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Introduction Quantum Gravity EFT of Gravity Potential Summary Backup Effective Field Theory of Gravity: Leading Quantum Gravitational Corrections to Newtons and Colulombs Law Sven Faller Theoretische Physik 1 Universität Siegen Theorieseminar Universität Köln – 02.02.2009

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Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Effective Field Theory of Gravity:Leading Quantum Gravitational Corrections to

Newtons and Colulombs Law

Sven Faller

Theoretische Physik 1Universität Siegen

Theorieseminar Universität Köln – 02.02.2009

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

table of contents

1 Introduction

2 Quantum Gravity

3 EFT of Gravity

4 Potential

5 Summary

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Motivation

all known field theories: quantum field theoriesgravity quantization – Feynman (1962)

Is it possible that gravity is not quantized and all the rest ofthe world is? – Answer: No!illustration: Feynman’s two-slit diffraction Gedankenexperiment→ quantum nature of a field could not be destroyed

gravity must be a quantum field theoryproblem: consistent quantization method unknownQuantum Gravity: De Witt, Feynman, ’t Hooft and Veltman

present energies: quantum gravity non-renormalizablebut: low-energy predictions independent of high-energyinfluence

possible solution:Effective Field Theory of Gravity (Donoghue 1994)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Motivation

simple dimensional analysis: lowest-order corrections toNewtonian potential

V (r) = −Gm1m2

r

(1+α

G(m1 + m2)

rc2︸ ︷︷ ︸relativistic correction

+ βG~r2c3︸ ︷︷ ︸

quantum correction

+ . . .

)

lot of papers with different results for coefficients α and β, e.g.Paper α β

Donoghue (1994) -1 − 12730π2

Akhundov et al. (1997) +1 − 10730π2

Bjerrum-Bohr et al. (2003) +3 4110π

correct numbers:sufficiently interesting from theoretical point of view

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Newton’s Gravity (1687)

gravity force

~F12(~r) = mi ~a = −G mG MG~r1 −~r2

|~r1 −~r2|3

three different masses: inertial mass mi ,passive gravitational mass mG and activegravitational mass MG

third law - “actio est reactio„- inert and activegravitational mass equal→ problem: no explanation for equality

– seems to be coincidental↪→ experimental measurements:

verification of equality, basic for↪→ Einstein’s Principle of Equivalence

Torsionfaden

Spiegel

Kugel derMasse m

Kugel derMasse M

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Einstein’s Special Relativity

Newton: Galilei transformations between ISEinstein (1905):Newton’s Theory must be specialized by universality of thevelocity 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

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

General Relativity

Einstein (1916):Die Grundlagen der allgemeinen Relativitätstheorie.Ann. d. Physik, 322(10):891-921Newton: 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)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Some Definitions

units: ~ = c = 1flat space: ηµν = diag(1,−1,−1,−1)

metric tensor gµν and g = detgµνRieman Space (R4)

metric definition: ds2 = gµν(x) dxµ dxν

affine connection: Γλµν = 12 gλσ

(∂µgνσ + ∂νgνσ − ∂σgµν

)Riemann curvature tensor:

Rλµσν = Γλµν,σ − Γλµσ,ν + ΓλτσΓτµν − ΓλτνΓτµσ

Ricci tensor: Rµν = Rλµλν ≡ Rνµ

Ricci scalar: R = gµν Rµν

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Background Field Method

introduced by ’t Hooft and Veltman (1974)gravitational field expanded about smooth backgroundmetric gµν

gµν = gµν + κhµνgµν = gµν − κhµν + κ2hµλhλν + . . .

classical equations of motion: gµν satifies

Rµν − 12

gµν R = −κ2

4Tµν (Einstein’s equation)

quantum field hµν : all dynamical information

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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 +O(R3) + Lm

}upper bound: constants c1, c2 < 1074 (Stelle 1978)λ ≡ −8πGΛ, cosmological constant Λ ≡ 0

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Scalar QED

general: scalar (charged or neutral) matterLm = 1

2

(gµν ∂µφ ∂νφ∗ −m2 |φ|2)

generally covariant Lagrangian for scalar QED

LSQED =√−g

[−1

4(gαµgβνFαµFβν

)+gµν DµφDνφ

∗−m2|φ|2]

photon field strength Fµν = DµAν − DνAµQED covariant derivative Dµ = ∂µ − ieqAµ(x)

