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Study in Noncommutative Geometry Inspired Physics

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Study in Noncommutative Geometry Inspired Physics STUDYINNONCOMMUTATIVEGEOMETRYINSPIRED PHYSICS thesis submitted for the degree of doctor of philosophy [sc.] in physics (theoretical) by YENDREMBAMCHAOBADEVI Department of Physics University of Calcutta 2016 Dedicated to all my Teachers in Manipur PUBLICATIONS The thesis is based on the following publications 1. Thermal effective potential in two- and three-dimensional non-commutative spaces Y.C.Devi, K.J.B.Ghosh, B.Chakraborty and F.G.Scholtz, J. Phys. A: Math. Theor. 47 (2014) 025302. 2. Connes distance function on fuzzy sphere and the connection between geometry and statistics Y.C.Devi, S.Prajapat, A.K.Mukhopadhyay, B.Chakraborty, F.G.Scholtz, J.Math.Phys. 56, 041707 (2015). 3. Revisiting Connes’ Finite Spectral Distance on Non-commutative spaces : Moyal plane and Fuzzy sphere Y.C.Devi, A.Patil, A.N.Bose, K.Kumar, B.Chakraborty and F.G.Scholtz arXiv:1608.0527. iii ACKNOWLEDGEMENTS I am thankful to my supervisor Professor Biswajit Chakraborty for his kind patience over my slow progress and his support in many ways. I am thankful to Professor Frederik G Scholtz and Professor A P Balachandran for many insightful discussions.. I am also thankful to all the staff of S N Bose National Centre for Basic Sciences for providing me all the necessary facilities needed to complete this work. I am also thankful to my collaborators: Kumar Jang Bahadur Ghosh, Aritra Narayan Bose, Shivraj Prajapat, Aritra K Mukhopadhyay, Alpesh Patil, Kaushlendra Kumar for useful dis- cussions and arguments. Thanks also goes to Arghya and Subhashish for many useful dis- cussions. Many thanks to all my friends at S N Bose National Centre for Basic Sciences specially Bivas bhaiya, Sandipa and Sayani for giving me healthy atmosphere and their emotional supports during my stay here. Special thanks to Sreeraj T P for helping and supporting me throughout the period of this work. Above all, I would like to thank my family for their support and encouragement and their faith in me without which the thesis would have been impossible. iv CONTENTS 1 introduction1 1.1 Non-commutative/Quantized space-time 1 1.1.1 Non-commutative field theory employing star products 2 1.1.2 Non-commutative quantum mechanics: Operator method 5 1.2 Non-commutative geometry 8 1.3 Plan of the thesis 15 I Non-relativistic quantum physics on Non-commutative spaces 17 2 hilbert-schmidt operator formalism 18 R2 2.1 2D Moyal plane ? [44, 45] 18 2.1.1 Position bases 20 2.2 3 D Moyal space [47] 22 2.3 Fuzzy Sphere [48] 27 3 voros position basis states on 3d moyal space 29 3.1 Review of Symplectic Invariant Uncertainty Relations 29 3.1.1 Variance matrix and Williamson theorem 29 3.2 Computation of Non-commutative Variance matrix 31 4 multi-particle system on 3d moyal space 37 4.1 Two-particle syatem 37 4.2 Multi-particle states 40 ( N ) 4.2.1 Momentum Basis on 3 41 Hq 4.3 Second Quantization 43 4.3.1 Creation and Annihilation operators in twisted momentum basis 44 4.3.2 Creation and Annihilation operators in quasi-commutative mometum basis 45 4.4 Field operators 46 4.5 Two-particle Correlation function 48 4.5.1 Twisted correlation function 48 4.5.2 Quasi-commutative Basis 52 4.5.3 Thermal Effective Potential 52 v contents vi II Spectral distances on Non-commutative spaces 55 5 spectral triple and connes’ spectral distance 56 5.1 Spectral triple method of Non-commutative geometry 56 5.2 Spin geometry 57 5.2.1 Clifford algebras 57 5.2.2 Spin Manifold 58 5.2.3 Spin Connection 59 5.3 Connes’ spectral triple of a compact spin manifold 60 5.4 Spectral distance on Real line R1 62 5.5 Spectral distance on two-point space, a discrete space [107, 109] 63 1 5.6 Spectral distance on the state space CP of M2(C) [82, 110] 64 5.7 Coherent states 67 5.7.1 Moyal plane 68 5.7.2 Fuzzy sphere 68 6 connes’ spectral distance on moyal plane 70 6.1 Eigenspinors of Dirac operator 71 R2 6.2 Spectral Triple for Moyal plane ? and the Spectral distance 72 6.3 Spectral distance on the configuration space of Moyal plane 75 6.3.1 Ball condition with the Eigenspinors of 76 DM 6.3.2 Distance between discrete/harmonic oscillator states 76 6.3.3 Distance between coherent states 79 6.4 Spectral Triple on and the connection of geometry with Statistics 85 Hq 7 spectral distances on fuzzy sphere 88 7.1 Spectral triple on the configuration space of fuzzy sphere 88 Fj 7.1.1 Eigenspinors of Dirac operator [89] 90 D f 7.1.2 Ball condition with Dirac eigenspinors 90 7.1.3 Spectral distance between discrete states 91 7.1.4 Spectral distance between