From Quantum Field Theory to Quantum Geometry

Transcription

1 From Quantum Field Theory to Quantum Geometry Institute for Theoretical Physics, University of Vienna

2 Introduction 1 Introduction QFT: Scalar fields, QED, Standard Model, Gravity (electro-weak, strong, gravitational) 2 Space-Time Concepts quantize, use noncommutative geometry 3 Regularization of QFT Fuzzy S 2, CP N, T 2,... 4 Noncommutative Quantum field Theory renormalization, scalar fields, φ 3, gauge models? 5 Induced Gauge Models 6 Conclusions

3 History Project Quantum Mechanics: (Planck, Bohr, Heisenberg, Schrödinger,...) 1905, 1915 Special and General Relativity Einstein perturbative Quantum Field Theory renormalization, scalar fields, fermions UV, IR, convergence problems renormalized gauge models Standard Model, W, Z, Higgs? add Gravity or deform Space-Time merge general relativity with quantum physics through noncommutative geometry

4 Requirements QFT: Wightman, Symanzik,... Minkowski-Euclidean, reflection positivity Require: Covariance, spectrum condition, locality, vacuum representation of Lorentz group, scalar, spinor; vector; tensor fields renormalization gives a calculus (Connes, Kreimer) IR, UV, convergence problems nonperturbative approches, renormalization group, β function Models: Dimension: 1, 2, 3, 4?,...10, 11 Attempts Supersymmetry, Superstrings,...Loop Quantum Gravity Noncommutative Quantum Field Theory!!! (there are relations among them)

5 Space-Time Concepts in Newton gr., QM, ED, GR, QFT and QG? G (or θ) Newton gr. gen. rel.?, or def. QM class. FT? nonrel. QM 1/c QFT

6 History Limited localisation of events in space-time D R ss = G/c 4 hc/λ G/c 4 hc/d (1) gives Planck lenght as a lower bound Riemann, Schrödinger, Heisenberg, Peierls, Pauli, Oppenheimer, Snyder, Connes: Noncommutative Geometry 1992 H. G. and Madore: Regularization using nc manifolds 1995 Filk: Feynman rules, Doplicher et al: Free fields H.G. and Klimcik, Presnajder 1999 Schomerus: obtains nc models from strings H. G. and Schweda, Wulkenhaar, nc Gauge models

7 Strategy Classical Standard Model Quantised Standard Model Unification: Noncommutative Geometry? General Relativity Loop Quantum Gravity Quantum Field Theory Approach 1 renormalisation of quantum field theories on noncommutative geometries

8 Main ideas Replace manifold by algebra deform it keep differential calculus derivations... Replace fields by projective modules Replace integrals by traces use renormalized perturbation expansion Connes, Majid, S.-Jabbari, Landi, Lizzi, Vitale, Szabo, Dabrowski, Piacitelli, Filk, Doplicher, Fredenhagen, Roberts, Bahns, Liao, Sibold, Schweda, O Connor, Madore, Steinacker, Dolan, Klimcik, Presnajder, Wess, Schupp, Jurco, Aschieri, Chaichian, Balachandran, Jochum, Gayral, Wohlgenannt, Gracia-Bondia, Ruiz-Ruiz, Lukierski,Rivasseau, Magnen, Seiberg, Vignes-Tourneret, Witten, Wulkenhaar, Varilly...

9 Regularization FUZZY SPHERE expand smooth functions on the sphere: use generators of su(2) f(x) = f 0 + f i x i + f ij x i x j +... A N = [0] [j] use differential calculus on matrices, integration is trace i=1,2,3 embed sequence of algebras... (XN i )2 = R 2, [XN i, X j N ] = iǫ ijkrxn k N 2 1

10 Regularization Scalar Field becomes regularized S N [φ] = Tr N [XN i,φ ][XN i,φ] + m2 φ φ + λ(φ φ) 2 < φ... φ > N = 1 [dφ] N e SN[φ] φ...φ Z N Quantization of sections of line bundles through sequence of embedded NxM matrices of functions on superspace through sequence of embedded graded matrices of representations of SU q (2) gives q-deformed sphere Regularization preserves symmetries

