FORCES OF THE QUANTUM VACUUM

Transcription

1 FORCES OF THE QUANTUM VACUUM An Introduction to Casimir Physics

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3 FORCES OF THE QUANTUM VACUUM An Introduction to Casimir Physics Editors William Simpson Weizmann Institute of Science, Israel Ulf Leonhardt Weizmann Institute of Science, Israel World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

4 Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore USA office: 27 Warren Street, Suite , Hackensack, NJ UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Forces of the quantum vacuum : an introduction to Casimir physics / editors, William M.R. Simpson, Ulf Leonhardt, The Weizmann Institute of Science, Israel. pages cm Includes bibliographical references and index. ISBN (hardcover : alk. paper) -- ISBN (pbk. : alk. paper) 1. Quantum theory. 2. Casimir effect. I. Simpson, William M. R. II. Leonhardt, Ulf, 1965 QC680.F '33--dc British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover design by Tanya Skrynnik Photograph of the pier in St Andrews, Scotland, by Spencer Bentley ( Proofread by Mrs Rachel Parks, St Andrews Copyright 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore

5 William Simpson dedicates this book to his family, and most especially his brother, John

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7 Contents Foreword 1 Introduction 3 Chapter 1 Normal mode quantum electrodynamics: the quantum vacuum and its consequences 7 1. Wave-particle duality Light as a particle Light as a wave The photon Classical electrodynamics in free space Normal modes in free space Normal modes inside a cuboid cavity Normal modes: general remarks Electromagnetic waves Energy of the electromagnetic field Normal mode quantum electrodynamics Basic elements of quantum theory The quantised electromagnetic field Quantum statistics and Fock states Energy of the electromagnetic field Quantum dynamics of the electromagnetic field The quantum vacuum Vacuum fluctuations Thermal fluctuations Quantum vacuum effects in free space Atom field interactions Lamb shift Spontaneous decay Electron magnetic moment Van der Waals potential Quantum vacuum effects near boundaries vii

8 viii Contents 6.1. Casimir Polder potential Purcell effect Electron magnetic moment Van der Waals potential The Casimir force Perfectly conducting plates Conducting and permeable plates Thermal effects The quantum vacuum revisited Vacuum energy Changes in vacuum energy Virtual photons Normal mode QED: discussion Advantages Limitations Problems Bibliography Chapter 2 Van der Waals and Casimir Polder dispersion forces Introduction Van der Waals interaction Quasistatic analysis Electrodynamic analysis Casimir-Polder atom-surface interaction Quasistatic analysis Electrodynamic analysis A few comments Non-additivity of dipolar interactions Example: non-additivity in atom-surface interaction Multiple scattering as the origin of non-additivity Problems Bibliography Chapter 3 The Casimir stress in real materials Dispersion and dissipation in real media Dipole model of the optical response The Kramers Kronig relations Generalised mode decomposition Open quantum systems and Langevin equations Linear fluctuation-dissipation theorem The stress tensor in vacuum Lifshitz theory of dispersion forces Rytov s theory of fluctuating electromagnetic fields Lifshitz s approach to the Casimir force

9 Contents ix Green function approach Thermal contribution to Casimir Lifshitz forces Problems Bibliography Chapter 4 Macroscopic QED and vacuum forces Preliminary remarks An introduction to macroscopic QED Macroscopic QED in one dimension Macroscopic QED in three dimensions Vacuum forces between moving bodies Moving bodies in 1D macroscopic QED Computing classical forces Quantum theory of radiation pressure The vacuum force A simple case of quantum friction Moving bodies in 3D macroscopic QED Quantum friction between sliding plates Problems Bibliography Chapter 5 Measuring Casimir phenomena General experimental considerations Representative experimental results Early results Beginning of the modern Casimir era Effect of material properties Effects of temperature Recent directions Techniques for measurement Minimum detectable interaction Measurement of the interaction Static method Dynamic method Measurement of the separation Calibration of the apparatus Comparison with theory Problems Bibliography Chapter 6 Casimir forces at the cutting edge Experimental issues in Casimir physics Drude or plasma? Patch potentials

10 x Contents 39. Theoretical questions in Casimir physics Repulsive shells and Casimir s electron Divergence of the Casimir stress Maxwell s fisheye and renormalisation The Casimir effect and cosmology Bibliography Further reading 247 Appendix: Regularisation techniques 251 A. Point-splitting regularisation B. Dimensional regularisation C. Zeta-function regularisation Index 257 About the authors 263

