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A Course in Mathematical Physics 1 and 2: Classical Dynamical Systems and Classical Field Theory
1 Introduction.- 1.1 Equations of Motion.- 1.2 The Mathematical Language.- 1.3 The Physical Interpretation.- 2 Analysis on Manifolds.- 2.1 Manifolds.- 2.2 Tangent Spaces.- 2.3 Flows.- 2.4 Tensors.- 2.5 Differentiation.- 2.6 Integrals.- 3 Hamiltonian Systems.- 3.1 Canonical Transformations g.- 3.2 Hamilton's Equations.- 3.3 Constants of Motion.- 3.4 The Limit t ? I ± ?.- 3.5 Perturbation Preliminaries.- 3.6 Perturbation The Iteration.- 4 Nonrelativistic Motion.- 4.1 Free Particles.- 4.2 The Two-Body Problem.- 4.3 The Problem of Two Centers of Force.- 4.4 The Restricted Three-Body Problem.- 4.5 The N-body Problem.- 5 Relativistic Motion.- 5.1 The Hamiltonian Formulation of the Electrodynamic Equations of Motions.- 5.2 The Constant Field.- 5.3 The Coulomb Field.- 5.4 The Betatron.- 5.5 The Traveling Plane Disturbance.- 5.6 Relativistic Motion in a Gravitational Field.- 5.7 Motion in the Schwarzschild Field.- 5.8 Motion in a Gravitational Plane Wave.- 6 The Structure of Space and Time.- 6.1 The Homogeneous Universe.- 6.2 The Isotropic Universe.- 6.3 Me according to Galileo.- 6.4 Me as Minkowski Space.- 6.5 Me as a Pseudo-Riemannian Space.- 1. Introduction.- 1.1 Physical Aspects of Field Dynamics.- 1.2 The Mathematical Formalism.- 1.3 Maxwell's and Einstein's Equations.- 2. The Electromagnetic Field of a Known Charge Distribution.- 2.1 The Stationary-Action Principle and Conservation Theorems.- 2.2 The General Solution.- 2.3 The Field of a Point Charge.- 2.4 Radiative Reaction.- 3. The Field in the Presence of Conductors.- 3.1 The Superconductor.- 3.2 The Half-Space, the Wave-Guide, and the Resonant Cavity.- 3.3 Diffraction at a Wedge.- 3.4 Diffraction at a Cylinder.- 4. Gravitation.- 4.1 Covariant Differentiation and the Curvature of Space.- 4.2 Gauge Theories and Gravitation.- 4.3 Maximally Symmetric Spaces.- 4.4 Spaces with Maximally Symmetric Submanifolds.- 4.5 The Life and Death of Stars.- 4.6 The Existence of Singularities.
576 pages, Paperback
First published May 10, 1979
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April 3, 2023
Walter Thirring has done at last what John von Neumann would have had he lived longer! Every student of quantum physics will be familiar with the controversy between the latter and his counterpart, the distinguished pioneer of the quantum theory, P.A.M. Dirac, but despite his undoubted eminence as a pure mathematician and the justice of his concerns, few indeed will be partisans of von Neumann. In short, von Neumann criticizes Dirac for loose standards of rigor in his treatment of foundational results in the nascent theory of quantum mechanics (see Dirac’s Principles of Quantum Mechanics, first published in 1930; our review here), and not only this, seems to have been stimulated thereby to undertake his own investigations in order to derive everything in what, according to his lights, would be the right way. The outcome of this endeavor was the redoubtable Mathematische Grundlagen der Quantenmechanik (1932), reviewed by us here. For the first time, one arrived at a clear concept of what a Hilbert space is and what an unbounded self-adjoint operator on it should be. Now, in the post-war years von Neumann’s incipient program was taken up by a small band of followers, including inter alia Freeman Dyson, Eliot Lieb and Walter Thirring himself (then, as a brilliant young upstart).
In his maturity as a tenured professor at the University of Vienna, Thirring conceived the idea of setting down all that had been accomplished in his four-volume A Course in Mathematical Physics. The general feature of Thirring’s style is a willingness to set aside the encrustations of tradition and to develop the subject anew, always maintaining complete mathematical rigor. As is the case with volumes one and two on classical point particles and classical fields, respectively (reviewed by us here resp. here), Thirring develops the subject according to his own lights with a view to what is absolutely necessary to get to the results he wants to discuss later on. Thus, after a brief introduction highlighting the physical intuition, chapter two commences with a rapid review of the mathematical prerequisites in the theory of linear manifolds and their topologies (strong, weak, weak*), operator algebras, spectral theory, representations of C*-algebras in Hilbert space, the Gelfand-Naimark-Segal theorem, 1-parameter groups, the Hille-Yosida theorem, unbounded operators and quadratic forms, Friedrichs extension and the Kato-Rellich theorem. The expert will recognize in this listing many standard topics, collected here by Thirring with a view to the subsequent chapters on quantum mechanics proper. Despite its extreme condensation, everything Thirring himself says is clear if one reads slowly enough. Needless to say, one couldn’t learn very well a vast field such as this from Thirring alone and it certainly helps to have had some previous exposure to the ideas.
