Mathematics of quantization and quantum fields

MATHEMATICS OF QUANTIZATION AND QUANTUM FIELDS Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of some of the basic mathematical aspects of quantization and quantum field theory The authors present both elementary and more advanced subjects of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, such as tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance) Applications of these topics to physical problems are discussed in later chapters Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: the diagrammatic method and the Euclidean approach to constructive quantum field theory With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics ´ s k i is a Professor in the Faculty of Physics at the University Jan Derezin of Warsaw His research interests cover various aspects of quantum physics and quantum field theory, especially from the rigourous point of view ´ r a r d is a Professor at the Laboratoire de Math´ematiques Christian Ge at Universit´e Paris-Sud He was previously Directeur de Recherches at CNRS His research interests are the spectral and scattering theory in non-relativistic quantum mechanics and in quantum field theory www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 CAMBRIDGE MONOGRAPHS ON MATHEMATICAL PHYSICS General Editors: P V Landshoff, D R Nelson, S Weinberg S J Aarseth Gravitational N-Body Simulations: Tools and Algorithms J Amb jørn, B Durhuus and T Jonsson Quantum Geometry: A Statistical Field Theory Approach A M Anile Relativistic Fluids and Magneto-fluids: With Applications in Astrophysics and Plasma Physics J A de Azc´a rraga and J M Izquierdo Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics † O Bab elon, D Bernard and M Talon Introduction to Classical Integrable Systems F Bastianelli and P van Nieuwenhuizen Path Integrals and Anomalies in Curved Space V Belinski and E Verdaguer Gravitational Solitons J Bernstein Kinetic Theory in the Expanding Universe G F Bertsch and R A Broglia Oscillations in Finite Quantum Systems N D Birrell and P C W Davies Quantum Fields in Curved Space † K Bolejko, A Krasi´ n ski, C Hellaby and M -N C´e l´e rier Structures in the Universe by Exact Methods: Formation, Evolution, Interactions D M Brink Semi-Classical Methods for Nucleus-Nucleus Scattering † M Burgess Classical Covariant Fields E A Calzetta and B.-L B Hu Nonequilibrium Quantum Field Theory S Carlip Quantum Gravity in +1 Dimensions † P Cartier and C DeW itt-M orette Functional Integration: Action and Symmetries J C Collins Renormalization: An Introduction to Renormalization, the Renormalization Group and the Operator-Product Expansion † P D B Collins An Introduction to Regge Theory and High Energy Physics † M Creutz Quarks, Gluons and Lattices † P D D’Eath Supersymmetric Quantum Cosmology J Derezi´ n ski and C G´e rard Mathematics of Quantization and Quantum Fields F de Felice and D Bini Classical Measurements in Curved Space-Times F de Felice and C J S Clarke Relativity on Curved Manifolds B DeW itt Supermanifolds, nd edition P G O Freund Introduction to Supersymmetry † F G Friedlander The Wave Equation on a Curved Space-Time † Y Frishm an and J Sonnenschein Non-Perturbative Field Theory: From Two Dimensional Conformal Field Theory to QCD in Four Dimensions J A Fuchs Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory † J Fuchs and C Schweigert Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists † Y Fujii and K M aeda The Scalar-Tensor Theory of Gravitation J A H Futterm an, F A Handler, R A M atzner Scattering from Black Holes † A S Galp erin, E A Ivanov, V I O gievetsky and E S Sokatchev Harmonic Superspace R Gambini and J Pullin Loops, Knots, Gauge Theories and Quantum Gravity † T Gannon Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics M Gă o ckeler and T Sch u ¨ cker Differential Geometry, Gauge Theories, and Gravity † C G´o m ez, M Ruiz-Altaba and G Sierra Quantum Groups in Two-Dimensional Physics M B Green, J H Schwarz and E W itten Superstring Theory Volume 1: Introduction M B Green, J H Schwarz and E W itten Superstring Theory Volume 2: Loop Amplitudes, Anomalies and Phenomenology V N Grib ov The Theory of Complex Angular Momenta: Gribov Lectures on Theoretical Physics J B Griffiths and J Po dolsk´ y Exact Space-Times in Einstein’s General Relativity S W Hawking and G F R Ellis The Large Scale Structure of Space-Time † F Iachello and A Arim a The Interacting Boson Model F Iachello and P van Isacker The Interacting Boson-Fermion Model C Itzykson and J.