coherent states 96 7.1.5 Lower bound to the spectral distance between infinitesimally separated continuous pure states 97 S2 7.1.6 Spectral distance on the configuration space of fuzzy sphere 1 99 2 S2 7.1.7 Spectral distance on the configuration space of fuzzy sphere 1 103 7.2 Spectral triple and distance on quantum Hilbert space of fuzzy sphere 114 8 conclusion 120 contents vii III Appendices 122 a basic theory of hopf algebra 123 a.1 Algebra [115] 123 a.2 Co-algebra [115] 123 a.3 Bi-algebra [115] 123 a.4 Hopf algebra [115] 123 a.4.1 Co-commutative 124 a.4.2 Almost Co-commutative 124 a.4.3 Quasi-triangular 124 a.4.4 Twisted Hopf algebra 125 b robertson and schrödinger uncertainty relations 126 c some definitions 128 c.1 C∗-algebra [113, 116] 128 c.2 -homomorphism and functional of C -algebra 129 ∗ ∗ c.3 Representations of a C∗-algebra 130 c.4 Hilbert-Schimdt operators 130 c.5 Normal State 131 c.6 Module 131 d construction of dirac operators 133 d.1 Moyal Plane 133 d.2 Fuzzy sphere [88] 134 d.2.1 Dirac operator on S3 and S2 134 S2 d.2.2 Dirac operator on fuzzy sphere ? 139 LISTOFFIGURES Figure 1.1 Pictorial representation of Gelfand Naimark Theorem. 9 q Figure 4.1 Thermal effective potential vs distance for l2 = 0.1 for each case in three-dimension. q 2 2 Note that, in the twisted case, this depends functionally on r = rx + ry and r , ? k in contrast to quasi commutative case, where it depends only on r. 54 Figure 5.1 Computation of Connes’ spectral distance in 1D 62 Figure 7.1 Infinitesimal change on the surface of sphere with respect to the change in m 93 S2 S2 S2 Figure 7.2 Fuzzy spheres, 1 , 1, 3 95 2 2 1 Figure 7.3 Space of Perelomov’s SU(2) coherent states for j = 2 . 100 2 2 Figure 7.4 Infimum corresponding to the plots of M11/(dq) and M22/(dq) vs b1/dq 111 Figure A.1 Commutative diagram showing the axioms on the bi-algebra H that make it Hopf algebra. 124 viii LISTOFTABLES Table 1.1 Dictionary of duality between space and algebra 10 Table 7.1 Data set for various distances corresponding to different angles 114 ix 1 INTRODUCTION One of the most challenging problems persisting since 20th century is the quantum the- ory of gravity. It is expected that it will also unify the four fundamental forces: gravitation, electromagnetic, strong and weak forces. As, quantum mechanics is formulated in terms of operators acting on some Hilbert space realizing operator algebras and gravity, on the other hand, is the theory which describes the geometry and dynamics of the space-time manifold, it is quite conceivable that a future theory of quantum gravity would deal with some gen- eralized geometry containing aspects from both. Non-commutative geometry [1] is such a generalized geometry and is indeed a very promising candidate for a future formulation of quantum gravity. 1.1 non-commutative/quantized space-time The idea of quantization of space-time was proposed by Heisenberg in 1930’s, not much later than the quantization of phase space, to cure the ultraviolet divergences of quantum field theory. However, the first paper [2] on Lorentz invariant quantized space-time was given by Snyder in 1947. In his second paper [3], Snyder obtained the equation of motion for the electromagnetic field on the Lorentz invaraint quantized space-time. But it was Bronstein [4] before Snyder who observed that consideration of both quantum mechanics and Einstein’s general relativity imposes an upper bound on the density of test body while making a precise measurement of gravitational field and hence a fundamental length scale below which the notion of space-time point i.e. event doesn’t make sense. But the success of renormalization techniques rendered the ideas of quantized space-time dor- mant for many years. However, in 1994 [5, 6], Doplicher et.al independently refreshed the idea of quantizing space-time by proposing a Poincare covariant spacetime uncertainty rela- q tions at Planck length scales (l = hG¯ 1.616 10 33 cm), whereh ¯ = 6.582 10 16 eV.s is p c3 ≈ × − × − the reduced Planck constant, G = 6.674 10 8 cm3g 1s 2 is the gravitational constant and × − − − c = 2.997 1010 cm.s 1 is the speed of light in vacuum. This was suggested by quantum × − mechanics and Einstein’s theory of gravity as quantum measurement of a spacetime event with accuracy l need a probing particle with Compton wavelength l . This means par- ∼ p ≤ p 2phc¯ 28 ticle with energy El 2.557 10 eV which will generate so strong a gravitational p ≥ lp ≈ × 1 1.1 non-commutative/quantized space-time 2 2GE field with the Schwarzschild radius R = lp 4.215 10 33 cm > l so that a black s c4 ≈ × − p hole will form and no information will come out.
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