Background independent quantum gravity

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I give a brief overview of background independent approaches to quantum gravity culminating with causal dynamical triangulations.

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Background independent quantum gravity

  1. 1. Background Independent Quantum Gravity Adam Getchell
  2. 2. Why Quantum Gravity? General Relativity and Quantum Mechanics are Fundamentally Incompatible • • GR violates the Uncertainty Principle QM violates Causality
  3. 3. Why Quantum Gravity? General Relativity and Quantum Field Theory give different answers • • • Black Hole Information Paradox Wormholes, Time Machines, and Warp Drives Origins and structure of the universe
  4. 4. A brief history of Quantum Gravity • • • • • • 1916: Einstein points out that quantum effects must modify GR • 1949: Peter Bergmann and group begin the canonical approach to quantum gravity. Bryce Dewitt applies Schwinger’s covariant quantization method to the gravitational field for his thesis. • 1952: Gupta develops the covariant approach to quantum gravity. 1927: Oskar Klein suggests quantum gravity should modify concepts of spacetime 1930s: Rosenfeld applies Pauli method to linearized Einstein field equations 1934: Blokhintsev and Gal’perin discuss the spin-two quantum of the gravitational field, naming it the graviton 1936: Bronstein realizes need for background independence, issues with Riemannian geometry 1938: Heisenburg points out issues with the dimensionality of the gravitational coupling constant for a quantum theory of gravity
  5. 5. A brief history of Quantum Gravity • 1957: Charles Misner introduces the “Feynman quantization of general relativity”, or sum over histories approach. His paper outlines the three approaches, canonical, covariant, and sum over history in modern terms • • • 1961: Arnowit, Deser, and Misner complete the ADM formulation of GR • • 1966: Faddeev and Popov independently come up with DeWitt’s “Faddeev-Popov” ghosts. • 1969: Charles Misner starts quantum cosmology by truncating the Wheeler-DeWitt equation to a finite number of degrees of freedom 1962: Using ADM, Peres writes the Hamilton-Jacobi formulation of GR 1964: Feynman starts loop corrections to GR, but notices unitarity was lost for naive Feynman rules. DeWitt adds correction terms involving loops of fictitious fermionic particles, solving the problem. 1967: DeWitt’s seminal paper, completed in 1966 but not published until 1967 due to publishing charges, develops canonical quantum gravity and the Wheeler-DeWitt equation. DeWitt and Feynman complete the Feynman rules for GR.
  6. 6. Quick intro to GR Tp p manifold M
  7. 7. Quick intro to GR 2 2 ds = e dt Metric gµ⌫ 0 R µ⌫ 2 e B 0 =B @ 0 0 µ⌫ ⇢ 2 e 2(⌫ ) 2 dr + dz 0 e2(⌫ 0 0 ) e2(⌫ 0 = @⌫ ⇢ µ 2 r e 0 0 + 0 0 0 r2 e ) Connection 1 = g (@µ g⌫ + @⌫ g 2 @µ ⇢ ⌫ 2 ⇢ µ 2 d Line Element 2 1 2 Rµ⌫ = Rµ C C A µ R = Rµ = g µ⌫ Rµ⌫ Gµ⌫ ⌘ Rµ⌫ @ gµ⌫ ) µ ⌫ ⇢ ⌫ ⌫ Curvature µ 1 Rgµ⌫ = 8⇡GTµ⌫ 2 Matter
