Dynamical polyhedra and the atoms of space in quantum gravity

Transcription

1 Dynamical polyhedra and the atoms of space in quantum gravity Hal Haggard Bard College June 2 nd, 2016 Helsinki Workshop Helsinki, Finland

2 A quite revolution in the study of polyhedra is brewing along the road to a quantum theory of gravity. Stories, like people and butterflies and songbirds eggs and human hearts and dreams, are also fragile things, made up of nothing stronger or more lasting than twenty-six letters and a handful of punctuation marks. Or they are words on the air, composed of sounds and ideas-abstract, invisible, gone once they ve been spoken-and what could be more frail than that? But some stories, small, simple ones about setting out on adventures or people doing wonders, tales of miracles and monsters, have outlasted all the people who told them, and some of them have outlasted the lands in which they were created. I have a simple story for you. Neil Gaiman 1

3 A first example in quantum mechanics......but, what of the quantum mechanics of the box? 2

4 The gravito-electric analogy: q E = 1 4πɛ 0 qq r 2 ˆr m g = GmM r 2 ˆr V E da = Q ɛ 0 V g da = 4πGM 3

5 The two roles of the gravitational Gauß law... E ibc = ɛ ijk e j b ek c (b, c = 1, 2, 3) (i, j, k = 1, 2, 3)... the constraint and the generator of gauge E = 0 local choice of frame 4

6 In 1897 H. Minkowski proved A A N = 0, with Ai = A iˆn i, A i = area face i ˆn i = to face i Physical proof for one direction: A A N = 0 = 5

7 Combine this with Kepler s elegant realization: Angular momentum can be used to encode areas Can define dynamical polyhedra! If F ( A) and G( A) then: {F, G} = ( A F ) A G A (e.g. {A x, A y } = ( A ˆx) ŷ = (ˆx ŷ) A = A z ) 6

8 Minkowski s theorem is a discrete version of the two roles of the gravitational Gauß law... E ibc = ɛ ijk e j b ek c (b, c = 1, 2, 3) (i, j, k = 1, 2, 3)... the constraint and the generator of gauge E = 0 local choice of frame A A N = 0 diagonal rotation of all the vecs 7

9 99 years after Minkowski, in 1996, M. Kapovich and J. J. Millson found a reduced phase space for fixed areas of the polyhedron faces Put the area vectors end-to-end to form a non-planar polygon p = A 1 + A 2, rotates A 1 and A 2 leaving others fixed q = Angle of rotation generated by p: {q, p} = 1 8

10 Remarkably, the volume of the tetrahedron is easily expressed in terms of these variables Fun to discover that, with e.g. A 2 = 1 2 e 1 e 3 etc., V = 1 6 e 1 ( e 2 e 3 ) V 2 = 2 9 A 1 ( A 2 A 3 ) 9

11 With this phase space, take the volume as a Hamiltonian 2 H = V = A 3 1 ( A 2 A 3 ) 10

12 We can now quantize the system semiclassically Require Bohr-Sommerfeld quantization condition, S = pdq = (n )h. γ Area of orbits given in terms of complete elliptic integrals, ( S(V ) = ak(m) + ) 4 b i Π(αi 2, m) V i=1 11

13 V Tet = A 1 ( A 2 A 3 ) A 1 = j + 1/2 A 2 = j + 1/2 A 3 = j + 1/2 A 4 = j + 3/2 4 2 = Numerical = Bohr-Som [PRL 107, ]

14 Table j 1 j 2 j 3 j 4 Loop gravity Bohr- Accuracy Sommerfeld % % % % % % % % % % % % 13

15 This text is for space. Is space discrete like a mosaic? It may be, and so far......we understand this as a spectral discreteness.

16 One Is space a spectral mosaic? Two The shadow of the cosmological constant Three Projecting polyhedra 15

17 In the presence of Λ, we match the symmetry if we construct a manifold out of homogeneously curved building blocks. Quantum mechanically homogenous curvature is implemented via BF Λ 6 BB dynamics, and defects are created as in the flat case Λ-GR = BF Λ 6 BB + geometricity constraints 16

18 For boundary connection functionals, ΛBF in the bulk is equivalent to CS on the boundary i B F[A] 2l Z(ψ Γ ) := DBDA e 2 Λ 6 B B ( ) P fγ ψ Γ (G[A]) 3i F[A] F[A] 4Λl = DA e 2 ( ) P fγ ψ Γ (G[A]) = DA e 3πi Λl 2 P CS[A] ( fγ ψ Γ ) (G[A]), where the Chern-Simons functional is Dual to 4-simplex boundary Γ CS[A] := 1 da A + 2 4π S 3 3 A A A [1D lower drawing] S 3 17

