combinations: κ2 m2 , κ2 q 2 expansion: s « (−q 2 ) m2 F1 (q ) = 1 + d1 q + κ q l1 + l2 ln + l + . . . , 3 µ2 −q 2 s „ « (−q 2 ) m2 2 2 2 2 F2 (q ) = −4(d2 − d3 )m + κ m l4 + l5 ln + l6 + ... µ2 −q 2 2 2 2 2 „ (2) di : Lm contributions li : one-loop contributions l1 , l4 : divergent high enery contributions l2 , l3 , l5 , l6 : finite non-analytic low energy contributions Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Renormalization combination l1 , l4 and di → renormalized values d1 (r) (µ2 ) = d1 + κ2 l1 d2 (r) (µ2 ) + d3 (r) (µ2 ) = d2 + d3 − κ2 l4 4 experiments: measure renormalized values di(r) (µ2 ) → measured values depend on µ2 choice in logarithms but all physics independent of µ2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Renormalization combination l1 , l4 and di → renormalized values d1 (r) (µ2 ) = d1 + κ2 l1 d2 (r) (µ2 ) + d3 (r) (µ2 ) = d2 + d3 − κ2 l4 4 experiments: measure renormalized values di(r) (µ2 ) → measured values depend on µ2 choice in logarithms but all physics independent of µ2 Sven Faller General Relativity as an Effective Field Theory Introduction Quantum Gravity Effective Field Theory of Gravity Leading Quantum Corrections Evaluation of the Vertex Corrections Gravitational Potential Potential Definitions Summary Effective Gravity Renormalization combination l1 , l4 and di → renormalized values d1 (r) (µ2 ) = d1 + κ2 l1 d2 (r) (µ2 ) + d3 (r) (µ2 ) = d2 + d3 − κ2 l4 4 experiments: measure renormalized values di(r) (µ2 ) → measured values depend on µ2 choice in logarithms but all physics independent of µ2 Sven Faller General Relativity as an Effective Field Theory