Chapter three presents quantum mechanics from Thirring’s point of view. He starts with Weyl’s representation by means of bounded operators, culminating in a proof of commutativity of the diagram relating quantum dynamics to its corresponding classical limit. Along the way, a detour on angular momentum in which – in contrast to what is usual – the angular momentum operators are treated seriously as unbounded and their domain of essential self-adjointness defined. Unitary time evolution of a few example systems is studied in explicit detail. Then, the limit as time tends to plus or minus infinity is treated in a completely mathematically satisfactory fashion, in which the Møller operators are defined and asymptotic completeness can be proved. Especially good for its discussion of channels in many-particle scattering. Next comes rigorous perturbation theory under an assumption of relative compactness. Here one will find succinct derivations of many isolated results such as the Brillouin-Wigner formula, the min-max principle, concavity of the perturbed eigenvalues, Wienhold’s and Duffin’s criteria, the projection method, Temple’s inequality and the Birman-Schwinger bound. Chapter three concludes with a treatment of scattering theory (Lippmann-Schwinger equation, spectral representation of the S-matrix, Low equation, phase shifts, singularity structure of the S-matrix, scattering amplitudes and cross-sections, optical theorem, Born’s approximation with error estimate and Kohn’s variational principle). All this is excellent material. The sections on scattering and perturbation theory are notably more coherent than in Arno Bohm’s advanced textbook (reviewed by us a while ago here): Thirring supplies rigorous definitions of everything, convergence proofs (for instance for the Born series), clarity on the sense in which convergence holds (strong, weak etc.), a few worked examples, concepts such as relative compactness and preservation of continuous spectrum, when the Hamiltonian is self-adjoint, and so on.
Unlike what is the case with customary textbook accounts of quantum mechanics, any analysis of actual atomic systems is deferred to chapter four, beginning with the hydrogen atom – a staple topic, needless to say, but few will have seen it treated in complete and rigorous detail as Thirring does (for instance, proof of the absence of singular spectrum and derivation of the asymptotic constants of motion). Then, a nice discussion of the hydrogen atom in an external field containing, among other things, a complete justification of why perturbation theory works even though not strictly applicable in this context (see Theorem 4.2.9) and derivation of the Stark effect, despite non-existence of the point spectrum for any non-zero electric field. §4.3 on helium and helium-like systems is amazing: compare with what Archimedes does in bounding π to lie within a fairly narrow interval. But §§4.4, 4.5, 4.6 on scattering theory in simple atoms and on complex atoms can be increasingly hard to take, as Thirring descends into detailed computations with explicit bounds. In the final section, §4.6, a rigorous presentation of the Born-Oppenheimer approximation, again focusing on the mathematical situation to the apparent neglect of the physics. Would a textbook on quantum chemistry be easier to follow as first pass?
The 116 problems with which Thirring furnishes the text turn out not to be very conceptually important in themselves, but offer a good opportunity to consolidate ideas just introduced in the text. Solving them proves to be laborious in that one is faced with having to master both conceptual and computational aspects at the same time. This means repeatedly going back to the text to look up precise definitions and some indication of a method of calculation to follow, then one has to devote a considerable effort to pin down all the details – if one has the patience to. After working on the problems a while, one does gain a good sense of what has to be done even if one lacks the stamina to tidy up every last numerical coefficient. Recommended not to refer to the answers until done with the problem oneself, else one’s learning will suffer – but somehow Thirring’s solutions always turn out better and more elegant!
To conclude, the first three chapters cover general theory, then chapter four becomes highly application-oriented. Notwithstanding Thirring’s customary clarity of exposition, the text will still be hard going for most because of the extreme concision of his style – notably more demanding in this respect than volumes one and two in the series. Indeed, he will race through in a paragraph what could constitute an entire chapter in another presentation of the material. Even if one responsibly works the problems, he may be left rattled and, to fill in Thirring’s rapid overview with something more comprehensive, want to go on say, to Reed-Simon, which appears to be more gently paced (Methods of Modern Mathematical Physics , vols. 2-4 on Fourier analysis, self-adjointness, analysis of operators and scattering theory).
