Tai Lieu Chat Luong LIGHT The Physics of the Photon LIGHT The Physics of the Photon Ole Keller Aalborg University, Denmark Cover image: Courtesy of Esben Hanefelt Kristensen, based on a painting entitled “A Wordless Statement.” Taylor & Francis Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor & Francis Group, LLC Taylor & Francis is an Informa business No claim to original U.S Government works Printed on acid-free paper Version Date: 20140428 International Standard Book Number-13: 978-1-4398-4043-6 (Hardback) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com In memory of my mother, Cecilie Marie Keller Contents Preface xiii Acknowledgments xix About the author xxi I Classical optics in global vacuum Heading for photon physics Fundamentals of free electromagnetic fields 2.1 Maxwell equations and wave equations 2.2 Transverse and longitudinal vector fields 2.3 Complex analytical signals 2.4 Monochromatic plane-wave expansion of the electromagnetic field 2.5 Polarization of light 2.5.1 Transformation of base vectors 2.5.2 Geometrical picture of polarization states 2.6 Wave packets as field modes 2.7 Conservation of energy, moment of energy, momentum, and angular momentum 2.8 Riemann–Silberstein formalism 2.9 Propagation of analytical signal 7 10 13 14 14 15 18 Optics in the special theory of relativity 3.1 Lorentz transformations and proper time 3.2 Tensors 3.3 Four-vectors and -tensors 3.4 Manifest covariance of the free Maxwell equations 3.5 Lorentz transformation of the (transverse) electric and magnetic fields Duality 3.6 Lorentz transformation of Riemann–Silberstein vectors Inner-product invariance 27 27 30 31 33 II Light rays and geodesics Maxwell theory in general relativity 21 22 24 35 38 39 The light-particle and wave pictures in classical physics 41 Eikonal theory and Fermat’s principle 5.1 Remarks on geometrical optics Inhomogeneous vacuum 5.2 Eikonal equation Geometrical wave surfaces and rays 5.3 Geodetic line: Fermat’s principle 45 45 47 52 vii viii Contents Geodesics in general relativity 6.1 Metric tensor Four-dimensional Riemann space 6.2 Time-like metric geodesics 6.3 The Newtonian limit: Motion in a weak static gravitational field 6.4 Null geodesics and “light particles” 6.5 Gravitational redshift Photon in free fall 55 55 56 59 61 62 The 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 67 67 69 70 71 73 74 76 76 Electromagnetic theory in curved space-time 8.1 Vacuum Maxwell equations in general relativity 8.2 Covariant curl and divergence in Riemann space 8.3 A uniform formulation of electrodynamics in curved and flat space-time 8.3.1 Maxwell equations with normal derivatives 8.3.2 Maxwell equations with E, B, D, and H fields 8.3.3 Microscopic Maxwell–Lorentz equations in curved space-time 8.3.4 Constitutive relations in curved space-time 8.3.5 Remarks on the constitutive relations in Minkowskian space 8.3.6 Permittivity and permeability for static metrics 8.4 Permittivity and permeability in expanding universe 8.5 Electrodynamics in potential description Eikonal theory and null geodesics 8.6 Gauge-covariant derivative 79 79 80 81 81 83 84 85 87 88 89 91 95 III Photon wave mechanics 97 The elusive light particle 99 space-time of general relativity Tensor fields Covariant derivative Parallel transport Riemann curvature tensor Algebraic properties of the Riemann curvature tensor Einstein field equations in general relativity Metric compatibility Geodesic deviation of light rays 10 Wave mechanics based on transverse vector potential 10.1 Gauge transformation Covariant and noncovariant gauges 10.2 Tentative wave function and wave equation for transverse photons 10.3 Transverse photon as a spin-1 particle 10.4 Neutrino wave mechanics Massive eigenstate neutrinos 105 105 107 110 113 11 Longitudinal and scalar photons Gauge and near-field light quanta 11.1 L- and S-photons Wave equations 11.2 L- and S-photon neutralization in free space 11.3 NF- and G-photons 11.4 Gauge transformation within the Lorenz gauge 119 119 120 122 124 Contents 12 Massive photon field 12.1 Proca equation 12.2 Dynamical equations for E and A 12.3 Diamagnetic interaction: Transverse photon mass 12.4 Massive vector boson (photon) field 12.5 Massive photon propagator ix 13 Photon energy wave function formalism 13.1 The Oppenheimer light quantum theory 13.2 Interlude: From spherical to Cartesian representation 13.3 Photons and antiphotons: Bispinor wave functions 13.4 Four-momentum and spin of photon wave packet 13.5 Relativistic scalar product Lorentz-invariant integration energy shell 127 127 129 130 132 136 on the 143 143 146 150 153 155 IV Single-photon quantum optics in Minkowskian space 159 14 The photon of the quantized electromagnetic field 161 15 Polychromatic photons 15.1 Canonical quantization of the transverse electromagnetic field 15.2 Energy, momentum, and spin operators of the transverse field 15.3 Monochromatic plane-wave photons Fock states 15.4 Single-photon wave packets 15.5 New T-photon “mean” position state 15.6 T-photon wave function and related dynamical equation 15.7 The non-orthogonality of T-photon position states 16 Single-photon wave packet correlations 16.1 Wave-packet basis for one-photon states 16.2 Wave-packet photons related to a given t-matrix 16.3 Integral equation for the time evolution operator in the interaction picture 16.4 Atomic and field correlation matrices 16.5 Single-photon correlation matrix: The wave function fingerprint 165 165 168 171 173 177 179 181 183 183 184 186 189 194 17 Interference phenomena with single-photon states 197 17.1 Wave-packet mode interference 197 17.2 Young-type double-source interference 198 17.3 Interference between transition amplitudes 201 17.4 Field correlations in photon mean position state 201 17.4.1 Correlation supermatrix 202 17.4.2 Relation between the correlation supermatrix and the transverse photon propagator 203 18 Free-field operators: Time evolution and commutation relations 18.1 Maxwell operator equations Quasi-classical states 18.2 Generalized Landau–Peierls–Sudarshan equations 18.3 Commutation relations 18.3.1 Commutation relations at different times (τ 6= 0) 18.3.2 Equal-time commutation relations 205 205 207 208 209 210 = ζk αjk βjk + k X ζl α∗jl βjl l X k ! ζk |αjk |2 + ! X ∗ ζk βjk βjk k X k ζk |βjk |2 ! ! X ζl αjl α∗jl l Finally, it thus appears that the squared standard deviation is given by X X X ∗ (∆ng,i ) = ζj αij βij ζj |αij |2 ζj |βij |2 +