引力的量子效應(yīng)導(dǎo)論

前言

This book is an expanded and reorganized version of the lecture notes for a course taught at the Ludwig-Maximilians University, Munich, in the spring semester of 2003. The course is an elementary introduction to the basic concepts of quantum field theory in classical backgrounds. A certain level of familiarity with general relativity and quantum mechanics is required, although .many of the necessary concepts are introduced in the text.The audience consisted of advanced undergraduates and beginning graduate students. There were 11 three-hour lectures. Each lecture was accompanied by exercises that were an integral part of the exposition and encapsulated longer but straightforward calculations or illustrative numerical results. Detailed solutions were given for all the exercises. Exercises marked by an asterisk （*） are more difficult or cumbersome.The book covers limited but essential material: quantization of free scalar fields; driven and time-dependent harmonic oscillators; mode expansions and Bogolyubov transformations; particle creation by classical backgrounds; quantum scalar fields in de Sitter spacetime and the growth of fluctuations; the Unruh effect;Hawking radiation; the Casimir effect; quantization by path integrals; the energy-momentum tensor for fields; effective action and backreaction; regularization of functional determinants using zeta functions and heat kernels. Topics such as quantization of higher-spin fields or interacting fields in curved spacetime, direct renormalization of the energy-momentum tensor, and the theory of cosmological perturbations are left out.The emphasis of this course is primarily on concepts rather than on compu-tational results. Most of the required calculations have been simplified to the barest possible minimum that still contains all relevant physics. For instance,only free scalar fields are considered for quantization; background spacetimes are always chosen to be conformally flat; the Casimir effect, the Unmh effect,and the Hawking radiation are computed for massless scalar fields in suitable 1 + 1-dimensional spacetimes. Thus a fairly modest computational effort suffices to explain important conceptual issues such as the nature of vacuum and parti-cles in curved spacetimes, thermal effects of gravitation, and backreaction. This should prepare students for more advanced and technically demanding treatments suggested below.The authors are grateful to Josef GalBner and Matthew Parry for discussions and valuable comments on the manuscript. Special thanks are due to Alex Vikman who worked through the text and prompted important revisions, and to Andrei Barvinsky for his assistance in improving the presentation in the last chapter.The entire book was typeset with the excellent LyX and TEX document prepa-ration system on computers running Debian GNU/Linux. We wish to express our gratitude to the creators and maintainers of this outstanding free software.Suggested literatureThe following books offer a more extensive coverage of the subject and can be studied as a continuation of this introductory course.N. D. BIRRELL and P. C. W. DAVIES, Quantum Fields in Curved Space （Cambridge University Press, 1982）.S. A. FULLING, Aspects of Quantum Field Theory in Curved Space-Time （Cambridge University Press, 1989）.

內(nèi)容概要

本書系統(tǒng)地介紹了引力中的量子場論方法，適合用作理論天體物理，宇宙學(xué)，粒子物理和超弦理論等專業(yè)的本科生和研究生教材。本書首先介紹了量子場論中的一般概念，重點(diǎn)講解了利用量子場論方法研究引力，同時(shí)還介紹了基本且必要的不斷擴(kuò)張宇宙中的場量子化和膨脹時(shí)空中的量子漲落。此外，本書還詳細(xì)討論了Casimir效應(yīng)，Unruh效應(yīng)和霍金效應(yīng)，并介紹了計(jì)算外部重力場中量子系統(tǒng)的反向反應(yīng)的有效作用量。作者從最初的基本原理出發(fā)，經(jīng)過演繹得出最終結(jié)果，并對最終結(jié)果做詳細(xì)的解釋，幫助讀者對該領(lǐng)域建立完整的物理圖像。本書內(nèi)容全面，講解深刻，附有習(xí)題及其答案。閱讀本書只需要基本的廣義相對論知識。前言；(第一部分)正則量子化和粒子產(chǎn)生：概述：量子場；回顧：經(jīng)典和量子理論；受迫諧振子；從諧振子到場；回顧：經(jīng)典場；擴(kuò)張宇宙中的量子場；Sitter宇宙中的量子場；Unruh效應(yīng)；Hawking效應(yīng)，黑洞熱力學(xué)；Casimir效應(yīng)；（第二部分）路徑積分和真空極化：路徑積分；有效作用量；熱核計(jì)算；從有效作用量導(dǎo)出的結(jié)論；附錄1：數(shù)學(xué)補(bǔ)充；附錄2：反向反應(yīng)的有效作用量推導(dǎo)；附錄3：模展開要點(diǎn)；附錄4：習(xí)題答案；索引?！　∽x者對象：理論天體物理，宇宙學(xué)，粒子物理和超弦理論等專業(yè)的高年級本科生、研究生和相關(guān)領(lǐng)域的科研人員。

