IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
General Relativity as an Effective Field Theory
Sven Faller
Theoretical Physics 1University of Siegen
Theory Seminar 18.12.2006
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
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
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation
all known field theories: quantum field theoriesgravity quantization - Feynman´s Gedankenexperiment
gravity must be a quantum field theoryproblem: consistent quantization method unknown
Quantum Gravity: De Witt, Feynman, ’t Hooft and Veltmanpresent energies, quantum gravity non-renormalizable
low-energy predictions independent of high-energyinfluenceDonoghue:possible solution Effective Field Theory of Gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation of Quantization
Feynman´s Gedankenexperiment: two-slit diffractionexperiment with gravity detector
characteristic for a quantum field⇒ should be describedby an amplitude rather than a probability
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Motivation of Quantization
Feynman´s Gedankenexperiment: two-slit diffractionexperiment with gravity detector
characteristic for a quantum field⇒ should be describedby an amplitude rather than a probability
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Laws (1687)
law of inertia
no external force : d~rdt = ~v = const .
⇒ inertial frame of reference (IS)
second law
force ∝ inertia mass mi ⇒ ~F = mi · ~a.
third lawactio est reactio
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Laws (1687)
law of inertia
no external force : d~rdt = ~v = const .
⇒ inertial frame of reference (IS)
second law
force ∝ inertia mass mi ⇒ ~F = mi · ~a.
third lawactio est reactio
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Laws (1687)
law of inertia
no external force : d~rdt = ~v = const .
⇒ inertial frame of reference (IS)
second law
force ∝ inertia mass mi ⇒ ~F = mi · ~a.
third lawactio est reactio
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Relativity
three different masses: inertia mass mi , passivegravitational mass mG and active gravitational mass MG
third law: passive and active gravitational mass equalforce of gravity
~F12(~r) = −G m1 m2~r1 −~r2
|~r1 −~r2|3
problem: equality of inertia and passive massesexperimental measurements: verification of equality,bases for Einstein’s Principle of Equivalence
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Relativity
three different masses: inertia mass mi , passivegravitational mass mG and active gravitational mass MG
third law: passive and active gravitational mass equalforce of gravity
~F12(~r) = −G m1 m2~r1 −~r2
|~r1 −~r2|3
problem: equality of inertia and passive massesexperimental measurements: verification of equality,bases for Einstein’s Principle of Equivalence
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Relativity
three different masses: inertia mass mi , passivegravitational mass mG and active gravitational mass MG
third law: passive and active gravitational mass equalforce of gravity
~F12(~r) = −G m1 m2~r1 −~r2
|~r1 −~r2|3
problem: equality of inertia and passive massesexperimental measurements: verification of equality,bases for Einstein’s Principle of Equivalence
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Relativity
three different masses: inertia mass mi , passivegravitational mass mG and active gravitational mass MG
third law: passive and active gravitational mass equalforce of gravity
~F12(~r) = −G m1 m2~r1 −~r2
|~r1 −~r2|3
problem: equality of inertia and passive massesexperimental measurements: verification of equality,bases for Einstein’s Principle of Equivalence
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Newton’s Relativity
three different masses: inertia mass mi , passivegravitational mass mG and active gravitational mass MG
third law: passive and active gravitational mass equalforce of gravity
~F12(~r) = −G m1 m2~r1 −~r2
|~r1 −~r2|3
problem: equality of inertia and passive massesexperimental measurements: verification of equality,bases for Einstein’s Principle of Equivalence
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Einstein’s Special Relativity
Newton: Galilei transformations between ISEinstein 1905: Newton’s Theory must be specialized byuniversality of the velocity of light in all frames
x 7−→ x ′ = ΛΛΛx + a (Lorentz transformation)
Postulategeneral transformation for the line element must satisfy
ds2 = ηαβ dxµ dxν = c2 dt2 − d~x2
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Einstein’s Special Relativity
Newton: Galilei transformations between ISEinstein 1905: Newton’s Theory must be specialized byuniversality of the velocity of light in all frames
x 7−→ x ′ = ΛΛΛx + a (Lorentz transformation)
Postulategeneral transformation for the line element must satisfy
ds2 = ηαβ dxµ dxν = c2 dt2 − d~x2
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Einstein’s Special Relativity
Newton: Galilei transformations between ISEinstein 1905: Newton’s Theory must be specialized byuniversality of the velocity of light in all frames
x 7−→ x ′ = ΛΛΛx + a (Lorentz transformation)
Postulategeneral transformation for the line element must satisfy
ds2 = ηαβ dxµ dxν = c2 dt2 − d~x2
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral 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 mannifoldcurvature of space = measure for mass:
„matter tells space how to curve, and space tells matterhow to move“ (Miesner, 1973)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral 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 mannifoldcurvature of space = measure for mass:
„matter tells space how to curve, and space tells matterhow to move“ (Miesner, 1973)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral 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 mannifoldcurvature of space = measure for mass:
„matter tells space how to curve, and space tells matterhow to move“ (Miesner, 1973)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral 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 mannifoldcurvature of space = measure for mass:
„matter tells space how to curve, and space tells matterhow to move“ (Miesner, 1973)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral 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 mannifoldcurvature of space = measure for mass:
„matter tells space how to curve, and space tells matterhow to move“ (Miesner, 1973)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Principle of Equivalence
„At every space-time point in an arbitrary gravitational field it is possible tochoose a „locally inertial coordinate system“ such that, within sufficientlysmall region of the point in question, the laws of nature take the same form asin unaccelerated Cartesian coordinate systems in absence of gravitation.“
relation between accelerated local IS xα and static frameof reference xµ described by metric tensor, which leavesline element ds2 invariant:
gµν = ηαβ∂xα
∂xµ∂xβ
∂xν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Principle of Equivalence
„At every space-time point in an arbitrary gravitational field it is possible tochoose a „locally inertial coordinate system“ such that, within sufficientlysmall region of the point in question, the laws of nature take the same form asin unaccelerated Cartesian coordinate systems in absence of gravitation.“
relation between accelerated local IS xα and static frameof reference xµ described by metric tensor, which leavesline element ds2 invariant:
gµν = ηαβ∂xα
∂xµ∂xβ
∂xν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Principle of Equivalence
„At every space-time point in an arbitrary gravitational field it is possible tochoose a „locally inertial coordinate system“ such that, within sufficientlysmall region of the point in question, the laws of nature take the same form asin unaccelerated Cartesian coordinate systems in absence of gravitation.“
relation between accelerated local IS xα and static frameof reference xµ described by metric tensor, which leavesline element ds2 invariant:
gµν = ηαβ∂xα
∂xµ∂xβ
∂xν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Principle of General Covariance
General laws of nature should be expressed in terms ofequations which are true in all frames of reference andtransform covariantly by arbitrary substitutions.
general coordinate transformation: x 7−→ x ′ = f (x)
Principle of General Covariance is not an invarianceprinciple like Principle of Galilean or Special Relativity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Principle of General Covariance
General laws of nature should be expressed in terms ofequations which are true in all frames of reference andtransform covariantly by arbitrary substitutions.
general coordinate transformation: x 7−→ x ′ = f (x)
Principle of General Covariance is not an invarianceprinciple like Principle of Galilean or Special Relativity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Principle of General Covariance
General laws of nature should be expressed in terms ofequations which are true in all frames of reference andtransform covariantly by arbitrary substitutions.
general coordinate transformation: x 7−→ x ′ = f (x)
Principle of General Covariance is not an invarianceprinciple like Principle of Galilean or Special Relativity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Recall: Lorentz Invariance
global coordinate change: xµ 7−→ x ′µ = Λµν xν
Minkowski metric ηµν invariantfields transform as scalars, vectors, etc.
φ(x) 7−→ φ′(x ′) = φ(x)
Aµ(x) 7−→ A′µ(x) = Λµν(x) Aν(x)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Recall: Lorentz Invariance
global coordinate change: xµ 7−→ x ′µ = Λµν xν
Minkowski metric ηµν invariantfields transform as scalars, vectors, etc.
φ(x) 7−→ φ′(x ′) = φ(x)
Aµ(x) 7−→ A′µ(x) = Λµν(x) Aν(x)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Recall: Lorentz Invariance
global coordinate change: xµ 7−→ x ′µ = Λµν xν
Minkowski metric ηµν invariantfields transform as scalars, vectors, etc.