Feynman Rules at 1 loop order→ expansion LSQED tosecond order in hphoton: Lorenz gauge LC = −1

2 (∂µAµ)2

this talk: only scalar theory discussed

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Quantization Problems

nonlinear nature of the theorydimensionful coupling constant κ =

√32πG

↪→ divergences appear which can not absorbed by introducedparameters

coupling grows with energy↪→ strong energy coupling at very high energies E > MPl

possible solution: Effective Field Theoryseparate high enery fluctuations from small quantumfluctuations at ordinary energies

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Effective Quantum Gravity

quantization process: first order curvaturerenormalization higher order needed→ value of c1, c2 shifted↪→ renormalization effects absorbed by c(r)

1 and c(r)2

↪→ one-loop Feynman rules: no contribution from higher order

gravitational action at one loop

S = Svac + Sm + Sgf + Sghost

=

∫d4x

√−g{

2R

κ2 + Lgf + Lghost︸ ︷︷ ︸= Lgr

+Lm

}

quantum degrees of freedom:gravitation field hµν and ghostfields ηµ, η∗µ

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Effective Lagrangian

low-energy d.o.f.: hµν + ghost fields + matter fields

Z[J] =

∫[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 + . . .

→ Feynman rules from Leff

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

S-Matrix

two particle scattering process: momentum transfer qFeynman 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-spacepotential

V (~r) = − 12m1

12m2

∫d3~q

(2π)3 ei~q·~r M

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Expansion: Gravitational Potential

lowest order: V (r) = −G m1·m2r

higher order effects: O(v2/c2), O(Gm/rc2)

pure gravity interaction general form:

V (r) = −G m1 m2

r

[1 + a · G(m1 + m2)

r c2 . . .

]dimensional analysis:loop diagrams→ extra power of κ2 ∼ G, factor ~

pure gravitational potential: general form

V (r) = −Gm1m2

r

(1 + α

G(m1 + m2)

rc2 + βG~

r2c3 + . . .

)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

One Particle Irreducible Diagrams - Scalar Theory

k1 k2

k′1 k′

2q

= + +q

+set

of+ +

+

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

One Particle Irreducible Diagrams - SQED

(m1, e1) (m2, e2) =

+set

of

{+ +

}

+set

of+

set

of

{+

}

+ +vacuum polarization

diagrams

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Vacuum Polarization Diagrams

scalar theory:

self-energy ghost massless scalar particle photon

k1 k2

k′1 k′

2

′t Hooft and Veltman (1974) Capper et al. (1974) Capper et al. (1974)

Hamber, Liu (1995)

SQED: only one diagram: Bjerrum-Bohr (2002)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Result: Scalar Theory

Scattering Potential

V (r) = −G m1 m2

r

[1 + 3

G(m1 + m2)

r c2 +131 + 6Nν

30πG ~r2c3

]

new result for massless scalar loopphoton loop calculation:in agreement with Capper et al. (1974)results from previous publications could be verified, e.g.Bjerrum-Bohr (2003)non-relativistic potential:~Gc3 ∼ 10−70m2 corrections should be unmeasureable

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Result: SQED

Scattering Potential

VSQED(r) =VGR(r)

+αe1e2

r

(1 + 3

G (m1 + m2)

c2r+

G~c3r2

)+

12

(m1e22 + m2e

21) Gα

c2r2 − 43π

(m2

2e21 + m2

1e22

m1m2

)Gα~c3r3

rescaled by α = ~c/137charges e1, e2 normalized in units of elementary chargesagreement with previous publication: Bjerrum-Bohr (2002)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Summary

full theory of quantum gravity unknowneffective field theory of gravity

low energy effects separated from high-energy effectsone-loop order quantum predictionsEFT only valid at energies below Planck scale ∼ 1019 GeVand long distances

evaluate leading quantum corrections→ effective potentialand Schwarzschild-metricunderstanding quantum nature of gravityfurther interesting papers:

