Lorentz Violation: mechanisms and models

Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models

Lorentz Violation:

mechanisms and models

Robertus Potting

Physics Department, FCTUniversity of the Algarve, Faro, Portugal, and

CENTRA, Instituto Superior TecnicoUniversity of Lisbon, Lisbon, Portugal

SME2021 Summer School,30 May 2021


Outline

Models of quantum gravityString Field TheoryLoop Quantum GravitySpacetime foam

Noncommutative field theory

Varying space-time constantsA supergravity-inspired model

Bumblebee and cardinal modelsSymmetry vs. Broken SymmetryThe bumblebeeThe cardinal


String theory (1)

String theory:

• Fundamental theory of nature in which basic object isvibrating string

• Vibrational string states correspond to different particles

• can either consider open + closed strings, or only closedstrings

• Massless string spectrum includes graviton

• world sheet reparametrization invariance: 2d conformal fieldtheory


String theory (2)

String scattering amplitude:

String amplitude involves sum over all intermediate states:


String Field Theory

Action of Witten’s open string field theory (OSFT)

I (Φ) =1

2

∫Φ ⋆ QΦ+

g

3

∫Φ ⋆ Φ ⋆ Φ

Legenda:

• Φ ≡ Φ (xµ(σ), b(σ), c(σ)) is the string field

• ⋆: gauge invariant string field product;

• kinetic operator = open string BRST operator Q;

Gauge invariance:

δΦ = QΛ + gα′[Λ ⋆ Φ−Φ ⋆ Λ]


String field Theory (2)

Cubic vertex ”glues” free string propagators:

Perturbation theory around canonical vacuum Φ = 0 yieldsamplitudes of first-quantized string theory.



Mode decomposition:

xµ(σ) = xµ0 +

√2

∞∑

n=1

xµn cos(nσ)

• x0: ”center of mass” of string

Can now expand string field in Fourier series of string modes:

|Φ〉 =[φ(x0) + Aµ(x0)α

µ−1 + i Bµ(x0)α

µ−2 + Bµν(x0)α

µ−1α

ν−1 + . . .

]|0〉

• αµ−n: string mode creation operators;

• |0〉 harmonic oscillator vacuum for xn coordinates

• φ: Scalar field (tachyon); Aµ: massless vector field; etc.


Solutions of String field Theory

Nontrivial static solutions of the SFT equations of motion:

1. ”canonical vacuum” Φ = 0• not local minimum ⇒ unstable• tachyonic mode in spectrum

2. Kostelecky and Samuel (1989): new numerical solution (leveltruncation)

• no physical open string excitations• ”true” stable vacuum

3. Kostelecky and Potting (1996): additional LV solutions


Interpretation of the solutions

Interpretation in terms of D-branes

1. ”True” vacuum interpreted by Sen (1999) as absence of anyD-brane

• canonical vacuum: space-filling D-25 brane

2. LV solutions presumably correspond to solutions involvinglower-dimensional D-branes

Boundary string field theory

• Alternative method to study D-brane solitons based on singlefield tachyon condensation

• Hashimoto and Murata (2012) numerically found large(infinite?) class of LV soliton solutions in BSFT

• physical interpretation still not clear



Cubic string field theory indicates couplings of the typeφTµ1...µnT

µ1...µn .

• φ acts as type of Higgs field;

• φ acquires vacuum expectation value;

• Could imply non-zero vacuum expectation values for thetensor fields Tµ1...µn ;

• Such energetically favorable configurations are Lorentzviolating


Superstring field theory

Lorentz-violating solutions may occur the same way in the contextof superstring field theory.

• two candidate theories for SUSY SFT;

• String field contains fermionic as well as bosonic sector;

• Both can be expanded in terms of component string fields.

Could expect solutions in which bosonic tensor componentsTµ1...µn acquire v.e.v., leading to effective LV interactions:

LI ⊃ λ

Mkpl

T · ψΓ(i∂)kχ+ h.c.

