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Dirac Fermions on Rotating Space-Times

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Dirac Fermions on Rotating Space-Times DIRAC FERMIONS ON ROTATING SPACE-TIMES VICTOR EUGEN AMBRUS, PhD thesis Supervisor: Prof. Elizabeth Winstanley School of Mathematics and Statistics University of Sheffield December 2014 Abstract Quantum states of Dirac fermions at zero or finite temperature are investigated using the point-splitting method in Minkowski and anti-de Sitter space-times undergoing rotation about a fixed axis. In the Minkowski case, analytic expressions presented for the thermal expectation values (t.e.v.s) of the fermion condensate, parity violating neutrino current and stress-energy tensor show that thermal states diverge as the speed of light surface (SOL) is approached. The divergence is cured by enclosing the rotating system inside a cylinder located on or inside the SOL, on which spectral and MIT bag boundary conditions are considered. For anti-de Sitter space-time, renormalised vacuum expectation values are calcu- lated using the Hadamard and Schwinger-de Witt methods. An analytic expression for the bi-spinor of parallel transport is presented, with which some analytic expres- sions for the t.e.v.s of the fermion condensate and stress-energy tensor are obtained. Rotating states are investigated and it is found that for small angular velocities Ω of the rotation, there is no SOL and the thermal states are regular everywhere on the space-time. However, if Ω is larger than the inverse radius of curvature of adS, an SOL forms and t.e.v.s diverge as inverse powers of the distance to it. ii Contents Abstract ii Table of contents iii List of figures viii List of tables xvi Preface xvii 1 Introduction 1 2 General concepts 4 2.1 The quantised scalar field . 4 2.1.1 Second quantisation . 4 2.1.2 Stress-energy tensor . 6 2.1.3 Fock space . 7 2.1.4 Finite temperature field theory . 10 2.1.5 Green’s functions . 11 2.2 The quantised Dirac field . 14 2.2.1 Gamma matrices . 14 2.2.2 Second quantisation . 16 2.2.3 Stress-energy tensor and conserved current . 17 2.2.4 Fock space . 19 2.2.5 Field theory at finite temperature . 21 2.2.6 Green functions . 21 2.3 Summary . 24 iii iv CONTENTS 3 Minkowski space-time 25 3.1 Space-time characteristics . 25 3.2 Scalar field theory in cylindrical coordinates . 26 3.2.1 Modes in cylindrical coordinates . 26 3.2.2 Thermal expectation values . 28 3.3 Dirac fermions in cylindrical coordinates . 32 3.3.1 Modes in cylindrical coordinates . 33 3.3.2 Dirac’s equation using a cylindrical tetrad . 37 3.3.3 Finite temperature expectation values . 40 3.4 Summary . 47 4 Rotating Minkowski space-time 48 4.1 Space-time characteristics . 49 4.2 Scalar field theory in a rotating background . 53 4.2.1 Rigidly rotating modes . 53 4.2.2 Rigidly rotating thermal states . 54 4.2.3 Summary . 65 4.3 Polarised rotating fermions . 65 4.3.1 Construction of modes . 65 4.3.2 Thermal expectation values . 67 4.3.3 Numerical results . 78 4.3.4 Summary . 81 4.4 Chapter summary . 82 5 Bounded rotating Minkowski space 84 5.1 Scalars in a cylinder . 85 5.1.1 Modes and field operator . 85 5.1.2 Bounded rigidly rotating thermal states . 87 5.1.3 Casimir effect . 88 5.1.4 Casimir divergence near the boundary . 91 5.1.5 Summary . 95 5.2 Dirac fermions obeying spectral boundary conditions . 96 5.2.1 Boundary conditions and mode solutions . 96 CONTENTS v 5.2.2 Thermal expectation values . 100 5.2.3 Casimir effect . 103 5.2.4 Casimir divergence near the boundary . 110 5.3 Dirac fermions obeying MIT bag boundary conditions . 115 5.3.1 Boundary conditions and mode solutions . 115 5.3.2 Thermal expectation values . 123 5.3.3 Casimir effect . 134 5.3.4 Casimir divergence near the boundary . 140 5.4 Estimates of the energy density . 145 5.4.1 Boundary inside the SOL . 145 5.4.2 Boundary outside the speed of light surface . 152 5.4.3 Summary . 163 6 Quasi-Euclidean QFT on rotating space 165 6.1 Equivalence to Lorentzian formulation . 166 6.1.1 Quasi-Euclidean formulation of non-rotating thermal distri- butions . 166 6.1.2 Quasi-Euclidean formulation of unbounded rotating thermal distributions . 168 6.1.3 Quasi-Euclidean formulation of bounded rotating thermal dis- tributions . 169 6.2 Analysis of bounded thermal states near the SOL . 172 6.3 Summary . 175 7 Anti-de Sitter space 176 7.1 Geometric structure . 176 7.1.1 Metric, tetrad and connection . 176 7.1.2 Geodesic structure . 180 7.1.3 Bi-vector of parallel transport . 