nLab books and reviews in mathematical physics

The intention of this page is to list a wide choice of main books and comprehensive reviews in mathematical physics. The irrelevant repetitions and minor, too specialized and obsolete books in any major respect should be avoided. We avoid references for quantum groups as they are many and the main ones can be found at the quantum group entry; similarly we avoided the relevant books on Kac-Moody algebras and groups but included the books on related VOAs and the Pressley-Segal book.

Classical mathematical physics

Here PDEs, ODEs, and integral equation of mathematical physics, special functions, generalized functions, analytic functions, basic functional analysis, potential theory:

• R. Courant, David Hilbert, Methods of mathematical physics, 2 vols.

• P. M. Morse, H. Feshbach, Methods of theoretical physics I, II, publisher

• Michael Reed, Barry Simon, Methods of modern mathematical physics, 4 vols. (emphasis on functional analysis)

• V. Vladimirov, Equations of mathematical physics, Moscow, Izdatel’stvo Nauka, (1976. 528 p. Russian; English edition, Mir 198x); Generalized functions in mathematical physics, Moscow

Focus on differential geometry:

• Yvonne Choquet-Bruhat, Cécile DeWitt-Morette, Analysis, manifolds and physics, North Holland (1982, 2001) [ISBN:9780444860170]

• Mikhail O. Katanaev, Geometrical methods in mathematical physics (in Russian) [arXiv1311.0733]

Classical mechanics

• Vladimir Arnold, Mathematical methods of classical mechanics, Springer (1989).

• Michael Spivak, Physics for Mathematicians, Mechanics I, xvi + 733 pages, Publish or Perish 2010 contents, amazon; Elementary mechanics from a mathematician’s viewpoint, pdf

• R. Abraham, J. Marsden, The Foundations of Mechanics, Benjamin Press, 1967, Addison-Wesley, 1978; large pdf 86 Mb free at CaltechAuthors

• William Lawvere, Stephen Schanuel (eds.), Categories in Continuum Physics, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986

• Vicente Cortés, Alexander S. Haupt, Lecture Notes on Mathematical Methods of Classical Physics, (arXiv:1612.03100)

Mathematical introduction to quantum mechanics

On quantum mechanics:

• Anthony Sudbery, Quantum mechanics and the particles of nature: An outline for mathematicians

• Leon A. Takhtajan, Quantum mechanics for mathematicians, Graduate Studies in Mathematics 95, Amer. Math. Soc. 2008.

• Brian C. Hall, Quantum theory for mathematicians, Springer GTM 267 (has also a big chapter on geometric quantization)

Geometry and symmetries in classical and QM, quantization (but no QFT)

In addition to the geometrically written titles under classical mechanics above,

• Bernard F. Schutz, Geometrical methods of mathematical physics (elementary intro) amazon, google

• Peter Olver, Equivalence, invariants, and symmetry, Cambridge Univ. Press 1995; Applications of Lie groups to differential equations, Springer.

• Shlomo Sternberg, Group theory and physics, Cambridge University Press 1994.

• Victor Guillemin, Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press (1990)

• Theodore Frankel, The Geometry of Physics - An Introduction, Cambridge Univ. Press

• Chris Isham, Modern Differential Geometry for Physicists

• Klaas Landsman, Mathematical topics between classical and quantum mechanics, Springer Monographs in Mathematics 1998. xx+529 pp.

• Sean Bates, Alan Weinstein, Lectures on the geometry of quantization, pdf

• A. Cannas da Silva, A. Weinstein, Geometric models for noncommutative algebras, 1999, pdf

• Nikolai Reshetikhin, Lectures on quantization of gauge systems, arxiv/1008.1411

Lorentzian geometry and general relativity

Global aspects of the geometry of spacetimes:

• John K. Beem, Paul E. Ehrlich, Kevin L. Easley, Global Lorentzian geometry (ZMATH) (global aspects)

• Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1987, xii + 510 pp. (differential geometry of solutions to Einstein equations, constant negative curvature, classification results, encyclopaedic; for a review see Bull. AMS

After the introduction emphasis on asymptotics of spacetimes far from gravitation objects:

• John Stewart, Advanced general relativity (ZMATH entry)

Despite its title the next monograph does not just present the Kerr spacetime, it illustrates many core features of GR with the Kerr spacetime as the prominent example:

• Barrett O’Neill, The geometry of Kerr black holes. (ZMATH)

Here is an introduction to spinors in GR:

• Peter O’Donnell, Introduction to 2-spinors in general relativity. (ZMATH)

while the classic reference for this is:

• Roger Penrose, Wolfgang Rindler, Spinors and spacetimes (2 vols, vol 1, ZMATH)

• Eric Poisson, A relativist’s toolkit. The mathematics of black-hole mechanics. (ZMATH) (computationally oriented)

See also the above book by Ward and Wells; and mainstream theoretical physics gravity textbooks by Misner, Thorne and Wheeler; Schutz; Landau-Lifschitz vol. 2; Wald; Chandrasekhar etc. For the supergravity see the appropriate chapters in the above listed collection by Deligne et al. or the references listed at supergravity.

Integrable systems and solitons

On integrable systems and solitons:

• O. Babelon, D. Bernard, M. Talon, Introduction to classical integrable systems, Cambridge Univ. Press 2003.

• T. Miwa, M. Jimbo, E. Date, Solitons: Differential equations, symmetries and infinite dimensional algebras, Cambridge Tracts in Mathematics 135, translated from Japanese by Miles Reid

• V.E. Korepin, N. M. Bogoliubov, A. G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge Univ. Press 1997.

