Institute for Mathematics, Astrophysics and Particle Physics

Transcription

1 arxiv: Compact Phase Space for Loop Quantum Gravity Institute for Mathematics, Astrophysics and Particle Physics

2 THE PROBLEM Fact: our universe has a positive cosmological constant Does this affect the kinematics of LQG? (Dupuis-Girelli '13) Replacing flat cells with uniformly curved cells (Bahr-Dittrich 09) Classical kinematics: su(2) SU(2) 8` Idea: replace the algebra with the group > finiteness (Haggard-Han-Kaminski-Riello 14) Classically: compact phase space > finite Liouville volume Quantum: finite # of Planck cells, finite # orthogonal states > finite dim Hilbert space

3 CONSTANT CURVATURE GEOMETRY R =1 ` = L`/R k l : rotation associated to the curved arc l h l : holonomy of the 3d connection k`,h` 2 SO(3) SU(2) SU(2) SU(2) small triangles: Standard LQG phase space: local isometry group, or Chern-Simon gauge group (Meusburger-Schroers 08) k` = e ~ J` ~ 1 + J` =1+ ~ J` ~ SU(2) SU(2) R1 su(2) SU(2) = T SU(2) (k, h) 7 R1 (J, h)

4 U(1)xU(1) h h = e i,k = e i 2 C Symplectic 2-form: = h 1 dh ^ k 1 dk in the limit in which the radius of one of the two circles can be considered large we want to recover the symplectic form of the cotangent space = d ^ d Poisson brakets: {k, h} = hk QUANTIZATION Hilbert space: ni n =1,...,N =dimh Operators: k ni = e i 2 N n ni h ni = n +1i cyclic: h Ni = 1i Commutator: [h, k] = e i 2 N 1 hk discrete spectrum [â, ˆb] =i~ \ {a, b} ~ = 2 N [Planck length] [cosmological constant]

5 SU(2)xSU(2) (k h = e J,h) 2 SU(2) SU(2) limit: arc R where = Tr[kh 1 dh] Symplectic 2-form: Poisson brakets? = Tr[dk ^ h 1 dh kh 1 dh ^ h 1 dh] quasi Poisson-Lie groups QUANTIZATION Hilbert space: L 2 [SU(2)] 1 j=0(h j H j ) Operators: hu jmni = Dmn(U) j 1 h (U) =U (U) h AB jmni = 2 A m m B n n 0 j 0 m 0 n 0 i J i (U) =L i (U) J i jmni = (j) mk jkni

6 SU(2)xSU(2) (k h = e J,h) 2 SU(2) SU(2) limit: arc R where = Tr[kh 1 dh] Symplectic 2-form: Poisson brakets? QUANTIZATION Hilbert space: = Tr[dk ^ h 1 dh kh 1 dh ^ h 1 dh] quasi Poisson-Lie groups QUANTUM GROUPS j max L 2 [SU(2)] 1 j=0(h j H j ) Operators: hu jmni = Dmn(U) j 1 h (U) =U (U) h AB jmni = 2 A m m q B n n 0 j 0 m 0 n 0 i q J i (U) =L i (U) J i jmni = (j) mk jkni q r = 1 j max = r 2 2

7 KNOTS 1 h AB jmni = 2 A m m q B n n 0 j 0 m 0 n 0 i q does not commute any more 9hreps Wigner symbols as trivalent nodes (h AB ) mn m 0 n 0 = Acting with two operators: h AB h CD = Inverse order: h CD h AB = crossing operators h AB h CD = R A0 C 0 AC R B0 D 0 BD h C 0 D 0 h A 0 B 0 _ Expanding in h so that R 1+r : {h AB,h CD } = r B0 D 0 BD h CD 0h AB 0 + r A0 C 0 AC h C 0 Dh A 0 B

8 PHYSICS AND CONCLUSIONS We have introduced a modification of LQG kinematics compact phase space allows to introduce a positive cosmological constant finite dimensional Hilbert space dim determined by the ratio between the two constants: quantization (physically: Planck constant scale) simplex curvature / deformation of Poisson algebra (physically: cosmological constant) Hilbert space reduces to usual LQG one for triangles small compared to curvature radius A q-deformation of the dynamics: renders quantum gravity finite (Turaev-Viro 92, Han 10) amount to introduce the cosmological constant (Mizoguchi-Tada 91, Han 10) q = e ip ~G Compactness: discretization of the intrinsic and extrinsic geometry Time discreteness: K ab dq ab /dt where q ab ( t) q ab (0) + dq ab /dt t minimum proper time Planckian, full discrete spectrum depends on cosmological constant Euclidean 2+1 to do: Lorentzian 3+1