11 Renormalization Formulation φ 4 on nc R 4, [x µ, x ν ] = iθ µν or equivalently (a b)(x) = dy dka(x + θk )b(x + y)eiky 2 φ 4 action S = dp(p 2 +m 2 )φ p φ p +λ Feynman rules 4 ( ) 4 dpj φ pj δ( p j )e i 2 j=1 j=1 i<j pµ i p ν j θ µν 1 p 2 +m 2 λ i 4! e 2 i<j p µ p ν i j θ µν cyclic order of momenta leads to ribbon graphs

12 IR/UV mixing One-loop two-point function planar and nonplanar contributions: k p = λ 1 dk 6 k 2 + m 2 planar graphs: renormalize BPHZ... k p = λ 12 dk eipµ k ν θ µν k 2 + m 2 p 0 1 p 2

13 IR/UV mixing non-planar graph finite (noncommutativity serves as regulator), but behaves p 2 for small momenta (renormalisation not possible) this leads to non-integrable integrals when inserted as subgraph into bigger graphs: IR/UV-mixing Minwalla, van Raamsdonk & Seiberg, 1999 more rigorous treatment: power-counting theorem for ribbon graphs Chepelev & Roiban 1999/2000 proposals: resummation, supersymmetry (all integrals exist, but unbounded as momenta 0),... satisfactory solution: modify action

14 Main result H. G. and R. Wulkenhaar φ 4 model modified IR/UV mixing: short and long distances related Theorem: Action S = ( 1 d 4 x 2 µφ µ φ+ Ω2 2 ( x µφ) ( x µ φ)+ µ2 ) 0 2 φ φ+λφ φ φ φ (x) for x µ := 2(θ 1 ) µν x ν is perturbativly renormalizable to all orders in λ p µ x µ, ˆφ(p) π 2 detθ φ(x) Fourier transformation ˆφ(p a ) = d 4 x e ( 1)a ip a,µx a µ φ(xa ), leads to Langmann-Szabo duality S [ φ;µ 0,λ,Ω ] Ω 2 S [ φ; µ 0 Ω, λ, 1 ] Ω 2 Ω

15 Unitary Transform operator base φ(x) = m 1,m 2, N φ m1 m 2 b m1 m 2 (x) where b m1 n 1 (x 1, x 2 ) = (x 1+ix 2 ) m 1 m1!(2θ) m 1 (2e 1 θ (x2 1 +x2)) 2 (x 1 ix 2 ) n 1 n1!(2θ) n 1 (b mn b kl )(x) = δ nk b ml (x), d 4 x b mn (x) = (2πθ) 2 δ mn interaction becomes matrix product no oscillations ( 1 ) S = (2πθ) 2 2 φ mng mn;kl φ kl + λφ mn φ nk φ kl φ lm m,n,k,l N

16 Decay Properties Propagator complicated! Decay exponents decide on renormalization! θ/8 mm mm(0) 00; 00 m 1 m 1 m 2 m 2 ; π (m+1) +Ω2 (m+1) 2 (0) = θ 2(1+Ω) 2 (m 1 +m 2 +1) ( 1 Ω 1+Ω) m1 +m 2 exact results for special values has equidistant spectrum, study Jacobi matrix using Meixner polynomials closed formula (finite sum) due to identity for Meixner polynomials Propagator has asymmetric decays, is quasilocal proof power counting rule for ribbon graphs renormalization by flow equation

17 Renormalization Group Wilson-Polchinski approach to nonlocal matrix models Define QFT by cutoff partition function S[φ,Λ] = (2πθ) 2( m,n,k,l 1 ) 2 φ mngmn;kl K (Λ)φ kl + L[φ,Λ] startfrom interaction L[φ, ] = λ m,n,k,l φ mnφ nk φ kl φ lm require cutoff independence, add power series of interactions graphs drawn on Riemann surface of genus g 1 2g = L I + V with B holes L...single line loops for closed external lines I...double line propagators B...loops which cary external legs

18 Polchinski L[φ, Λ] Λ = 1 Λ 2 Λ K nm;lk (Λ) ( L[φ, Λ] L[φ, Λ] 1 2 L[φ, Λ] ) Λ φ mn φ kl (2πθ) 2 φ mn φ kl m,n,k,l Modified nc φ 4 model is renormalizable adjust four parameters: coupling, mass, frequency, field amplitude Proof Power counting rule use quasilocality of propagator to estimate ribbon graphs example 3 loop diagram, four independent sums! proof: all nonplanar graphs irrelevant planar graphs with more than 4 legs irrelevant 4leg graph log. divergent, 2leg graph quadr. divergent need 4 rel/marginal parameters!

