on lorentz violation in superluminal theories · de felice et al., 2013 mikhail kuznetsov on...

On Lorentz violation in superluminal theories

Mikhail Kuznetsov

in collaboration with Sergey Sibiryakov

INR RAS & MIPT

Universite Libre de Bruxelles, September 2013

Mikhail Kuznetsov On Lorentz violation in superluminal theories

Outline

Motivation: modified gravity theories

A toy model with superluminality

Instabilities and Lorentz violation

Pathologies in stress-energy tensor

Conclusions

Modified gravity theories: examples

Fierz–Pauli linearized massive gravity

S = M2pl

∫d4x

(2)EH(hµν) +

4hµνh

µν +β

]; gµν = ηµν + hµν

Fierz, Pauli, 1939

Ghost free only for α = −βvan Dam–Veltman–Zakharov discontinuity — GR is not restored as m→ 0

Non-linear effects allow to avoid discontinuity at small distancesVainshtein, 1972

On non-flat background Boulware–Deser ghost appearsBoulware, Deser, 1972

S = M2pl

∫d4x

(2)EH(hµν) +

4hµνh

µν +β

Fierz, Pauli, 1939

S = M2pl

∫d4x

(2)EH(hµν) +

4hµνh

µν +β

Fierz, Pauli, 1939

New covariant non-linear generalization of Fierz–Pauli theory

S = M2Pl

∫d4x√−g[R

g (L2 + α3L3 + α4L4)

Li = Li (δµν −

√gµρηab∂ρφa∂νφb)

de Rham, Gabadadze, 2010de Rham, Gabadadze, Tolley, 2010

Free of Boulware-Deser ghostsHassan, Rosen, 2011

Provides self-accelerated cosmological solutions, Λ = c(αi )m2g

Gumrukcuoglu, Lin, Mukohyama, 2011de Felice et al., 2013

New covariant non-linear generalization of Fierz–Pauli theory

S = M2Pl

∫d4x√−g[R

g (L2 + α3L3 + α4L4)

Li = Li (δµν −

√gµρηab∂ρφa∂νφb)

de Rham, Gabadadze, 2010de Rham, Gabadadze, Tolley, 2010

Free of Boulware-Deser ghostsHassan, Rosen, 2011

Provides self-accelerated cosmological solutions, Λ = c(αi )m2g

Gumrukcuoglu, Lin, Mukohyama, 2011de Felice et al., 2013

Decoupling limit of new massive gravity: MPl →∞ ;mg → 0 ;MPlm2g = const

⇓Galileon theory

∫d4x

2(∂µπ)2 − ∂2

µπ(∂µπ)2 + ...

]; ∂µπ → ∂µπ + cµ invariance

Nicolis, Rattazzi, Trincherini, 2009

Also as generalizations of Dvali–Gabadadze–Porrati model

Features

Ghost-free

Late time cosmic acceleration

Alternative to inflation (Galilean Genesis)Creminelli, Nicolis, Trincherini, 2010

⇓Galileon theory

∫d4x

2(∂µπ)2 − ∂2

µπ(∂µπ)2 + ...

Features

Ghost-free

⇓Galileon theory

∫d4x

2(∂µπ)2 − ∂2

µπ(∂µπ)2 + ...

Features

Ghost-free

Modified gravity theories: problems

One and the same feature of all these theories — presence of superluminalmodes

GalileonNicolis, Rattazzi, Trincherini, 2009

Massive GravityBurrage et al., 2011

Gruzinov, 2011de Rham, Gabadadze, Tolley, 2011

Superluminality itself looks not so bad...

⇓Possible problems with causality

Chronology protection conjecture — there are no closed timelike curves.Proven for some cases.

de Rham et al., 2011

One and the same feature of all these theories — presence of superluminalmodes

GalileonNicolis, Rattazzi, Trincherini, 2009

Massive GravityBurrage et al., 2011

Gruzinov, 2011de Rham, Gabadadze, Tolley, 2011

Superluminality itself looks not so bad...

⇓Possible problems with causality

Chronology protection conjecture — there are no closed timelike curves.Proven for some cases.

de Rham et al., 2011

More general problem — UV-completionAdams et al., 2006

Key moment: number of derivatives in self-coupling should be greater than 2.Take for example Galileon second term.

L = ∂µπ ∂µπ +

Λ4∂2µπ(∂νπ)2 + ... ; c < 0

Superluminality

Principal possibility to make closed timelike curves

Problems with Lorentz notion of causality

Scattering amplitudes do not satisfy S-matrix analyticity axioms

Superluminal effective theories could not be UV-completed to local QFT orunitary string theory!

