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1. Deformation Quantization and the Fedosov Star-Product on Constant Curvature Manifolds of Codimension One Ph.D. Defense Philip Tillman University of Pittsburgh Thursday, March 15, 2007 P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 1 / 38

2. Outline (1) 1. The rules of a general quantum theory are unclear I Con‡icts with general relativity (GR) 2. How deformation quantization (DQ) solves some problems: I It’s both manifestly covariant and independent of Lagrangians, Hamiltonians, etc. 3. The conceptual advantages of quantum theory on phase-space: I Observables remain unchanged in "quantization" I Ordering ambiguities are clari…ed P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 2 / 38

5. Outline (2) 4. The results of this thesis: Construction of the Fedosov star-product (the fundamental object in DQ) on a class of constant curvature manifolds 5. Final remarks, open questions, and future directions P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 3 / 38

6. Outline (2) 4. The results of this thesis: Construction of the Fedosov star-product (the fundamental object in DQ) on a class of constant curvature manifolds 5. Final remarks, open questions, and future directions P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 3 / 38

7. Summary of Results Quantization of a class of constant curvature manifolds: 1 is covariant and independent of dynamics 2 does not rely on any symmetries 3 is an exact construction P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 4 / 38

10. The Problem The general rules of quantization are unclear This problem is exempli…ed by "quantum gravity" I Loop Quantum Gravity, String Theory, etc. Q: There are several to choose from, which one should we use? To answer, …rst isolate the fundamental assumptions of physics retained in a general quantum theory I The starting place is easy... Start with the principles and assumptions of general relativity (GR) and standard quantum theory P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 5 / 38

13. The Principles The Principles Underlying General Relativity (GR): 1 Mach’s Principle I "State" of rest of universe determines local inertial laws 2 The Equivalence Principle I Gravity and inertial forces are the same I Space-time is curved 3 Di¤eomorphism Covariance I If related by a di¤eomorphism, two models are di¤erent mathematical descriptions of the same physical system The Quantum Principle: All matter and energy both come in chunks and are waves, called quanta P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 6 / 38

17. Beyond GR and Quantum Theory The problem Finding the correct theory that subsumes both GR and quantum theory I There are con‡icts between the two (Tillman P. 2006a): 1 Canonical Quantization at best hides covariance and, at worst, breaks it 2 Path Integral Methods are covariant, but dependent on a Lagrangian (the dynamics) Deformation Quantization is both independent of all dynamical structures (including the metric) and is manifestly covariant P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 7 / 38

21. Birth of Deformation Quantization In 1927 Herman Weyl wrote his quantization rule W as an integral transform I W associates a unique linear Hilbert space operator to each phase-space function Shortly afterwards, Eugene Wigner wrote the inverse W 1 Groenewold in 1946 (and later Moyal) found: W 1 (W (f ) W (g)) = f exp " i h 2 ∂ ∂xµ ! ∂ ∂pµ ∂ ∂pµ ! ∂ ∂xµ !# g P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 8 / 38

24. Birth of Deformation Quantization The Groenewold-Moyal star-product is born. For any two phase-space functions f and g: f g def = f exp " i h 2 ∂ ∂xµ ! ∂ ∂pµ ∂ ∂pµ ! ∂ ∂xµ !# g = fg + i h 2 ∂f ∂xµ ∂g ∂pµ ∂f ∂pµ ∂g ∂xµ + 1 2! i h 2 2 f ∂ ∂xµ ! ∂ ∂pµ ∂ ∂pµ ! ∂ ∂xµ !2 g + P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 9 / 38

25. Birth of Deformation Quantization Observations of Groenewold-Moyal star-product The product is linear and associative Star-algebra is isomorphic to algebra of observables in quantum mechanics: [xµ , pν] = i hδµ ν , [xµ , xν ] = 0 = pµ, pν where [f , g] = f g g f , xµ is position, and pµ is momentum It’s coordinate independent. P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 10 / 38

28. Birth of Deformation Quantization Groenewold and Moyal found a completely equivalent formulation of quantum mechanics "We suggest that quantization be understood as a deformation of the structure of the algebra of classical observables, rather than as a radical change in the nature of the observables"–Bayen, Flato, Frønsdal, Lichnerowicz, Sternheimer 1978 A star-product is a complicated object containing, in general, derivatives of all orders I We need fancy tools in their construction and classi…cation P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 11 / 38

