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The Attractiveness of Loops and Ribbons There would appear to be strong reasons for believing that the con- tinuum concept may eventually have to be abandoned as one of the basic ingredients of a fundamental physical theory. { R. Penrose,Theory of Quantized Directions Seth Major SIAM Colloquium RPI 4 February 2008 Outline (Some of) the story so far from loop quantum gravity • How can a graph represent geometry? • Are such graphs related to gravity? • Adding a cosmological constant • Possible observational consequences 1 Spin networks Graphs for 3 dimensional spatial geometry Edges A finite set feig embedded and labeled with \spin" 1 3 j = ; 1; ; 2; ::: 2 2 Vertices finite set fvig Trivalent vertex spins sat- isfy the triangle inequalities j1 + j2 ≥ j3; j2 + j3 ≥ j1; j1 + j3 ≥ j2 and the sum j1 + j2 + j3 is an integer. Higher valence vertices are labeled with \intertwiners" - decomposition in terms or of trivalent vertices These are spin networks (Penrose) states of geometry 2 Primer on Quantum Mechanics (i) Complex vector space of \states" j i with scalar product (Hilbert space) In basis fj iig X j i = i j ii i Normed states - \The system exists in some state" X jhi j ij2 = 1 i (ii) Physical measurements are represented by self-adjoint operators A^ hi j A^y j ji ≡ hj j A^ j ii∗ = hi j A^ j ji Results of measurement are eigenvalues of self-adjoint operators (no degen- eracy) 2 A^ j i = a j i with Prob(A = aij ) = j ij (iii) System evolves in time according to Sch¨odinger'sequation d i~ j si = H^ j si dt 3 Discrete Spatial Geometry Familiar geometric quantities arise through measurement - operators - on the spin network states Area: spin network lines are “flux lines of area" Volume arises only at vertices Angle is defined at vertices 4 Discrete spectra for Geometry ∗ Area : A^S j si = a j si N 2 X q a = `P jn(jn + 1) with n=1 s G ` = ~ = 10−35m P c3 Angley: θ^ j si = θ j si ! j (j + 1) − j1(j1 + 1) − j2(j2 + 1) θ = arccos r r 1=2 2 [j1(j1 + 1) j2(j2 + 1)] ∗ Rovelli,Smolin Nuc. Phys. B 422 (1995) 593; Asktekar,Lewandowski Class. Quant. Grav. 14 (1997) A43 y SM Class. Quant. Grav. 16 (1999) 3859 5 Discrete spectra for Geometry Area: A^S j si = a j si Suppose a single edge, with j = 3=2, passes through a surface A measurement of area would yield p q 15 a = `2 j(j + 1) = `2 ≈ 10−70m2 P P 2 Small! 6 Discrete Spatial Geometry What if the Planck length were `P = 2 m ? Suppose you observed a growing whale... 7 Discrete Spatial Geometry What if the Planck length were `P = 2 m ? Suppose you observed a growing whale... with surface area (all edges with j = 1=2) 9 p 2 X q 2 3 2 a = `P jn(jn + 1) = `P 9 ≈ 31:2 m n=1 2 8 Discrete Spatial Geometry What if the Planck length were `P = 2 m ? Suppose you observed a growing whale... who grew the minimum amount to 10 p 2 X q 2 3 2 a = `P jn(jn + 1) = `P 10 ≈ 34:6 m n=1 2 9 Discrete Spatial Geometry A link constructed of a set of j = 1=2 loops (key rings!). What geometry does it have? C. Rovelli, Physics World Nov. 2003 10 Attractiveness of loops? In Einstein's general relativity the attractiveness of gravity arises through curvature. Gravity: Einstein described the universe (a smooth manifold M = S × R) with a metric gab satisfying the non-linear partial differential equation (a,b,... = 0,1,2,3) Gab = 8πG Tab • Spatial part on S: Gij = 8πG Tij Solutions give possible spatial geometry - \Space can curve". Matter and light follow geodesics in the curved geometry attraction of gravity • Temporal part on R G0i = 8πG T0i tells us how space evolves This talk will focus on vacuum solutions without cosmological constant Gab = 0 and with cosmological constant Gab + Λgab = 0 11 Attractiveness of loops? In Einstein's general relativity the attractiveness of gravity arises through curvature. Gravity: Einstein described the universe (a smooth manifold M = S × R) with a metric gab satisfying the non-linear partial differential equation (a,b,... = 0,1,2,3) Gab = 8πG Tab • Spatial part on S: Gij = 8πG Tij Solutions give possible spatial geometry - \Space curves". Matter and light follow geodesics in the curved geometry attraction of gravity • Temporal part on R G0i = 8πG T0i tells us how space evolves Finding quantum states of \Gab = 0" is hard! It is easier with variables other than the (spatial) metric (and extrinsic curvature). 