φ(x) 7−→ φ′(x ′) = φ(x)
Aµ(x) 7−→ A′µ(x) = Λµν(x) Aν(x)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Covariant Derivative
local coordinate changes require covariant derivative:
Dµ Aν = ∂µAν + ΓνµλAλ = Aν,µ + ΓνµλAλ ≡ Aν;µ
affine connection Γλµν (geometric interpretation)for scalar fields: Φ;µ ≡ Φ,µ = ∂µΦ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Covariant Derivative
local coordinate changes require covariant derivative:
Dµ Aν = ∂µAν + ΓνµλAλ = Aν,µ + ΓνµλAλ ≡ Aν;µ
affine connection Γλµν (geometric interpretation)for scalar fields: Φ;µ ≡ Φ,µ = ∂µΦ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Covariant Derivative
local coordinate changes require covariant derivative:
Dµ Aν = ∂µAν + ΓνµλAλ = Aν,µ + ΓνµλAλ ≡ Aν;µ
affine connection Γλµν (geometric interpretation)for scalar fields: Φ;µ ≡ Φ,µ = ∂µΦ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Riemann Space (R4)
metric definition: ds2 = gµν(x) dxµ dxν
affine connection: Γλµν = 12gλσ
(∂µgνσ + ∂νgνσ − ∂σgµν
)Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rλ
µσν
Riemann curvature tensor:
Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ
Ricci tensor: Rµν = Rλµλν ≡ Rνµ
Ricci scalar: R = gµν Rµν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Riemann Space (R4)
metric definition: ds2 = gµν(x) dxµ dxν
affine connection: Γλµν = 12gλσ
(∂µgνσ + ∂νgνσ − ∂σgµν
)Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rλ
µσν
Riemann curvature tensor:
Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ
Ricci tensor: Rµν = Rλµλν ≡ Rνµ
Ricci scalar: R = gµν Rµν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Riemann Space (R4)
metric definition: ds2 = gµν(x) dxµ dxν
affine connection: Γλµν = 12gλσ
(∂µgνσ + ∂νgνσ − ∂σgµν
)Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rλ
µσν
Riemann curvature tensor:
Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ
Ricci tensor: Rµν = Rλµλν ≡ Rνµ
Ricci scalar: R = gµν Rµν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Riemann Space (R4)
metric definition: ds2 = gµν(x) dxµ dxν
affine connection: Γλµν = 12gλσ
(∂µgνσ + ∂νgνσ − ∂σgµν
)Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rλ
µσν
Riemann curvature tensor:
Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ
Ricci tensor: Rµν = Rλµλν ≡ Rνµ
Ricci scalar: R = gµν Rµν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Riemann Space (R4)
metric definition: ds2 = gµν(x) dxµ dxν
affine connection: Γλµν = 12gλσ
(∂µgνσ + ∂νgνσ − ∂σgµν
)Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rλ
µσν
Riemann curvature tensor:
Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ
Ricci tensor: Rµν = Rλµλν ≡ Rνµ
Ricci scalar: R = gµν Rµν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
FundamentalsGeneral Relativity
Riemann Space (R4)
metric definition: ds2 = gµν(x) dxµ dxν
affine connection: Γλµν = 12gλσ
(∂µgνσ + ∂νgνσ − ∂σgµν
)Ricci identity: Aµ;σν − Aµ;νσ ≡ Aλ Rλ
µσν
Riemann curvature tensor:
Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ
Ricci tensor: Rµν = Rλµλν ≡ Rνµ
Ricci scalar: R = gµν Rµν
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
General relativity as a gauge theory (Sabbata 1985)
Poincaré group is non abeliancf. Yang-Mills theory
Lgauge = −14
F aµν F aµν = −1
2trF 2
gravity: introduction of vierbein- or tetrad fields e λµ
Lgauge = − e2g
eµλeνσR λσ
µν (ωωω) ≡ 2κ2
√−g R
with g = det[gµν ] and κ2 = 32πG.
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
General relativity as a gauge theory (Sabbata 1985)
Poincaré group is non abeliancf. Yang-Mills theory
Lgauge = −14
F aµν F aµν = −1
2trF 2
gravity: introduction of vierbein- or tetrad fields e λµ
Lgauge = − e2g
eµλeνσR λσ
µν (ωωω) ≡ 2κ2
√−g R
with g = det[gµν ] and κ2 = 32πG.
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
General relativity as a gauge theory (Sabbata 1985)
Poincaré group is non abeliancf. Yang-Mills theory
Lgauge = −14
F aµν F aµν = −1
2trF 2
gravity: introduction of vierbein- or tetrad fields e λµ
Lgauge = − e2g
eµλeνσR λσ
µν (ωωω) ≡ 2κ2
√−g R
with g = det[gµν ] and κ2 = 32πG.
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
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Background Field Method
introduced by ’t Hooft and Veltmann (1974)gravitational field expanded about smooth backgroundmetric gµν
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
classical equations of motion: gµνquantum field hµν : all dynamical information
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
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Quantization
Background Field Method
introduced by ’t Hooft and Veltmann (1974)gravitational field expanded about smooth backgroundmetric gµν
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
classical equations of motion: gµνquantum field hµν : all dynamical information
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
Background Field Method
introduced by ’t Hooft and Veltmann (1974)gravitational field expanded about smooth backgroundmetric gµν
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
classical equations of motion: gµνquantum field hµν : all dynamical information
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
Background Field Method
introduced by ’t Hooft and Veltmann (1974)gravitational field expanded about smooth backgroundmetric gµν
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
classical equations of motion: gµνquantum field hµν : all dynamical information
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
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Quantization
Gravitational Action
Einstein-Hilbert action Svac =∫
d4x√−g 2
κ2 R
matter action Sm =∫
d4x Lm
Sgr = Svac + Sm =
∫d4x
[√−g
2κ2 R + Lm
]further gauge invariant terms
L =√−g{λ+
2κ2 R + c1R
2 + c2RµνRµν + . . .
}+ Lm
upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0
Sven Faller General Relativity as an Effective Field Theory
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Gravitational Action
Einstein-Hilbert action Svac =∫
d4x√−g 2
κ2 R
matter action Sm =∫
d4x Lm
Sgr = Svac + Sm =
∫d4x
[√−g
2κ2 R + Lm
]further gauge invariant terms
L =√−g{λ+
2κ2 R + c1R
2 + c2RµνRµν + . . .
}+ Lm
upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0
Sven Faller General Relativity as an Effective Field Theory
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Gravitational Action
Einstein-Hilbert action Svac =∫
d4x√−g 2
κ2 R
matter action Sm =∫
d4x Lm
Sgr = Svac + Sm =
∫d4x
[√−g
2κ2 R + Lm
]further gauge invariant terms
L =√−g{λ+
2κ2 R + c1R
2 + c2RµνRµν + . . .
}+ Lm
upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
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Quantization
Gravitational Action
Einstein-Hilbert action Svac =∫
d4x√−g 2
κ2 R
matter action Sm =∫
d4x Lm
Sgr = Svac + Sm =
∫d4x
[√−g
2κ2 R + Lm
]further gauge invariant terms
L =√−g{λ+
2κ2 R + c1R
2 + c2RµνRµν + . . .
}+ Lm
upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0
Sven Faller General Relativity as an Effective Field Theory
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Effective Field Theory of GravityLeading Quantum Corrections
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Quantization
Gravitational Action
Einstein-Hilbert action Svac =∫
d4x√−g 2
κ2 R
matter action Sm =∫
d4x Lm
Sgr = Svac + Sm =
∫d4x
[√−g
2κ2 R + Lm
]further gauge invariant terms
L =√−g{λ+
2κ2 R + c1R
2 + c2RµνRµν + . . .
}+ Lm
upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0
Sven Faller General Relativity as an Effective Field Theory
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Quantization
Gravitational Action
Einstein-Hilbert action Svac =∫
d4x√−g 2
κ2 R
matter action Sm =∫
d4x Lm
Sgr = Svac + Sm =
∫d4x
[√−g
2κ2 R + Lm
]further gauge invariant terms
L =√−g{λ+
2κ2 R + c1R
2 + c2RµνRµν + . . .
}+ Lm
upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0
Sven Faller General Relativity as an Effective Field Theory
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Quantization
Expansion: Vacuum Lagrangian
metric expansion:p−g =
p−g
1−κ
2hαα −
κ2
4hαβhβα +
κ2
8
`hαα´2
+O(h3)
ffLagrangian expansion
2κ2
p−gR =
p−g»
2κ2
R + L(1)gr + L(2)
gr + . . .
–,
L(1)gr =
1κ
hµνˆgµνR − 2Rµν
˜,
L(2)gr =
12
Dαhµν Dαhµν −12
Dαh Dαh + Dαh Dβhαβ − Dαhµβ Dβhµα
+ R
„14
h2 −12
hµνhµν«
+ Rµν`2hλµhνλ − h hµν
´.
Sven Faller General Relativity as an Effective Field Theory
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Expansion: Vacuum Lagrangian
metric expansion:p−g =
p−g
1−κ
2hαα −
κ2
4hαβhβα +
κ2
8
`hαα´2
+O(h3)
ffLagrangian expansion
2κ2
p−gR =
p−g»
2κ2
R + L(1)gr + L(2)
gr + . . .