-M Drouffe Statistical Field Theory Volume 1: From Brownian Motion to Renormalization and Lattice Gauge Theory † C Itzykson and J M Drouffe Statistical Field Theory Volume 2: Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems † C V Johnson D-Branes P S Joshi Gravitational Collapse and Spacetime Singularities J I Kapusta and C Gale Finite-Temperature Field Theory: Principles and Applications, nd edition V E Korepin, N M Bogoliub ov and A G Izergin Quantum Inverse Scattering Method and Correlation Functions † M Le Bellac Thermal Field Theory † www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 Y M akeenko Methods of Contemporary Gauge Theory N M anton and P Sutcliffe Topological Solitons N H M arch Liquid Metals: Concepts and Theory I M ontvay and G M u ă nster Quantum Fields on a Lattice L O ’Raifeartaigh Group Structure of Gauge Theories † T Ort´ın Gravity and Strings A M Ozorio de Alm eida Hamiltonian Systems: Chaos and Quantization † L Parker and D Tom s Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity R Penrose and W Rindler Spinors and Space-Time Volume 1: Two-Spinor Calculus and Relativistic Fields † R Penrose and W Rindler Spinors and Space-Time Volume 2: Spinor and Twistor Methods in Space-Time Geometry † S Pokorski Gauge Field Theories, nd edition † J Polchinski String Theory Volume 1: An Introduction to the Bosonic String J Polchinski String Theory Volume 2: Superstring Theory and Beyond J C Polkinghorne Models of High Energy Processes † V N Pop ov Functional Integrals and Collective Excitations † L V Prokhorov and S V Shabanov Hamiltonian Mechanics of Gauge Systems R J Rivers Path Integral Methods in Quantum Field Theory † R G Rob erts The Structure of the Proton: Deep Inelastic Scattering † C Rovelli Quantum Gravity † W C Saslaw Gravitational Physics of Stellar and Galactic Systems † R N Sen Causality, Measurement Theory and the Differentiable Structure of Space-Time M Shifm an and A Yung Supersymmetric Solitons H Stephani, D Kram er, M M acCallum , C Ho enselaers and E Herlt Exact Solutions of Einstein’s Field Equations, nd edition † J Stewart Advanced General Relativity † J C Taylor Gauge Theories of Weak Interactions † T Thiem ann Modern Canonical Quantum General Relativity D J Tom s The Schwinger Action Principle and Effective Action A Vilenkin and E P S Shellard Cosmic Strings and Other Topological Defects † R S Ward and R O Wells, Jr Twistor Geometry and Field Theory † E J Weinb erg Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics J R W ilson and G J M athews Relativistic Numerical Hydrodynamics † Issued as a pap erback www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 Mathematics of Quantization and Quantum Fields ´ JAN DEREZI NSKI University of Warsaw ´ CHRISTIAN G ERARD Universit´ e Paris-Sud www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ a o Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9781107011113 C J Derezi´ n ski and C G´erard 2013 This publication is in copyright Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published 2013 Printed and bound in the United Kingdom by the MPG Books Group A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data Derezi´ n ski, Jan, 1957– Mathematics of quantization and quantum fields / Jan Derezi´ n ski, University of Warsaw, Poland; Christian G´erard, Universite de Paris-Sud, France pages cm – (Cambridge monographs on mathematical physics) Includes bibliographical references and index ISBN 978-1-107-01111-3 Quantum theory – Mathematics I G´erard, Christian, 1960– II Title QC174.17.G46D47 