Resources for aspiring mathematical physicists
Basic online resources
MIT OpenCourseWare: Physics | Mathematics
The Feynman lectures in physics
t'Hooft's How to become a good theoretical physicist
David Tong: Lectures on Theoretical Physics
Leonard Susskind's Theoretical minimum
Perimeter Institute Recorded Seminar Archive: PIRSA
Note: For mathematical subjects mentioned below, please refer to the MSC classification.
General references on "geometry and physics"
Introductory to intermediate
Jost, Geometry and physics
Fecko, Differential geometry and Lie groups for physicists
Isham, Modern differential geometry for physicists
Burke, Applied differential geometry
Bleecker, Gauge theory and variational principles
Baez and Muniain, Gauge fields, knots and gravity
Frankel, The geometry of physics
Hamilton, Mathematical gauge theory
Advanced
Boss and Bleecker, Topology and analysis: the Atiyah-Singer index formula and gauge-theoretic physics
Nash, Differential topology and quantum field theory
Dunajski, Solitons, instantons and twistors
Deigne et al, Quantum fields and strings: a course for mathematicians
Recommended books and review articles by subject
Classical mechanics
Introductory
Goldstein et al, Classical Mechanics
Arnold, Mathematical Methods of Classical Mechanics
Intermediate
Jose and Saletan, Classical dynamics: A contemporary approach
Landau and Lifschitz, Mechanics
Advanced
Marsden and Ratiu, Introduction to mechanics and symmetry
Abraham and Marsden, Foundations of Mechanics
Spivak, Physics for mathematicians: Mechanics
Liebermann and Marle, Symplectic geometry and analytical mechanics
Souriau et al, Structure of dynamical systems: a symplectic view of physics
Guillemin & Sternberg, Symplectic techniques in physics
Relevant mathematics:
- Lagrangian formulation: basic differential geometry, variational calculus
- Hamiltonian formulation and symplectic reduction: symplectic and Poisson geometry
- Noether theorems: Jet bundles, Lie algebras and Lie groups, Lie group actions on manifolds
- ODEs and basic theory of geometric dynamical systems
Special relativity
Gourgoulhon, Special relativity in general frames
Naber, The geometry of Minkowski spacetime
Relevant mathematics:
- Linear and multi-linear algebra and tensor calculus
- Quadratic forms and quadratic spaces
- Affine spaces
- Lie groups and Lie algebras
- Basic theory of linear PDEs
Classical electrodynamics
Introductory
Purcell and Morrin, Electricity and magnetism
Zangwill, Modern Electrodynamics
Brau, Modern problems in classical electrodynamics
Greiner, Classical electrodynamics
Schwartz, Principles of electrodynamics
Landau & Lifschitz, The classical theory of fields
Intermediate
Jackson, Classical electrodynamics
Franklin, Classical electromagnetism
Advanced
Jentschura, Advanced classical electrodynamics
Hehl and Obukhov, Foundations of classical electrodynamics
Garrity, Electricity and magnetism for mathematicians
Scheck: Classical field theory: On Electrodynamics, Non-Abelian Gauge Theories and Gravitation
Relevant mathematics:
- ODEs and some PDEs
- some theory of distributions
- some basic differential geometry, including the geometry of Abelian principal bundles
Classical field theory
Soper, Classical field theory
Rubakov, Classical theory of gauge fields
Bleecker, Gauge theory and variational principles
Gockeler & Schucker, Differential geometry, gauge theory and gravity
Choquet-Bruhat and DeWitt-Morette, Analysis, manifolds and physics
Atiyah, Geometry of Yang-Mills fields
Atiyah, Gauge theories
Relevant mathematics:
- fiber bundles
- connections on principal and associated bundles
- characteristic classes and Chern-Weyl theory
- geometric PDEs
- spin geometry
- variational calculus
General relativity
Wald, General relativity
Kriele, Spacetime
Weinberg, Gravitation and cosmology
Straumann, General relativity
Hawking and Ellis, The large scale structure of space-time
Joshi, Gravitational collapse and spacetime singularities
Poisson, A relativist's toolkit
Choquet-Bruhat, General Relativity and the Einstein Equations
Relevant mathematics:
- Riemannian and semi-riemannian geometry
- basic differential topology
- geometric PDEs
- fiber bundles
- spin geometry
- Dirac operators and index theory
Quantum mechanics
Introductory to intermediate
Shankar, Principles of Quantum mechanics
Sakurai, Modern quantum mechanics
Messiah, Quantum mechanics, vols 1 & 2
Schwabl, Advanced quantum mechanics
Weinberg, Lectures on quantum mechanics
Jackson, Mathematics for quantum mechanics
Gustafson and Sigal, Mathematical concepts of quantum mechanics
Advanced and special topics
Tannoudji et al, Quantum mechanics, vols 1--3
Marino, Advanced topics in quantum mechanics
Sternberg, A mathematical companion to quantum mechanics
Prugovecki, Quantum mechanics in Hilbert space
Moretti, Spectral theory and quantum mechanics
Mackey, Mathematical foundations of quantum mechanics
Strocchi, An introduction to the mathematical structure of quantum mechanics
Faddeev and Yakubovskii, Lectures on quantum mechanics for mathematics students
Takhtajan, Quantum mechanics for mathematicians
Hall, Quantum theory for mathematicians
Glimm and Jaffe: Quantum physics: a functional integral point of view
Relevant mathematics:
- Basic functional analysis
- operator theory
- linear PDEs
- some measure and probability theory
- basic distribution theory
- C* algebras
- von Neumann algebras
- representation theory of Lie groups (including finite and countable groups).
