From Matrices to Quantum Geometry

Transcription

1 From Matrices to Quantum Geometry Harold Steinacker University of Vienna Wien, june 25, 2013

2 Geometry and physics without space-time continuum aim: (toy-?) model for quantum theory of all fund. interactions (gravity!) pre-geometric geometry, gravity emerge at low energies quantum structure of space-time at L P candidates: string theory: here:... vast landscape of possible vacua... (??) Matrix Models related to string theory, more predictive dynamical NC space-time, matrix geometry accessible, novel tools!

3 Geometry and physics without space-time continuum aim: (toy-?) model for quantum theory of all fund. interactions (gravity!) pre-geometric geometry, gravity emerge at low energies quantum structure of space-time at L P candidates: string theory: here:... vast landscape of possible vacua... (??) Matrix Models related to string theory, more predictive dynamical NC space-time, matrix geometry accessible, novel tools!

4 Quantized phase space in quantum mechanics classical mechanics phase space R 2 functions f (q, p) A, A = C(R 2 )...commutative algebra quantum mechanics quantized phase space R 2 observables f (Q, P) A A = L(H)...noncommutative algebra (Heisenberg algebra) Poisson bracket {q, p} = 1 [Q, P] = i 1l minimal area = 2π quantization map: Q : A A Q(f ) Q(g) = Q(fg) + O( ) [Q(f ), Q(g)] = Q(i{f, g}) + O( 2 )

5 Quantization of Poisson (symplectic) manifolds (M, {.,.})... 2n-dimensional manifold with Poisson structure quantization map: such that {f, g} = iθ µν (x) µ f ν g Q : C(M) L(H) Q(f ) Q(g) = Q(fg) + O(θ) [Q(f ), Q(g)] = Q(i{f, g}) + O(θ 2 ) ( nice ) Φ L(H) = Mat(, C) quantized function on M here: assume M compact (torus, sphere,...) Hilbert space H = C N finite-dimensional, A θ = Mat(N, C) (2π )Tr Q(φ) M d 2 x φ(x) (Bohr-Sommerfeld) 2πN = 2πTr 1l M ω = Vol(M) ω... sympletic (volume) form on M

6 Quantization of Poisson (symplectic) manifolds (M, {.,.})... 2n-dimensional manifold with Poisson structure quantization map: such that {f, g} = iθ µν (x) µ f ν g Q : C(M) L(H) Q(f ) Q(g) = Q(fg) + O(θ) [Q(f ), Q(g)] = Q(i{f, g}) + O(θ 2 ) ( nice ) Φ L(H) = Mat(, C) quantized function on M here: assume M compact (torus, sphere,...) Hilbert space H = C N finite-dimensional, A θ = Mat(N, C) (2π )Tr Q(φ) M d 2 x φ(x) (Bohr-Sommerfeld) 2πN = 2πTr 1l M ω = Vol(M) ω... sympletic (volume) form on M

7 NC (matrix) geometry: same math as Q.M., Poisson (sympletic) manifold M interpreted as physical space(-time) quantized space(-time) M θ NC algebra of functions A θ beyond Q.M: need metric structure, dynamical (from M.M.!)

8 math background, NC geometry: Gelfand-Naimark theorem: commutative C - algebra A with 1 is isomorphic to C - algebra of (continuous) functions on compact Haussdorf-space M. NC geometry: A = NC (operator-) algebra ˆ= functions on quantum space + additional structure (metric,...) many possibilities (A. Connes; matrix models;...) guideline: physically relevant models of QFT, gauge theory here = matrix-models

9 Example: the fuzzy torus T 2 N The embedded 2D torus via x a : T 2 R 4, a = 1, 2, 3, 4 x 1 + ix 2 = e iϕ, x 3 + ix 4 = e iψ so that x a = x a (ϕ, ψ)... functions on T 2, (x 1 ) 2 + (x 2 ) 2 = 1, (x 3 ) 2 + (x 4 ) 2 = 1 U(1) L U(1) R symmetry ϕ ϕ + α, invariant Poisson structure ψ ψ + β {ϕ, ψ} = θ, {e iϕ, e iψ } = θ e iϕ e iψ symplectic form ω = θ 1 dϕ dψ