11 Foreword Peter Milonni In 1947 Lamb and Retherford reported the measurement of a difference in the 2s 1/2 and 2p 1/2 energy levels of the hydrogen atom, in contrast to the equal energies calculated using the Dirac equation for the electron in the Coulomb field of the nucleus. This difference comes predominantly from the interaction of the electron with the zero-point electromagnetic field. The energies calculated by standard methods were divergent, owing to the fact that all field frequencies ν and zero-point energies 1 2 hν contribute to the interaction. The Lamb shift not only stimulated the development of new renormalisation techniques; it also inspired, less famously, consideration of Some observable consequences of the quantum-mechanical fluctuations of the electromagnetic field, the title of a paper published in 1948 by Theodore Welton. On May 29, 1948, six days before Welton s paper was received by Physical Review, Hendrik Casimir communicated to the Royal Netherlands Academy of Arts and Sciences his paper On the attraction between two perfectly conducting plates. Although he deduced this attraction from the change, caused by the plates, in the (infinite) zero-point energy of the electromagnetic field, it would be false to the facts to think that Casimir took a cue from the Lamb-Retherford work and its implications for the quantum fluctuations and zero-point energy of the field. In reply to a query of mine many years ago, Casimir wrote that he was not at all familiar with the work of Welton and others and that he went [his] own, somewhat clumsy way. He was, however, influenced by a discussion with Niels Bohr concerning the simple expression for the retarded van der Waals interaction that Casimir and Dirk Polder had derived (and submitted for publication in February 1947). Casimir remarked that Bohr mumbled something about zero-point energy. That was all, but it put me on a new track. In the introduction to his 1948 paper he wrote that he had recently shown that the retarded van der Waals interaction could be derived by consideration of a change in zero-point field energy, and that he would now take the same approach to calculate the force between two perfectly conducting plates. Casimir s prediction that there is an attraction between the plates due to the change in zero-point electromagnetic energy was, to say the least, a bold one. Although the concept of zero-point energy played a small role early on in the development of quantum theory, the notion that zero-point energy and quantum fluctu- 1

12 2 Foreword ations of vacuum fields should be taken seriously was not very popular. One reads in Pauli s Nobel Lecture (1945), for example, that... the zero-point energy of the vacuum derived from the quantised field becomes infinite, a result which is directly connected with the fact that the system considered has an infinite number of degrees of freedom. It is clear that this zero-point energy has no physical reality... And thirty years after Casimir s paper, Julian Schwinger called the Casimir force one of the least intuitive consequences of quantum electrodynamics. Casimir s papers on the retarded van der Waals interaction and the force between two conducting plates played a role in the interpretation of experiments carried out, most notably in the Soviet Union, in the 1950s. Both papers are cited in the influential paper by Lifshitz (1955) on The Theory of Molecular Attractive Forces between Solids. Lifshitz calculated the interaction between two dielectric media, in thermal equilibrium and occupying half-spaces with plane-parallel boundaries, considering the interaction to come about through... the fluctuating electromagnetic field. Casimir s results were rederived as special cases of Lifshitz s general expression for the force per unit area between the dielectrics. A few years later Sparnaay measured an attractive force between conductors that did not contradict Casimir s theoretical prediction. It was not until the mid-1990s that the Casimir force was very accurately measured, a few years after the first experimental confirmation of the long-range interaction between a neutral atom and a conducting surface that Casimir and Polder derived in their paper on the retarded van der Waals interaction. Aside from its basic importance as an interaction between any two bodies, and more generally as a consequence of quantised fields, the Casimir (or Casimir Lifshitz) effect may take on increasingly practical importance. One reason was recognised more than half a century ago by Feynman (1959) in a talk on the physics involved in making micromachines: there is the problem that materials stick together by the molecular (van der Waals) attractions. In more recent years considerable efforts have been devoted to controlling this stiction. Casimir s 1948 paper three pages long! was written with a clarity and a focus on essential physics that was characteristic of papers by many prominent physicists of the period. The more recent literature on Casimir physics, unfortunately, seldom emulates Casimir s clarity, and it often presumes considerable familiarity with quantum electrodynamics, many-body theory, dyadic Green functions, different approaches to open quantum systems, and other theoretic tools, let alone sophisticated experimental techniques. This slender book, in contrast, is a refreshingly readable introduction to Casimir physics. The contributing experts have introduced the concepts and techniques needed for students and nonspecialists to learn the essential physics of van der Waals and Casimir forces, to appreciate what is and what is not well understood, and perhaps to make contributions of their own to this fascinating field. P. W. Milonni September 2014