19 Twisting the previous construction using the γ-holst action Z ΛEPRL (ψ Γ ) := DADA e i k 2 CS[A]+i k 2 CS[ A ] ( f γ ψ Γ ) (G[A, A]) where (A, A) are the self- and antiself-dual parts of A ( ) and k := 12π 1 Λl 2 γ + i is the complex CS level P ΛEPRL = SL(2, C)-CS eval of a specific Wilson graph operator This functional involves only quantities living on the boundary and the equations of motion of CS-theory lead to flat connections 18

20 The semiclassical ΛRegge limit is ( ) k := 12π 1 Λl 2 γ + i P l P 0, j, with a phys γl 2 P j = cnst l P 0 means k, which corresponds to CS classical flat limit, however j makes the Wilson graph operator stand out and act as a distributional source for (A, A), thus avoiding flatness Semiclassical limit = study of flat connections on the graph complement M = S 3 \ Γ 19

21 ( ) k := 12π 1 Λl 2 γ + i P Remarkably, the WKB approximation to this Chern-Simons action for a 4-simplex gives { ΛIm(k) ( ) } Z(u, ū) cos Σa ab Θ ab ΛV4 Λ 12π the Regge action of simplicial General Relativity with a cosmological constant the two branches of opposite parity combine I have suppressed an overall phase depending on the choice of lift of the complex variables used to arrive at this result 20

22 The Chinese invented an extraordinary mechanical compass The south-seeking chariot 21

23 A sequence of differential gear mechanisms mechanically compute the difference in travel distance of the two wheels and keep the pointer in the same direction while the cart turns. 22

24 When placed on a curved surface, the chariot computes holonomies Along the small circle, the pointer rotates through an angle α θ = k g (s)ds = 2π α For a surface or region of constant curvature, Gauss-Bonnet gives KdA + k g ds = 2π = α = a r 2 cap 23

25 A spherical tetrahedron is 4 points of S 3 connected by geodesics Each face is a triangular portion of a great 2-sphere Great spheres are flatly embedded in S 3 (i.e. K ij = 0) The normal to a face is well-defined and invariant under parallel transport 24

26 We prove O 4 O 3 O 2 O 1 = 1l Here ( ) a O = exp ˆn J, O SO(3) r2 The closure relation is the automatic homotopy constraint. One immediate check: for r O 4 O 3 O 2 O 1 = 1l + r 2 (a 1ˆn 1 + a 2ˆn 2 + a 3ˆn 3 + a 4ˆn 4 ) J + = 1l 25

27 Define simple paths to determine a geometrically meaningful curved Gram matrix. The geometrical dot product ˆn 1 ˆn 3 is well defined at vertex 4, but we have to rotate ˆn 4 to give ˆn 2 ˆn 4 meaning at 4. The Gram matrix is 1 ˆn 1 ˆn 2 ˆn 1 ˆn 3 ˆn 1 ˆn 4 1 ˆn Gram = 2 ˆn 3 ˆn 2 O 1ˆn 4 1 ˆn 3 ˆn 4. sym 1 26

28 One Is space a spectral mosaic? Two The shadow of the cosmological constant Three Projecting polyhedra 27

29 At this point in the talk, I paused the slides and illustrated Desargues theorem from projective geometry using the computer aided drafting program GeoGebra: geogebra.org/ Wikipedia has an article on the theorem here: wikipedia.org/wiki/desargues s_theorem 28

30 If Minkowski s theorem is only an existance and uniqueness theorem, what good is it to us really? A A n = 0. Minkowski reconstruction is hard; algebraic closure encodes the adjacency of faces in a highly non-trivial manner 29

31 Can solve pentahedral reconstruction by completing to a tetrahedron α, β, γ > 1 found from, α A 1 + β A 2 + γ A 3 + A 4 = 0 e.g. = α = A 4 ( A 2 A 3 )/ A 1 ( A 2 A 3 ) 30

32 This is just Desargues theorem in disguise! 31

33 Moduli Space of Flat Connections Curved Polyhedra Quasi-Poisson Spaces 32

34 Open Questions: Does Minkowski s theorem hold for all constantly curved polyhedra? Can projective invariants be used to classify the adjacency of the faces in all cases? Can simple closed formulas always be found? Cuboids are qualitatively different, why? What other surprises are hidden behind treating polyhedra as dynamical? 33