–,
L(1)gr =
1κ
hµνˆgµνR − 2Rµν
˜,
L(2)gr =
12
Dαhµν Dαhµν −12
Dαh Dαh + Dαh Dβhαβ − Dαhµβ Dβhµα
+ R
„14
h2 −12
hµνhµν«
+ Rµν`2hλµhνλ − h hµν
´.
Sven Faller General Relativity as an Effective Field Theory
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Expansion: Matter Lagrangian
e.g. scalar particle: Lm =√−g[1
2gµν∂µφ∂νφ− 12m2φ2]
Lagrangian expansion:
Lm =p−g˘L(0)
m + L(1)m + L(2)
m + . . .¯
L(0)m =
12
`∂µφ∂
µφ−m2φ2´L(1)
m = −κ
2hµνTµν
Tµν ≡ ∂µφ∂µφ−12
gµν`∂λφ∂
λφ−m2φ2´ (energy-momentum-tensor)
L(2)m = κ2
„12
hµλhνλ −14
hhµν«∂µφ∂νφ−
κ2
8
„hλσhλσ −
12
hh«ˆ∂µφ∂
µφ−m2φ2˜
Sven Faller General Relativity as an Effective Field Theory
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Expansion: Matter Lagrangian
e.g. scalar particle: Lm =√−g[1
2gµν∂µφ∂νφ− 12m2φ2]
Lagrangian expansion:
Lm =p−g˘L(0)
m + L(1)m + L(2)
m + . . .¯
L(0)m =
12
`∂µφ∂
µφ−m2φ2´L(1)
m = −κ
2hµνTµν
Tµν ≡ ∂µφ∂µφ−12
gµν`∂λφ∂
λφ−m2φ2´ (energy-momentum-tensor)
L(2)m = κ2
„12
hµλhνλ −14
hhµν«∂µφ∂νφ−
κ2
8
„hλσhλσ −
12
hh«ˆ∂µφ∂
µφ−m2φ2˜
Sven Faller General Relativity as an Effective Field Theory
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Einstein Equation
Rµν −12
gµν R =κ2
4Tµν
gµν satisfies Einstein equationLagrangian terms linear in quantum field hµν vanishone-loop order:
L0 =p−g
2 R
κ2 + L(0)m
ffLquad =
p−g˘L(2)
g + Lgf + Lghost + L(2)m¯
Sven Faller General Relativity as an Effective Field Theory
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Einstein Equation
Rµν −12
gµν R =κ2
4Tµν
gµν satisfies Einstein equationLagrangian terms linear in quantum field hµν vanishone-loop order:
L0 =p−g
2 R
κ2 + L(0)m
ffLquad =
p−g˘L(2)
g + Lgf + Lghost + L(2)m¯
Sven Faller General Relativity as an Effective Field Theory
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Einstein Equation
Rµν −12
gµν R =κ2
4Tµν
gµν satisfies Einstein equationLagrangian terms linear in quantum field hµν vanishone-loop order:
L0 =p−g
2 R
κ2 + L(0)m
ffLquad =
p−g˘L(2)
g + Lgf + Lghost + L(2)m¯
Sven Faller General Relativity as an Effective Field Theory
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Einstein Equation
Rµν −12
gµν R =κ2
4Tµν
gµν satisfies Einstein equationLagrangian terms linear in quantum field hµν vanishone-loop order:
L0 =p−g
2 R
κ2 + L(0)m
ffLquad =
p−g˘L(2)
g + Lgf + Lghost + L(2)m¯
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
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Quantization Problems
field equations non linearcoupling constant κ has mass dimensioncoupling grows with energypossible solution: Effective Field Theoryseparate high enery fluctuations from small quantumfluctuations at ordinary energies
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
Quantization Problems
field equations non linearcoupling constant κ has mass dimensioncoupling grows with energypossible solution: Effective Field Theoryseparate high enery fluctuations from small quantumfluctuations at ordinary energies
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
Quantization Problems
field equations non linearcoupling constant κ has mass dimensioncoupling grows with energypossible solution: Effective Field Theoryseparate high enery fluctuations from small quantumfluctuations at ordinary energies
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Quantization
Quantization Problems
field equations non linearcoupling constant κ has mass dimensioncoupling grows with energypossible solution: Effective Field Theoryseparate high enery fluctuations from small quantumfluctuations at ordinary energies
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Introduction
Effective Lagrangian
low-energy d.o.f.: hµν + ghost fields + matter fields
Z[J] =
Z[dφ][dhµν ]eiSeff(φ,g,h,J)
Seff =∫
d4x√−g Leff, Leff = Lgr + Lm
effective Lagrangian = expansion in powers of hµν
Lgr = L(0)gr + L(2)
gr + L(4)gr + . . .
Lm = L(0)m + L(2)
m + . . .
Sven Faller General Relativity as an Effective Field Theory
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Introduction
Effective Lagrangian
low-energy d.o.f.: hµν + ghost fields + matter fields
Z[J] =
Z[dφ][dhµν ]eiSeff(φ,g,h,J)
Seff =∫
d4x√−g Leff, Leff = Lgr + Lm
effective Lagrangian = expansion in powers of hµν
Lgr = L(0)gr + L(2)
gr + L(4)gr + . . .
Lm = L(0)m + L(2)
m + . . .
Sven Faller General Relativity as an Effective Field Theory
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Introduction
Effective Lagrangian
low-energy d.o.f.: hµν + ghost fields + matter fields
Z[J] =
Z[dφ][dhµν ]eiSeff(φ,g,h,J)
Seff =∫
d4x√−g Leff, Leff = Lgr + Lm
effective Lagrangian = expansion in powers of hµν
Lgr = L(0)gr + L(2)
gr + L(4)gr + . . .
Lm = L(0)m + L(2)
m + . . .
Sven Faller General Relativity as an Effective Field Theory
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Graviton Progpagator
second order Lagrangian Lgr
harmonic gauge→ gauge fixing Lagrangian Lgf
quantum field hµν bilinear Lagrangian Lfreegr = − 1
2 hαβ ∆−1αβγδ hγδ
graviton propagator in harmonic gauge
�qαβ µν =
12
iq2 + iε
(ηαµηβν + ηανηβµ − ηαβηµν
)
Sven Faller General Relativity as an Effective Field Theory
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Graviton Progpagator
second order Lagrangian Lgr
harmonic gauge→ gauge fixing Lagrangian Lgf
quantum field hµν bilinear Lagrangian Lfreegr = − 1
2 hαβ ∆−1αβγδ hγδ
graviton propagator in harmonic gauge
�qαβ µν =
12
iq2 + iε
(ηαµηβν + ηανηβµ − ηαβηµν
)
Sven Faller General Relativity as an Effective Field Theory
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Graviton Progpagator
second order Lagrangian Lgr
harmonic gauge→ gauge fixing Lagrangian Lgf
quantum field hµν bilinear Lagrangian Lfreegr = − 1
2 hαβ ∆−1αβγδ hγδ
graviton propagator in harmonic gauge
�qαβ µν =
12
iq2 + iε
(ηαµηβν + ηανηβµ − ηαβηµν
)
Sven Faller General Relativity as an Effective Field Theory
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Graviton Progpagator
second order Lagrangian Lgr
harmonic gauge→ gauge fixing Lagrangian Lgf
quantum field hµν bilinear Lagrangian Lfreegr = − 1
2 hαβ ∆−1αβγδ hγδ
graviton propagator in harmonic gauge
�qαβ µν =
12
iq2 + iε
(ηαµηβν + ηανηβµ − ηαβηµν
)
Sven Faller General Relativity as an Effective Field Theory
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Vertex Factors
vertex factors at one-loop order
�−→q
p
p′
�`′ ↗ p
`↖ p′
�−→k
↘ q
↗ `
Sven Faller General Relativity as an Effective Field Theory
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Scalar-Graviton-Vertex
vertex factor
τµν = iZ
d4x d4x1 d4x2 d4x3 ei(px1−p′x2+qx3) · ∂
∂φ(x1)
∂
∂φ(x2)
∂
∂hµν(x3)
·−κ
2hαβ ·
»∂αφ(x)∂βφ(x)− 1
ηαβ`∂γφ(x)∂γφ(x)−m2φ(x)2´–ff
scalar-graviton-vertex
�−→q
p
p′
µν = − iκ2
{pµp′ν + pνp′µ − ηµν
[(p · p′
)−m2]}
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Scalar-Graviton-Vertex
vertex factor
τµν = iZ
d4x d4x1 d4x2 d4x3 ei(px1−p′x2+qx3) · ∂
∂φ(x1)
∂
∂φ(x2)
∂
∂hµν(x3)
·−κ
2hαβ ·
»∂αφ(x)∂βφ(x)− 1
ηαβ`∂γφ(x)∂γφ(x)−m2φ(x)2´–ff
scalar-graviton-vertex
�−→q
p
p′
µν = − iκ2
{pµp′ν + pνp′µ − ηµν
[(p · p′
)−m2]}
Sven Faller General Relativity as an Effective Field Theory
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S-Matrix
Feynman diagrams→ invariant matrix element iMlong range interaction:Mfull = Aq2(1+ακ2q2 +βκ2q2 ln(−q2)+γκ2q2 m√
−q2+ . . . )
R-matrix: R = S− 1p, pT incoming, p′T , p′ outgoing momentum:
〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM
Born approximation: nonrelativistic limit position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
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S-Matrix
Feynman diagrams→ invariant matrix element iMlong range interaction:Mfull = Aq2(1+ακ2q2 +βκ2q2 ln(−q2)+γκ2q2 m√
−q2+ . . . )
R-matrix: R = S− 1p, pT incoming, p′T , p′ outgoing momentum:
〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM
Born approximation: nonrelativistic limit position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
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S-Matrix
Feynman diagrams→ invariant matrix element iMlong range interaction:Mfull = Aq2(1+ακ2q2 +βκ2q2 ln(−q2)+γκ2q2 m√
−q2+ . . . )
R-matrix: R = S− 1p, pT incoming, p′T , p′ outgoing momentum:
〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM
Born approximation: nonrelativistic limit position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
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S-Matrix
Feynman diagrams→ invariant matrix element iMlong range interaction:Mfull = Aq2(1+ακ2q2 +βκ2q2 ln(−q2)+γκ2q2 m√
−q2+ . . . )
R-matrix: R = S− 1p, pT incoming, p′T , p′ outgoing momentum:
〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM
Born approximation: nonrelativistic limit position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
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S-Matrix
Feynman diagrams→ invariant matrix element iMlong range interaction:Mfull = Aq2(1+ακ2q2 +βκ2q2 ln(−q2)+γκ2q2 m√
−q2+ . . . )
R-matrix: R = S− 1p, pT incoming, p′T , p′ outgoing momentum:
〈p′|R|p〉 = (2π)4δ4(p′ + p′T − p − pT ) iM
Born approximation: nonrelativistic limit position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
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Expansion: Gravitational Potential
lowest order: V (r) = −G m1·m2r (Newton)
higher order effects: O(v2/c2), O(Gm/rc2)general form:
V (r) = −G m1 m2
r
[1 + a · G(m1 + m2)
r c2 . . .