12 Attractiveness of loops? General relativity with new variables (A; E). [A is a smooth connection on principal su(2) bundle. E is a weighted frame field.] Einstein equations in the new variables: • Spatial part \G = 0" solutions a ij DaE = 0 give possible spatial geometry. b E Fab = 0 • Temporal part \G0i = 0" Λ Ea · (Eb × F + Eb × Ec)+::: = 0 tells us how space evolves ab 6 abc • We can find solutions to the spatial part! 13 Attractiveness of loops? General relativity with new variables (A; E). [A is a smooth connection on principal su(2) bundle. E is a weighted frame field.] Einstein equations in the new variables: • Spatial part \G = 0" solutions a ij DaE = 0 give possible spatial geometry. b E Fab = 0 a \DaE = 0" (gauge freedom) divergence-free vector field loops! A linear combination of these constraints im- ply that the state is invariant under smooth, invertible maps of space into itself - diffeomor- phisms! 14 Attractiveness of loops? General relativity with new variables (A; E). • Loops tell us how much a vec- tor rotates due to the curvature of space: ∼ U(j) Z U(j) = P exp −i A(j) e http://torus.math.uiuc.edu/jms/java/dragsphere/ • Spin networks are linear combi- nations of j = 1=2 loops that are the eigenvectors of geometric oper- ators, e.g. j1 = 3=2; j2 = 3=2; j3 = 1 15 Attractiveness of loops? General relativity with new variables (A; E). • Loops tell us how much a vec- tor rotates due to the curvature of space: ∼ U(j) Z U(j) = P exp −i A(j) e http://torus.math.uiuc.edu/jms/java/dragsphere/ • Thm: (Baez) Spin networks are a basis of the gauge invariant Hilbert space L2(A=G). 16 Adding a cosmological constant General relativity with new variables (A; E). [complex sl(2; C)-valued connection] Einstein equations in the (old) new variables: • Spatial part \G = 0" solutions a ij DaE = 0 give possible spatial geometry. b E Fab = 0 • Temporal part \G0i = 0" Λ Ea · (Eb × F + Eb × Ec) = 0 tells us how space evolves ab 6 abc • To recover GR we must implement \reality conditions" • Seek wavefunctions (A) such that these constraints are satisfied. Can we find a solution? Sure, Kodama did. 17 Adding a cosmological constant With the Chern-Simons form Z 2 3 SCS = (A ^ dA + A ^ A ^ A) d x Σ 3 Let 3 ! = exp hA j i N − 2SCS Λ`p N possibly topology-dependent norm (Soo gr-qc/0109046). The handy fact δ abc SCS = Fbc δAa ensures that the Kodama state satisfies the Hamiltonian constraint. Λ Ea · (Eb × F + Eb × Ec) = 0 ab 6 abc Also (small) gauge and diffeomorphism invariant. 18 Adding a cosmological constant What is this state in the spin network representation? What would be the transform of a spin network state s(A)? hs j i = X hs j AihA j i \A" Z = dµ(A)s(A) (A) Z " 3 # = ( ) ( ) exp ? N dµ A s A 2SCS Λ`p Witten showed that the path integral Z ik Ψ(L) = dµ(A)L(A) exp S 4π CS is, for knots and links L, and real-valued connections equivalent to an in- variant, the Kauffman bracket! Key point: The invariant is sensitive to twists of the spin net edges. PI only defined for framed links - \tubes with stripes" or \ribbons" 19 Quantum Gravity with cosmological constant? Beautiful Picture: • State(s) of Quantum Gravity! • Includes the cosmological constant! • Knot classes are label the states! Ψ(s) = K(s) • Has the DeSitter cosmology as a semiclassical limit! • Cosmological constant - particle statistics connection? - composite particle statistics determined by framing in theory of fractional QHE Key new feature: apparently depends on framed spin networks. But, we do not know what Z " 3 # ( ) ( ) exp dµ A s A 2SCS Λ`p means for spin networks s(A) and sl(2; C)-valued connections. 20 Quantum Gravity with cosmological constant? Obviously this is too good to actually hold. • Kodama state is in Lorenztian framework. While Witten's result is in YM theory, with real-valued connections Z ik K(L) = dµ(A)W (L; A) exp S (A) ; 4π CS does not (obviously) hold for a complex connection. Like defining the inverse Laplace transform due care is required in the choice on contour. • Is the state normalizable? Not in linearized Lorentzian case [Freidel-Smolin CQG 21 (2004) 3831] • Violates CPT (relevance? NPT vs. QFT) • Using the variational calculus methods, the \invariant" for graphs acquires tangent space sensitivity (SM hep-th/9810071) 21 Quantum Gravity with cosmological constant? • Due to invariance under large gauge transformations, k is an integer • Equating YM and Kodama coefficients ik 3i 12π = = = 2 ) k 2 4π Λ`p Λ`p 12π So 2 is an integer. Note: Small Λ means large k. Λ`p • The \deformation parameter", a measure of twist, is 0 21 πi i Λ `p q = exp ∼ exp @ A k + 2 6 a root of unity.
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