Γ: γ-matrix structure; Lorentz indices on T , Γ and (i∂)k havebeen suppressed.


LQG

Loop Quantum Gravity is an attempt to reconcile standardquantum mechanics and standard general relativity.

• Ashtekar reformulation of GR (1986) admits loop solutions forWheeler-DeWitt eq.

• Loop solutions form basis of nonperturbativebackground-independent theory of quantum gravity

• quantum operators for area and volume have discretespectrum ⇒ spin networks: basis of states of quantumgeometry

• Canonical formulation with anomaly-free Hamiltonian(Thiemann)

• Presumably LQG should have GR as semiclassical limit


LQG (2)

Physical consequences

• physical picture of space as a consequence of quantization:discrete, ”granular” space

• Planck size constitutes minimum distance

• Black hole entropy from LQG (S = A/4)• specific prediction for spectrum of evaporating BH

• Loop quantum cosmology• prediction of ”Big Bounce”, with observable consequences• cosmological perturbations around FLRW solution: quantum

background• predictions from LQG for primordial power spectrum (sources

of CMB anisotropies)

• possible violations of Lorentz invariance


LQG (3)

Lorentz violation in LQG

• Discreteness of geometric operators might result in modifieddispersion relations for high-energy particles:E ≃ p +m2/(2p) ± ξp(p/Mpl)

n. Amelino-Camelia et.al., 1998

• Polymer-like structure of spacetime at microscales may alsolead to photon birefringence Gambini and Pullin, 1998

• Helicity-independent corrections to neutrino propagation. Alfaro

et.al., 1999


Spacetime foam

”Qualitative” idea: Wheeler, 1955

• As distance/time scales under consideration become smaller,the energy of virtual particles increases;

• According to GR, these virtual particles must curve spacetime

• At the Planck scale, one expects fluctuations to be largeenough to cause departures from smooth spacetime: foamyspacetime

• Without complete theory of quantum gravity, precise effectsnot clear.


Spacetime foam

Possible effects of spacetime foam:

• Non-deterministic motion of certain particles (i.e., photons)on Planck scale:

• Might expect energy-dependent stochastic fluctuations inparticle speed

• nontrivial Lorentz-violating effects on dispersion relation?

• Recent searches concentrated on looking for variations inmoment of arrival of photons of different energies emitted bya gamma ray burst



Noncommutative field theory: application of noncommutativemathematics to the spacetime of quantum field theory in which thecoordinate functions are noncommutative.

Commonly studied version has the ”canonical” commutationrelation:

[xµ, xν ] = θµν

θµν : antisymmetric tensor of dimension −2.

⇒ uncertainty relation for the coordinates similar to theHeisenberg uncertainty relation.



• Heisenberg and Hartland Snyder (1947) suggested usingnoncommutative field theory in order to renormalize UVinfinities.

• 1980’s: development of noncommutative geometry by A.Connes and other mathematicians.

• Connes, Douglas and Schwarz (1997): certaincompactifications of M-theory involve NC FT

• Seiberg and Witten (1999): open strings on D-branes inpresence of NS B-field satisfy noncommutative algebra.

• Minwalla et. al. (2000): IR/UV mixing phenomenon:low-energy expansion problematic. Possible solution bysupersymmetry.


Noncommutative QED

Define Moyal product:

f ⋆ g(x) ≡ exp(12 iθµν∂xµ∂yν )f (x)g(y)

∣∣x=y

.

One can now define noncommutative QED:

L = 12 i ψ ⋆ γ

µ↔

Dµ ψ −mψ ⋆ ψ − 1

4q2Fµν ⋆ F

µν .

where Fµν = ∂µAν − ∂νAµ − i [Aµ, Aν ]⋆, and

Dµψ = ∂µψ − i Aµ ⋆ ψ, with f ⋆↔

Dµ g ≡ f ⋆ Dµg − Dµf ⋆ g .


Lorentz violation

θµν : fixed background tensor

• transforms under observer Lorentz transformations;

• does not transform under particle Lorentz transformations;

• thus any NC field theory violates Lorentz invariance.

