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Causal Dynamical Triangulations Israel Garc´ ıa @renegarxia 20 de mayo de 2015
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1. Causal Dynamical Triangulations Israel Garca @renegarxia 20 de mayo de 2015 2. Table of contents The classical action Quantum mechanics Path Integrals in relativity 3. Path Integrals 4. Some waves 5. More waves (click to watch them) 6. Path Integrals With xa, xb xed, how do we minimize S? S = tb ta L(x, x, t)dt (action) L = m 2 x2 V (x, t) (Lagrangian) 7. Path Integrals Set x x + x, then L m 2 (x + x)2 V (x + x, t), = m 2 x2 + m x x + m 2 x2 V (x, t) x V (x, t)x O(x2 ), Therefore, L L + m x x + m 2 x2 x V (x, t)x O(x2 ), What happens to the action? (S) 8. The Action S tb ta L + m x x + m 2 x2 x V (x, t)x O(x2 ) dt, = S + tb ta m x x x V (x, t)x dt (up to rst order) With xed extremes, xa = xb = 0. Integration by parts follows: S S tb ta (mx + x V (x, t)) x dt 9. Least action The true x is such that, if x x + x, then S is the same, to rst order. The true trajectory obeys this equation: mx = x V (x, t), which is the second law of mechanics in disguise. 10. Quantum mechanics `a la Feynman The path integral: K(a, b) = b a ei/ S Dx(t) 11. A relativistic example See: counting-paths-in-spacetime, random-walks-in-a-lattice, and corners-distribuition 12. Path integrals in relativity Theres an action principle for general relativity: SEH = 1 G d4 x det g (R 2) (Einstein-Hilbert action) Problem: Make sense of the path integral: gG Dg eiSEH 13. So, you want to quantize gravity? String theory. Loop quantum gravity. Euclidean quantum gravity. Causal dynamical triangulations. 14. What is curvature? 15. Discrete curvature 16. Euclidean Gravity This is Wicks rotation: dt2 + dx2 d(i t)2 + dx2 It makes gravity euclidean. And turns amplitudes into probabilities! 17. First attempt: failed! 18. Causality (you cant kill your parents...) ...before you are born... 19. Global hyperbolicity This is acceptable: This is not: Wormhole, baby universes. 20. Causal triangulations 21. Quantum gravity in your desktop 22. References I J. Ambjorn, A. Goerlich, J. Jurkiewicz, and R. Loll, Quantum Gravity via Causal Dynamical Triangulations. Jan Ambjorn, J. Jurkiewicz, and R. Loll, Dynamically triangulating Lorentzian quantum gravity, Nucl.Phys. B610 (2001), 347382. J. Ambjorn, J. Jurkiewicz, and R. Loll, The Universe from scratch, Contemp.Phys. 47 (2006), 103117. R. Loll, The Emergence of spacetime or quantum gravity on your desktop, Class.Quant.Grav. 25 (2008), 114006.