  8. 8. Parallel Transport @µTµ⌫ =?? ⌫ Not a tensor Tensor Parallel Transport ⌫ rµ V = @ µ V + D d = dxµ rµ d ⌫ µ V = 0 along xµ ( ) That is, the directional covariant derivative is equal to zero along the curve xµ parameterized by . For a vector this can be simply written as: ⌫ Parallel Transport of the tangent vector ⌫ rµ V = @ µ V + D dxµ d2 xµ = + d d d 2 ⌫ µ µ ⇢ V =0 Geodesic equation dx⇢ dx =0 d d
  9. 9. ADM 3+1 formalism t Ni N x, y, z gµ⌫ ij space at a fixed time 0 N B 0 =B @ 0 0 N1 11 0 0 N2 N3 22 23 0 33 12 Ni =(4) g0i ⇣ ⌘ (4) N = g00 13 1 2 1 C C A
  10. 10. Quantum gravity Hamilton-Jacobi equations SH @S H+ =0 @t Z p 1 4 = gRd x 2 k k gij (t, x ) ! gij (t, x ) ˆ ⇡ij (t, xk ) ! ˆ i gij (t, xk )
  11. 11. Wheeler-DeWitt equation  2 Gijkl Gijkl ij 3 R( ) 1/2 + 2⇤ 1/2 kl 1 = 2 1/2 ( ik jl + il jk [ ij ] ij kl ) =0
  12. 12. Semiclassical Universe from First Principles • Ambjorn, Jurkiewicz, and Loll get quantum cosmology without the Wheeler-DeWitt equation using the sum-over-histories approach of Causal Dynamical Triangulations • They “observe” a bounce, or state of vanishing spatial extension which tunnels to a universe of finite linear extension • What is Causal Dynamical Triangulations?
  13. 13. Causal Dynamical Triangulations • Causal, or Lorentzian, signature for the metric (which avoids problems of degenerate quantum geometries which occur with Euclidean metrics) • Non-perturbative path integral without a fixed background. Spacetime is determined dynamically as part of the theory • Spacetime is approximated by tiling flat simplicial building blocks (Regge calculus) without coordinates (simplicial manifold) 1.Any face of a simplex in T is a simplex in T 2.Two simplices in T are either disjoint or share a common face • Well-defined Wick rotation (analytic continuation)
  14. 14. Causal Dynamical Triangulations y y (b) (a) (a) Name Vertex Edge Triangle Tetrahedron Pentatope Dim 0 1 2 3 4 0-faces 1 2 3 4 5 x x 1-faces 1 3 6 10 (b) 2-faces 1 4 10 3-faces 1 5 4-faces 1 Table 1: Types and causal structures of simplices Causal Structure {1,1} {2,1} {1,2} {3,1} {2,2} {1,3} {4,1} {3,2} {2,3} {1,4}
  15. 15. Causal Dynamical Triangulations G(gi , gf ; ti , tf ) = SH 1 = 2 Z p 1 X SR = Vh 8⇡G h Z g(R h D[g]e iSEH 4 2⇤)d x ⇤ X Vd 8⇡G d
  16. 16. CDT 1 S= 8⇡G X space-like h 0 1 V ol(h) @2⇡ i X d-simplices 1 ⇥A 1 8⇡G ↵! Z Z= X ⇣ D[g]e iS ! exp (k0 + 6 )N0 T p p X time-like h 0 V ol(h) @2⇡ X d-simplices 1 ⇥A ⇤ 8⇡G time-like d 7 ↵, ↵ > 12 X T 1 iS(T ) e C(T ) k4 N4 (4,1) (2N4 p ✓ 3 ⇤ 5 3 5 k0 = , k4 = + cos 8G 768⇡G 8G 2⇡ 1 + N (3,2) ) (1/4) ◆ 1 X ⌘ V ol(d)
  17. 17. References Carlip, S. Is Quantum Gravity Necessary? ArXiv e-print, March 24, 2008. http://arxiv.org/abs/0803.3456. ! Gambini, Rodolfo, and Jorge Pullin. Loop Quantization of the Schwarzschild Black Hole. ArXiv e-print, February 21, 2013. http://arxiv.org/abs/1302.5265. ! Rovelli, Carlo. “Notes for a Brief History of Quantum Gravity.” Gr-qc/0006061 (June 16, 2000). http:// arxiv.org/abs/gr-qc/0006061. ! Carroll, Sean M. “Lecture Notes on General Relativity.” Gr-qc/9712019 (December 2, 1997). http:// arxiv.org/abs/gr-qc/9712019. ! Hartle, J. B., and S. W. Hawking. “Wave Function of the Universe.” Physical Review D 28, no. 12 (December 15, 1983): 2960–2975. doi:10.1103/PhysRevD.28.2960.

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