Physical interpretation?

• not obvious how to identify of physical quantities with NCoperators

• possible approach: Seiberg-Witten map. Maps NC field theoryto ordinary field theory. Seiberg, Witten, 1999


NC field theories and the SME

Seiberg-Witten map yields usual commuting LV field theory withconstant background θµν tensor. Thus

• Any realistic noncommutative fied theory is physicallyequivalent to subset of SME.

• CPT is preserved.

Other features:

• θµν always accompanied by two derivatives ⇒ minimumdimension of LV operators is 5 or 6 (rather than 3 or 4).

• no difficulties with perturbative unitarity provided θµνθµν > 0


Seiberg-Witten mapSeiberg-Witten map to lowest order:

Aµ = Aµ − 1

2θαβAα(∂βAµ + Fβµ),

ψ = ψ − 1

2θαβAα∂βψ.

This yields the NC QED Lagrangian Carroll et.al., 2001

L =1

2iψγµ

↔

Dµ ψ −mψψ − 1

4FµνF

µν

−1

8iqθαβFαβψγ

µ↔

Dµ ψ +1

4iqθαβFαµψγ

µ↔

Dβ ψ

+1

4mqθαβFαβψψ

−1

2qθαβFαµFβνF

µν +1

8qθαβFαβFµνF

µν .


Varying space-time constants

Consider the supergravity-inspired Lagrangian with dilaton field A

and axion field B Kostelecky, Lehnert, Perry (2002); Bertolami, Lehnert, R.P., Ribeiro (2004)

Lsg = −1

2

√gR − 1

4

√gMFµνF

µν − 1

4

√gNFµν F

µν

+√g(∂µA∂

µA+ ∂µB∂µB)/4B2 ,

where

M =B(A2 + B2 + 1)

(1 + A2 + B2)2 − 4A2,

N =A(A2 + B2 − 1)

(1 + A2 + B2)2 − 4A2.


Time-dependent couplings

• This model has been integrated in a cosmological model(Friedman-Robertson-Walker universe) coupled to dust.

• Equations of motion for A and B define their timedevelopment. See example in figure.


Time-dependent couplings

• Time development of A and B defines time development ofthe effective coupling constants M and N.

• Figure shows relative time variation of electromagneticcoupling.


Apparent Lorentz violation

Compare SG lagrangian with conventional electrodynamicslagrangian

Lem = − 1

4e2FµνF

µν − θ

16π2Fµν F

µν .

with e2 ≡ 1/M and θ ≡ 4π2N. Eqs. of motion:

1

e2∂µFµν −

2

e3(∂µe)Fµν +

1

4π2(∂µθ)Fµν = 0 .

For spacetime-dependent e and θ, obtain effectiveLorentz-violating lagrangian!


Masslessness from symmetry or broken symmetry

Gauge Symmetries

Generator of unbroken gauge symmetry ⇒ massless vectorboson

General Relativity

Diffeomorphism invariance ⇒ massless gravitons

Spontaneously Broken Global Symmetry

Spontaneously broken global symmetry ⇒ masslessNambu-Goldstone boson


The Nambu-Goldstone theorem

Nambu-Jona-Lasinio model (1961)

L = i ψ /∂ψ +λ

4

((ψψ)(ψψ)− (ψγ5ψ)(ψγ5ψ)

)

invariant under ordinary and chiral phase rotations:

ψ → e iαψ, ψ → ψe−iα

ψ → e iαγ5ψ, ψ → ψe iαγ5 .