What is the particular mechanism of Lorentz breaking?

Lets look for a physical difference between static and boosted classicalsolutions.

L = ∂µπ ∂µπ +

Λ4∂2µπ(∂νπ)2 + ... ; c < 0

Superluminality

L = ∂µπ ∂µπ +

Λ4∂2µπ(∂νπ)2 + ... ; c < 0

Superluminality

L = ∂µπ ∂µπ +

Λ4∂2µπ(∂νπ)2 + ... ; c < 0

Superluminality

Toy model

Toy-model in two dimensions.

∫d2x

2(∂µφ ∂

µφ)− m2

2φ2− 1

4Λ2(∂µφ ∂

µφ)2 + Mδ(x)φ

Consider classical perturbations on a background of a static solution. It isenough to consider e.o.m. without non-linear φ term.

φ′′ −m2φ = −Mδ(x) ⇒ φ =M

2me−m|x|

Toy model

Toy-model in two dimensions.

∫d2x

2(∂µφ ∂

µφ)− m2

2φ2− 1

4Λ2(∂µφ ∂

µφ)2 + Mδ(x)φ

]Consider classical perturbations on a background of a static solution. It isenough to consider e.o.m. without non-linear φ term.

φ′′ −m2φ = −Mδ(x) ⇒ φ =M

2me−m|x|

Superluminality

Consider quadratic action for classical perturbations φ = φ+ ξ.

∫d2x

2Zµν∂µξ ∂νξ −

Zµν =

(1 + M2

4Λ2 e−2m|x| 0

0 −1− 3M2

4Λ2 e−2m|x|

Dispersion relation

ω± = ±√−Z 11

⇓∣∣∣∣dωdk

∣∣∣∣ > 1

Group velocity of the excitations exceeds the speed of light.

Superluminality

Consider quadratic action for classical perturbations φ = φ+ ξ.

∫d2x

2Zµν∂µξ ∂νξ −

Zµν =

(1 + M2

4Λ2 e−2m|x| 0

0 −1− 3M2

4Λ2 e−2m|x|

Dispersion relation

ω± = ±√−Z 11

⇓∣∣∣∣dωdk

∣∣∣∣ > 1

Group velocity of the excitations exceeds the speed of light.

Classical stability

Looking for a classical instability

detZµν < 0 ⇒ the solution φ is classically stable.

In a boosted frame (x → γ|x + βt|)

Zµν =

(1− M2

4Λ2 γ2(3β2 − 1) e−2mγ|x+βt| M2

2Λ2 γ2β e−2mγ|x+βt|

2Λ2 γ2β e−2mγ|x+βt| −1− M2

4Λ2 γ2(3− β2) e−2mγ|x+βt|

However, detZµν = const under Lorentz transformation.

⇓The solution φ is classically stable in any frame.

Classical stability

Looking for a classical instability

detZµν < 0 ⇒ the solution φ is classically stable.

In a boosted frame (x → γ|x + βt|)

Zµν =

(1− M2

4Λ2 γ2(3β2 − 1) e−2mγ|x+βt| M2

2Λ2 γ2β e−2mγ|x+βt|

2Λ2 γ2β e−2mγ|x+βt| −1− M2

4Λ2 γ2(3− β2) e−2mγ|x+βt|

However, detZµν = const under Lorentz transformation.

⇓The solution φ is classically stable in any frame.

Quantum instability

Looking for a quantum instability

Z 00 change its sign in a boosted frame ⇒ Ghosts appear!

Z 00 = 1− M2

4Λ2γ2(3β2 − 1) e−2mγ|x+βt| < 0

⇓Critical boost factor γ > Λ

Memγ|x+βt|

⇓The trajectory of φ should be close to x = −βt

Looking for UV dynamics we can take m = 0 and regard boosted backgroundas a constant

∫d2x

2Zµν∂µξ ∂νξ

]; Zµν =

(Z 00 Z 01

Z 01 Z 11

)= const

Quantum instability

Looking for a quantum instability

Z 00 change its sign in a boosted frame ⇒ Ghosts appear!

Z 00 = 1− M2

4Λ2γ2(3β2 − 1) e−2mγ|x+βt| < 0

⇓Critical boost factor γ > Λ

Memγ|x+βt|

⇓The trajectory of φ should be close to x = −βt

Looking for UV dynamics we can take m = 0 and regard boosted backgroundas a constant

∫d2x

2Zµν∂µξ ∂νξ

]; Zµν =

(Z 00 Z 01

Z 01 Z 11

)= const

Quantum instability

Looking for evolution of negative energy modes.