2012 530.1201 51 – dc23 2012032862 ISBN 978-1-107-01111-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 Since my high school years, I have kept in my memory the following verses: Profesor Otto Gottlieb Schmock Pracuje ju˙z dziesi¸ aty rok Nad dzielem co zadziwi´c ma ´swiat: Der Kaiser, Gott und Proletariat As I checked recently, it is a somewhat distorted fragment of a poem by Julian Tuwim from 1919 I think that it describes quite well the process of writing our book Jan Derezi´ nski Je d´edie ce livre `a mon pays Que diront tant de Ducs et tant d’hommes guerriers Qui sont morts d’une plaie au combat les premiers, Et pour la France ont souffert tant de labeurs extrˆemes, La voyant aujourd’hui d´etruire par soi-mˆeme? Ils se repentiront d’avoir tant travaill´e, Assailli, d´efendu, guerroy´e, bataill´e, Pour un peuple mutin divis´e de courage Qui perd en se jouant un si bel h´eritage (Pierre de Ronsard, 1524–1585) Christian G´erard www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 132.166.62.90 on Mon Nov 17 13:49:40 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 Contents Dedication page vii Introduction 1 1.1 1.2 1.3 1.4 1.5 Vector spaces Elementary linear algebra Complex vector spaces Complex structures Groups and Lie algebras Notes 8 17 23 32 35 2.1 2.2 2.3 2.4 2.5 Operators in Hilbert spaces Convergence and completeness Bounded and unbounded operators Functional calculus Polar decomposition Notes 36 36 38 45 53 56 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Tensor algebras Direct sums and tensor products Tensor algebra Symmetric and anti-symmetric tensors Creation and annihilation operators Multi-linear symmetric and anti-symmetric forms Volume forms, determinant and Pfaffian Notes 57 57 64 65 73 78 85 91 4.1 4.2 4.3 4.4 Analysis in L2 (Rd ) Distributions and the Fourier transformation Weyl operators x, D-quantization Notes 92 92 100 106 110 5.1 5.2 Measures General measure theory Finite measures on real Hilbert spaces 111 111 121 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:50:16 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 References Araki, H., 1963: A lattice of von Neumann algebras associated with the quantum theory of free Bose field, J Math Phys 4, 1343–1362 Araki, H., 1964: Type of von Neumann algebra associated with free field, Prog Theor 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http://dx.doi.org/10.1017/CBO9780511894541.024 Cambridge Books Online © Cambridge University Press, 2014 References 665 Leibfried, D., Meekhof, D M., King, B E., et al., 1996: Experimental determination of the motional quantum state of a trapped atom, Phys Rev Lett 77, 4281–4285 Leray, J., 1978: Analyse Lagrangienne et M´ ecanique Quantique: Une Structure Math´ ematique Apparent´ ee aux D´ eveloppements Asymptotiques et a ` l’Indice de Maslov, S´erie de Math´ematiques Pures et Appliqu´ees, IRMA, Strasbourg Lundberg, L E., 1976: Quasi-free “second-quantization”, Comm Math Phys 50, 103–112 Manuceau, J., 1968: C ∗ -alg`ebres de relations de commutation, Ann Henri Poincar´e Sect A 8, 139–161 Maslov, V P., 1972: Th´ eorie de Perturbations et M´ ethodes Asymptotiques, Dunod, Paris Mattuck, R., 1967: A Guide to Feynman Diagrams in the Many-Body Problem, McGraw-Hill, New York Moyal, J E., 1949: Quantum mechanics as a statistical theory, Proc Camb Phil Soc 45, 99–124 Nelson, E., 1965: A quartic 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of the canonical anti-commutation relations, Comm Math Phys 16, 1–33 Racah, G., 1927: Symmetry between particles and anti-particles, Nuovo Cimento 14, 322– 328 Reed, M., Simon, B., 1975: Methods of Modern Mathematical Physics, II: Fourier Analysis, Self-Adjointness, Academic Press, London Reed, M., Simon, B., 1978a: Methods of Modern Mathematical Physics, III: Scattering Theory, Academic Press, London Reed, M., Simon, B., 1978b: Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, London Reed, M., Simon, B., 1980: Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, London Rieffel, M A., van