M. S. Butt: Leading quantum gravitational corrections toQED, [arXiv:gr-qc/0605137] (2006)B. R. Holstein and A. Ross: Long Distance Effects in MixedElectromagnetic-Gravitational Scatterin,[arxiv:0802.0717] (2008)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Feynmans Gedankenexperiment (Feynman Lectures on Gravitation 1962-1963)

postulate of quantum mechanical behavior: there is anamplitude ψ for different processes↪→ particle described by ψ cannot have interaction described

by a probabilitytwo-slit diffraction experiment with an electron

classical gravity detector: information about which slit theelectron has passed

Intensity

gravity

detector

wavegravitational

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Feynmans Gedankenexperiment (Feynman Lectures on Gravitation 1962-1963)

no signal: electron position described by an

Amplitude =12ψ(upper slit) +

12ψ(lower slit)

if gravitational interaction transmitted by a field→ gravity field must have an amplitude:

12 of the amplitude corresponds to gravity field of anelectron which went through either slit

↪→ precisely characteristic of a quantum field

Back

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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.

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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)) .

Back

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Expansion: Curvature

Riemann curvature tensor:

Rβαµν = 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

Back

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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

Back

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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, . . . cn

two different typs of quantum field theoriesasympotically free theories - ultraviolet stable theoriesultraviolet unstable theories

ultraviolet unstable theories low energy limit offundamental theory - no difference between renormalizableand effective non-renormalizable theory

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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

p−g Rαβ

γδRγδρσRρσ

αβ

Back

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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, . . . , Jn

external and internal gravity field: Hµ1ν11 , . . . ,Hµmνm

m

incoming p1, . . . ,pn, outgoing q1, . . .qm momentum

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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)

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Vertex Factors

vertex factors at one-loop order

−→q

p

p′

ℓ′ ր p

ℓտ p′

−→k

ց q

ր ℓ

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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]}

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Graviton-Graviton-Scalar-Vertex

ℓրρσ

p

ℓ′ տ p′ηλ

Vηλρσ =iκ2

2

»1ηλαδ1

δρσβ −

14

`ηηλ1ρσαβ + ηρσ1ηλαβ

´–`pαp′β + pβp′α

´−

12

»1ηλρσ −

12ηηληρσ

–`(p · p′)−m2´ff

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Three-Graviton-Vertex

τµναβγδ(k , q) = − iκ2

Pαβγδ

»kµkν + (k − q)µ(k − q)ν + qµqν +

32ηµνq2

–+ 2qλqσ

ˆ1

σλαβ 1

µνγδ + 1

σλγδ 1

µναβ − 1 µσ

αβ 1νλ

γδ − 1 µσγδ 1

νλαβ

˜+ˆqλqµ

`ηαβ1

νλγδ + ηγδ1

νλαβ

´+ qλqν

`ηαβ1

µλγδ + ηγδ1

µλαβ

´− q2`ηαβ1 µν

γδ + ηγδ1µν

αβ

´− ηµνqλqσ

`ηαβ1

σλγδ + ηγδ1

σλαβ

´˜+ˆ2qλ

˘1

λσαβ 1

νγδσ (k − q)µ + 1

λσαβ 1

µγδσ (k − q)ν

− 1 λσγδ 1

ναβσ kµ − 1 λσ

γδ 1µ

αβσ kν¯

+ q2`1

µαβσ 1

νσγδ + 1

νσαβ 1

µγδσ

´+ ηµνqσqλ

`1λραβ1

σγδρ + 1

λργδ 1

σαβρ

´˜+

`k2 + (k − q)

´»1

µσαβ 1

νγδσ + 1

νσγδ 1

µαβσ −

12ηµνPαβγδ

–−`1

µνγδ ηαβk2 − 1 µν

αβ ηγδ(k − q)2´ffffffff

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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

Back

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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 contributions

li : one-loop contributions

l1, l4 : divergent high enery contributions

l2, l3, l5, l6 : finite non-analytic low energy contributions

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

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 =

hµνˆgµνR − 2Rµν

˜,

L(2)gr =

12

Dαhµν Dαhµν −12

Dαh Dαh + Dαh Dβhαβ − Dαhµβ Dβhµα

+ R

„14

h2 −12

hµνhµν«

+ Rµν`2hλµhνλ − h hµν

´.

Introduction Quantum Gravity EFT of Gravity Potential Summary Backup

Expansion: Matter Lagrangian

e.g. scalar particle: Lm =√−g

[12gµν∂µφ∂νφ− 1

2m2φ2]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˜