Add interaction with matter scalar field χ.

∫d2x

2(∂µχ ∂

µχ)−m2

2χ2 +

2ξχχ

Triples ξχχ created from vacuum as Z 00 < 0

Required hierarchy of parameters:

mχ < mγ � k � Λ ;

m� M ; gM

m� m2

⇓Estimations for background decay rate and energy loss rate

Integral saturated in IR

dt∼ Γ ∼ g 2

2πm2γ2.

Small, but non-zero quantity ⇒ Lorentz invariance is violated!

Quantum instability

∫d2x

2(∂µχ ∂

µχ)−m2

2χ2 +

2ξχχ

]Triples ξχχ created from vacuum as Z 00 < 0

mχ < mγ � k � Λ ;

m� M ; gM

m� m2

dt∼ Γ ∼ g 2

2πm2γ2.

Quantum instability

∫d2x

2(∂µχ ∂

µχ)−m2

2χ2 +

2ξχχ

mχ < mγ � k � Λ ;

m� M ; gM

m� m2

dt∼ Γ ∼ g 2

2πm2γ2.

Quantum instability

∫d2x

2(∂µχ ∂

µχ)−m2

2χ2 +

2ξχχ

mχ < mγ � k � Λ ;

m� M ; gM

m� m2

dt∼ Γ ∼ g 2

2πm2γ2.

Quantum instability

Naive estimation for 4D

∫d2x

2(∂µχ ∂

µχ)−m2

2χ2 +

2ξ(∂µχ∂

]This simulates coupling g′

2ξ2χ2 in 4D as it have the same dimensionality.

Estimation for background decay rateIntegral saturated in UV

Γ ∼ g ′2

8πΛ2

Estimation for energy loss rate

dt∼ g ′

This can be very high and probably could destroy the classical solution.

Quantum instability

Naive estimation for 4D

∫d2x

2(∂µχ ∂

µχ)−m2

2χ2 +

2ξ(∂µχ∂

]This simulates coupling g′

2ξ2χ2 in 4D as it have the same dimensionality.

Estimation for background decay rateIntegral saturated in UV

Γ ∼ g ′2

8πΛ2

Estimation for energy loss rate

dt∼ g ′

This can be very high and probably could destroy the classical solution.

Stress-energy tensor pathology

Expect some features of T φµν(ξ) as γ → γcritical

Below γcritical , Tφboosted

µν (ξ) = T φstatic

µν (ξ)

Looking for accelerated solution φ, (β = t/τ ' 1)

Zµν =

(1− t2/α2 −t2/α2

−t2/α2 −1− t2/α2

); α2 =

2Λ2(τ 2 − t2)

M2= const

After quantization of ξ on this “curved” background we get

〈T00〉 = −〈T11〉 =α2

α2 − t2

∞∫0

Indication for pathology as t → α, although renormalization of Tµν is needed

Work in progress...

µν (ξ)

Zµν =

(1− t2/α2 −t2/α2

−t2/α2 −1− t2/α2

); α2 =

2Λ2(τ 2 − t2)

M2= const

〈T00〉 = −〈T11〉 =α2

α2 − t2

∞∫0

Work in progress...

µν (ξ)

Zµν =

(1− t2/α2 −t2/α2

−t2/α2 −1− t2/α2

); α2 =

2Λ2(τ 2 − t2)

M2= const

〈T00〉 = −〈T11〉 =α2

α2 − t2

∞∫0

Work in progress...

Conclusions & Discussion

Physical solution in the simple superluminal quantum field theory showsdifferent behavior in static and boosted frames. I.e. Lorentz invariance isbroken.

Theory is pathological and we don’t need to consider it.

As the theory is non-Lorentzian we need to complete a Lagrangian withother Lorentz-braking terms.

It is a spontaneous breaking of Lorentz invariance, so we need to consider aphase with a broken symmetry.

This analysis seem to be suitable for other theories that admitsuperluminal propagation on classical backgrounds.

Merci!

Conclusions & Discussion

Physical solution in the simple superluminal quantum field theory showsdifferent behavior in static and boosted frames. I.e. Lorentz invariance isbroken.

Theory is pathological and we don’t need to consider it.

As the theory is non-Lorentzian we need to complete a Lagrangian withother Lorentz-braking terms.

It is a spontaneous breaking of Lorentz invariance, so we need to consider aphase with a broken symmetry.

This analysis seem to be suitable for other theories that admitsuperluminal propagation on classical backgrounds.

Merci!

on lorentz violation in superluminal theories · de felice et al., 2013 mikhail kuznetsov on...

Documents