Statistical physics
Cardy, Scaling and renormalization in statistical physics
Dorlas, Statistical mechanics
Sethna, Statistical mechanics: entropy, order parameters and complexity
Gallavotti, Statistical mechanics: a short treatise
Huang, Statistical mechanics
Georgii, Gibbs measures and phase transitions
Ellis, Entropy, large deviations and statistical mechanics
Lavis and Bell, Statistical mechanics of lattice systems, vols 1 & 2
Simon, The statistical mechanics of lattice gases
Friedli and Venenik, Statistical mechanics of lattice systems
Khinchin and Gamow, Mathematical foundations of statistical mechanics
Minlos, Introduction to mathematical statistical physics
Ruelle, Statistical mechanics: rigorous results
Thompson, Mathematical statistical mechanics
Goldenfeld, Lectures on phase transitions and the renormalization group
Yeomans, Statistical mechanics of phase transitions
McCoy and Wu, The two-dimensional Ising model
Bovier, Statistical mechanics of disordered systems
Lanford, Entropy and equilibrium states in classical statistical mechanics
Petersen, Ergodic theory
Bratelli and Robinson, Operator algebra and quantum statistical mechanics, vols 1 & 2
Manybody theory
Fetter and Walecka, Quantum theory of many-particle systems
Altland and Simons, Condensed matter quantum field theory
Rammer, Quantum field theory of non-equilibirum states
Negele and Orland, Quantum many-particle systems
Mahan, Many-particle physics
Coleman, Introduction to many-body physics
Abrikosov et al, Methods of quantum field theory in statistical physics
Fradkin, Field theories of condensed matter physics
Danielewicz, Quantum theory of nonequilibrium processes (review paper, parts I and II)
Lehmann, Mathematical methods of many-body quantum field theory
Quantum Field Theory
Srednicki, Quantum field theory
Schwartz, Quantum field theory and the standard model
Weinberg, The quantum theory of fields, vols 1 & 2
Itzikson and Zuber, Quantum field theory
Fradkin, Quantum field theory: an integrated approach
Folland, Quantum field theory: a tourist guide for mathematicians
Talagrand, What is a quantum field theory ?
Relevant mathematics:
- functional analysis, operator theory and PDEs
- distribution theory
- measure and probability theory
- representations of Lie groups and Lie algebras
- some combinatorics
- some representation theory of associative algebras.
Quantum field theory on curved spacetimes
Birrell and Davies, Quantum fields in curved space
Parker and Toms, Quantum field theory in curved spacetime
Fulling, Aspects of quantum field theory in curved spacetime
Wald, Quantum field theory in curved spacetime and black hole thermodynamics
Bar and Fredenhagen, Quantum field theory on curved spacetimes
Hack, Cosmological applications of algebraic quantum field theory in curved spacetimes
Relevant mathematics: Everything needed for QFT and GR.
Algebraic and axiomatic QFT
Wightman and Streater, PCT, spin and statistics and all that
Haag, Local quantum physics
Araki, Mathematical theory of quantum fields
Brunetti et al, Advances in algebraic quantum field theory
Relevant mathematics: Everything needed for QFT.