Daele, A., 1977: A bounded operator approach to Tomita–Takesaki theory, Pacific J Math 69, 187–221 Robert, D., 1987: Autour de l’Approximation Semiclassique, Progress in Mathematics 68, Birkhă a user, Basel Robinson, D., 1965: The ground state of the Bose gas, Comm Math Phys 1, 159–174 Roepstorff G., 1970: Coherent photon states and spectral condition, Comm Math Phys 19, 301–314 Rosen, L., 1970: A λφ n field theory without cutoffs, Comm Math Phys 16, 157–183 Rosen, L., 1971: The (φ n )2 quantum field theory: higher order estimates, Comm Pure Appl Math 24, 417–457 Ruijsenaars, S N M., 1976: On Bogoliubov transformations for systems of relativistic charged particles, J Math Phys 18, 517–526 Ruijsenaars, S N M., 1978: On Bogoliubov transformations, II: the general case Ann Phys 116, 105–132 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:13 GMT 2014 http://dx.doi.org/10.1017/CBO9780511894541.024 Cambridge Books Online © Cambridge University Press, 2014 666 References Sakai, S., 1971: C ∗ -Algebras and W ∗ -Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete 60, Springer, Berlin Sakai, T., 1996: Riemannian Geometry, Translations of Mathematical Monographs 149, AMS, Providence, RI ă Schră o dinger, E., 1926: Der stetige Ubergang von der Mikro- zur Makromechanik, Naturwissenschaften 14, 664–666 Schwartz, L., 1966: Th´ eorie des Distributions, Hermann, Paris Schweber, S S., 1962: Introduction to Non-Relativistic Quantum Field Theory, Harper & Row, New York Segal, I E., 1953a: A non-commutative extension of abstract integration, Ann Math 57, 401–457 Segal, I E., 1953b: Correction to “A non-commutative extension of abstract integration”, Ann Math 58, 595–596 Segal, I E., 1956: Tensor algebras over Hilbert spaces, II, Ann Math 63, 160–175 Segal, I E., 1959: Foundations of the theory of dynamical systems of infinitely many degrees of freedom (I), Mat Fys Medd Danske Vid Soc 31, 1–39 Segal, I E., 1963: Mathematical Problems of Relativistic Physics, Proceedings of summer seminar on applied mathematics, Boulder, CO, 1960, AMS, Providence, RI Segal, I E., 1964: Quantum fields and analysis in the solution manifolds of differential equations In Analysis in Function Space, Proceedings of a conference on the theory and applications of analysis in function space, Dedham, MA, 1963, M.I.T Press, Cambridge, MA Segal, I E, 1970: Construction of non-linear local quantum processes, I, Ann Math 92, 462– 481 Segal, I E., 1978: The complex-wave representation of the free boson field, Suppl Studies, Adv Math 3, 321–344 Shale, D., 1962: Linear symmetries of free boson fields, Trans Amer Math Soc 103, 149– 167 Shale, D., Stinespring, W F., 1964: States on the Clifford algebra, Ann Math 80, 365–381 Simon, B., 1974: The P (φ)2 Euclidean (Quantum) Field Theory, Princeton University Press, Princeton, NJ Simon, B., 1979: Trace Ideals and Their Applications, London Math Soc Lect Notes Series 35, Cambridge University Press, Cambridge Simon, B., Høgh-Krohn, R., 1972: Hyper-contractive semi-groups and two dimensional selfcoupled Bose fields, J Funct Anal 9, 121–180 Skorokhod, A V., 1974: Integration in Hilbert Space, Springer, Berlin Slawny, J., 1971: On factor representations and the C ∗ -algebra of canonical commutation relations, Comm Math Phys 24, 151–170 Srednicki, M., 2007: Quantum Field Theory, Cambridge University Press, Cambridge Stratila, S., 1981: Modular Theory in Operator Algebras, Abacus Press, Tunbridge Wells Streater, R F., Wightman, A S., 1964: PCT, Spin and Statistics and All That, W A Benjamin, New York Symanzik, K., 1965: Application of functional integrals to Euclidean quantum field theory In Mathematical Theory of Elementary Particles, W T Martin and I E Segal, eds, MIT Press, Cambridge, MA Takesaki, M., 1979: Theory of Operator Algebras I, Springer, Berlin Takesaki, M., 2003: Theory of Operator Algebras II, Springer, Berlin Tao, T., 2006: Local and Global Analysis of Non-Linear Dispersive and Wave Equations, CMBS Reg Conf Series in Mathematics 106, AMS, Providence, RI Tomonaga, S., 