10 Example: the fuzzy torus T 2 N The embedded 2D torus via x a : T 2 R 4, a = 1, 2, 3, 4 x 1 + ix 2 = e iϕ, x 3 + ix 4 = e iψ so that x a = x a (ϕ, ψ)... functions on T 2, (x 1 ) 2 + (x 2 ) 2 = 1, (x 3 ) 2 + (x 4 ) 2 = 1 U(1) L U(1) R symmetry ϕ ϕ + α, invariant Poisson structure ψ ψ + β {ϕ, ψ} = θ, {e iϕ, e iψ } = θ e iϕ e iψ symplectic form ω = θ 1 dϕ dψ

11 The fuzzy torus TN 2: unitary matrices U =..., V = e 2πi 1 N e 2πi 2 N... UV = qvu, U N = V N = 1, q = e 2πi N satisfy N 1 2πi e N generate A = Mat(N, C)... quantiz. algebra of functions on T 2 N Z N Z N action: Z N A A similar other Z N (ω k, φ) U k φu k A = N 1 n,m=0 Un V m... harmonics

12 quantization map: satisfies in particular Q : C(T 2 ) A = Mat(N, C) e inϕ e imψ { U n V m, n, m < N/2 0, otherwise Q(fg) = Q(f )Q(g) + O( 1 N ), Q(i{f, g}) = [Q(f ), Q(g)] + O( 1 N 2 ) [U, V ] = (q 1)UV 2πi N UV Poisson structure {e iϕ, e iψ } = 2π N eiϕ e iψ on T 2 {ϕ, ψ} = 2π N θ T 2 N... quantization of (T 2, Nω)

13 need something like a metric on T 2 N several possibilities: metric encoded in Laplacian g φ = 1 µ ( g g µν ν φ) g define Laplace operator (or Dirac operator... (Connes)) on A θ can recover (almost...) metric g µν from spectrum of g ( can you hear the shape of a drum?) induced by embedding ( matrix models!) X a x a : M R 10 more transparent, work with Poisson manifolds (M, θ µν, g µν ) differential calculus (Madore),...

14 metric on T 2 N : encoded in embedding X a = Q(x a ) : T 2 R 4 U = X 1 + ix 2 = Q(x 1 + ix 2 ) = Q(e iϕ ), V = X 3 + ix 4 = Q(x 3 + ix 4 ) = Q(e iψ )... quantization of embedding maps Laplace operator: φ = [X a, [X b, φ]]δ ab = [U, [U, φ]] + [V, [V, φ]] = 2φ UφU U φu (%V ) (U n V m ) ([n] 2 q + [m] 2 q) U n V m (n 2 + m 2 ) U n V m where [n] q = qn/2 q n/2 sin(nπ/n) q 1/2 = q 1/2 sin(π/n) n ( q-number ) spec spec T 2 below cutoff geometry = flat torus UV cutoff n N/2

15 The fuzzy sphere classical S 2 : } x a : S 2 R 3 x a x a A = C = 1 (S 2 ) fuzzy sphere S 2 N : (Hoppe, Madore) let X a Mat(N, C)... 3 hermitian matrices [X a, X b ] = i ε abc X c, C N = 1 CN 4 (N2 1) X a X a = 1l, realized as X a = 1 J a... N dim irrep of su(2) on C N, CN generate A = Mat(N, C)... alg. of functions on S 2 N SO(3) action: su(2) A A (J a, φ) [X a, φ]

