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perform renormalized perturbation in quantum gauge field with mass generation a` la Higgs. Mathematical Aspects The fundamental importance of QFT as a framework for modern was further underlined when it was found in of Quantum Theory the 1960s that large parts of could by Edson de Faria and Welington de Melo profitably be formulated in the language of QFT (albeit in its nonrelativistic version and with far less emphasis on infinities NEW YORK: CAMBRIDGE UNIVERSITY PRESS, 2010, XIII + 298 and renormalization) [2], a trend that has continued ever since PP., £ 43 HARDBACK, ISBN-13: 9780521115773 [23]. Furthermore, in the 1970s phase transitions in classical (!) REVIEWED BY N. P. (KLAAS) LANDSMAN nonrelativistic (!) statistical were spectacularly brought within the scope of relativistic quantum field theory [13]. From a mathematical point of view, however, the successes he discovery of the Higgs boson at CERN in 2012 of QFT and especially the are rather puzzling. (made public on July 4th) put the finishing touch on The point is that practically all successful (i.e., empirically TTthe so-called Standard Model of high-energy physics. validated) predictions of QFT are based on renormalized This model describes all (known) elementary particles and , which is a recipe (i.e., a set of rules) rather their interactions (except gravity), and is an extension of the than a theory. What is more, one may validly doubt whether theory of , the quantum theory of the Standard Model actually has a formulation as a mathe- and electrically charged particles. Apart from those, the matical theory! Hence, as is often remarked, insistence on Standard Model also incorporates the weak and strong nuclear mathematical during the development of QFT would forces through which subatomic particles such as quarks and probably have blocked all progress; in this sense it is almost gluons interact. When the model reached its present form in fortunate that the subject was largely developed precisely the early 1970s, it successfully retrodicted the outcomes of all during a period (approximately 1945–1975) when the cross- relevant past experiments, and it made a number of highly fertilisation of and physics, which had enriched nontrivial predictions (such as the existence of neutral during the 250-year time span between say Newton currents, the W ± and Z bosons, the top quark, and the and Einstein, had subsided. Yet it can hardly be overstated Higgs boson), all of which were subsequently verified how strange and novel it is to have an incredibly successful experimentally. Thus the Standard Model is an undisputed physical ‘‘theory’’ without an apparent mathematical founda- triumph of modern science (see [25]forasuperbphysics tion; all other great theories of physics have meanwhile been textbook by one of its creators and [14] for history). (re)formulated with utmost mathematical clarity and rigour, The Standard Model is formulated in the framework of including [1], classical quantum field theory (QFT), whose beginnings may be [18], [4], [16, 17], and traced back to papers by Dirac, Heisenberg, Jordan, Pauli, quantum statistical mechanics [3]. and a few others during 1926–1928, that is, immediately This peculiar situation has not gone unnoticed. One atti- after the birth of quantum mechanics. Unfortunately, pro- tude, held by Nobel Laureate Veltman [24], is to claim that QFT gress was almost immediately blocked by multitudes of simply is the set of rules for renormalized perturbation theory, infinities that were encountered if one tried naively to or, in other words, that the theory coincides with the recipe. adapt the perturbation methods of quantum mechanics to This extreme stance is exceptional even among ; the relativistic field theories, which are systems with an infinite more typical attitude, represented throughout the physics lit- number of degrees of freedom compatible with Einstein’s erature, is to start from a classical field theory and attempt theory of from 1905. These problems were to construct ‘‘the corresponding quantum theory’’ (scare only overcome about two decades later, through the quotes). In complicated (gauge) theories such as the Standard development of renormalized perturbation theory by Model, this can only be done in practice using Feynman’s path Feynman, Schwinger, and Tomonaga in the late 1940s (see integral, supplemented with some method for dealing with [21] for a delightful history of this era). gauge invariance (e.g., the so-called Faddeev–Popov trick or This technique combined a systematic way of doing cal- the BRST formalism). Somewhere along the way, mathemat- culations (since then usually formulated in the language of ical rigour (if not mere describability) turns out to get lost, but so-called Feynman diagrams) with a method of removing the at the end of the day one does arrive at the recipe. By itself, the infinities that plagued the earlier work, and in due course