]dimensional analysis: loop diagrams→ extra power ofκ2 ∼ G, factor ~
gravitational potential: general form
V (r) = −Gm1m2
r
(1 + α
G(m1 + m2)
rc2 + βG~
r2c3 + . . .
)Sven Faller General Relativity as an Effective Field Theory
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Expansion: Gravitational Potential
lowest order: V (r) = −G m1·m2r (Newton)
higher order effects: O(v2/c2), O(Gm/rc2)general form:
V (r) = −G m1 m2
r
[1 + a · G(m1 + m2)
r c2 . . .
]dimensional analysis: loop diagrams→ extra power ofκ2 ∼ G, factor ~
gravitational potential: general form
V (r) = −Gm1m2
r
(1 + α
G(m1 + m2)
rc2 + βG~
r2c3 + . . .
)Sven Faller General Relativity as an Effective Field Theory
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Feynman RulesScatteringPotentialVertex Corrections
Expansion: Gravitational Potential
lowest order: V (r) = −G m1·m2r (Newton)
higher order effects: O(v2/c2), O(Gm/rc2)general form:
V (r) = −G m1 m2
r
[1 + a · G(m1 + m2)
r c2 . . .
]dimensional analysis: loop diagrams→ extra power ofκ2 ∼ G, factor ~
gravitational potential: general form
V (r) = −Gm1m2
r
(1 + α
G(m1 + m2)
rc2 + βG~
r2c3 + . . .
)Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Expansion: Gravitational Potential
lowest order: V (r) = −G m1·m2r (Newton)
higher order effects: O(v2/c2), O(Gm/rc2)general form:
V (r) = −G m1 m2
r
[1 + a · G(m1 + m2)
r c2 . . .
]dimensional analysis: loop diagrams→ extra power ofκ2 ∼ G, factor ~
gravitational potential: general form
V (r) = −Gm1m2
r
(1 + α
G(m1 + m2)
rc2 + βG~
r2c3 + . . .
)Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Expansion: Gravitational Potential
lowest order: V (r) = −G m1·m2r (Newton)
higher order effects: O(v2/c2), O(Gm/rc2)general form:
V (r) = −G m1 m2
r
[1 + a · G(m1 + m2)
r c2 . . .
]dimensional analysis: loop diagrams→ extra power ofκ2 ∼ G, factor ~
gravitational potential: general form
V (r) = −Gm1m2
r
(1 + α
G(m1 + m2)
rc2 + βG~
r2c3 + . . .
)Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Tree Level
iM =�kpk ′
q
p′
m1 m2 = ταβ(k , k ′) ·(
iPαβγδ
q2 + iε
)· τγδ(p,p′)
nonrelativitstic position space potential
V (~r) = −κ2
8m1m2
∫d3~q
(2π)3 ei~q·~r 1~q2 = −κ
2
8m1m2
14πr
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Tree Level
iM =�kpk ′
q
p′
m1 m2 = ταβ(k , k ′) ·(
iPαβγδ
q2 + iε
)· τγδ(p,p′)
nonrelativitstic position space potential
V (~r) = −κ2
8m1m2
∫d3~q
(2π)3 ei~q·~r 1~q2 = −κ
2
8m1m2
14πr
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Vertex Corrections - Overview
� =�(a)
+�(b)
+�(c)
+�(d)
+�(e)
+�(f)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
General Form
QED: Ward identiy⇔ vertex: energy conservation∂µTµν = 0momentum conservation: qµVµν ≡ 0
general vertex form
�q k1
k2 Vµν = 〈 k2 |Tµν | k1 〉 (1)
= F1(q2)
[kµ1 kν2 + kν1 kµ2 +
12
q2 gµν]
(2)
+ F2(q2)[qµqν − gµνq2] . (3)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
General Form
QED: Ward identiy⇔ vertex: energy conservation∂µTµν = 0momentum conservation: qµVµν ≡ 0
general vertex form
�q k1
k2 Vµν = 〈 k2 |Tµν | k1 〉 (1)
= F1(q2)
[kµ1 kν2 + kν1 kµ2 +
12
q2 gµν]
(2)
+ F2(q2)[qµqν − gµνq2] . (3)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
General Form
QED: Ward identiy⇔ vertex: energy conservation∂µTµν = 0momentum conservation: qµVµν ≡ 0
general vertex form
�q k1
k2 Vµν = 〈 k2 |Tµν | k1 〉 (1)
= F1(q2)
[kµ1 kν2 + kν1 kµ2 +
12
q2 gµν]
(2)
+ F2(q2)[qµqν − gµνq2] . (3)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Form Factors
�tree-level limit−−−−−−−−→�
normalization condition: F1 ≡ 0F2 no normalization conditionform factors dimensionless
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Form Factors
�tree-level limit−−−−−−−−→�
normalization condition: F1 ≡ 0F2 no normalization conditionform factors dimensionless
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Form Factors
�tree-level limit−−−−−−−−→�
normalization condition: F1 ≡ 0F2 no normalization conditionform factors dimensionless
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Feynman RulesScatteringPotentialVertex Corrections
Form Factors
�tree-level limit−−−−−−−−→�
normalization condition: F1 ≡ 0F2 no normalization conditionform factors dimensionless
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
First Loop Diagram
�←−q
k2
k1
mµν
x`
= iPσραβ iPγδλκi∫
d4`
(2π)41
`2(`− q)2[(`− k2)2 −m2]
· τρσ(k2 − `, k2,m) τλκ(k1, k2 − `,m) τµναβγδ(`,q) .