書籍目錄

Preface Part I Canonical quantization and particle production 　1 Overview: a taste of quantum fields 　　1.1 Classical field 　　1.2 Quantum field and its vacuum state 　　1.3 The vacuum energy 　　1.4 Quantum vacuum fluctuations 　　1.5 Particle interpretation of quantum fields 　　1.6 Quantum field theory in classical backgrounds 　　1.7 Examples of particle creation 　2 Reminder: classical and quantum theory 　　2.1 Lagrangian formalism 　　2.1.1 Functional derivatives 　　2.2 Hamiltonian formalism 　　2.3 Quantization of Hamiltonian systems 　　2.4 Hilbert spaces and Dirac notation 　　2.5 Operators, eigenvalue problem and basis in a Hilbert space 　　2.6 Generalized eigenvectors and basic matrix elements 　　2.7 Evolution in quantum theory 　3 Driven harmonic oscillator 　　3.1 Quantizing an oscillator 　　3.2 The "in" and "out" states 　　3.3 Matrix elements and Green's functions 　4 From harmonic oscillators to fields 　　4.1 Quantum harmonic oscillators 　　4.2 From oscillators to fields 　　4.3 Quantizing fields in a flat spacetime 　　4.4 The mode expansion 　　4.5 Vacuum energy and vacuum fluctuations 　　4.6 The Schr'odinger equation for a quantum field 　5 Reminder: classical fields 　　5.1 The action functional 　　5.2 Real scalar field and its coupling to the gravity 　　5.3 Gauge invariance and coupling to the electromagnetic field 　　5.4 Action for the gravitational and gauge fields 　　5.5 Energy-momentum tensor 　6 Quantum fields in expanding universe 　　6.1 Classical scalar field in expanding background 　　6.1.1 Mode expansion 　　6.2 Quantization 　　6.3 Bogolyubov transformations 　　6.4 Hilbert space; "a- and b-particles" 　　6.5 Choice of the physical vacuum 　　6.5.1 The instantaneous lowest-energy state 　　6.5.2 Ambiguity of the vacuum state 　　6.6 Amplitude of quantum fluctuations 　　6.6.1 Comparing fluctuations in the vacuum and excited states 　　6.7 An example of particle production 　7 Quantum fields in the de Sitter universe 　　7.1 De Sitter universe 　　7.2 Quantization 　　7.2.1 Bunch-Davies vacuum 　　7.3 Fluctuations in inflationary universe 　8 Unruh effect 　　8.1 Accelerated motion 　　8.2 Comoving frame of accelerated observer 　　8.3 Quantum fields in inertial and accelerated frames 　　8.4 Bogolyubov transformations 　　8.5 Occupation numbers and Unmh temperature 　9 Hawking effect. Thermodynamics of black holes 　　9.1 Hawking radiation 　　9.1.1 Schwarzschild solution 　　9.1.2 Kruskal-Szekeres coordinates 　　9.1.3 Field quantization and Hawking radiation 　　9.1.4 Hawking effect in 3 + 1 dimensions 　　9.2 Therroodynamics of black holes 　　9.2.1 Laws of black.hole thermodynamics 　10 The Casimir effect 　　10.1 Vacuum energy betw.een plates 　　10.2 Regularization and renormalization Part II Path integrals and vacuum polarization 　11 Path integrals 　　11.1 Evolution operator. Propagator 　　11.2 Propagator as a path integral 　　11.3 Lagrangian path integrals 　　11.4 Propagators for free particle and harmonic oscillator 　　11.4.1 Free particle 　　11.4.2 Quadratic potential 　　11.4.3 Euclidean path integral 　　11.4.4 Ground state as a path integral 　12 Effective action 　　12.1 Driven harmonic oscillator (continuation) 　　12.1.1 Green's functions and matrix elements 　　12.1.2 Euclidean Green's function 　　12.1.3 Introducing effective action 　　12.1.4 Calculating effective action for a driven oscillator 　　12.1.5 Matrix elements 　　12.1.6 The effective action "recipe" 　　12.1.7 Backreaction 　　12.2 Effective action in external gravitational field 　　12.2.1 Euclidean action for scalar field 　　12.3 Effective action as a functional determinant 　　12.3.1 Reformulation of the eigenvalue problem 　　12.3.2 Zeta function 　　12.3.3 Heat kernel 　13 Calculation of heat kernel 　　13.1 Perturbative expansion for the heat kernel 　　13.1.1 Matrix elements 　　13.2 Trace of the heat kernel 　　13.3 The Seeley-DeWitt expansion 　14 Results from effective action 　　14.1 Renormalization of the effective action 　　14.2 Finite terms in the effective action 　　14.2.1 EMT from the Polyakov action 　　14.3 Conformal anomaly Appendix 1 Mathematical supplement 　A1.1 Functionals and distributions (generalized functions) 　A1.2 Green's functions, boundary conditions, and contours 　A1.3 Euler's gamma function and analytic continuations Appendix 2 Backreaction derived from effective action Appendix 3 Mode expansions cheat sheet Appendix 4 Solutions to exercises Index

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