Conformal field theory
Di Francesco et al, Conformal field theory
Schottenloher, A mathematical introduction to conformal field theory
Frenkel and Ben-Zvi, Vertex algebras and algebraic curves
Fuchs, Affine Lie algebras and quantum groups
Kohno, Conformal field theory and topology
Lepowski and Li, Introduction to vertex operator algebras and their representations
Kac, Infinite-dimensional Lie algebras
Kac and Raina, Bombay lectures on infinite-dimensional Lie algebras and highest weight representations
Carter, Lie algebras of finite and affine type
Wakimoto, Lectures on infinite-dimensional Lie Algebra
Relevant mathematics:
- representations of infinite-dimensional Lie algebras
- representations of associative algebras
- Riemann surfaces
- some operator theory and theory of distributions
- quantum groups
- some category theory
Supersymmetry
Wess and Bagger, Supersymmetry and supergravity
Ceccotti, Supersymmetric field theories
Frappat et al, Dictionary on Lie superalgebras
Scheunert, The theory of Lie superalgebras: an introduction
Gorelik and Papi, Advances in Lie superalgebras
Tuynman, Supermanifolds and supergroups
Rogers, Supermanifolds
De Witt, Supermanifolds
Varadarajan, Supersymmetry for mathematicians
Carmeli et al, Mathematical foundations of supersymmetry
Cattaneo and Schaetz, Introduction to supergeometry
Kapranov, Supergeometry in mathematics and physics
Kessler, Supergeometry, super Riemann surfaces and and the superconformal action functional
Westra, Superrings and supergroups
Relevant mathematics:
- Lie superalgebras
- supermanifolds
- basic commutative and supercommutative algebra
- basic notions of scheme theory
- basic theory of quantum groups
Supergravity
Ortin, Gravity and strings
Freedman and van Proeyen, Supergravity
Castellani, D'Auria and Fre, Supergravity and superstrings: a geometric perspective
Sezgin, Survey of supergravities
Basic string theory
Polchiski, String theory, vols 1 & 2
Hatfield, Quantum field theory of point particles and strings
West, Introduction to strings and branes
Becker, Becker and Schwarz, String theory and M-theory: a modern introduction
Ibanez et al, String theory and particle physics: an introduction to string phenomenology
Deligne et al, Quantum fields and strings: a course for mathematicians
Jost, Bosonic strings: a mathematical treatment
Albeveiro et al, A mathematical introduction to string theory
Relevant mathematics: Everything from the above plus:
- theory of (pseudo-)harmonic maps
- some Teichmuller theory
- moduli spaces of Riemann surfaces
- index theory
- some topological K-theory
- some complex and Kahler geometry
- some algebraic geometry
- basic theory of G-structures
Advanced string theory
A. (Super)string perturbation theory
Witten, Superstring perrturbation theory via super Riemann surfaces: an overview
B. Calabi-Yau compactifications
Yau and Nadis, The shape of inner space
Greene, String theory on Calabi-Yau manifolds
Yau, A survey of Calabi-Yau manifolds
Joyce, Compact manifolds with special holonomy
Gross, Joyce & Huybrechts, Calabi-Yau manifolds and related geometries
C. Topological string theory
Marino, Chern Simons theory, matrix models and topological strings
Alim, Lectures on mirror symmetry and topological string theory
Nietzke and Vafa, Topological strings and their physcal applications
Marino, An introduction to topological string theory
D. Mirror Symmetry (ordinary and homological)
Abramovich et al, Enumerative invariants in algebraic geometry and string theory
Cox and Katz, Mirror symmetry and algebraic geometry
Hori et al, Mirror Symmetry
Bocklandt, A gentle introduction to homological mirror symmetry
Wang Yao, A glimpse into homological mirror symmetry
Aspinwall et al, Dirichlet branes and mirror symmetry
Huybrechts, Fourier-Mukai transforms in algebraic geometry
Auroux, A beginner's introduction to Fukaya categories
Smith, A symplectic prolegomenon
Seidel, Lectures on categorical dynamics and symplectic topology
Seidel, Fukaya categories and Picard-Lefschetz theory
Fukaya et al: Lagrangian intersection Floer theory: anomaly and obstruction, vols 1 & 2
Jarvis and Priddis (eds), Singularities, Mirror Symmetry and the Gauged Linear Sigma Model
E. Mathematical foundations of string and supergravity theories
Relevant mathematics:
- Complex and Kahler geometry
- Algebraic geometry (including toric geometry)
- noncommutative algebraic geometry
- Symplectic geometry and topology
- Gromow-Witten theory
- G-manifolds
- G-structures
- theory of foliations
- noncommutative geometry in the sense of Connes
- singularity theory and Thom-Mather theory (a.k.a. "catastrophe theory")
- Triangulated and derived categories
- Floer homology and Fukaya categories
- algebraic K-theory
- differential topology
- moduli spaces (differential and algebraic)
- basic theory of differential and algebraic stacks
- algebraic homotopy theory
- theory of gerbes
- simplicial methods and higher category theory
- geometric analysis