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Results
form factor F1(q2)
result from ln(−q2) π2 m√−q2
our - 3/4 1/16Donoghue - 3/4 1/16Akhundov et al. - 5/4 - 1/16
form factor F2(q2)
result from ln(−q2) π2 m√−q2
our 7/3 7/8Donoghue 3 7/8Akhundov et al. -7/3 - 7/8
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Results
form factor F1(q2)
result from ln(−q2) π2 m√−q2
our - 3/4 1/16Donoghue - 3/4 1/16Akhundov et al. - 5/4 - 1/16
form factor F2(q2)
result from ln(−q2) π2 m√−q2
our 7/3 7/8Donoghue 3 7/8Akhundov et al. -7/3 - 7/8
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
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Potential DefinitionsSummary
Second Loop Diagram
�k1
k2
µν
x` m
= iPαβλκiPγδρσVαβγδ∫
d4`
(2π)4
τµνλκρσ(`,q)
`2(`− q)2
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Results
form factor F1(q2)
result from ln(−q2) π2 m√−q2
our 0 0Donoghue 0 0Akhundov et al. 0 0
form factor F2(q2)
result from ln(−q2) π2 m√−q2
our -13/3 0Donoghue -13/3 0Akhundov et al. 7/3 0
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Results
form factor F1(q2)
result from ln(−q2) π2 m√−q2
our 0 0Donoghue 0 0Akhundov et al. 0 0
form factor F2(q2)
result from ln(−q2) π2 m√−q2
our -13/3 0Donoghue -13/3 0Akhundov et al. 7/3 0
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Result: Vertex Correctionform factors
F1(q2) = 1 +κ2q2
32π2
»−
34
ln(−q2) +1
16π2 mp−q2
–,
F2(q2) =κ2 m2
32π2
»−2 ln(−q2) +
78
π2 mp−q2
–.
tree-level normalized→ factor κ/2i
Vµν = −iκ2
»1 +
κ2 q2
32π2
„−
34
ln(−q2) +1
16π2 mp−q2
«–„kµ1 kν2 + kν1 kµ2 +
12
q2ηµν«
−κ3 m2
64π2
»−2 ln(−q2) +
78
π2 mp−q2
–„qµqν −
12
q2ηµν«.
Sven Faller General Relativity as an Effective Field Theory
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Effective Field Theory of GravityLeading Quantum Corrections
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Result: Vertex Correctionform factors
F1(q2) = 1 +κ2q2
32π2
»−
34
ln(−q2) +1
16π2 mp−q2
–,
F2(q2) =κ2 m2
32π2
»−2 ln(−q2) +
78
π2 mp−q2
–.
tree-level normalized→ factor κ/2i
Vµν = −iκ2
»1 +
κ2 q2
32π2
„−
34
ln(−q2) +1
16π2 mp−q2
«–„kµ1 kν2 + kν1 kµ2 +
12
q2ηµν«
−κ3 m2
64π2
»−2 ln(−q2) +
78
π2 mp−q2
–„qµqν −
12
q2ηµν«.
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Vacuum Polarisation - Diagrams
�
=
�(a)
+
�(b)
+
�(c)
vacuum polarisation tensor Παβγδ → graviton propagatorcorrection
∆αβγδ + ∆αβµν iΠµνρσ∆ρσγδ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Vacuum Polarization
counter-term, graviton self energy and ghost→ Veltman and ’t Hooft (1974)matter loop
Παβγδ =i
32π2
κ2
4
»−15
`qαqβ − q2ηαβ
´`qγqδ − q2ηγδ
´−
130
`qαqγ − q2ηαγ
´·`qβqδ − q2ηβδ
´−
130
`qαqδ − q2ηαδ
´`qβqγ − q2ηβγ
´–− 2m4
3ηαβηγδ
−23
m23ηαβ`ηγδq2 − qγqδ
´−
23
m23ηγδ`ηαβq2 − qαqβ
´ffln(−q2)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Vacuum Polarization
counter-term, graviton self energy and ghost→ Veltman and ’t Hooft (1974)matter loop
Παβγδ =i
32π2
κ2
4
»−15
`qαqβ − q2ηαβ
´`qγqδ − q2ηγδ
´−
130
`qαqγ − q2ηαγ
´·`qβqδ − q2ηβδ
´−
130
`qαqδ − q2ηαδ
´`qβqγ − q2ηβγ
´–− 2m4
3ηαβηγδ
−23
m23ηαβ`ηγδq2 − qγqδ
´−
23
m23ηγδ`ηαβq2 − qαqβ
´ffln(−q2)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
One Particle Irreduzible Diagrams
�k1
k ′1k2
k ′2
q=�+�
+�+�q
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Gravitational Potential
1PI-diagrams→ S-matrix contributionsiM = Vµν1 (k1, k2, q,m1)
ˆ444µνρσ+444µναβ iΠαβγδ444γδρσ
˜Vρσ2 (k ′1, k
′2,−q,m)
position space gravitational potential
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 − 16730π
G ~r2c3
]include massless Neutrino-loop
gravitational potential
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 −(
16730π
+Nν
40π
)G ~r2c3
]Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Gravitational Potential
1PI-diagrams→ S-matrix contributionsiM = Vµν1 (k1, k2, q,m1)
ˆ444µνρσ+444µναβ iΠαβγδ444γδρσ
˜Vρσ2 (k ′1, k
′2,−q,m)
position space gravitational potential
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 − 16730π
G ~r2c3
]include massless Neutrino-loop
gravitational potential
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 −(
16730π
+Nν
40π
)G ~r2c3
]Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Gravitational Potential
1PI-diagrams→ S-matrix contributionsiM = Vµν1 (k1, k2, q,m1)
ˆ444µνρσ+444µναβ iΠαβγδ444γδρσ
˜Vρσ2 (k ′1, k
′2,−q,m)
position space gravitational potential
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 − 16730π
G ~r2c3
]include massless Neutrino-loop
gravitational potential
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 −(
16730π
+Nν
40π
)G ~r2c3
]Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Gravitational Potential (so far)
our result:
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 −(
16730π
+Nν
40π
)G ~r2c3
]Donoghue
V (r) = −G m1 m2
r
[1− G(m1 + m2)
r c2 − (135 + 2Nν)
30π2G ~r c3
].
Akhundov et al.
V (r) = −G m1 m2
r
[1 +
G(m1 + m2)
r c2 − 10730π2
G ~r2 c2
]Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Further Potential Definition
Hamber and Liu (1995)
���m1
�m2�m1 m2�q
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Further Contributions
double-seagull diagramm:
�V4 = −G m1 m2
r
[2G(m1 + m2)
r c2 − 14π
G ~r2 c3
]
triangle diagrams:
�V〉◦〈 = −G m1 m2
r
[112π
G ~r2 c3
]
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Further Contributions
double-seagull diagramm:
�V4 = −G m1 m2
r
[2G(m1 + m2)
r c2 − 14π
G ~r2 c3
]
triangle diagrams:
�V〉◦〈 = −G m1 m2
r
[112π
G ~r2 c3
]
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Further Contributions
cross-boxed diagramms (Bjerrum-Bohr 2003)
�+�V�(r) = −473
m1 m2 G2
π r3 .
Gravitational Potential
V (r) = −G m1 m2
r
[1 +
G(m1 + m2)
r c2 − 64 + Nν
40πG ~r2c3
]
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Further Contributions
cross-boxed diagramms (Bjerrum-Bohr 2003)
�+�V�(r) = −473
m1 m2 G2
π r3 .
Gravitational Potential
V (r) = −G m1 m2
r
[1 +
G(m1 + m2)
r c2 − 64 + Nν
40πG ~r2c3
]
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Summary
full theory of quantum gravity unknowneffective field theory of gravity
low energy effects separated from high-energy effectsone-loop order quantum predictions
evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Summary
full theory of quantum gravity unknowneffective field theory of gravity
low energy effects separated from high-energy effectsone-loop order quantum predictions
evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Summary
full theory of quantum gravity unknowneffective field theory of gravity
low energy effects separated from high-energy effectsone-loop order quantum predictions
evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Summary
full theory of quantum gravity unknowneffective field theory of gravity
low energy effects separated from high-energy effectsone-loop order quantum predictions
evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Summary
full theory of quantum gravity unknowneffective field theory of gravity
low energy effects separated from high-energy effectsone-loop order quantum predictions
evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Summary
full theory of quantum gravity unknowneffective field theory of gravity
low energy effects separated from high-energy effectsone-loop order quantum predictions
evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravity
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Thanks
supervisor: Prof. Dr. T. Mannelsecond supervisor: Dr. A. Khodjamirianfor usefull tips and discussions
Dr. Th. FeldmannDr. E. Bjerrum-BohrProf. Dr. F. DonoghueDipl.-Phys. M. JungDipl.-Phys. N. OffenDipl.-Phys. K. Grybel
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion - Affine Connection
background field method: metric expansion
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
affine connection: Γλµν = Γλµν +−Γλµν +
=Γλµν
with
Γλµν =12
gλσ(∂µgσν + ∂ν gσµ − ∂σgµν
)(O(h0)) ,
−Γλµν =
κ
2gλσ(Dµhσν + Dνhσµ − Dσhµν
)(O(h1)) ,
=Γλµν = −κ
2
2hλγ(Dµhγν + Dνhµγ − Dγhµν
)(O(h2)) .