1946: On the effect of the field reactions on the interaction of mesotrons and nuclear particles, I, Prog Theor Phys 1, 83–91 Trautman, A., 2006: Clifford algebras and their representations In Encyclopedia of Mathematical Physics 1, Elsevier, Amsterdam, pp 518–530 van Daele, A., 1971: Quasi-equivalence of quasi-free states on the Weyl algebra, Comm Math Phys 21, 171–191 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:13 GMT 2014 http://dx.doi.org/10.1017/CBO9780511894541.024 Cambridge Books Online © Cambridge University Press, 2014 References 667 van Hove, L., 1952: Les difficult´es de divergences pour un mod`ele particulier de champ quantifi´e, Physica 18, 145–152 Varilly, J C., Gracia-Bondia, J M., 1992: The metaplectic representation and boson fields, Mod Phys Lett A7, 659–673 Varilly, J C., Gracia-Bondia, J M., 1994: QED in external fields from the spin representation, J Math Phys 35, 3340–3367 von Neumann, J., 1931: Die Eindeutigkeit der Schră o dingerschen Operatoren, Math Ann 104, 570578 Wald, R M., 1994: Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics, University of Chicago Press, Chicago, IL Weil, A., 1964: Sur certains groupes d’op´erateurs unitaires, Acta Math 111, 143–211 Weinberg, S., 1995: The Quantum Theory of Fields, Vol I: Foundations, Cambridge University Press, Cambridge Weinless, M., 1969: Existence and uniqueness of the vacuum for linear quantized fields, J Funct Anal 4, 350–379 Weyl, H., 1931: The Theory of Groups and Quantum Mechanics, Methuen, London Wick, G C., 1950: The evaluation of the collision matrix, Phys Rev 80, 268–272 Widder, D., 1934: Necessary and sufficient conditions for the representation of a function by a doubly infinite Laplace integral, Bull AMS 40, 321326 ă Wigner, E., 1932a: Uber die Operation der Zeitumkehr in der Quantenmechanik, Gă o tt Nachr 31, 546–559 Wigner, E., 1932b: On the quantum correction for thermodynamic equilibrium, Phys Rev 40, 749–759 Wilde, I F, 1974: The free fermion field as a Markov field J Funct Anal 15, 12–21 Williamson J., 1936: On an algebraic problem concerning the normal forms of linear dynamical systems, Amer J Math 58, 141–163 Yafaev, D., 1992: Mathematical Scattering Theory: General Theory, Translations of Mathematical Monographs 105, AMS, Providence, RI www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:13 GMT 2014 http://dx.doi.org/10.1017/CBO9780511894541.024 Cambridge Books Online © Cambridge University Press, 2014 Symbols index Ifi n , 36 A c l , 39 A c n l , 563 A c o n , 562 A lin k , 563 A n l , 563 A , 145 A(O), 521, 548, 553 a(Φ), 78 a(w), 75 a ∗ , 19 a ∗ (Φ), 78 a ∗ (w), 75 a ∗γ , l , 437 a ∗γ , r , 437 a π ∗ (z), 184, 317 a π (z), 184, 317 a # , 10 a πch∗ (z), 186 a πch (z), 186 a γ , l (z), 437 AL(Y), 33 AO(R1 , d ), 514 B(H), 39 B (H1 , H2 ), 43 B (H1 , H2 ), 43 B B a r (z1 , z ), 237 B fi n Γ a (Z) , 348 B fi n Γ s (Z) , 233 B p (H1 , H2 ), 43 B s H, 47 B ∞ (H1 , H2 ), 42 BY , 121 Bc y l , 121 C (X ), 37 C (X , Y), 37 C c∞ (Ω), 92 ∞ (X ), 540 C sc ∞ (R1 , d ), 514 C sc C ∞ (X ), 96 C c (X , K), 37 C m , 524 ± Cm , 524 cc (X , K), 36 CAR(Y), 324 ∗ CAR C (Y), 332 W∗ (Y), 333 CAR CAR γ , l , 454 CAR γ , r , 454 CAR j (Y), 325 CCR(Y), 191 CCR We y l (Y), 195 CCR S (X # ⊕ X ), 99 CCR S (Y), 192 CCR p o l (Y), 189 CCR re g (Y), 193 CCR S (X # ⊕ X ), 99 CCR S (Y), 192 CCR γ , l , 437 CCR γ , r , 437 CCR p o l (X # ⊕ X ), 96 C hSp(Z), 23 chsp(Z), 23 C l(H, K), 39 C la (H, H), 41 C ls (H, H), 41 ∗ Cliff C (Y), 333 ∗ Cliff W (Y), 334 Cliff (Y), 325 CX , 24 D π , 186 D (Ω), 92 D(Ω), 92 D, 550 D(m, A), 527 det ζ, 88 det a, 88 Dom A, 39 dΓ(h), 65 E R , 158 E c p l , 37 E S , 117 eJ , 70 ek , 69 eJ , 69 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:23 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 Symbols index N , 65 N π , 293, 421 n π , 294, 421 ek , 69 ess sup, 115 F (Fourier transform), 96 Fin(X ), 121 G ± , 499 G ± (x, y), 518 G E , 501 G F , 499 G F , 499 G E , β , 502 G P v , 499 GL(Y), 32 gl(Y), 32 gl(Y, ), 144 Gr A, 39 H∞ , 248 Hπ , 291, 420 Ht , 248 H−∞ , 248 H∞, π , 181 π , 304 H[f ] I, 65 inert ν, 12 J (U ), 513, 539 J μ (ζ1 , ζ2 , x), 543 j μ (ζ1 , ζ2 , x), 520 Kπ , 288, 417 Kfπ , 301 K(n), 368 Ker a, L h (Z, Z ∗ ), 21 L fd (Y), L p (R, ω), 156 L a (Y, Y # ), 13 L s (Y, Y # ), 11 L 2C (Z, e−z z dzdz), 139 O(Y), 13 O(R1 , d ), 514 O j (Y), 395 O p (Y), 353 O (Y), 351 O j, a f (Y), 396 o(Y), 13 o1 (Y), 351 op (Y), 353 oj, a f (Y), 396 oj (Y), 395 Op, 198 Op c t , 206 Op c v , 208 Op D , x , 107 ∗ Op a , a , 227, 346 ∗ Op a , a , 227, 346 x Op , ∇x , 171 Op x , D , 107 P v , 204 P z , 216 Pair2 d , 90 Pf(ζ), 91 P in(Y), 359 P in(Rq , p ), 384 P in c (Y), 358 P in (Y), 360 P in c1 (Y), 360 P in (Y), 364 P in cj (Y), 406 P in j, a f (Y), 413 Pola (Y # ), 79 Pols (Y # ), 79 Prn , 558 Prc , 558 Pra , 558 Prs , 558 L (X , e− x dx), 134 Lg(D), 560 Lg(F ), 559 Lg ± (W ), 576 Ln(D), 560 M(Q), 115 M(R, ω), 156 M+ (Q), 115 M p(Y), 252 M p c (Y), 251 M p cj (Y), 283 M p j, a f (Y), 286 Q π , 320 Q πX , 316 Rq , p , 374 Ran a, res A, 39 S, 572 S(x, y), 528 S ± , 507, 572 S ± (x, y), 528 S G L , 574 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:23 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 669 670 Symbols index Y(O), 520, 530, 548, 553 Y C , 24 YR , 24 S E , 508 S F , 507 S F , 507 S E , β , 509 S P v , 507 S (X ), 94 S(X ), 93 SL(Y), 32 sl(Y), 32 SO ↑ (R1 , d ), 514 SO j (Y), 395 SO p (Y), 353 SO (Y), 351 SO j, a f (Y), 396 Sp(Y), 14 Sp j, a f (Y), 271 Sp j (Y), 271 sp(Y), 14 sp j (Y), 271 sp j, a f (Y), 272 Span U , Span c l (U ), 37 spec A, 39 Spin(Y), 359 Spin c (Y), 358 Spin (Y), 360 Spin c1 (Y), 360 Spin (Y), 364 Spin cj (Y), 406 Spin j, a f (Y), 413 Z, 18 Z ∗ , 19 Zch , 186 T rw , 222 T F B I , 204 T c w , 217, 343, 344 T m o d , 84 Texp(·), 571 U (exponential identification of Fock spaces), 70 U (Z), 22 U m o d , 82 U p e rp , 465 U c l , 36 U H F (2 ∞ ), 150 Γ(p), 65 o d (Z), 84 Γm a Γ rw (a), 224 o d (Z), 84 Γm s Γ a (Y), 67 Γ s (Y), 67 δ(τ ), 93 Θ(σ), 61 Θ a , 66 Θ na , 66 Θ s , 66 Θ ns , 66 θ(t) (Heaviside function), 499 θ (Hodge star operator), 86 Λ, 77 Ξ d u a l , 85 Ξ L io u v , 86 Ξ, 89 re v Ξ , 89 l φ (y), 340 φ π (y), 180 φ r (y), 340 φ s (g), 614 Ψ v , 204 ψ π ∗ (y), 185 ψ π (y), 185 Ω c , 278, 402 Ω z , 216 ✷, 514 ✷(m ), 517 ✷(m , A), 517 A j , 558 j ∈J ⊗ (Hi , Ω i ), 62 i ∈I ⊗ A i , 63 i ∈I ⊗ Φ i , 63 i ∈I W π (y), 174 W γ , l (z), 437 W γ , r (z), 437 W f (y), 303 X a n , 10 X ω ⊥ , 14 Xa n , 10 xπ , 186 ∇, 44 ∇Γμ , 542 (2 ) ∇x , 94 ∇v , 161 ⊗Y, 64 ⊗a , 67 ⊗s , 67 ← − ∇ v , 81 ∧ (wedge), 67 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:23 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 Subject index adiabatic scattering operator, 574 adiabatic dynamics, 574 advanced Green’s function, 499, 507, 518 affiliated, 149 annihilation operators, 75 annihilator, 10 anomalies, 413 anomaly-free orthogonal group, 396 anomaly-free symplectic group, 271 anti-dual pair, 20 anti-Feynman Green’s function, 499, 507 anti-Hermitian form, 21 anti-holomorphic derivative, 94 anti-holomorphic polynomials, 83 anti-holomorphic subspace, 26 anti-intertwining, 316 anti-involution, 23 anti-orthochronous, 514 anti-self-adjoint operator, 13 anti-symmetric, 13, 41 anti-symmetric calculus, 159, 163 anti-unitary, 22 anti-Wick ordering, 226 anti-Wick quantization, 227, 346 anti-Wick symbol, 227, 346 Araki–Woods representation, 437 Araki–Wyss representation, 453 asymptotic fields, 657 Baker–Campbell formula, 214 Bargmann kernel, 169, 236 Bargmann representation, 238 Berezin calculus, 163 Berezin integral, 162 Bogoliubov automorphism, 191 Bogoliubov rotation, 179, 319 Bogoliubov transformation, 179, 406 Bogoliubov translation, 179 Bose gas, 441 bosonic Fock space, 67 bosonic Gaussian vector, 278 bosonic harmonic oscillator, 261 Brownian motion, 607 CAR algebra, 324, 331–333 CAR representation, 314 Cauchy hypersurface, 539 causal, 513 causally separated, 514 Cayley transform, 34 CCR representation, 174 characteristic functional of a measure, 122 charge symmetry, 31 charge operator, 485 charge reversal, 375, 488 charged