183 7.1.4 Bi-spinor of parallel transport . 184 7.1.5 Coincidence limit expansions . 190 7.1.6 Generators of isometries and conserved operators . 195 7.2 Mode solutions of the Dirac equation . 197 vi CONTENTS 7.3 Two-point functions . 204 7.3.1 Using mode sums . 205 7.3.2 Small distance behaviour of two-point functions . 211 7.3.3 Using the spinor parallel propagator . 213 7.4 Renormalised vacuum expectation values . 214 7.4.1 Schwinger-de Witt method . 215 7.4.2 Hadamard renormalisation . 217 7.5 Thermal expectation values . 224 7.5.1 Using the Feynman propagator . 224 7.5.2 Mode sum approach . 228 7.5.3 Numerical results . 231 7.6 Summary . 232 8 Rotating fermions on adS 235 8.1 Space-time characteristics . 235 8.2 The Dirac equation in rotating coordinates . 237 8.3 Thermal expectation values . 238 8.3.1 Mode sums . 239 8.3.2 The geometric approach . 247 8.4 Numerical results . 257 8.5 Geometric approach on Minkowski . 264 8.5.1 Minkowski propagator and thermal expectation values . 264 8.5.2 Rotating thermal states . 266 8.5.3 Summary . 270 8.6 Summary . 271 9 Conclusion 272 A Properties of Bessel functions 274 A.1 Definition . 274 A.2 Asymptotic forms . 277 A.3 Orthogonality relations and some integrals . 280 A.4 Summation formulae . 281 CONTENTS vii B Jacobi Polynomials 282 C Spherical harmonics 284 C.1 Properties of Legendre polynomials . 284 C.2 Properties of associated Legendre functions . 285 C.3 Properties of spherical harmonics . 286 C.4 Useful summation formulae . 288 m C.5 Contractions of the ψ 1 bi-spinors . 289 j± 2 C.5.1 Contractions of ψ± ........................ 289 C.5.2 Contractions of ψ± sandwiching a σ ............... 290 D Gauss’ hypergeometric function 294 References 296 List of Figures P 3.1 The density ρ, pressure P and equation of state ρ for a thermal distri- bution of scalar particles are plotted as functions of the mass µ of the quanta (on the left) and as functions of the logarithm of the temperature T = β−1, on the right. The solid black curve shows numerical results obtained by integrating (3.2.25) while the dashed blue and purple curves show the asymptotic expressions (3.2.29) and (3.2.30) for large and small values of βµ, respectively. ......................... 32 3.2 On the left hand side, the density ρ, pressure P and equation of P state ρ per degree of freedom (4 for Dirac fermions) for a thermal distribution of fermions are plotted as functions of the mass µ of the quanta with β = 1 (first two plots) and as functions of the logarithm of the temperature with µ = 1 (last plot). The solid black curve shows numerical results, the dashed blue curve is the large µ approximation and the dashed purple curve is the small µ approximation. The plots on the right compare numerical results for the density, pressure and equation of state of fermions (blue) and bosons (red). The values for the pressure and density are always higher for bosons, but the equation of state decreases with temperature slower for fermions. 46 4.1 The filled shapes represent anti-particle modes while the unfilled shapes represent particle states. The circles represent Vilenkin’s quantisation, which defines particle modes as modes with positive energy E. Iyer’s quantisation is represented by squares, particle modes having Ee ≥ 0. The two schemes differ in regions I, where Iyer-type antiparticles are Vilenkin- type particles and vice-versa in region III. ................. 67 viii LIST OF FIGURES ix 4.2 The curves show the contributions Ttˆtˆ[m] made by each value of m (to- gether with −m − 1) to h: Ttˆtˆ(x):I iβ for a rigidly rotating Dirac field with µ = 0 (thin solid black lines) and µ = 2Ω (thick lines) at inverse tempera- tures (a) βΩ = 2.0 and (b) βΩ = 0.8, at four distances from the rotation axis. The value of Ttˆtˆ[m] increases up to a maximum value at mρ after which it decreases monotonically to 0. The value mρ increases with the distance from the rotation axis, which is why the further the point is, the more values of m must be considered. However, mρ does not seem to depend on β or µ. The curves terminate according to the algorithm described in the main text. ......................... 78 4.3 The logarithm of the fermion condensate h: ψψ :I iβ (first line), neu- trino current h: J : izˆ and − 1 h: T : i (bottom line) against ρΩ ν I ρΩ ϕˆtˆ I β 1 on the left and ln(1/ε) on the right. The prefactor − ρΩ has been in- troduced for to render the argument − 1 h: T : i of the logarithm ρΩ ϕˆtˆ I β positive and non-zero on the rotation axis.
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