16 decompose A = Mat(N, C) into irreps of SO(3): A = Mat(N, C) = (N) ( N) = (1) (3)... (2N 1) = {Ŷ 0 0} {Ŷ m} 1... {Ŷ m N 1 }.... fuzzy spherical harmonics (polynomials in X a ); UV cutoff! quantization map: Q : C(S 2 ) { A = Mat(N, C) Ym l Ŷ l m, l < N 0, l N satisfies Q(fg) = Q(f )Q(g) + O( 1 N ), Q(i{f, g}) = [Q(f ), Q(g)] + O( 1 N 2 ) SO(3)-inv. Poisson structure {x a, x b } = 2 N εabc x c on S 2

17 decompose A = Mat(N, C) into irreps of SO(3): A = Mat(N, C) = (N) ( N) = (1) (3)... (2N 1) = {Ŷ 0 0} {Ŷ m} 1... {Ŷ m N 1 }.... fuzzy spherical harmonics (polynomials in X a ); UV cutoff! quantization map: Q : C(S 2 ) { A = Mat(N, C) Ym l Ŷ l m, l < N 0, l N satisfies Q(fg) = Q(f )Q(g) + O( 1 N ), Q(i{f, g}) = [Q(f ), Q(g)] + O( 1 N 2 ) SO(3)-inv. Poisson structure {x a, x b } = 2 N εabc x c on S 2 S 2 N... quantization of (S2, Nω)

18 decompose A = Mat(N, C) into irreps of SO(3): A = Mat(N, C) = (N) ( N) = (1) (3)... (2N 1) = {Ŷ 0 0} {Ŷ m} 1... {Ŷ m N 1 }.... fuzzy spherical harmonics (polynomials in X a ); UV cutoff! quantization map: Q : C(S 2 ) { A = Mat(N, C) Ym l Ŷ l m, l < N 0, l N satisfies Q(fg) = Q(f )Q(g) + O( 1 N ), Q(i{f, g}) = [Q(f ), Q(g)] + O( 1 N 2 ) SO(3)-inv. Poisson structure {x a, x b } = 2 N εabc x c on S 2 S 2 N... quantization of (S2, Nω)

19 metric structure of fuzzy sphere metric encoded in NC Laplace operator : A A, φ = [X a, [X b, φ]]δ ab Ŷ m l = 1 C N l(l + 1)Ŷ m l spectrum identical with classical case g φ = 1 µ ( g g µν ν φ) g effective metric of = round metric on S 2

20 geometry from matrices: given a suitable set of matrices X a Mat(, C) = L(H) { algebra A = f (X a ) Mat(, C) defines quantized embedding X a x a : M R 10 Lemma: f (X) := [X a, [X a, f (X)]] e σ G f (x)... Matrix Laplace- operator, effective metric (H.S. Nucl.Phys. B810 (2009) ) G µν (x) = e σ θ µµ (x)θ νν (x) g µ ν (x) effective metric (cf. open string m.) g µν (x) = µ x a ν x b η ab induced metric on M 4 θ (cf. closed string m.) e 2σ = θ 1 µν g µν

21 classical geometry NC geometry Poisson manifolds (M, {, }) NC space (A, H) C (M) = {f : M C} NC algebra A, rep. on H {x µ, x ν } = θ µν z.b. [X µ, X ν ] = iθ µν 1l additional geom. structures NC diff. calculus (A. Connes) diff. calculus, metric... Dirac operator /D, Laplacian g embedded manifolds x a : M R D matrices X a, a = 1,..., D field theorie: e.g. g φ = λφ NC field theorie: φ = λφ, φ C (M) φ A QFT NC QFT dφ e S(φ) dφ e S(φ) C(M) quantum gravity: e.g. (?) matrix models NC embedded manif. M R D, dg µν e S EH [g] dx e S YM [X] geometries A matrices

22 classical geometry NC geometry Poisson manifolds (M, {, }) NC space (A, H) C (M) = {f : M C} NC algebra A, rep. on H {x µ, x ν } = θ µν z.b. [X µ, X ν ] = iθ µν 1l additional geom. structures NC diff. calculus (A. Connes) diff. calculus, metric... Dirac operator /D, Laplacian g embedded manifolds x a : M R D matrices X a, a = 1,..., D field theorie: e.g. g φ = λφ NC field theorie: φ = λφ, φ C (M) φ A QFT NC QFT dφ e S(φ) dφ e S(φ) C(M) quantum gravity: e.g. (?) matrix models NC embedded manif. M R D, dg µν e S EH [g] dx e S YM [X] geometries A matrices

