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Quantum Gravity: Infinities, Loops, and Strings Vsevolod Ivanov 2014 • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? Introduction • Gravitation and three other fundamental forces have non-compatible formalism • Each breaks down outside its domain • Quantized theory of gravity is desired • Secondary goal: “theory of everything” Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? Quick Review of Feynman Diagrams • Electron or positron: • Photon: −𝑖𝑔𝜇𝜈 p2 • Vertex: 𝑖𝛼 𝛾 𝜇 • Loop: 𝑑𝑑𝑝 (2𝜋)^4 𝑖 𝛾𝜇 𝑝𝜇 +𝑚 𝑝2 −𝑚2 QED in d Dimensions • For a typical diagram we get: ~ 𝑑 𝑑 𝑝1 𝑑 𝑑 𝑝2 ⋯𝑑 𝑑 𝑝𝐿 (𝛾𝜇 𝑝𝑖 −𝑚)⋯ 𝑝𝑖2 −𝑚2 ⋯𝑝𝑗2 ⋯ • This will diverge when 𝐷 = 𝑑𝐿 − 𝑃𝑒 − 2𝑃𝛾 ≥ 0 • Loops: 𝐿 = 𝑃𝑒 + 𝑃𝛾 − 𝑉 + 1 • Vertices: 𝑉 = 2 𝑃𝛾 + 𝑁𝛾 = • Eliminating: 𝐷=𝑑+ 𝑑−4 2 𝑉 − 1 2 2𝑃𝑒 + 𝑁𝑒 𝑑−2 2 𝑁𝛾 − • Some definitions: • Super-Renormalizable theory • Renormalizable theory • Non-Renormalizable theory 𝑑−1 2 𝑁𝑒 Field Theory for Gravity • ℒ= 1 2 𝜕𝜇 𝜙 • Propagator: 2 − 1 2 1 𝑝2 +𝑚2 𝑚2 𝜙 2 − 𝜆 𝑛! 𝜙𝑛 , 𝜙 𝑛 vertex: −𝑖𝜆 • Dimensionless action: 𝑆 = 𝑑𝑑 𝑥 ℒ • ℒ~ 𝑚𝑎𝑠𝑠 𝑑 , so 𝜆~ 𝑑 − 𝑛(𝑑 − 2)/2 • For diagram with N external lines, V vertices: 𝑑−2 𝑑−2 𝑑−𝑁 =𝑉 𝑑−𝑛 +𝐷 2 2 Mass Dimension of the Coupling Constant • 𝐷 =𝑑 −𝑁 𝑑−2 2 −𝑉𝑑 𝜆 • For QED, coupling is 𝛼 ~ • Gravity: 𝐺 = 1 137 , so renormalizable! 𝑐𝑚^3 𝑔 ∙𝑠^2 • With ℏ = 𝑐 = 1, we find s~ 1 , 𝑚𝑎𝑠𝑠 𝑐𝑚~ • 𝐺~𝑚𝑎𝑠𝑠 −2 → Non-Renormalizalbe! 1 , 𝑚𝑎𝑠𝑠 𝑔~𝑚𝑎𝑠𝑠 Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? SUper GRAvity • Discovery: 1976 (Freedman, Ferrara, Nieuwnhuizen) • 1985 → minimal SUGRA (Chamseddine, Arnowitt, Nath) • 11d maximal SUGRA (4 pillars) • 11d upper limit • 11d lower limit • Classical action for 11d SUGRA • SUSY preserving compactification to 4d • Two pillars broken (embedding and compactification) • Problems! • Avoided by 10d superstring theories! Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? SUper GRAvity • First superstring revolution (1984-1986) • Only three 10d SUGRA models (Green, Schwarz, Gross) • 6 extra dimensions removed through compactification • Too many Calabi-Yaus, still no SM • Second superstring revolution (1994-2000) • D-branes and dualities • 11d M-theory (Witten) • Long-𝜆 limit → SUGRA! • SUGRA framework used to understand string theory Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? Loop Quantum Gravity • Quantization of gravity without unification • States in LQG Hilbert space are described by spin networks: • 𝑎 + 𝑏 + 𝑐 ≥ 2 max 𝑎, 𝑏, 𝑐 • 𝑎 + 𝑏 + 𝑐 → 𝑒𝑣𝑒𝑛 • Operators act on the area 𝐴 of spin network on surface Σ: 𝐴Σ = 8𝜋ℓ2𝑃𝐿 𝛾 𝑖 𝑗𝑖 𝑗𝑖 + 1 • Can also compute probabilities: 𝑃∝ ± −2 𝑐 𝑛! Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? Experimental Tests of Quantum Gravity • Quantum gravitational effects are too weak to test! • Recent proposals: • QG effects in gamma ray bursts • Violation of Lorentz invariance • Polarization of the CMB CMB Polarization 2 • Inflation phase creates gravity waves (𝐻 ∝ 𝐸𝑖𝑛𝑓 , WMAP: 𝐸𝑖𝑛𝑓 < 3 ∙ 1016 𝐺𝑒𝑉) • Temperature creates quadrupole anisotropy • Other effects: • Density fluctuations (scalar) • Vorticity from defects (vector), damped out by inflation • Gravitational waves (tensor) • Two Polarization Types: Curl-free E-mode and Grad-free B-mode BICEP 2 Measures B-mode Polarization! • NSF South Pole Telescope: B-modes due to lensing • BICEP 2 measures non-zero B-modes due to GW at 𝜎 = 7.7 significance • First evidence of quantum fluctuations of gravity in early universe • Ken Olum “This is the only observational evidence that we have that actually shows that gravity is quantized… It’s probably the only evidence of this we will ever have.” Quantum Gravity: Infinities, Loops, and Strings • What is THE question (and why do we care)? • A quantum history • The trouble with gravity • SUGRA, almost sugar but not as sweet • A tale of two revolutions • Loopy nets • Experimentalists flex their muscles • The future? Future Experimental Tests • Lorentz invariance violation: future experiments will have better precision for higher order terms • Planck probe will measure anisotropies to higher precision • GW-superconductor interactions? References “Quantum Gravity Notes” M. Spiropilu (2004) "Cosmic Microwave Background Polarization: The Next Key Toward the Origin of the Universe", Y. D. Takahashi "Angular momentum: an approach to combinatorial space-time", R. Penrose "Introduction to Loop Quantum Gravity", Simone Mercuri "Quantum gravity: an introduction to some recent results", Enrique Alvarez "Detection of B-Mode Polarization in the Cosmic Microwave Background with Data from the South Pole Telescope", D. Hanson et al., PRL 111, 141301 (2013) "Experimental Search for Quantum Gravity", Sabine Hossenfelder An Introduction to Modern Astrophysics, B.W. Carroll, D. A. Ostlie (1996) Images: http://astrophysics.pro/wp-content/uploads/2013/03/Quantum-Gravity.jpg http://upload.wikimedia.org/wikipedia/commons/d/d2/Spinnetwork.jpg Questions?