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion - Affine Connection
background field method: metric expansion
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
affine connection: Γλµν = Γλµν +−Γλµν +
=Γλµν
with
Γλµν =12
gλσ(∂µgσν + ∂ν gσµ − ∂σgµν
)(O(h0)) ,
−Γλµν =
κ
2gλσ(Dµhσν + Dνhσµ − Dσhµν
)(O(h1)) ,
=Γλµν = −κ
2
2hλγ(Dµhγν + Dνhµγ − Dγhµν
)(O(h2)) .
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion - Affine Connection
background field method: metric expansion
gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .
affine connection: Γλµν = Γλµν +−Γλµν +
=Γλµν
with
Γλµν =12
gλσ(∂µgσν + ∂ν gσµ − ∂σgµν
)(O(h0)) ,
−Γλµν =
κ
2gλσ(Dµhσν + Dνhσµ − Dσhµν
)(O(h1)) ,
=Γλµν = −κ
2
2hλγ(Dµhγν + Dνhµγ − Dγhµν
)(O(h2)) .
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion: CurvatureRiemann curvature tensor:Rβ
αµν = DµΓβαν − DνΓβαµ + ΓλανΓβλµ − ΓλαµΓβλν ≡ Rβαµν +−Rβαµν +
=Rβαµν
Ricci scalar:R = gαµ
=Rαµ − κhαµ
−Rαµ + κ2hαγ hγµRαµ
= κ2−
12
Dµ`hβγDµhγβ
´+
12
Dβˆhβν`2Dµhνµ − Dνhµµ
´˜+
14
`Dµhνβ + Dβhνµ − Dνhµβ
´`Dµhβν + Dνhβµ − Dβhµν
´−
14
`2Dµhνµ − Dνhµµ
´Dνhββ −
12
hαµDµDαhββ
+12
hµαDβ`Dαhβµ + Dµhβα − Dβhαµ
´+ κ2hβµhαβ Rµ
α
ff
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion: CurvatureRiemann curvature tensor:Rβ
αµν = DµΓβαν − DνΓβαµ + ΓλανΓβλµ − ΓλαµΓβλν ≡ Rβαµν +−Rβαµν +
=Rβαµν
Ricci scalar:R = gαµ
=Rαµ − κhαµ
−Rαµ + κ2hαγ hγµRαµ
= κ2−
12
Dµ`hβγDµhγβ
´+
12
Dβˆhβν`2Dµhνµ − Dνhµµ
´˜+
14
`Dµhνβ + Dβhνµ − Dνhµβ
´`Dµhβν + Dνhβµ − Dβhµν
´−
14
`2Dµhνµ − Dνhµµ
´Dνhββ −
12
hαµDµDαhββ
+12
hµαDβ`Dαhβµ + Dµhβα − Dβhαµ
´+ κ2hβµhαβ Rµ
α
ff
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Gauge Fixing and Ghost Field
Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields
gauge fixing Lagrangian (’t Hoof and Veltman 1974)
Lgf =√−g{(
Dνhµν −12
Dµh)(
Dλhµλ − 12
Dµh)]
ghost field Lagrangian (ebd.)
Lghost =√−g η?µ
[DλDλgµν − Rµν
]ην
complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Gauge Fixing and Ghost Field
Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields
gauge fixing Lagrangian (’t Hoof and Veltman 1974)
Lgf =√−g{(
Dνhµν −12
Dµh)(
Dλhµλ − 12
Dµh)]
ghost field Lagrangian (ebd.)
Lghost =√−g η?µ
[DλDλgµν − Rµν
]ην
complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Gauge Fixing and Ghost Field
Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields
gauge fixing Lagrangian (’t Hoof and Veltman 1974)
Lgf =√−g{(
Dνhµν −12
Dµh)(
Dλhµλ − 12
Dµh)]
ghost field Lagrangian (ebd.)
Lghost =√−g η?µ
[DλDλgµν − Rµν
]ην
complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Gauge Fixing and Ghost Field
Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields
gauge fixing Lagrangian (’t Hoof and Veltman 1974)
Lgf =√−g{(
Dνhµν −12
Dµh)(
Dλhµλ − 12
Dµh)]
ghost field Lagrangian (ebd.)
Lghost =√−g η?µ
[DλDλgµν − Rµν
]ην
complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Gauge Fixing and Ghost Field
Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields
gauge fixing Lagrangian (’t Hoof and Veltman 1974)
Lgf =√−g{(
Dνhµν −12
Dµh)(
Dλhµλ − 12
Dµh)]
ghost field Lagrangian (ebd.)
Lghost =√−g η?µ
[DλDλgµν − Rµν
]ην
complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Gauge Fixing and Ghost Field
Yang-Mills field theorygauge fixingintroduce Fadeev-Popov ghost fields
gauge fixing Lagrangian (’t Hoof and Veltman 1974)
Lgf =√−g{(
Dνhµν −12
Dµh)(
Dλhµλ − 12
Dµh)]
ghost field Lagrangian (ebd.)
Lghost =√−g η?µ
[DλDλgµν − Rµν
]ην
complex ghostfield η: only contribution from vacuumpolarization to the graviton propagator
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories
asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories
asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories
asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories
asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories
asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories
asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Field Theories - Overview
renormalizabel and non-renormalizable field theoriesgeneral form of the Lagrangian
L = L(c1, c2, . . . , cn)
low energy structure determined by finite parametersc1, c2, . . . cntwo different typs of quantum field theories
asympotically free theories - ultraviolet stable theoriesultraviolet unstable theories
ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Field Theory
two different types of effecive field theoriesdecoupling effective field theories
heavy degrees of freedom integrated outeffective level no light particlesLagrangian general form
Leff = LD≤4 +∑D>4
1ΛD−4
∑iD
giD OiD
non-decoupling effective field theoriesfundamental to effective level by phase transistionspontaneously broken symmetry→ light pseudo-Goldstonebosons
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective General Relativity
quantization process: first order curvaturerenormalization higher order neededone-loop Feynman rules→ higher order curvature nocontributiongravitational action
S = Svac + Sm + Sgf + Sghost
=
∫d4x√−g{
2R
κ2 + Lm + Lgf + Lghost
}quantum degrees of freedom: gravitation field hµν ,ghostfields ηµ, η∗µ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective General Relativity
quantization process: first order curvaturerenormalization higher order neededone-loop Feynman rules→ higher order curvature nocontributiongravitational action
S = Svac + Sm + Sgf + Sghost
=
∫d4x√−g{
2R
κ2 + Lm + Lgf + Lghost
}quantum degrees of freedom: gravitation field hµν ,ghostfields ηµ, η∗µ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective General Relativity
quantization process: first order curvaturerenormalization higher order neededone-loop Feynman rules→ higher order curvature nocontributiongravitational action
S = Svac + Sm + Sgf + Sghost
=
∫d4x√−g{
2R
κ2 + Lm + Lgf + Lghost
}quantum degrees of freedom: gravitation field hµν ,ghostfields ηµ, η∗µ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective General Relativity
quantization process: first order curvaturerenormalization higher order neededone-loop Feynman rules→ higher order curvature nocontributiongravitational action
S = Svac + Sm + Sgf + Sghost
=
∫d4x√−g{
2R
κ2 + Lm + Lgf + Lghost
}quantum degrees of freedom: gravitation field hµν ,ghostfields ηµ, η∗µ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective General Relativity
quantization process: first order curvaturerenormalization higher order neededone-loop Feynman rules→ higher order curvature nocontributiongravitational action