31. Star-Quantization Procedure Let’s assume that we have a star-product made-to-order. 1 Observables stay as the same function f ! f 2 The eigenvalue equation for f : f ρc = ρc f = cρc where c 2 C is the eigenvalue 3 Given any ρ = ∑c ac ρc (ac 2 C) the expectation of f is: hf i = 1 (2π h)n Z dn pdn x ρ f 4 Introduce a Lagrangian or Hamiltonian P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 12 / 38

35. Advances in Deformation Quantization 1. All star-products fall into equivalence classes parametrized by B = B0 + hB1 + h2 B2 + with coe¢ cients Bi 2 H2 dR (T M) for all i (Dito G. and Sternheimer D. 2002) 2. Fedosov’s construction of a star-product on arbitrary symplectic manifold, a generalized phase-space (Fedosov B. 1996) 3. Dito’s construction of a star-product on free covariant …elds and covariant …elds with a class of interaction terms (Dito G. 2002) I Renormalization: Divergences are singular coboundaries of Hochschild cohomology P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 13 / 38

38. Advances in Deformation Quantization 4. Kontsevich’s construction of a star-product on arbitrary Poisson manifold (a generalized symplectic manifold) in Kontsevich M. (2003) 5. Applications to statistical mechanics including the KMS condition in Basart H. et al (1984) and Bordemann M. et al (1998) 6. A GNS construction of a Hilbert space in Bordemann M. and Waldmann S. (1998) 7. Fermionic and bosonic star-products in Hirshfeld A. and Henselder P. (2002) and Hirshfeld A. et al (2005) P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 14 / 38

42. Main Problem in Deformation Quantization (DQ) Standard treatments use formal series in h: f = f0 + hf1 + h2 f2 + I Convergence in general may be problematic P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 15 / 38

43. Quantum Theory on Phase-Space A New Perspective of Quantum Theory Let’s assume that we have a star-product made-to-order. Q: What advantages does this have? A: Observables are the same phase-space functions I The conceptual break with classical theory is less severe than operator methods Operators Methods: Observables are fundamentally di¤erent objects (e.g., Hilbert space operators) P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 16 / 38

46. Quantum Theory on Phase-Space A New Perspective of Quantum Theory Let’s assume that we have a star-product made-to-order. Q: What advantages does this have? A: Observables are the same phase-space functions Concepts of topology, coordinate transformations, derivatives, integrals, etc. extend much more naturally than in operator methods Just see the texts on Noncommutative Geometry: Connes A. (1992), Brattelli O. and Robinson D. (1979) E.g. Topology: To get a local quantum theory restrict all observables/functions to regions of compact support P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 17 / 38

49. Quantum Theory on Phase-Space A New Perspective of Quantum Theory Let’s assume that we have a star-product made-to-order: Q: What advantages does this have? A2: Also, ordering ambiguities in quantization: x2 p ! Wλ x2 p = ˆx2 ˆp, ˆx ˆpˆx, ˆpˆx2 or something else? are understood in a natural way as di¤erent star-products: f λ g := W 1 λ (Wλ (f ) Wλ (g)) The Weyl transform W is symmetric ordering: W x2 p = ˆx2 ˆp + ˆx ˆpˆx + ˆpˆx2 /3 *Not all orderings correspond to star-products P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 18 / 38

50. Equations, States, and Probabilities Q: What about equations and states in DQ? Start with the well-known equation: ˆH jψ (t)i = i h ∂ ∂t jψ (t)i where hψ (t) jψ (t)i = 1 With ˆρ := jψ (t)i hψ (t)j, an equivalent equation is: [ ˆH, ˆρ] = i h∂ˆρ/∂t where Tr (ˆρ) = 1, ˆρ2 = ˆρ, and ˆρ† = ˆρ This equation can then be mapped to phase-space by W 1 to get... P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 19 / 38

54. Equations, States, and Probabilities Q: What about equations and states in DQ? A: Time-Dependendent Schrödinger Equation [H, ρ] = i h∂ρ/∂t where Tr (ρ) = 1, ¯ρ = ρ, and ρ ρ = ρ The function ρ (t), called a Wigner function, represents the state jψ (t)i Also, the expectation value of any observable f is: Tr (ρ f ) := 1 (2π h)n Z dn pdn x ρ f P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 20 / 38

57. Equations, States, and Probabilities This same basic procedure can be done for others: 1 Time-Independendent Schrödinger Equation H ρn = ρn H = Enρn 2 The Heisenberg Time-Evolution Equation [H, f ] = i h∂f /∂t 3 The Klein-Gordon Equation in Minkowski Space H ρ = ρ H = m2 ρ where Tr (ρ) = 1, ¯ρ = ρ, and ρ ρ = ρ for all P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 21 / 38