In his maturity as a tenured professor at the University of Vienna, Thirring conceived the idea of setting down all that had been accomplished in his four-volume A Course in Mathematical Physics. The general feature of Thirring’s style is a willingness to set aside the encrustations of tradition and to develop the subject anew, always maintaining complete mathematical rigor. As is the case with volumes one and two on classical point particles and classical fields, respectively (reviewed by us here resp. here), Thirring develops the subject according to his own lights with a view to what is absolutely necessary to get to the results he wants to discuss later on. Thus, after a brief introduction highlighting the physical intuition, chapter two commences with a rapid review of the mathematical prerequisites in the theory of linear manifolds and their topologies (strong, weak, weak*), operator algebras, spectral theory, representations of C*-algebras in Hilbert space, the Gelfand-Naimark-Segal theorem, 1-parameter groups, the Hille-Yosida theorem, unbounded operators and quadratic forms, Friedrichs extension and the Kato-Rellich theorem. The expert will recognize in this listing many standard topics, collected here by Thirring with a view to the subsequent chapters on quantum mechanics proper. Despite its extreme condensation, everything Thirring himself says is clear if one reads slowly enough. Needless to say, one couldn’t learn very well a vast field such as this from Thirring alone and it certainly helps to have had some previous exposure to the ideas.
Chapter three presents quantum mechanics from Thirring’s point of view. He starts with Weyl’s representation by means of bounded operators, culminating in a proof of commutativity of the diagram relating quantum dynamics to its corresponding classical limit. Along the way, a detour on angular momentum in which – in contrast to what is usual – the angular momentum operators are treated seriously as unbounded and their domain of essential self-adjointness defined. Unitary time evolution of a few example systems is studied in explicit detail. Then, the limit as time tends to plus or minus infinity is treated in a completely mathematically satisfactory fashion, in which the Møller operators are defined and asymptotic completeness can be proved. Especially good for its discussion of channels in many-particle scattering. Next comes rigorous perturbation theory under an assumption of relative compactness. Here one will find succinct derivations of many isolated results such as the Brillouin-Wigner formula, the min-max principle, concavity of the perturbed eigenvalues, Wienhold’s and Duffin’s criteria, the projection method, Temple’s inequality and the Birman-Schwinger bound. Chapter three concludes with a treatment of scattering theory (Lippmann-Schwinger equation, spectral representation of the S-matrix, Low equation, phase shifts, singularity structure of the S-matrix, scattering amplitudes and cross-sections, optical theorem, Born’s approximation with error estimate and Kohn’s variational principle). All this is excellent material. The sections on scattering and perturbation theory are notably more coherent than in Arno Bohm’s advanced textbook (reviewed by us a while ago here): Thirring supplies rigorous definitions of everything, convergence proofs (for instance for the Born series), clarity on the sense in which convergence holds (strong, weak etc.), a few worked examples, concepts such as relative compactness and preservation of continuous spectrum, when the Hamiltonian is self-adjoint, and so on.
Unlike what is the case with customary textbook accounts of quantum mechanics, any analysis of actual atomic systems is deferred to chapter four, beginning with the hydrogen atom – a staple topic, needless to say, but few will have seen it treated in complete and rigorous detail as Thirring does (for instance, proof of the absence of singular spectrum and derivation of the asymptotic constants of motion). Then, a nice discussion of the hydrogen atom in an external field containing, among other things, a complete justification of why perturbation theory works even though not strictly applicable in this context (see Theorem 4.2.9) and derivation of the Stark effect, despite non-existence of the point spectrum for any non-zero electric field. §4.3 on helium and helium-like systems is amazing: compare with what Archimedes does in bounding π to lie within a fairly narrow interval. But §§4.4, 4.5, 4.6 on scattering theory in simple atoms and on complex atoms can be increasingly hard to take, as Thirring descends into detailed computations with explicit bounds. In the final section, §4.6, a rigorous presentation of the Born-Oppenheimer approximation, again focusing on the mathematical situation to the apparent neglect of the physics. Would a textbook on quantum chemistry be easier to follow as first pass?
The 116 problems with which Thirring furnishes the text turn out not to be very conceptually important in themselves, but offer a good opportunity to consolidate ideas just introduced in the text. Solving them proves to be laborious in that one is faced with having to master both conceptual and computational aspects at the same time. This means repeatedly going back to the text to look up precise definitions and some indication of a method of calculation to follow, then one has to devote a considerable effort to pin down all the details – if one has the patience to. After working on the problems a while, one does gain a good sense of what has to be done even if one lacks the stamina to tidy up every last numerical coefficient. Recommended not to refer to the answers until done with the problem oneself, else one’s learning will suffer – but somehow Thirring’s solutions always turn out better and more elegant!
To conclude, the first three chapters cover general theory, then chapter four becomes highly application-oriented. Notwithstanding Thirring’s customary clarity of exposition, the text will still be hard going for most because of the extreme concision of his style – notably more demanding in this respect than volumes one and two in the series. Indeed, he will race through in a paragraph what could constitute an entire chapter in another presentation of the material. Even if one responsibly works the problems, he may be left rattled and, to fill in Thirring’s rapid overview with something more comprehensive, want to go on say, to Reed-Simon, which appears to be more gently paced (Methods of Modern Mathematical Physics , vols. 2-4 on Fourier analysis, self-adjointness, analysis of operators and scattering theory).
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