Mass term breaks chiral symmetry.But: possibility of chiral condensate 〈ψψ〉:

• Spontaneously breaks chiral symmetry

• Yields effective mass term

• Broken symmetry leads to massless Goldstone boson



Bjorken (1963): ”emergent photons”

L = ψ(i /∂ −m)ψ +G

2(ψγµψ)(ψγ

µψ)

• Nonvanishing fermion condensate carrying vacuum currentpossible

• Dynamics can be interpreted in terms of photon in temporalgauge

• Lorentz-violating effects assumed unphysical, can besuppressed by taking G very large

• Lattice simulations suggest that Lorentz-breaking fermioniccondensates can form in large N strongly-coupled latticegauge theories. (Tomboulis ’10, ’11)



Nambu (1968): QED in nonlinear gauge

L = −1

4FµνF

µν + ψ(i /∂ −m)ψ − eAµψγµψ

with Aµ subject to the constraint

A2µ = M2

• M 6= 0 implies Lorentz-violating expectation value for Aµ

• No Lorentz-violating physical effects assumed: constraintmerely implies Lorentz-violating choice of gauge


(Effective) field theory without gauge invarianceAssume nonderivative potential for vector field: (Kostelecky,Samuel ’89; Krauss,

Tomboulis ’02; Bluhm, Kostelecky PRD’05; Nielsen et.al. ’07)

L = −1

4FµνF

µν − V (AµAµ) + Lmatter (ψ,Aµ)

Here V is a potential that has a local minimum for either timelikeor spacelike Aµ, at which we have the constraint

A2µ = ±M2

Example 1: Mexican hat potentialV (AµA

µ) = −µ2AµAµ + κ(AµA

µ)2

Example 2: Lagrange multiplier potentialV (AµA

µ) = λ(AµAµ ±M2)

Consequence: Aµ acquires vacuum expectation value Aµ


photons as Nambu-Goldstone modes

Fluctuations of Aµ around vacuum expectation value:Aµ = Aµ + aµ Lagrangian for fluctuations:

L = −1

4fµν − V (Aµ, aµ)

Goldstone bosons identified by inifitesimal Lorentz transformationson vector vev’s:

aµ = −Θµν(x)Aν

with

Θµν =

0 β1 β2 β3−β1 0 θ3 −θ2−β2 −θ3 0 θ1−β3 θ2 −θ1 0

βi = βie

ik·x , θi = θieik·x



Example: purely timelike vector with only A0 6= 0Three Goldstone bosons:

aµ = −Θµ0 =

0β1β2β3

Every choice of vev corresponds to a different gauge: temporal,axial, ...Three Goldstone bosons can be decomposed in:

• 2 transverse modes: kµǫtransµ = 0

• 1 longitudinal mode: ǫlongµ = kµ − AαaαAαAα

Aµ


Stability issues

Stability issues

• Hamiltonian analysis Bluhm, Gagne, R.P., Vrublevskis (2008); Carroll et.al. (2009)

shows instabilities can occur when A2µ 6= M2 that involve the

longitudinal mode

• Constraining phase space such that A2µ = M2 can be done

consistently. Resulting model equivalent to electrodynamics infixed gauge Escobar, Martın-Ruiz (2017)



Bumblebee Lagrangian:

• has no gauge invariance

• Lorentz invariance is spontaneously broken

• Masslessness of vector field is direct consequence of Lorentzbreaking (Goldstone boson)

• Maxwell theory is “emergent” phenomenon

• Stability issues for non-Maxwell longitudinal mode

• Can be coupled to gravity → “eather” field


Spontaneous Lorentz violation: other models (1)

antisymmetric tensor field Bµν Altschul et. al. (2010)

L = − 1

12HλµνH

λµν − V (X1,X2)

• Hλµν = 3∂[λBµν]

• X1 = BµνBµν , X2 =12xǫ

λµντBλµBντ

• V breaks gauge invariance Bµν → Bµν + ∂[µΛν]

• V (X1,X2) has minimum for nonzero X1, X2: spontaneous LV

• Propagating Nambu-Goldstone modes: “phon” modes

• Hamiltonian analysis reveals singular behavior of the DOF onthe vacuum manifold Seifert (2019)


Spontaneous Lorentz violation: other models (2)

Nonlinear Electrodynamics (Plebanski’s first-order formulation)

L = −Pµν∂[µAν] − V (P ,Q)