58. The Fedosov Star-Product Fedosov’s algorithm Constructing a star-product on an arbitrary symplectic manifold by means of a Weyl-like quantization map σ 1 : f g def = σ σ 1 (f ) σ 1 (g) The algorithm constructs σ 1 order by order in h I Convergence may be problematic in general P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 22 / 38

59. The Fedosov Star-Product Fedosov’s algorithm Constructing a star-product on an arbitrary symplectic manifold by means of a Weyl-like quantization map σ 1 : f g def = σ σ 1 (f ) σ 1 (g) The algorithm constructs σ 1 order by order in h I Convergence may be problematic in general P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 22 / 38

60. The Fedosov Star-Product on the Two-Sphere We embed the two sphere in R3 : δµνxµ xν = 1/C , xµ pµ = A δµν = diag (1, 1, 1) = 0 @ 1 0 0 0 1 0 0 0 1 1 A where δ is the embedding metric, and C and A are arbitrary constants P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 23 / 38

61. The Fedosov Star-Product on the Two-Sphere In an exact calculation: [xµ , xν ] = 0 , [xµ , Lν] = i hε µ νρxρ Lµ, Lν = i hε ρ µνLρ where Lµ = εν ρµxρ pν Observations 1 L’s generate SO (3) 2 L’s and x’s generate SO (3, 1) P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 24 / 38

62. The Fedosov Star-Product on the Two-Sphere In an exact calculation: [xµ , xν ] = 0 , [xµ , Lν] = i hε µ νρxρ Lµ, Lν = i hε ρ µνLρ where Lµ = εν ρµxρ pν Observations 1 L’s generate SO (3) 2 L’s and x’s generate SO (3, 1) P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 24 / 38

63. The Fedosov Star-Product on dS/AdS We embed them in R5 : ηµνxµ xν = 1/C , xµ pµ = A ηµν = diag (1, 1, 1, 1, 1) , AdS (C > 0) diag (1, 1, 1, 1, 1) , dS (C < 0) where η is the embedding metric, and C and A are arbitrary constants P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 25 / 38

64. The Fedosov Star-Product on dS/AdS In an exact calculation: [xµ , xν ] = 0 , xµ, Mνρ = i hx[νηρ]µ Mµν, Mρσ = i h Mσ[µην]ρ Mρ[µην]σ where 2Mµν = xµpν xνpµ Observations 1 M’s generate SO (1, 4) and SO (2, 3) for dS and AdS respectively 2 M’s and x’s generate SO (2, 4) for both dS and AdS respectively P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 26 / 38

65. The Fedosov Star-Product on dS/AdS In an exact calculation: [xµ , xν ] = 0 , xµ, Mνρ = i hx[νηρ]µ Mµν, Mρσ = i h Mσ[µην]ρ Mρ[µην]σ where 2Mµν = xµpν xνpµ Observations 1 M’s generate SO (1, 4) and SO (2, 3) for dS and AdS respectively 2 M’s and x’s generate SO (2, 4) for both dS and AdS respectively P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 26 / 38

66. The Fedosov Star-Product on Constant Curvature Manifolds of Codimension One We embed them in Rp+q+1 : ηµνxµ xν = 1/C , xµ pµ = A ηµν = 8 >>< >>: diag(1, . . . , 1| {z } p+1 , 1, . . . , 1| {z } q ) , C > 0 diag(1, . . . , 1| {z } p , 1, . . . , 1| {z } q+1 ) , C < 0 9 >>= >>; where η is the embedding metric, and C and A are arbitrary constants P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 27 / 38

67. The Fedosov Star-Product on Constant Curvature Manifolds of Codimension One The pattern holds in general... In an exact calculation: [xµ , xν ] = 0 , xµ, Mνρ = i hx[νηρ]µ Mµν, Mρσ = i h(Mσ[µην]ρ Mρ[µην]σ) where 2Mµν = xµpν xνpµ Observations 1 M’s generate SO (p, q + 1) and SO (p + 1, q) for C < 0 and C > 0 respectively 2 M’s and x’s generate SO (p + 1, q + 1) for both C < 0 and C > 0 respectively P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 28 / 38

