Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models Lorentz Violation: mechanisms and models Robertus Potting Physics Department, FCT University of the Algarve, Faro, Portugal, and CENTRA, Instituto Superior T´ ecnico University of Lisbon, Lisbon, Portugal SME2021 Summer School, 30 May 2021
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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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
String theory (2)
String scattering amplitude:
String amplitude involves sum over all intermediate states:
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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]
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
String field Theory (2)
Cubic vertex ”glues” free string propagators:
Perturbation theory around canonical vacuum Φ = 0 yieldsamplitudes of first-quantized string theory.
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
String field Theory (2)
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.
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
String field Theory (3)
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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.
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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.
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
Spacetime foam
Possible effects of spacetime foam:
• Non-deterministic motion of certain particles (i.e., photons)on Planck scale:
• 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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
Noncommutative field theory
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.
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
Noncommutative field theory
• 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.
• 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)
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
The Nambu-Goldstone theorem
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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
(Effective) field theory without gauge invarianceAssume nonderivative potential for vector field: (Kostelecky,Samuel ’89; Krauss,
• 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
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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” !
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
Cardinal model Kostelecky, R.P., 2005,2009; Carroll et.al. 2009
Consider symmetric 2-tensor h in flat Minkowski space:
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models
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.
Models of quantum gravity Noncommutative field theory Varying space-time constants Bumblebee and cardinal models