• P = 14P

µνPµν , Q = 18ǫ

λµντPλµPντ

• Maxwell theory: V = −P ; Born-Infeld theory corresponds tononlinear V

• Idea: consider V with nontrivial local minima: spontaneousLVAlfaro, Urrutia (2010)

• Gauge invariance maintained → big advantage

• Choices of V exist corresponding to energetically stablesystem with spontaneous LVC. Escobar, RP (2020)

• Hamiltonian analysis yields singular behavior of the DOF onthe vacuum manifold C. Escobar, RP (2020)

• see talk by Carlos Escobar


Renormalization

Fixed points of Renormalization Group

• Interesting to consider behaviour of theory under (Wilson)renormalization group

• Gaussian fixed point exists that is UV stable in certaindirections of linearized RG flow (Altschul, Kostelecky ’05)

• These relevant directions of RG flow correspond toasymptotically free theory with nonpolynomial interactions,similar to behaviour for scalar fields (Halpern, Huang ’95)

• These potentials exhibit stable nontrivial minima for AµAµ,

implying ”spontaneous bumblebee potential” !


Cardinal model Kostelecky, R.P., 2005,2009; Carroll et.al. 2009

Consider symmetric 2-tensor h in flat Minkowski space:

L = 12 h

µνKµναβ hαβ + V (hµν , ηµν)

Kµναβ = ηµν∂α∂β + ηαβ∂µ∂ν + (−ηµνηαβ + 12ηµαηνβ + 1

2ηµβηνα)∂2

− 12ηµα∂ν∂β − 1

2ηνα∂µ∂β − 12ηµβ∂ν∂α − 1

2ηνβ∂µ∂α

• V : scalar potential built out of the 4 independent scalarsX1 = hµνηνµ, X2 = (h · η · h · η)µµ,. . .

• kinetic term invariant under hµν → hµν − ∂µΛν − ∂νΛµ;invariance broken by V !

• V acquires minimum for hµν = Hµν : spontaneous breaking ofLorentz symmetry

• Generally all six Lorentz generators are broken; Specialsituation may arise with three or five broken generators


Linearized “Cardinal” dynamics

At low energy, assume V can be approximated by sum ofdelta-functions that fix the 4 independent scalars: V =

∑4n=1

λn

nXn

Fluctuations around vev: hµν = Hµν + hµν

equation of motion:

Kµναβhαβ = GL(h)µν = 0

cardinal constraints:

hµµ = 0, Hµνhµν = 0, (HηH)µνhµν = 0, (HηHηH)µνhµν = 0

Low-energy dynamics of hµν -fluctuations around vev equal tolinearized general relativity (in axial-type “cardinal” gauge)!


Counting degrees of freedom

Propagating massless degrees of freedom

• Can be considered Nambu-Goldstone modes of spontanouslybroken Lorentz generators Eµα:

hµν = EµαHαν + EναHµα

• Equations of motion imply masslessness ∂2hµν = 0 andLorenz conditions ∂µhµν = 0

• Number of propagating massless degrees of freedom:6− 4 = 2


Comparison between bumblebee and cardinal model

Photon Graviton

# # massive modes 1 4

Equivalent gauge condition Temporal / axial Cardinal

Goldstone modes 3 6

# transverse modes 2 2

# longitudinal modes 1 4

Kinetic term Maxwell Einstein-Hilbert


Cardinal bootstrap Kostelecky, R.P., 2009

bootstrap procedure to nonlinear theory

• Proper theory of gravity should include coupling to theenergy-momentum tensor of gravitons to matter but also tothe energy-momentum tensor of gravitons itself.

• Leads to recursive “bootstrap” procedure, forcing theinclusion of cubic, quartic, ... graviton terms of the kineticterm. Resummation can be shown to lead to theEinstein-Hilbert action. Deser 1970

• Bootstrap procedure applied to the potential leads tointegrability conditions restricting the potential to set of veryparticular expressions.


Thanks for your attention!

Lorentz Violation: mechanisms and models

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