latter is completely rigorous and as such has been the subject made quantum electrodynamics the experimentally best ver- of satisfactory mathematical treatments such as [5, 19, 20]. ified theory in all of science (its precision has meanwhile In mathematical physics, three large research programs reached twelve decimals, cf. [26]). During the 1950s and 1960s, attempting to provide a mathematical foundation of QFT Feynman, Schwinger, Gell-Mann, Yang and Mills, Higgs, should be mentioned (with their leaders and authoritative Glashow, Weinberg, Salam, and others gradually understood monographs). In historical order, these were axiomatic QFT how the weak and strong nuclear forces could be described in (Wightman [22]), algebraic QFT, later renamed as local an analogous way using so-called gauge field theories (based quantum physics (Haag [11]), and constructive QFT (Glimm on an idea of Weyl from the 1920s, originally proposed and Jaffe [10]). The first two programs revolved around axioms however in the context of gravity), on which ’t Hooft and (whose models were supposed to be quantum field theories) Veltman completed the Standard Model by showing how to and their consequences (thought of as properties any QFT

Ó 2013 Springer Science+Business Media New York, Volume 35, Number 3, 2013 85 DOI 10.1007/s00283-013-9392-6 ought to possess). The former uses the language of (unboun- to discretize the (mathematically undefined) ‘‘path integral’’ of ded) -valued distributions, whereas the latter employs the physicists and proceed to derive the ‘‘recipe.’’ This operator (i.e., C*-algebras and von Neumann alge- approach sidesteps the problem of defining an interacting bras). The third program, then, had the outspoken goal of QFT (in the continuum, that is), but otherwise it is a remark- constructing physically relevant models for the axioms. ably efficient way of deriving the Feynman rules, also for The legacy of these programs (of which the first is no longer gauge theories (which are treated essentially using the Fad- active) is controversial. Physicists simply ignored them, partly deev–Popov gauge-fixing trick of the physicists, with a brief because of a lack of interest in the problem of mathematically addendum on the BRST technique). From this point onward, it justifying physics per se, and partly because each program had would have been possible to provide a complete and rigorous difficulties precisely with realistic theories such as the Stan- account of renormalization theory, but the corresponding dard Model (and more generally, with gauge theories in chapter is disappointingly brief, restricting itself to an expla- space–time dimension four). Indeed, the construction of pure nation of Zimmermann’s famous ‘‘forest formula’’ and some SU(3) Yang–Mills theory in d = 4 (which is the theory of the comments on modern proofs of renormalizability using flow strong interaction without quarks) is one of the million-dollar equations (see [20] for a detailed treatment). Instead, the last Clay Millennium Problems! Consequently, constructive field chapter provides a reasonably detailed treatment of the theorists turned their attention toward the easier (yet still Standard Model, alas only as a classical field theory, so that the challenging) problems of condensed matter physics, whereas problem of the perturbative renormalization of spontaneously local quantum physics has became a research field by itself, broken gauge theories (whose historical solution by ’t Hooft in with its own goals and results (meanwhile somewhat decou- the early 1970s really launched the Standard Model) is not pled from the initial aim of clarifying the heuristic QFT models even mentioned. The book ends with two appendices on of the physicists). Mathematicians showed more interest; in Hilbert spaces and C*-algebras (which are hardly used in the the past , operator algebras, and infinite- book). dimensional (stochastic) analysis fruitfully interacted with Overall, this book provides a brief but careful, reasonably axiomatic, algebraic, and constructive QFT, respectively. balanced, and representative introduction for mathematicians More recently, mathematical disciplines such as Monstrous to QFT as it is used in physics. It does not live up to its title and Moonshine [9] and algebraic [15] received specta- even within its scope has some gaps (a few of which have cular and completely unexpected inspiration from various been mentioned), but it does a good job of preparing readers mathematical approaches to QFT (viz. conformal QFT and who want more for either mathematical physics books like [10, topological QFT, respectively). 