68. The Fedosov Star-Product on Constant Curvature Manifolds of Codimension One The pattern holds in general... In an exact calculation: [xµ , xν ] = 0 , xµ, Mνρ = i hx[νηρ]µ Mµν, Mρσ = i h(Mσ[µην]ρ Mρ[µην]σ) where 2Mµν = xµpν xνpµ Observations 1 M’s generate SO (p, q + 1) and SO (p + 1, q) for C < 0 and C > 0 respectively 2 M’s and x’s generate SO (p + 1, q + 1) for both C < 0 and C > 0 respectively P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 28 / 38

69. Remarks and Open Questions Remark 1 This result is con…rmed by (Frønsdal C. 1965, 1973) 2 At this point, we can introduce any Lagrangian, Hamiltonian, or some other dynamical rule Open Question 1 Is this SO (p + 1, q + 1) the conformal group? (SO (3, 1) for two-sphere, SO (2, 4) for dS/AdS) I A clear interpretation of the generators is needed P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 29 / 38

72. Conclusions Previous methods relied crucially on symmetries (Frønsdal C. 1965, 1973) Results are exact addressing convergence of σ 1 : f Fed g def = σ σ 1 (f ) σ 1 (g) f GM g def = W 1 (W (f ) W (g)) The quantization method DQ is independent of dynamics and manifestly covariant Advantages of working on phase-space: 1 All observables are the same phase-space function as in classical theory 2 Clari…cation of ordering ambiguities P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 30 / 38

81. Future Directions 1 Apply current results to QFT using Dito’s star-products (Dito G. 2002) and Hirshfeld A. and Henselder P. 2002 and Hirshfeld A. et al 2005 2 Construct, for example, star-products for Maxwell-Dirac …elds 3 Explore connections between standard QFT, Algebraic QFT and DQ (Haag R. 1992) I Applications to quantum thermodynamics (KMS states) P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 32 / 38

84. Acknowledgements I would like to thank my advisors George Sparling and E. Ted Newman I would like to thank my other committee members Adam Leibovich, Donna Naples, and Gordon Belot. P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 33 / 38

85. The End FIN P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 34 / 38

86. References Basart H. et al 1984 Lett. Math. Phys. 8, 483. Bayen F. et al 1978 Ann. Physics 111, 61. Bordemann M. et al 1998 Lett. Math. Phys. 45, 49. Bordemann M. and Waldmann S. 1998 Commun. Math. Phys. 195, 549. Brattelli O. and Robinson D. 1979 Operator Algebras and Quantum Statistical Mechanics (New York: Springer-Verlang). Connes A. 1992 Noncommutative Geometry (San Diego: Academic Press). P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 35 / 38

87. References Dito G. 2002 Proc. Int. Conf. of 68ème Rencontre entre Physiciens Théoriciens et Mathématiciens on Deformation Quantization I (Strasbourg), (Berlin: de Gruyter) p 55 Preprint math.QA/0202271. Dito G. and Sternheimer D. 2002 Proc. Int. Conf. of 68ème Rencontre entre Physiciens Théoriciens et Mathématiciens on Deformation Quantization I (Strasbourg), (Berlin: de Gruyter) p 9 Preprint math.QA/0201168. Dütsch M. and Fredenhagen K. 2000 Proc. Conf. Math. Physics in Mathematics and Physics (Siena), Preprint hep-th/0101079. P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 36 / 38

88. References Fedosov B. 1996 Deformation Quantization and Index Theory (Berlin: Akademie). Frønsdal C. 1965 Rev. Mod. Phys. 37, 221. Frønsdal C. 1973 Phys. Rev. D 10, 2, 589. Haag R. 1992 Local Quantum Physics (New York: Springer-Verlang). Hirshfeld A. and Henselder P. 2002 Annals Phys. 302, 59. Hirshfeld A. et al 2005 Annals Phys. 317, 107. Kontsevich M. 2003 Lett. Math. Phys. 66, 157. P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 37 / 38

89. References Tillman P. and Sparling G. 2006a Fedosov Observables on Constant Curvature Manifoldsand the Klein-Gordon Equation, Preprint gr-qc/0603017, Accepted with revisions to J. Geom. Phys. Tillman P. and Sparling G. 2006b J. Math. Phys. 47, 052102. Tillman P. 2006a to appear in Proc. of II Int. Conf. on Quantum Theories and Renormalization Group in Gravity and Cosmology, Preprint gr-qc/0610141. Tillman P. 2006b Deformation Quantization: From Quantum Mechanics to Quantum Field Theory, Preprint gr-qc/0610159. P. Tillman (University of Pittsburgh) Fedosov Star on Constant Curvature Manifolds 03/15/07 38 / 38

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