• Ludwig D. Faddeev, Leon Takhtajan, Hamiltonian methods in the theory of solitons, Springer

Modern mathematical approaches to QFT and strings

On quantum field theory and string theory:

• Yvonne Choquet-Bruhat, Cecile Dewitt-Morette, Analysis, manifolds and physics, 1982 and 2001

• Albert Schwartz, Quantum field theory and topology, Grundlehren der Math. Wissen. 307, Springer 1993. (translated from Russian original)

• Howard Georgi: Lie Algebras in Particle Physics. From isospin to unified theories. (ZMATH entry)

• Eberhard Zeidler, Quantum field theory. A bridge between mathematicians and physicists. I: Basics in mathematics and physics. , II: Quantum electrodynamics

• Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press 1992.

• Charles Nash, Differential topology and quantum field theory, Acad. Press 1991.

• P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison and E. Witten, eds. Quantum fields and strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

• Gregory L. Naber, Topology, Geometry and Gauge fields – Foundations, Texts in Applied Mathematics 25 (2011) [doi:10.1007/978-1-4419-7254-5]

• Gregory L. Naber, Topology, Geometry and Gauge fields – Interactions, Applied Mathematical Sciences 141 (2011) [doi:10.1007/978-1-4419-7895-0]

• Mikio Nakahara, Geometry, Topology and Physics

• Marian Fecko, Differential geometry and Lie groups for physicists

• V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, AMS and Courant Institute, 2004.

• R. S. Ward, R. O. Wells, Twistor geometry and field theory (CUP, 1990)

• R. E. Borcherds, A. Barnard, Lectures on QFT, arxiv:math-ph/0204014

• Anton Kapustin, Topological field theory, higher categories, and their applications, survey for ICM 2010, arxiv/1004.2307

• Siddhartha Sen, Kumar Sankar Gupta, Many-body physics, topology and geometry

Branes (mathematical aspects)

• Paul Aspinwall, Tom Bridgeland, Alastair Craw, Michael R. Douglas, Mark Gross, Dirichlet branes and mirror symmetry, Amer. Math. Soc. Clay Math. Institute 2009.

• Hisham Sati, Geometric and topological structures related to M-branes, part I (arXiv:1001.5020), part II: Twisted $String$ and $String^c$ structures (arxiv/1007.5419); part III: Twisted higher structures (http://arxiv.org/abs/1008.1755)

• A. N. Kapustin, D. O. Orlov, Lectures on mirror symmetry, derived categories, and $D$-branes, Russian Mathematical Surveys, 2004, 59:5, 907–940 (Russian version: pdf, [arxiv version: arxiv:math.AG/0308173).]

Conformal field theory and vertex algebras

On conformal field theory and its chiral part (vertex operator algebras, chiral algebras):

• Edward Frenkel, David Ben-Zvi: Vertex algebras and algebraic curves, Math. Surveys and Monographs 88, AMS 2001, xii+348 pp. (Bull. AMS. review, ZMATH entry)

• Martin Schottenloher, A mathematical introduction to conformal field theory (CFT on the plane)

• Philippe Di Francesco, Pierre Mathieu, David Sénéchal, Conformal field theory, Springer 1997 (comprehensive textbook for theoretical physicists)

• Ralph Blumenhagen, Erik Plauschinn, Introduction to conformal field theory: with applications to string theory, Springer Lecture Notes in Physics (2011)

• V. Kac, Vertex algebras for beginners, Amer. Math. Soc.

• B. Bakalov, A. Kirillov, Lectures on tensor categories and modular functors, AMS, University Lecture Series, (2000) (web)

• Alexander Beilinson, Vladimir Drinfeld, Chiral algebras, Colloqium Publications 51, Amer. Math. Soc. 2004, gbooks

The related subject of positive energy representations for loop groups is represented in unavoidable reference

• A. Pressley, G. Segal, Loop groups, Oxford Math. Monographs, 1986.

Axiomatic quantum/statistical field theory and rigorous approaches to path integral

• N. N. Bogoliubov, A. A. Logunov, I. T. Todorov, Introduction to axiomatic quantum field theory, 1975

• R. Haag, Local quantum physics: fields, particles, algebras, Springer 1992 MR94d:81001, 1996 MR98b:81001

• Huzihiro Araki: Mathematical theory of quantum fields. Oxford University Press 1999 ZMATH entry.

• James Glimm, Arthur Jaffe, Quantum physics: a functional integral point of view, Springer

• Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn, Nonstandard methods in stochastic analysis and mathematical physics, Acad. Press 1986 (there is also a Dover 2009 edition, and a 1990 Russian translation)

Other reference lists

• http://en.wikibooks.org/wiki/Introduction_to_Mathematical_Physics/References (that list has mostly references in the area of theoretical physics and just a minority with rigor of mainstream mathematical physics)

• www.stringwiki.org – wiki contains entries on several dozens of string theory related topics, where each entry has several references, mainly papers available online

• Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics 83, Amer. Math. Soc. (2011)

• MathOverflow discussion: Companion to theoretical physics for working mathematicians

• reference list of Movshev’s sunysb course on QM

• geometrical background for Anton Kapustin‘s course at Caltech: list

• John Baez: How to Learn Math and Physics (here)

• Frédéric Paugam, Towards the mathematics of quantum field theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 59, Springer (2014), (a draft formerly available here).

• books about string theory

• various lectures notes in mathematical physics, Ulrich Theis’ list at Jena

category: reference