CAR representation, 318 charged CCR representation, 184, 185 charged dynamics, 484 charged field operators, 185, 319 charged symplectic dynamics, 485 charged symplectic space, 22 Clifford algebra, 325, 369 Clifford relations, 374 Clifford representation, 368 co-isotropic, 14 coherent representation, 303 coherent sector, 304 coherent state, 204 coherent states transformation, 204 commutant, 145 commutant theorem, 149 complete Kăa hler space, 29 complete lattice, 468 complete measure space, 113 complex-wave CCR representation, 217 complex-wave CAR representation, 342, 344 concrete C ∗ -algebra, 145 conditional expectation, 116, 150, 158 conjugate vector space, 18 conjugation in a complex vector space, 17 conjugation in a Kă a hler space, 30 conjugation in a symplectic space, 16 connected diagram, 562 conserved current, 541, 543, 550 convergence in measure, 117 covariance (of a measure), 124 covariance (of a quasi-free state), 424 covariant quantization, 208 creation operators, 75 cyclic set, 146 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:30 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 672 Subject index Glauber’s coherent vector, 210, 216 globally hyperbolic manifold, 540 GNS representation, 148 grading, 15 Green’s function, 499, 507 grounded Hilbert space, 62 cylinder function, 122 cylinder sets, 121 det-implementable, 361 det-implementing, 359 det-intertwining, 361 determinant, 88 Dirac equation, 504, 525, 527 Dirichlet form, 178 doubly Markovian map, 116 dressing operator, 309 dual pair, 10 Dyson time-ordered product, 571 equi-integrable family, 118 essentially self-adjoint, 51 Euclidean Green’s function, 501, 508 exponential law, 70 external legs, 563 factor, 148 factorial representation, 146 faithful representation, 145 faithful state, 147 FBI transform, 204 Feldman–Hajek theorem, 136 Fermi gas, 460 fermionic Gaussian vector, 402 fermionic harmonic oscillator, 354 Feynman Green’s function, 499 Feynman’s phase space 2-point function, 591 Feynman–Kac–Nelson kernel, 619 Fourier transform, 96 Fr´echet differentiable, 45 Fredholm determinant, 44 Friedrichs diagram, 577 fundamental solution, 499, 507 future oriented, 513 Gˆ a teaux differentiable, 44 Gaussian coherent vector, 209, 216 Gaussian FBI transform, 218 Gaussian integrals, 97 Gaussian integrals for complex variables, 97 Gaussian measure on a complex Hilbert space, 137 Gaussian measure on a real Hilbert space, 133 Gaussian path space, 612, 614 Gelfand–Najmark–Segal theorem, 148 Gell-Mann–Low scattering operator, 574 generalized path space, 610 generating function, 260 Gibbs state, 433, 449 Hermite polynomial, 223 Hermitian form, 21 higher-order estimates, 657 Hilbert–Schmidt operator, 43 Hodge star operator, 86 Hă o lders inequality, 115, 157 holomorphic derivative, 94 holomorphic polynomials, 83 holomorphic subspace, 26 hyper-contractive map, 116 implementable, 149, 283, 361 inextensible curve, 537 infinitesimally symplectic, 14 intertwiner, 146 irreducible algebra, 145 irreducible representation, 146 isotropic, 11 JordanWigner representation, 321 Kă a hler anti-involution, 29 Kă a hler space, 29 Kaplansky density theorem, 149 Kato–Heinz theorem, 48 Klein–Gordon equation, 493, 495–497 KMS condition, 153 KMS state, 153 Kohn–Nirenberg quantization, 107 Lagrangian, 518, 527 Lagrangian subspace, 13 Lebesgue–Vitali theorem, 118 light-like vector, 513 linked cluster theorem, 569, 585 Liouville volume form, 86 Liouvillean, 152 local Hermitian semi-group, 49 Majorana subspace, 489 Markov path space, 610, 624 Matsubara coefficients, 503, 510 maximal accretive, 49 metaplectic group, 252 Minlos–Sazonov theorem, 128 modified Fock spaces, 84 modular conjugation, 153 modular operator, 153 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:30 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 Subject index modular theory, 152 moments of a measure, 120 multi-degree, 559 multiple annihilation operators, 78 multiple creation operators, 78 Nelson’s analytic vectors theorem, 52 Nelson’s commutator theorem, 51 Nelson’s