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Book Reviewsunbounded self-adjoint operators in Chapter 9. The succeedingfour chapters are concerned with the canonicalcommutation relations in some way or other. Chapter11 deals with the harmonic oscillator, introducing thealgebraic approach to quantum mechanics in place ofanalysis as the way to solve quantum systems. Harmonicoscillators are rampant in physics. The Heisenberg uncertaintyprinciple and the Stone-von Neumann theoremare two important consequences of the commutationrelations between the position and momentum operators,the former being dealt with in Chapter 12 and thelatter being dealt with in Chapter 14. Both results areexamined carefully with due regard to certain technicaldomain conditions. Due to Groenewold’s no-go theorem[12], there is no single perfect quantization scheme,but the Weyl quantization, on which the author spendsa large portion of Chapter 13, is regarded as having thebest properties. The succeeding four chapters are concernedwith less elementary properties of quantum theory.Chapter 15 is devoted to so-called WKB (standing forGregor Wentzel, Hendrik Kramers and Léon Brillouin)approximation, which gives an approximation to the eigenfunctionsand eigenvalues of the Hamiltonian operatorin one dimension. Chapter 16 gives a brief introductionto Lie groups, Lie algebras and their representations,which is put to use in Chapter 17 for studying angularmomentum and spin in terms of representation of SO(3),where it is noted that the notion of fractional spin is tobe understood as a representation of the Lie algebra ofSO(3), which has no corresponding representation ofSO(3) itself. I note gladly in passing that the author is theauthor of a successful book on Lie groups and Lie algebras[13]. Chapter 18 is devoted to describing the energylevels of the hydrogen atom, including some discussionon its hidden symmetries. Chapter 19 is concerned withcomposite systems in terms of tensor products of Hilbertspaces, bosons and fermions, the Pauli exclusion principle,and so on. The coda of the book consists of threechapters, dealing with some advanced topics on classicaland quantum mechanics. Chapter 20 develops the pathintegral formulation of quantum mechanics rigorouslyby using the Wiener measure. The chapter begins withthe Trotter product formula, then turning to the heuristicformulation of Feyman himself and finally obtaining theso-called Feyman-Kac formula. Chapter 22 considers thegeometric quantization program from a symplectic viewpoint,paving the way to Chapter 23, which lays out in anorderly fashion all the ingredients (bundles, connections,polarizations, etc.) needed to do geometric quantizationgenerally.References[1] Takhtajan, Leon A. Quantum mechanics for mathematicians. GraduateStudies in Mathematics 95. Providence, RI: American MathematicalSociety (2008).[2] Ticciati, Robin. Quantum field theory for mathematicians. Encyclopediaof Mathematics and Its Applications. 72. Cambridge: CambridgeUniversity Press (1999).[3] de Faria, Edson; de Melo, Welington. Mathematical aspects of quantumfield theory. Cambridge Studies in Advanced Mathematics 127.Cambridge: Cambridge University Press (2010).[4] Grensing, Gerhard. Structural aspects of quantum field theory andnoncommutative geometry. Vol. 1. Hackensack, NJ: World Scientific(2013).[5] Grensing, Gerhard. Structural aspects of quantum field theory andnoncommutative geometry. Vol. 2. Hackensack, NJ: World Scientific(2013).[6] Folland, Gerald B. Quantum field theory: A tourist guide for mathematicians.Mathematical Surveys and Monographs 149. Providence,RI: American Mathematical Society (2008).[7] Costello, Kevin. Renormalization and effective field theory. MathematicalSurveys and Monographs 170. Providence, RI: AmericanMathematical Society (2011).[8] Deligne, Pierre; Etingof, Pavel; Freed, Daniel S.; Jeffrey, Lisa C.;Kazhdan, David; Morgan, John W.; Morrison, David R.; Witten,Edward (eds.). Quantum fields and strings: a course for mathematicians.Vol. 1, 2. Material from the Special Year on Quantum FieldTheory held at the Institute for Advanced Study, Princeton, NJ,1996–1997. Providence, RI: AMS, American Mathematical Society(1999).[9] Zeidler, Eberhard. Quantum field theory. II: Quantum electrodynamics.A bridge between mathematicians and physicists. Berlin:Springer (2009).[10] Zeidler, Eberhard. Quantum field theory III: Gauge theory. A bridgebetween mathematicians and physicists. Berlin: Springer (2011).[11] Teschl, Gerald. Mathematical methods in quantum mechanics. Withapplications to Schrödinger operators. Graduate Studies in Mathematics99. Providence, RI: American Mathematical Society (2009).[12] Groenewold, H.J. On the principles of elementary quantum mechanics.Physica 12, 405–460 (1946).[13] Hall, Brian C. Lie groups, Lie algebras, and representations. An elementaryintroduction. Graduate Texts in Mathematics. 222 (2003).Hirokazu Nishimura was born in Kyotoin 1953 and was raised there. He attendedKyoto University and receivedhis Bachelor of Science with a major inbiophysics in 1976. After graduation,he entered the Research Institute forMathematical Sciences at Kyoto Universityas a graduate. From November1979 through March 1986 he was enrolledas a regular member there. Since April 1986 he hasbeen enrolled at the Institute of Mathematics at the Universityof Tsukuba. Axiomatic methods are now the commonplace,but differential geometry has defied its axiomatization.His current ambition is to axiomatize differentialgeometry by emancipating synthetic differential geometryfrom topos theory, thereby giving a unified treatment tovarious differential geometries ranging from the orthodoxone to noncommutative geometry. He is the authorof more than 90 papers and editor of the book A LostMathematician, Takeo Nakasawa. His major speciality issurely mathematics, but his minor speciality is artistic design.He chose to present an artistic work of his in place ofhis picture. The title of the work is A Great Fuss of GreatMathematicians. He has displayed several works of hisdesign at the National Art Centre, Tokyo, Japan.56 EMS Newsletter December 2014