Spacetime Quantum Geometry

Transcription

1 Spacetime Quantum Geometry Peter Schupp Jacobs University Bremen 4th Scienceweb GCOE International Symposium Tohoku University 2012

2 Outline Spacetime quantum geometry Applying the principles of quantum mechanics to space & time Outline spacetime non-commutativity particle physics on non-commutative spaces non-commutative gravity and fuzzy black holes large scale application of small scale math (CMB analysis) higher geometric structures

3 Spacetime non-commutativity Planck scale quantum geometry Heuristic argument: quantum + gravity The gravitational field generated by the concentration of energy required to localize an event in spacetime should not be so strong as to hide the event itself to a distant observer. fundamental length scale, spacetime uncertainty x G c cm uncertainty principle noncommutative spacetime structure [ˆx i, ˆx j ] = iθ ij (x)

4 Spacetime non-commutativity Macroscopic non-commutativity Landau levels charged particle in a constant magnetic field B = A: quantize p = m x e A conjugate to x for m 0 (projection onto 1st Landau level): [ˆx i, ˆx j ] = 2i eb ɛij =: θ ij Fractional Quantum Hall Effect Non-commutative (NC) fluids, NC Chern Simons theory

5 Spacetime non-commutativity Loop quantum gravity, spacetime foam non-perturbative framework defines quantum spacetime in algebraic terms integrating out gravitational degrees of freedom appears to yield effective NC actions with Lie-type NC structure [ˆx i, ˆx j ] = iκ ɛ ijk ˆx k, κ = 4πG (κ-minkowski space) (feasible so far only for 2+1 dimensions)

6 Spacetime non-commutativity Non-commutativity in string theory effective dynamics of open strings ending on D-branes: non-abelian gauge theory in closed string background B-field non-commutative gauge theory string endpoints become non-commutative: f 1 (x(τ 1 ))... f n (x(τ n )) = dx f 1... f n, τ 1 <... < τ n with star product depending on B.

7 Spacetime non-commutativity Star product General x-dependent NC structure: f g = f g + i θ ij i f j g 2 θ ij θ kl i k f j l g ( θ ij j θ kl i k f l g k f i l g) on coordinates: [x i, x j ] = iθ ij

8 Spacetime non-commutativity How does a quantum space look like?

9 Spacetime non-commutativity How does a quantum space look like?

10 Spacetime non-commutativity How does a quantum space look like? (q-minkowski space; Cerchiai, Wess 1998)

11 Particle physics on non-commutative spaces Quantum/Noncommutative Spacetime model of quantum geometry, spacetime uncertainty Particle physics on non-commutative spaces construction of QFT on quantum spacetime experimental signatures? Features controlled Lorentz violation, non-locality, UV/IR mixing, mixing of internal & spacetime symmetries

12 Particle physics on non-commutative spaces Non-commutative Standard Model I Connes, Lott; Madore (+ many others) spacetime augmented by a discrete space Higgs field is NC gauge potential in the discrete direction beautiful geometrical interpretation of the full SM unlucky prediction of the Higgs mass

13 Particle physics on non-commutative spaces Non-commutative Standard Model II Calmet, Jurco, PS, Wess, Wohlgenannt (+ many others) deformed spacetime with -product particle content and structure group of undeformed SM enveloping algebra formalism, SW maps many new SM forbidden interactions

14 Particle physics on non-commutative spaces Star product and Seiberg-Witten map Star product: f g = fg θµν µ f ν g Corresponding expansion of fields via SW map: Â µ [A, θ] = A µ θξν {A ν, ξ A µ + F ξµ } +... Ψ[Ψ, A, θ] = Ψ θµν A ν µ Ψ θµν µ A ν Ψ +... Λ[Λ, A, θ] = Λ θξν {A ν, ξ Λ} +...

15 Particle physics on non-commutative spaces Deformed action Matrix multiplication is augmented by -products. e.g.: noncommutative Yang-Mills-Dirac action: ( Ŝ = d 4 x 1 ) 4 Tr( F µν F µν ) + Ψ i D/ Ψ with NC field strength F µν = µ Â ν ν Â µ i[âµ, Âν]

16 Particle physics on non-commutative spaces Non-perturbative effects at the quantum level Beta function of U(N) NC Yang-Mills theory: β(g 2 ) = g2 ln Λ = 22 3 g 4 N 2 8π 2 like ordinary SU(N) gauge theory but holds also for N = 1 This U(1) NC gauge theory is asymptotically free with strong coupling effects at large distance scales.

17 Particle physics on non-commutative spaces Photon neutrino interaction neutral particles can interact with photons via -commutator D µ Ψ = µ Ψ i[a µ, Ψ] 1-loop contributions to neutrino self-energy:

18 Particle physics on non-commutative spaces Superluminal neutrinos? deformed dispersion relation ( ) p 2 8π 2 e 2 1 ( 8π 2 ± 2 ) 1 2 e 2 1 } {{ } constant k, independent of θ direction dependent neutrino velocity E 2 = p 2 c 2 (1 + k sin 2 θ). pr 2, Horvat, Ilakovac, PS, Trampetic, You: arxiv: v1 [hep-th]

19 Non-commutative gravity Non-commutative gravity and exact solutions Gravity on non-commutative spacetime Fuzzy black hole solutions

20 Non-commutative gravity Motivation Quantum black holes nice theoretical laboratory for physics beyond QFT/GR information paradox, entropy, holography, singularities,... Major obstacle The existence of a fundamental length scale is a priori incompatible with spacetime symmetries The symmetry (Hopf algebra) must be deformed