hyper-contractivity theorem, 226 Nelson’s invariant domain theorem, 51 net, 36 non-commutative probability space, 155 normal operator, 46 number operator, 65 number quadratic form, 294 orientation, 89 orthochronous, 514 oscillator space, 247 pairings, 90 parity operator, 65, 201, 202 parity reversal, 523 partial conjugation, 392 partial isometry, 53 past oriented, 513 Pauli matrices, 320 Pauli–Jordan commutator function, 499, 519 Perron–Frobenius theorem, 116 Pfaffian, 91 Pin group, 358, 360, 364 Poincar´e group, 514 Poisson bracket, 240 polar decomposition, 53 polynomial CCR algebra, 189 positive energy quantization, 481, 483, 487, 488 positive symplectic transformation, 269 positivity improving map, 116 positivity preserving map, 116 pre-annihilator, 10 pre-measure, 113 probability measure, 112 pseudo-Euclidean space, 12 pseudo-Kă a hler space, 29 pseudo-quaternionic, 375, 532 pseudo-real, 375, 532 pseudo-Riemannian manifold, 539 pseudo-unitary space, 21 pullback of distributions, 93 quasi-free CCR representation, 424 quaternionic representation, 374 quaternionic spinor, 374 quaternions, 370 Racah time reversal, 491 Radon–Nikodym derivative, 120 real Hilbert space, 12 real-wave CAR representation, 329, 334, 340 real-wave CCR representation, 220, 221 reconstruction theorem, 611 regular CCR algebra, 193 regular CCR representation, 174 regular symplectic map, 241 regularized determinant, 44 restricted orthogonal group, 394 restricted symplectic group, 271 retarded Green’s function, 499, 507, 518 Riesz spectral projection, 45 right derivative, 81 scattering operator, 572 Schră o dinger representation, 176 Schwartz kernel theorem, 99 Schwartz space, 93, 181 Schwinger term, 277, 400 second quantization, 68 Segal field operators, 180 semi-direct product, 33 separating subset, 146 Shale theorem, 274, 283 Shale–Stinespring theorem, 407, 418 sharp-time fields, 614 Slater determinant, 339 space-compact, 513 space-like vector, 513 spinor structure, 549 standard representation, 152 state, 147 Stokes theorem, 538 Stone–von Neumann CCR algebra, 190 Stone–von Neumann theorem, 104 Sucher’s formula, 575 super-algebra, 144 super-Fock space, 72 super-Hilbert space, 15 super-space, 15 symplectic form, 14 symplectic space, 14 tensor product of C ∗ -algebras, 151 tensor product of W ∗ -algebras, 151 thermal Euclidean Green’s function, 502, 509 time orientation, 539 time reversal, 483 time-like vector, 513 time-zero fields, 611 Tomita–Takesaki theory, 152 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:30 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 673 674 Subject index trace-class operator, 43 tracial state, 147 transpose of an operator, 10 Trotter’s product formula, 52 two-fold covering, 257 unitary space, 21 vacua of a CAR representation, 417 vacua of a CCR representation, 288 van Hove Hamiltonian, 306, 309 volume form, 85 von Neumann algebra, 148 von Neumann density theorem, 149 wave operators, 572 weak characteristic functional, 128 weak distribution, 127 weakly stable, 479 Weyl CCR algebra, 194 Weyl operator, 174 Weyl–Wigner quantization, 198 Weyl–Wigner symbol, 200 Wick quantization, 227, 346 Wick rotation, 502, 509 Wick symbol, 227 Wick theorem, 198, 223, 328 Wick’s time-ordered product, 590 Wigner time reversal, 490 www.pdfgrip.com Downloaded from Cambridge Books Online by IP 150.244.8.152 on Mon Nov 17 13:54:30 GMT 2014 http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511894541 Cambridge Books Online © Cambridge University Press, 2014 www.pdfgrip.com .. .MATHEMATICS OF QUANTIZATION AND QUANTUM FIELDS Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of some of the... Cosmology J Derezi´ n ski and C G´e rard Mathematics of Quantization and Quantum Fields F de Felice and D Bini Classical Measurements in Curved Space-Times F de Felice and C J S Clarke Relativity... Quantized Fields and Gravity R Penrose and W Rindler Spinors and Space-Time Volume 1: Two-Spinor Calculus and Relativistic Fields † R Penrose and W Rindler Spinors and Space-Time Volume 2: Spinor and