23 IKKT (IIB) matrix model Ishibashi, Kawai, Kitazawa, Tsuchiya 1996 ( ) S[X] = Tr [X a, X b ][X a, X b ]η aa η bb + Ψγ a [X a, Ψ] X a = X a Mat(N, C), a = 0,..., 9 N gauge symmetry X a UX a U 1, SO(9, 1), SUSY { 1) nonpert. def. of IIB string theory (on R 10 ) (IKKT ) 2) N = 4 SUSY Yang-Mills gauge thy. on noncommutative R 4 θ dynamical NC branes M R 10 ( 4D gravity? H.S ff) brane-world scenarios

24 Space-time from matrix models: e.o.m.: δs = 0 [X a, [X a, X b ]]η aa = 0 solutions: [X a, X b ] = iθ ab 1l, quantum plane R 4 θ [X a, X b ] i{x a, x b } = iθ ab (x), generic quantum space space-time as 3+1-dim. brane solution X a x a : M 4 R 10 intersecting branes, stacks (as in string theory) compact extra dim M 4 T 2, etc.

25 ( ) X basic solution of [X a, [X a, X b ]] = 0: X a µ = X i 0 [ X µ, X ν ] = i θ µν 1l, µ, ν = 0,..., 3... Heisenberg algebra A = Mat(, C) = functions on (R 4 θ, θµν ) X µ Mat(, C)... coordinate functions on quantum plane R 4 θ X µ X ν θ µν quantization map (Weyl): Q : C(R 4 ) A f (x) = d 4 k f (k)e ikµx µ d 4 k f (k)e ikµ X µ =: F( X) derivatives: [ X µ, F( X)] =: θ µν ν F

26 tangential deformations: gauge fields ( ) X X a = X a + A a µ = 0 + ( A µ ( X ) µ ) 0 S = Tr([X a, X b ][X a, X b ]) is gauge-invariant: X a U 1 X a U fluctuations X µ = X µ + θ µν A ν transform as A µ U 1 A µ U + iu 1 µ U gauge fields! [X µ, X ν ] = iθ µν + iθ µµ θ νν ( µ A ν ν A µ + [A µ, A ν ]) = iθ µν + iθ µµ θ νν F µ ν field strength eff. action S = const + d 4 x G e σ G µµ G νν F µν F µ ν tangential perturbations gauge fields on R 4 θ, eff. metric Gµν

27 similarly: transversal deformations scalar fields ( ) ( ) X X a = X a + A a µ 0 = + 0 φ i ( X µ ) transversal fluctuations scalar fields on R 4 θ, eff. metric Gµν

28 nonabelian gauge theory: stack of coincident branes tangential fluctuation su(n) gauge fields background ( ) ( ) Y Y a µ X = Y i = µ 1l n φ i 1l n include fluctuations: Y a = (1 + A ρ ρ ) ( X µ 1l n φ i 1l n + Φ i ) effective action: S YM = d 4 x G e σ tr F, F G + 2 η(x) tr F F (H.S., JHEP 0712:049 (2007), JHEP 0902:044,(2009) ) IKKT model on stack of branes SU(n) N = 4 SYM coupled to metric G µν (x)

29 main results: universal effective metric G ab (x) on such branes, dynamical fluctuations of matrices X A around stack of branes SU(n) NC Yang-Mills gauge theory coupled to G ab (x) Poisson structure θ µν invisible (U(1) is sterile) prospects: intersecting branes chiral fermions A. Chatzistavrakidis, H.S., G. Zoupanos (2011) all ingredients for physics ( brane-world picture) well suited for quantization, predictive