S = Svac + Sm + Sgf + Sghost
=
∫d4x√−g{
2R
κ2 + Lm + Lgf + Lghost
}quantum degrees of freedom: gravitation field hµν ,ghostfields ηµ, η∗µ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Effective Quantum Gravity
Sgrav =∫
d4x√−g{ 2κ2 R + c1R
2 + c2RµνRµν + · · ·+ Lm}
parameter cifinite numbervalues unknownfree parameters
low-energy limit: effective quantum gravity
Sgrav =
∫d4x√−g{
2κ2 R + c1R
2 + c2RµνRµν + Lm
}higher energy: theory renormalizationvalue of c1, c2 shifted→ renormalization effects absorbedby parameters
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Contributions
gravitational contributions
L(0)gr (Λ) , L(2)
gr (R) , L(4)gr (R2)
matter fieldscalar matter field:
L(0)m (φ,m) , L(2)
m (φ,m,R)
massless matter field:
L(0)m = 0 , L(2)
m (φ,R) , L(0)m (φ,R,R2)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Contributions
gravitational contributions
L(0)gr (Λ) , L(2)
gr (R) , L(4)gr (R2)
matter fieldscalar matter field:
L(0)m (φ,m) , L(2)
m (φ,m,R)
massless matter field:
L(0)m = 0 , L(2)
m (φ,R) , L(0)m (φ,R,R2)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Contributions
gravitational contributions
L(0)gr (Λ) , L(2)
gr (R) , L(4)gr (R2)
matter fieldscalar matter field:
L(0)m (φ,m) , L(2)
m (φ,m,R)
massless matter field:
L(0)m = 0 , L(2)
m (φ,R) , L(0)m (φ,R,R2)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Contributions
gravitational contributions
L(0)gr (Λ) , L(2)
gr (R) , L(4)gr (R2)
matter fieldscalar matter field:
L(0)m (φ,m) , L(2)
m (φ,m,R)
massless matter field:
L(0)m = 0 , L(2)
m (φ,R) , L(0)m (φ,R,R2)
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Counter Terms
loop-diagrams→ UV-divergenciesUV-divergencies separated non-divergent parts byconstant ci and di regulationone-loop order: Veltman and ’t Hoof (1974)
L(1)M =
p−g
8π2ε
1
120R2 +
720
RµνRµν
ffwith ε = 4− D
MS-scheme
c(r)1 = c1 +
1960π2ε
and c(r)2 = c2 +
7160π2ε
two-loop order
L(2)M =
209κ2880(16π2)2
1ε
p−g Rαβ
γδRγδρσRρσ
αβ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Counter Terms
loop-diagrams→ UV-divergenciesUV-divergencies separated non-divergent parts byconstant ci and di regulationone-loop order: Veltman and ’t Hoof (1974)
L(1)M =
p−g
8π2ε
1
120R2 +
720
RµνRµν
ffwith ε = 4− D
MS-scheme
c(r)1 = c1 +
1960π2ε
and c(r)2 = c2 +
7160π2ε
two-loop order
L(2)M =
209κ2880(16π2)2
1ε
p−g Rαβ
γδRγδρσRρσ
αβ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Counter Terms
loop-diagrams→ UV-divergenciesUV-divergencies separated non-divergent parts byconstant ci and di regulationone-loop order: Veltman and ’t Hoof (1974)
L(1)M =
p−g
8π2ε
1
120R2 +
720
RµνRµν
ffwith ε = 4− D
MS-scheme
c(r)1 = c1 +
1960π2ε
and c(r)2 = c2 +
7160π2ε
two-loop order
L(2)M =
209κ2880(16π2)2
1ε
p−g Rαβ
γδRγδρσRρσ
αβ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Counter Terms
loop-diagrams→ UV-divergenciesUV-divergencies separated non-divergent parts byconstant ci and di regulationone-loop order: Veltman and ’t Hoof (1974)
L(1)M =
p−g
8π2ε
1
120R2 +
720
RµνRµν
ffwith ε = 4− D
MS-scheme
c(r)1 = c1 +
1960π2ε
and c(r)2 = c2 +
7160π2ε
two-loop order
L(2)M =
209κ2880(16π2)2
1ε
p−g Rαβ
γδRγδρσRρσ
αβ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Counter Terms
loop-diagrams→ UV-divergenciesUV-divergencies separated non-divergent parts byconstant ci and di regulationone-loop order: Veltman and ’t Hoof (1974)
L(1)M =
p−g
8π2ε
1
120R2 +
720
RµνRµν
ffwith ε = 4− D
MS-scheme
c(r)1 = c1 +
1960π2ε
and c(r)2 = c2 +
7160π2ε
two-loop order
L(2)M =
209κ2880(16π2)2
1ε
p−g Rαβ
γδRγδρσRρσ
αβ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Evaluation Vertex Factors
momentum space vertex factors
Vµ1ν1,...,µmνn = +iZ
d4x d4x1 . . . d4xn d4y1 . . . d4ym ei(p1x1+···+pnxn+q1y1+···+qmym)
· δ
δJ1(x1)· . . . · δ
δJn(xn)· δ
δHν1µ11 (y1)
· . . . · δ
δHµmνmm (ym)
· Lint`φ1, . . . , φn,H1, . . .Hm
´(x)
sources of gravity: J1, . . . , Jnexternal and internal gravity field: Hµ1ν1
1 , . . . ,Hµmνmm
incoming p1, . . . ,pn, outgoing q1, . . .qm momentumSven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Evaluation Vertex Factors
momentum space vertex factors
Vµ1ν1,...,µmνn = +iZ
d4x d4x1 . . . d4xn d4y1 . . . d4ym ei(p1x1+···+pnxn+q1y1+···+qmym)
· δ
δJ1(x1)· . . . · δ
δJn(xn)· δ
δHν1µ11 (y1)
· . . . · δ
δHµmνmm (ym)
· Lint`φ1, . . . , φn,H1, . . .Hm
´(x)
sources of gravity: J1, . . . , Jnexternal and internal gravity field: Hµ1ν1
1 , . . . ,Hµmνmm
incoming p1, . . . ,pn, outgoing q1, . . .qm momentumSven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Evaluation Vertex Factors
momentum space vertex factors
Vµ1ν1,...,µmνn = +iZ
d4x d4x1 . . . d4xn d4y1 . . . d4ym ei(p1x1+···+pnxn+q1y1+···+qmym)
· δ
δJ1(x1)· . . . · δ
δJn(xn)· δ
δHν1µ11 (y1)
· . . . · δ
δHµmνmm (ym)
· Lint`φ1, . . . , φn,H1, . . .Hm
´(x)
sources of gravity: J1, . . . , Jnexternal and internal gravity field: Hµ1ν1
1 , . . . ,Hµmνmm
incoming p1, . . . ,pn, outgoing q1, . . .qm momentumSven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Evaluation Vertex Factors
momentum space vertex factors
Vµ1ν1,...,µmνn = +iZ
d4x d4x1 . . . d4xn d4y1 . . . d4ym ei(p1x1+···+pnxn+q1y1+···+qmym)
· δ
δJ1(x1)· . . . · δ
δJn(xn)· δ
δHν1µ11 (y1)
· . . . · δ
δHµmνmm (ym)
· Lint`φ1, . . . , φn,H1, . . .Hm
´(x)
sources of gravity: J1, . . . , Jnexternal and internal gravity field: Hµ1ν1
1 , . . . ,Hµmνmm
incoming p1, . . . ,pn, outgoing q1, . . .qm momentumSven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Graviton-Graviton-Scalar-Vertex
Lagrangian O(h2)
L(2)m = κ2
„12
hµνhνλ−14
hhµν«∂µφ∂νφ−
κ2
8
„hλσhλσ−
12
hh«ˆ∂µφ∂
µφ−m2φ2˜vertex factor
Vηλρσ = +iZ
d4x d4x1 d4x2 d4x3 d4x4 ei(px1−p′x2+kx3−kx4)
· ∂
∂φ(x1)· ∂
∂φ(x2)· ∂
∂hηλ(x3)· ∂
∂hρσ(x4)
· κ2
2hηλ»
1ηλαδ1δ
ρσβ −14`ηηλ1ρσαβ + ηρσ1ηλαβ
´–∂αφ(x)∂βφ(x)
− 14
„1ηλρσ −
12ηηλ −
12ηηληρσ
«ˆ∂γφ(x)∂γφ(x)−m2φ(x)2˜ffhρσ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Graviton-Graviton-Scalar-Vertex
Lagrangian O(h2)
L(2)m = κ2
„12
hµνhνλ−14
hhµν«∂µφ∂νφ−
κ2
8
„hλσhλσ−
12
hh«ˆ∂µφ∂
µφ−m2φ2˜vertex factor
Vηλρσ = +iZ
d4x d4x1 d4x2 d4x3 d4x4 ei(px1−p′x2+kx3−kx4)
· ∂
∂φ(x1)· ∂
∂φ(x2)· ∂
∂hηλ(x3)· ∂
∂hρσ(x4)
· κ2
2hηλ»
1ηλαδ1δ
ρσβ −14`ηηλ1ρσαβ + ηρσ1ηλαβ
´–∂αφ(x)∂βφ(x)
− 14
„1ηλρσ −
12ηηλ −
12ηηληρσ
«ˆ∂γφ(x)∂γφ(x)−m2φ(x)2˜ffhρσ
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Graviton-Graviton-Scalar-Vertex
�`↗ ρσ
p
`′ ↖ p′ηλ
Vηλρσ =iκ2
2
»1ηλαδ1
δρσβ −
14
`ηηλ1ρσαβ + ηρσ1ηλαβ
´–`pαp′β + pβp′α
´−
12
»1ηλρσ −
12ηηληρσ
–`(p · p′)−m2´ff
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Three-Graviton-Vertex