11, 22] or books such as [25] (they should We now turn to the book under review. Despite its really read both). promising title, it is not intended to introduce its readers to Among its competitors, one should mention Folland [8] these mathematical programs, although a few comments are (which is quite similar in its goal as well as in its choice of made about each of them. Instead, the authors attempt to material, but, despite being written by a mathematician, is explain, to a mathematical audience, the way physicists use almost indistinguishable from a pseudocomprehensible QFT. This means that they have essentially written a physics physics text), Dimock [7](whocoverslessgroundthande textbook, with the difference that some mathematical preci- Faria and de Melo but is accordingly more detailed, to the sion has been added wherever this was felt to be possible and point of being complete in the material it does cover), and appropriate. Thus they start with basic physics in the form of finally the remarkable Princeton collection of lectures by a brief chapters on classical mechanics, quantum mechanics, parade of Fields medallists and similar folk [6](whichisbril- and special relativity (including the Dirac equation). To set the liant yet uneven). stage for the gauge field theories central to the Standard In conclusion, the ultimate QFT book for mathematicians Model, they subsequently introduce constructions from dif- remains to be written. ferential like fiber bundles, from which they define classical gauge field theories. This (geometric) story is inter- rupted in order to quantize free fields. This is done with some REFERENCES care, and it is mentioned (although, somewhat surprisingly, [1] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd not proved), that such fields satisfy the Wightman axioms. The ed., Addison Wesley, 1985. CPT theorem, a prime achievement of axiomatic QFT that [2] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzalosinski, Methods of shows invariance of any QFT under the combination of charge in , Prentice-Hall, 1963. conjugation, spatial reflection (parity), and time reversal, is [3] O. Bratteli and D. W. Robinson, Operator Algebras and Quantum also mentioned without proof. A somewhat of the fundamental LSZ reduction formulae, which Statistical Mechanics, Vols. I, II, Springer-Verlag, 1981–1987. provide the connection between formalism and experiment,is [4] Y. Choquet-Bruhat, General Relativity and the Einstein Equations, outlined, but this is no match for a serious proof such as Oxford University Press, 2009. Hepp’s [12] (which would not have been out of place in a book [5] K. Costello, Renormalization and , AMS, like this). 2011. So far (which is about halfway into the book), the discus- [6] P. Deligne, et al. (eds.), Quantum Fields and Strings: A Course for sion is, or easily could have been made, completely rigorous. Mathematicians, Vols. 1, 2, AMS, 1999. In the second half of the book, the authors do their best to [7] J. Dimock, Quantum Mechanics and Quantum Field Theory: A explain renormalized perturbation theory. Their approach is Mathematical Primer, Cambridge University Press, 2011.

86 THE MATHEMATICAL INTELLIGENCER [8] G. B. Folland, Quantum Field Theory: A Tourist Guide for Math- [19] V. Rivasseau, From Perturbative to Constructive Renormalization, ematicians, AMS, 2008. Princeton University Press, 1991. [9] T. Gannon, Monstrous moonshine and the classification of CFT, [20] M. Salmhofer, Renormalization: An Introduction, Springer-Verlag, arXiv:math/9906167. 1999. [10] J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral [21] S. S. Schweber, QED and the Men Who Made It, Princeton Point of View, Springer-Verlag, 1987. University Press, 1994. [11] R. Haag, Local Quantum Physics: Fields, Particles, Algebras, [22] R. F. Streater and A. S. Wightman, PCT, Spin and , and Springer-Verlag, 1992. All That, W. A. Benjamin, 1964. [12] K. Hepp, On the connection between the LSZ and Wightman [23] A. M. Tsvelik, Quantum Field Theory in Condensed Matter Quantum Field Theory, Commun. Math. Phys. 1: 95–111, 1965. Physics, Cambridge University Press, 1995. [13] C. Itzykson and J.-M. Drouffe, , Vols. 1–2, [24] M. Veltman, Diagrammatica: The Path to Feynman Diagrams, Cambridge University Press, 1991. Cambridge University Press, 1994. [14] L. Hoddeson, et al. (eds.), The Rise of the Standard Model: [25] S. Weinberg, The Quantum Theory of Fields, Vols. I–II, Cambridge in the 1960’s and 1970’s, Cambridge University University Press, 1996–1996. Press, 1997. [26] http://en.wikipedia.org/wiki/Precision_tests_of_QED. [15] J. Lurie, On the classification of topological field theories, Current developments in mathematics, International Press, pp. 129–280, Chair of Mathematical Physics 2008. Institute for Mathematics, [16] J. von Neumann, Mathematische Grundlagen der Quanten- , and Particle Physics mechanik, Springer, Berlin, 1932. Radboud Universiteit Nijmegen [17] M. Reed and B. Simon, Methods of Modern Mathematical Heyendaalseweg 135, 6525 AJ Nijmegen Physics. Vols. I–IV, Academic Press, 1972–1979. The Netherlands e-mail: [email protected] [18] D. Ruelle, Thermodynamic Formalism: The Mathematical Struc- tures of Classical Equilibrium Statistical Mechanics, Addison- Wesley, 1978.

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