30 Quantization Z = dx a dψ e S[X] S[Ψ] 2 interpretations: 1 on R 4 θ : NC gauge theory on R4 θ, UV/IR mixing in U(1) sector almost all models are sick (loops probe UV, too wild ) except IKKT model: N = 4 SYM, perturb. finite!(?) 2 on M 4 R 10 : U(1) absorbed in θ µν (x), G µν (x) induced E-H. action S eff d 4 x G ( Λ 4 + cλ 2 4 R[G] +... ) IKKT good quantization for theory with gravity! (SUSY) 4 noncompact dimensions preferred (higher dim unstable)

31 can be put on computer (Monte Carlo; Lorentzian)! can measure effective dimensions Kim, Nishimura, Tsuchiya PRL 108 (2012) result: 3 out of 9 spatial directions expand, 3+1 dims at late times

32 towards (emergent) gravity brane gravity (not bulk gravity); propagation on 4D brane complicated dynamics, not well understood Einstein equations not established several mechanisms: tang. modes NC U(1) gauge fields µ F µν = 0 δr µν = 0 Ricci-flat vacuum perturbations (around R 4 ) V.Rivelles (2003) similar for fluctuations of M 4 K R 10 A.Polychronakos, H.S., J.Zahn (2013) T µν induces perturbation of R µν in presence of extrinsic curvature (Newtonian) gravity (without E-H action!) H.S. (2009,2012) quantum effects induced gravity (?)

33 towards particle physics intersecting brane solutions chiral fermions at intersection = 4D space-times ( ) X a (11) ψ (12) ψ (21) X a (22) stacks of intersecting branes close to standard model A. Chatzistavrakidis, H.S., G. Zoupanos JHEP 1109 (2011) (cf. string theory) clear-cut, predictive framework 1-loop intersecting branes can form bound system!

34 Summary, conclusion matrix geometry: can describe space-time & geometry with (finite-dim.!) matrices quantum structure of space(time) matrix-models Tr[X a, X b ][X a, X b ] η aa η bb + fermions dynamical NC branes, emergent gauge theory (& gravity?!) background independent, all ingredients for physics not same as general relativity, but might be close enough (?) suitable for quantizing gauge theory & geometry (gravity?) (IKKT model, N = 4 SUSY in D = 4)

35 references H. Steinacker, Non-commutative geometry and matrix models, arxiv: [hep-th]. H. Steinacker, Emergent Geometry and Gravity from Matrix Models: an Introduction, arxiv: [hep-th]. topical review, Class. Quantum Grav. 27 (2010) N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A Large N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/ ]. H. Steinacker, Emergent Gravity from Noncommutative Gauge Theory. JHEP 12 (2007) 049. [arxiv: v1 (hep-th)] H. Steinacker, Emergent Gravity and Noncommutative Branes from Yang-Mills Matrix Models, Nucl. Phys. B 810:1-39,2009. arxiv: [hep-th]. A. Polychronakos, H. Steinacker, J. Zahn, compactified branes and 4-dimensional geometry in the IKKT model arxiv: D. N. Blaschke and H. Steinacker, Curvature and Gravity Actions for Matrix Models, arxiv: [hep-th].class. Quant. Grav. 27:165010,2010.

36 higher-order terms, curvature result: H ab := 1 2 [[X a, X c ], [X b, X c ]] + T ab := H ab 1 4 ηab H, H := H ab η ab = [X c, X d ][X c, X d ], X := [X b, [X b, X]] for 4-dim. M R D with g µν = G µν (Euclidean!): Tr ( ) 2T ab X a X b T ab H ab 2 (2π) d 4 x g e 2σ R 2 Tr([[X a, X c ], [X c, X b ]][X a, X b ] 2 X a X a ) 1 (2π) d 4 x g e ( σ e σ θ µη θ ρα R µηρα 2R + µ σ µ σ ) (Blaschke, H.S. arxiv: ) (cf. Arnlind, Hoppe, Huisken arxiv: ) Einstein-Hilbert- type action for gravity as matrix model pre-geometric version of (quantum?) gravity, background indep.!