Institute for Mathematics, Astrophysics and Particle Physics

Transcription

1 arxiv: discrete time in quantum gravity with Carlo Rovelli Institute for Mathematics, Astrophysics and Particle Physics

2 DISCRETE TIME 0 0 time keeps track of elementary discrete changes (cfr Heisenberg s S-matrix - Blum s talk)

3 DISCRETE TIME 0 4 time keeps track of elementary discrete changes (cfr Heisenberg s S-matrix - Blum s talk)

4 DISCRETE TIME Time is just one of the degrees of freedom of the system I am describing. Example: cosmology with anisotropies (Bianchi I) I describe one d.o.f. w.r.t. the others time keeps track of elementary discrete changes

5 ONTHOLOGY SPECIAL RELATIVITY: Space + Time = Spacetime QUANTUM FIELD THEORY: Fields have quantum properties DISCRETENESS: Measurements may give discrete spectral values GENERAL RELATIVITY: Spacetime is a field general-covariant fields SUBSTANTIVALISM: Spacetime exists even if there is no Matter QUANTUM GRAVITY: Fields have quantum properties general-covariant quantum fields RELATIONALITY: Observables are relational (QM, GR, gauge theories )

6 QUANTUM GEOMETRY Loop Quantum Gravity: 4D minimal eigenvalues, no minimal bricks (Lorentz invariance) the operators do not commute quantum superposition Extrinsic Curvature 3D 2D 1D Intrinsic Curvature h l Holonomy of the Ashtekar-Barbero connection along the link L l = {L i l},i=1, 2, 3 SU(2) generators gravitational field operator (tetrad) Achtung: the length operator is not naturally defined, same for a time operator

7 OBSERVABLES G ll l Gauge invariant operator X G ll = L l L l with G ll 0 =0 l Penrose s spin-geometry theorem (1971), and Minkowski theorem (1897) l2n A l Area: A = Lorentzian Area: l A = L i l Li l. Z R e o ^ e i = Z R K i = Z R L i t P 0 z P Spinfoam & Cosmology

8 HILBERT SPACE Abstract graphs: Γ={N,L} v n j l Group variables: Graph Hilbert space: { h l 2 SU(2) ~L l 2 su(2) H = L 2 [SU(2) L /SU(2) N ] s(l) l t(l) The space H admits a basis,j`,v n i Spinfoam & Cosmology

9 U(1) x IR h h = e i 2 C, 2 R Symplectic 2-form: Poisson brakets: = d ^ d {h,β}=ih QUANTIZATION The discreteness of " is a direct consequence of the fact that α is in a compact domain. Notice that [α, "] ih - because the derivative of the function α on the circle diverges at α = 0. Quantization must take into account the global topology of phase space. The correct elementary operator of this system is not α, but rather h = eiα discrete spectrum

10 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 [â, ˆb] =i~ \ {a, b} ~ = 2 N

11 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]

12 IDEA 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 New LQG kinematics (Borissov-Major-Smolin 96, Dupuis-Girelli '13) Replacing flat cells with uniformly curved cells (Bahr-Dittrich 09) Result: cosmological constant ( as the one our universe has )

13 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)

14 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: = Tr[dk ^ h 1 dh kh 1 dh ^ h 1 dh] but we are not going to use this QUANTIZATION Hilbert space: L 2 [SU(2)] 1 j=0(h j H j ) Operators: h (U) =U (U) J i (U) =L i (U)

15 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: = Tr[dk ^ h 1 dh kh 1 dh ^ h 1 dh] but we are not going to use this 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 j j 0 A m m j j 0 B n n 0 j 0 m 0 n 0 i J i (U) =L i (U) J i jmni = i (j) mk jkni

16 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: = Tr[dk ^ h 1 dh kh 1 dh ^ h 1 dh] but we are not going to use this QUANTIZATION Hilbert space: QUANTUM GROUPS j max L 2 [SU(2)] H = 1 j=0(h j H j ) Operators: hu jmni = Dmn(U) j 1 h (U) =U (U) h AB jmni = 2 j j 0 A m m j j 0 q B n n 0 j 0 m 0 n 0 i q J i (U) =L i (U) ( ) mk q J i jmni = i (j) jkni q r = 1 j max = r 2 2

17 KNOTS 1 h AB jmni = 2 j j 0 A m m j j 0 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 = 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 quasi Poisson-Lie groups crossing operators

18 PHYSICS 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

19 SUMMARY Treat space and time on equal foot Implement discreteness for all observables Treat the variables homogenelly: everything is holonomized New: extrinsic geometry turns out to be discrete New: time should be discrete too This seems to correctly capture the universe as we observe it