21 Non-commutative gravity Drinfel d twisted symmetry: F = exp ( i2 ) θab V a V b, [V a, V b ] = 0 (f ) = F (f )F 1 f g = F(f g) Twisted tensor calculus Two simple rules: The transformation of individual tensors is not deformed Tensors must be -multiplied Twisted quantization, Hopf algebra symmetry in string theory: Asakawa, Watamura (Tohoku University) afternoon session: T. Asakawa NC solitions of gravity

22 Non-commutative gravity Drinfel d twisted symmetry: F = exp ( i2 ) θab V a V b, [V a, V b ] = 0 (f ) = F (f )F 1 f g = F(f g) Twisted tensor calculus Two simple rules: The transformation of individual tensors is not deformed Tensors must be -multiplied Twisted quantization, Hopf algebra symmetry in string theory: Asakawa, Watamura (Tohoku University) afternoon session: T. Asakawa NC solitions of gravity

23 Non-commutative gravity Deformed Einstein equations with non-commutative sources T µν R µν (G, ) = T µν 1 2 G µνt Solutions? Solution = mutually compatible pair: algebra (twist) metric G µν

24 Non-commutative black hole Exact solution with rotational symmetry Star product (twist) for tensors: V W = VW + C n ( λ ρ )Ln ξ + V L n ξ W n=1 left invariant Killing vector fields: ξ ± = ξ 1 ± iξ 2 commutative radius: ρ 2 = x 2 + y 2 + z 2 C n ( λ ρ ) = B(n, ρ λ ) = λ n n! ρ(ρ λ)(ρ 2λ) (ρ (n 1)λ)

25 Non-commutative black hole Metric in isotropic coordinates ( ds 2 = 1 a ) dt 2 + r 2 ρ ρ 2 (dx 2 + dy 2 + dz 2 ) with r = (ρ + a/4) 2 /ρ, a = 2M ρ 2 = g ij x i x j = x 2 + y 2 + z 2 [x i, x j ] = 2iλɛ ijk x k quantized, quasi 2-dimensional onion -spacetime: ρ = 2jλ = nλ; n = 0, 1, 2,...

26 Non-commutative black hole

27 Non-commutative black hole

28 Non-commutative black hole

29 Non-commutative black hole

30 Hilbert Space L 2 (R 3 ) L 2 (S 2 ) We find the following surprising result: States describing events in 3 dim NC bulk are equivalent to wave functions on a sphere (minus a fuzzy sphere) H = n>n C n+1 = C n+1 H hidden NC Schwarzschild solution = Fuzzy Black Hole Holographic behavior appears quite naturally NC: bulk (3D) surface (2D) Heuristically: (1) coordinates are no longer independent: z [x, y] (2) number of commuting operators = two

31 Inside NC black hole two sequences of fuzzy spheres with limit points at r = 0 (singularity) and r = a (horizon)

32 Non-commutative black hole Fuzzy black hole inside and outside, in one figure:

33 Coherent States, Star Products, Entropy Generalized coherent state (SU(2), spin j representation) dω Ω = R Ω j, j, R Ω SU(2)/U(1); (2j+1) 4π Ω Ω = 1 j Star product For A(Ω) := Ω A Ω and B(Ω) := Ω B Ω define: A(Ω) B(Ω) = Ω AB Ω

34 Coherent States, Star Products, Entropy Von Neumann entropy S Q (ρ) = trρ ln ρ = (2j + 1) dω 4π ρ(ω) ln ρ(ω) Now switch off (or ignore) noncommutativity Wehrl entropy dω S W (ρ) = (2j + 1) ρ(ω) ln ρ(ω) 4π dω (2j + 1) 4π Ω Ψ 2 ln Ω Ψ 2 > 0 even for pure states

35 Coherent states and CMB Coherent state analysis of CMB pseudo entropy l

36 Coherent states and CMB CMB at j = 5 with multi-pole vectors ( coherent states)

37 Coherent states and CMB CMB at j = 28

38 Higher geometric structures Beyond non-commutative Recent work in M-theory reveales non-associative higher geometric structures featuring Nambu brackets: Basu-Harvey equations, fuzzy S 3 funnels Bagger-Lambert action for multiple M2 and M5-branes afternoon session: M. Sato 3-algebra Model of M-Theory

39 Higher geometric structures Nambu mechanics multi-hamiltonian dynamics with generalized Poisson brackets e.g. Euler s equations for the spinning top : d dt L i = {L i, L 2, T } i = 1, 2, 3 2 with angular momenta L 1, L 2, L 2, kinetic energy T = L 2 i 2I i Nambu-Poisson bracket [ ] (f, g, h) {f, g, h} det = ɛ ijk i f j g k h (L 1, L 2, L 3 ) and

40 Higher geometric structures Nambu-Poisson (NP) bracket more generally: {f, h 1,..., h p } = Π i j 1...j p (x) i f j1 h 1 jp h p + Fundamental Identity (FI) {{f 0,, f p }, h 1,, h p } = {{f 0, h 1,, h p }, f 1,, f p } {f 0,..., f p 1, {f p, h 1,, h p }}

41 Higher geometric structures Nambu-Dirac-Born-Infeld action (B Jurco & PS 2012) commutative non-commutative symmetry implies S DBI = d n x 1 p 1 [ det 2(p+1) [g] det 2(p+1) g + (B + F) g 1 (B + F) T ] g m = d n x 1 p [Ĝ] 1 [Ĝ+(ˆΦ+ˆF) G 1 det 2(p+1) det 2(p+1) (ˆΦ+ˆF) T ] G m This action interpolates between early proposals based on supersymmetry and more recent work featuring higher geometric structures.

42 Thank you for listening!