τµναβγδ
(k, q) = −iκ
2
Pαβγδ
»kµkν + (k − q)µ(k − q)ν + qµqν +
3
2η
µν q2–
+ 2qλqσˆ1
σλαβ
1µν
γδ+ 1
σλγδ
1µν
αβ− 1
µσαβ
1νλ
γδ− 1
µσγδ
1νλ
αβ
˜+
ˆqλqµ`
ηαβ1νλ
γδ+ ηγδ1
νλαβ
´+ qλqν `
ηαβ1µλ
γδ+ ηγδ1
µλαβ
´− q2`
ηαβ1µν
γδ+ ηγδ1
µναβ
´− ηµν qλqσ
`ηαβ1
σλγδ
+ ηγδ1σλ
αβ
´˜+
ˆ2qλ
˘1
λσαβ
1ν
γδσ(k − q)µ + 1
λσαβ
1µ
γδσ(k − q)ν
− 1λσ
γδ1
ναβσ
kµ − 1λσ
γδ1
µαβσ
kν ¯+ q2`
1µ
αβσ1
νσγδ
+ 1νσ
αβ1
µγδσ
´+ η
µν qσqλ`1
λραβ
1σ
γδρ+ 1
λργδ
1σ
αβρ
´˜+
`k2 + (k − q)
´»1
µσαβ
1ν
γδσ+ 1
νσγδ
1µ
αβσ−
1
2η
µνPαβγδ
–−
`1
µνγδ
ηαβk2 − 1µν
αβηγδ(k − q)2´ffffffff
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Born approximation
covariant normalized: 〈p′|p〉 = 2Epδ3(~p − ~p′), Ep =
√m2 + ~p2
Born approximation: 〈p′|p〉 = −i V (~q)(2π)δ(E~p′ − E~p)nonrelativistic limit: interaction potential
−V (~q) =M 12m1
12m2
∫d3~pT
(2π)3 δ3(~p′ + ~p)
nonrelativistic limit: V (~q) = − 12m1·2m2
MFourier transformation to position-space:
nonrelativistic limit: position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Born approximation
covariant normalized: 〈p′|p〉 = 2Epδ3(~p − ~p′), Ep =
√m2 + ~p2
Born approximation: 〈p′|p〉 = −i V (~q)(2π)δ(E~p′ − E~p)nonrelativistic limit: interaction potential
−V (~q) =M 12m1
12m2
∫d3~pT
(2π)3 δ3(~p′ + ~p)
nonrelativistic limit: V (~q) = − 12m1·2m2
MFourier transformation to position-space:
nonrelativistic limit: position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Born approximation
covariant normalized: 〈p′|p〉 = 2Epδ3(~p − ~p′), Ep =
√m2 + ~p2
Born approximation: 〈p′|p〉 = −i V (~q)(2π)δ(E~p′ − E~p)nonrelativistic limit: interaction potential
−V (~q) =M 12m1
12m2
∫d3~pT
(2π)3 δ3(~p′ + ~p)
nonrelativistic limit: V (~q) = − 12m1·2m2
MFourier transformation to position-space:
nonrelativistic limit: position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Born approximation
covariant normalized: 〈p′|p〉 = 2Epδ3(~p − ~p′), Ep =
√m2 + ~p2
Born approximation: 〈p′|p〉 = −i V (~q)(2π)δ(E~p′ − E~p)nonrelativistic limit: interaction potential
−V (~q) =M 12m1
12m2
∫d3~pT
(2π)3 δ3(~p′ + ~p)
nonrelativistic limit: V (~q) = − 12m1·2m2
MFourier transformation to position-space:
nonrelativistic limit: position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Born approximation
covariant normalized: 〈p′|p〉 = 2Epδ3(~p − ~p′), Ep =
√m2 + ~p2
Born approximation: 〈p′|p〉 = −i V (~q)(2π)δ(E~p′ − E~p)nonrelativistic limit: interaction potential
−V (~q) =M 12m1
12m2
∫d3~pT
(2π)3 δ3(~p′ + ~p)
nonrelativistic limit: V (~q) = − 12m1·2m2
MFourier transformation to position-space:
nonrelativistic limit: position-space potential
V (~r) = − 12m1
12m2
∫d3~q
(2π)3 ei~q·~r M
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion Form Factors
dimensionless combinations: κ2m2, κ2q2
expansion:
F1(q2) = 1 + d1q2 + κ2q2„
l1 + l2 ln(−q2)
µ2 + l3
sm2
−q2 + . . .
«,
F2(q2) = −4(d2 − d3)m2 + κ2m2„
l4 + l5 ln(−q2)
µ2 + l6
sm2
−q2 + . . .
«di : L(2)
m contributionsli : one-loop contributionsl1, l4 : divergent high enery contributionsl2, l3, l5, l6 : finite non-analytic low energy contributions
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion Form Factors
dimensionless combinations: κ2m2, κ2q2
expansion:
F1(q2) = 1 + d1q2 + κ2q2„
l1 + l2 ln(−q2)
µ2 + l3
sm2
−q2 + . . .
«,
F2(q2) = −4(d2 − d3)m2 + κ2m2„
l4 + l5 ln(−q2)
µ2 + l6
sm2
−q2 + . . .
«di : L(2)
m contributionsli : one-loop contributionsl1, l4 : divergent high enery contributionsl2, l3, l5, l6 : finite non-analytic low energy contributions
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion Form Factors
dimensionless combinations: κ2m2, κ2q2
expansion:
F1(q2) = 1 + d1q2 + κ2q2„
l1 + l2 ln(−q2)
µ2 + l3
sm2
−q2 + . . .
«,
F2(q2) = −4(d2 − d3)m2 + κ2m2„
l4 + l5 ln(−q2)
µ2 + l6
sm2
−q2 + . . .
«di : L(2)
m contributionsli : one-loop contributionsl1, l4 : divergent high enery contributionsl2, l3, l5, l6 : finite non-analytic low energy contributions
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion Form Factors
dimensionless combinations: κ2m2, κ2q2
expansion:
F1(q2) = 1 + d1q2 + κ2q2„
l1 + l2 ln(−q2)
µ2 + l3
sm2
−q2 + . . .
«,
F2(q2) = −4(d2 − d3)m2 + κ2m2„
l4 + l5 ln(−q2)
µ2 + l6
sm2
−q2 + . . .
«di : L(2)
m contributionsli : one-loop contributionsl1, l4 : divergent high enery contributionsl2, l3, l5, l6 : finite non-analytic low energy contributions
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion Form Factors
dimensionless combinations: κ2m2, κ2q2
expansion:
F1(q2) = 1 + d1q2 + κ2q2„
l1 + l2 ln(−q2)
µ2 + l3
sm2
−q2 + . . .
«,
F2(q2) = −4(d2 − d3)m2 + κ2m2„
l4 + l5 ln(−q2)
µ2 + l6
sm2
−q2 + . . .
«di : L(2)
m contributionsli : one-loop contributionsl1, l4 : divergent high enery contributionsl2, l3, l5, l6 : finite non-analytic low energy contributions
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Expansion Form Factors
dimensionless combinations: κ2m2, κ2q2
expansion:
F1(q2) = 1 + d1q2 + κ2q2„
l1 + l2 ln(−q2)
µ2 + l3
sm2
−q2 + . . .
«,
F2(q2) = −4(d2 − d3)m2 + κ2m2„
l4 + l5 ln(−q2)
µ2 + l6
sm2
−q2 + . . .
«di : L(2)
m contributionsli : one-loop contributionsl1, l4 : divergent high enery contributionsl2, l3, l5, l6 : finite non-analytic low energy contributions
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Renormalization
combination l1, l4 and di → renormalized values
d (r)1 (µ2) = d1 + κ2l1
d (r)2 (µ2) + d (r)
3 (µ2) = d2 + d3 − κ2 l44
experiments: measure renormalized values
d (r)i (µ2)→ measured values depend on µ2 choice in
logarithms but all physics independent of µ2
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Renormalization
combination l1, l4 and di → renormalized values
d (r)1 (µ2) = d1 + κ2l1
d (r)2 (µ2) + d (r)
3 (µ2) = d2 + d3 − κ2 l44
experiments: measure renormalized values
d (r)i (µ2)→ measured values depend on µ2 choice in
logarithms but all physics independent of µ2
Sven Faller General Relativity as an Effective Field Theory
IntroductionQuantum Gravity
Effective Field Theory of GravityLeading Quantum Corrections
Evaluation of the Vertex CorrectionsGravitational Potential
Potential DefinitionsSummary
Effective Gravity
Renormalization
combination l1, l4 and di → renormalized values
d (r)1 (µ2) = d1 + κ2l1
d (r)2 (µ2) + d (r)
3 (µ2) = d2 + d3 − κ2 l44
experiments: measure renormalized values
d (r)i (µ2)→ measured values depend on µ2 choice in
logarithms but all physics independent of µ2
Sven Faller General Relativity as an Effective Field Theory
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