19 RG functions evaluate β function ( lim N N N +Nγ+µ2 0 β µ 0 µ 2 0 +β λ λ +β ) Ω Γ[µ 0,λ,Ω, N] = 0 Ω β λ = N ( ) λ[n] = λ2 phys (1 Ω 2 phys ) N 48π 2 (1+Ω 2 + O(λ3 phys )3 phys ) β Ω = N ( ) Ω[N] = λ physω phys (1 Ω 2 phys ) N 96π 2 (1+Ω 2 + O(λ2 phys )3 phys ) Ω = 1 special...integrable?

20 A solvable model Scalar field φ 3 H. G. and Harold Steinacker recall: i φ = i[ x i,φ] S = ( x i φ x i φ x i x i φφ) + Ω 2 x i φ x i φ + µ2 2 φ2 + i λ 3! φ3 simplifies for Ω = 1 to S = ( x i x i + µ2 2 )φ2 + i λ ( 1 3! φ3 = Tr 2 Jφ2 + iλ 3! φ3). where J = 2(2πθ) 2 ( i x i x i + µ2 )... harmonic oscillator! 2 choose appropriate basis: in d = 2 : J n = 4π (n µ2 θ 2 ) n, n {0, 1, 2,...}

21 Free Energy for genus g Intersection theory of moduli space of Riemann surfaces of genus g with marked points Z Kont (M) = e F Kont (M) = = d φ exp F Kont 0 = 1 3 { dx exp Tr ( MX i X 3 6 { ( 1 i Tr 2iλ M2 φ 3! φ M3)} )} mi 3 1 (mi 2 2u 0 ) 3/2 u 0 (mi 2 2u 0 ) 1/2 3 i i i { (m 2 ln i 2u 0 ) 1/2 + (mk 2 2u } 0) 1/2 m i + m k + u i,k F Kont 1 = 1 24 ln(1 I 1),

22 Renormalize treated in dimension 2, 4, 6 I k = (2k 1)!! i u 0 = i 1, (mi 2 2u 0 ) k m 2i 2u 0 = I 0. all Fg Kont with g 2 are given by finite sums of polynomials in I k /(1 I 1 ) 2k+1 3. model nonperturbativly solvable also in 6 dimensions! (not superrenormalizable) agrees with pertubation expansion

23 Induced Gauge Models Couple gauge field to scalar field, H. G. and Michael Wohlgenannt S = ( 1 d D x 2 φ [B ν, [B ν, φ] ] + Ω2 2 φ {Bν, {B ν,φ} } + µ2 2 φ φ + λ ) 4! φ φ φ φ (x) use covariant coordinates gauge transformation B ν = x ν + A ν A µ iu µ u + u A µ u expand S calculate effective action

24 One-Loop Calculation in four dimensions quadratic divergent Γ ǫ 1l [φ] = 1 2 Use: Duhamel expansion ǫ dt t Tr (e th e th0) ( Γ ǫ 1 1l = 16π 2 d 4 1 x ǫθ (B ν B ν x 2 ) ( µ (B ν B ν x 2 ) + 1 ( (B µ B µ ) (B ν B ν ) ( x 2 ) 2) ) ) ln ǫ 2 for Ω = 1 (tbp) one loop calculation for general Ω Quantization? tadpole contribution shows up

25 Conclusions formulation of models on nc spaces possible gives symmetry preserving cutoffs removing cutoffs leads to IR/UV mixing not renormalizable modified actions for matter fields yields a calculus renormalons killed, constructive approach? Nontrivial? gauge fields? formulated gravity???