A new regularization procedure for calculating the Casimir energy By Bahman Ghadirian A thesis submitted as part of the requirement for the degree of Doctor of Philosophy University of Western Sydney 2008 Acknowledgements I would like to thank my supervisor, Dr Reynaldo Castillo, for the opportunity to work with him and for his invaluable help and supervision throughout this project. I would also like to thank my family specially my wife Roya for her continued support and encouragement throughout the time of my candidature. Statement of Authentication The work presented in this thesis is, to the best of my knowledge and belief, original e cept as acknowledged in the te t. I hereby declare that I have not submitted this material, either in full or in part, for a degree at this or any other institution. …………………………………………………………………………… Contents 1. Introduction......................................................................................................1 2. Linear Response Theory ................................................................................14 2.1. Quantum mechanics ................................................................................15 2.1.1. Wave functions................................................................................. 15 2.1.2. Eigenvalues and Eigenfunctions........................................................ 17 2.1.3. Linear operator ................................................................................. 17 2.1.4. Hermitian and Unitary operators....................................................... 19 2.1.5. Hilbert space..................................................................................... 19 2.1.6. Inner product and Norm.................................................................... 20 2.1.7. Expectation value ............................................................................. 21 2.1.8. Classical equation of motion............................................................. 22 2.1.9. Dirac’s compact notation.................................................................. 24 2.1.10. Transformation................................................................................. 25 2.1.11. Time-evolutions................................................................................ 26 2.2. Thermodynamics.....................................................................................29 2.2.1. The first law of thermodynamics....................................................... 29 2.2.2. Entropy............................................................................................. 30 2.2.3. Reversible and irreversible processes................................................ 33 2.2.4. The second law of thermodynamics .................................................. 34 2.2.5. The free energy functions ................................................................. 35 2.2.6. The Gibbs distribution and its free energy......................................... 36 2.3. Linear response theory.............................................................................39 2.3.1. Evolution operator............................................................................ 39 2.3.2. External perturbation ........................................................................ 41 2.3.3. Generalized susceptibility................................................................. 43 2.3.4. Thermal average............................................................................... 46 2.3.5. Correlation function.......................................................................... 47 2.3.6. Spectral function............................................................................... 49 2.3.7. Different forms of the Green functions.............................................. 50 2.3.8. Connections between various Green functions.................................. 56 2.3.9. Kramers-Kronig relation................................................................... 57 2.3.10. Dissipation........................................................................................ 58 2.3.11. Fluctuation-Dissipation theorem ....................................................... 59 3. Kubo-Martin-Schwinger (KMS) State .......................................................... 61 3.1. K.M.S Condition.....................................................................................62 3.2. Many-body problem................................................................................65 3.2.1. Microscopic property; Green’s function............................................ 65 4. Field Quantization..........................................................................................77 4.1. Canonical commutation relation ..............................................................79 4.2. Scalar field quantization ..........................................................................80 4.3. Complex scalar field quantization............................................................81 4.4. Fermion (Dirac) field quantization...........................................................83 4.5. Electromagnetic field quantization...........................................................88 4.6. Propagators .............................................................................................92 - i - 4.6.1. Propagator of a scalar field ............................................................... 93 4.6.2. Propagator of a Dirac field................................................................ 96 4.6.3. Propagator of the electromagnetic field............................................. 98 4.7. S-matrix ................................................................................................100 4.8. Wick’s theorem.....................................................................................103 4.9. Divergent results in quantum field theory ..............................................105 4.9.1. Regularization ................................................................................ 106 4.9.2. Renormalization ............................................................................. 108 5. Casimir Effect .............................................................................................. 110 5.1. Casimir effect in various field of physics...............................................111 5.2. Vacuum fluctuation...............................................................................112 5.2.1. An alternative look at the vacuum fluctuations................................ 112 5.2.2. Quantum Field Theory and vacuum fluctuation............................... 115 5.3. A simple model.....................................................................................118 5.4. Field energy ..........................................................................................121 5.5. Green’s function method for calculating the Casimir Force....................124 5.5.1. Casimir force on conducting parallel-plates .................................... 125 5.5.2. Casimir force on a conducting spherical shell ................................. 128 5.5.3. Casimir force on a conducting cylindrical shell............................... 138 5.5.4. Comparison of the Casimir force for different geometrical boundaries145 5.6. Casimir force as a limiting case of Van der Waals interaction................146 5.7. Zeta-function method for calculation of the Casimir force .....................154 5.7.1. Riemann zeta-function.................................................................... 155 5.7.2. Generalized zeta-function ............................................................... 156 5.7.3. The heat equation............................................................................ 157 5.7.4. Zeta function expansion.................................................................. 161 5.7.5. Zeta functional approach to the vacuum energy .......................... 161 5.7.6. An advanced look at the Casimir energy ......................................... 164 5.7.7. Various behaviours of the Casimir effect with different geometrical boundaries...................................................................................................... 166 5.7.8. One-loop effective action................................................................ 168 5.8. Casimir effect in the Kaluza-Klein model..............................................171 6. A New Approach to the Casimir Energy for a Massive Scalar Field......... 178 6.1. Casimir energy for a scalar field............................................................179 6.2. Casimir energy for a massive scalar field...............................................182 6.3. A new technique....................................................................................184 6.3.1. Casimir energy and renormalization................................................ 191 Conclusion............................................................................................................ 192 References ............................................................................................................ 197 Bibliography......................................................................................................... 202 Appendices ........................................................................................................... 204 Appendix A: Bessel’s functions ........................................................................204