Mathematical Physics – Sadri Hassani – 2nd Edition


The goal of this book is to expose the reader to the indispensable role that mathematics—often very abstract—plays in modern physics. Starting with the notion of vector spaces, the first half of the book develops topics as diverse as algebras, classical orthogonal polynomials, Fourier analysis, complex analysis, differential and integral equations, operator theory, and multi-dimensional Green’s functions. The second half of the book introduces groups, manifolds, Lie groups and their representations, Clifford algebras and their representations, and fiber bundles and their applications to differential geometry and gauge theories.

This second edition is a substantial revision of the first one with a complete rewriting of many chapters and the addition of new ones, including chapters on algebras, representation of Clifford algebras and spinors, fiber bundles, and gauge theories. The spirit of the first edition, namely the balance between rigor and physical application, has been maintained, as is the abundance of historical notes and worked out examples that demonstrate the “unreasonable effectiveness of mathematics” in modern physics.

Einstein has famously said, “The most incomprehensible thing about nature is that it is comprehensible.” What he had in mind was reiterated in another one of his famous quotes concerning the question of how ” … mathematics, being after all a product of human thought, is so admirably appropriate to the objects of reality.” It is a question that comes to everyone’s mind when encountering the highly abstract mathematics required for a deep understanding of modern physics. It is the experience that Eugene Wigner so profoundly described as “the unreasonable effectiveness of mathematics in the natural sciences.”

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    1. Mathematical Preliminaries

    2. Vectors and Linear Maps

    3. Algebras

    4. Operator Algebra

    5. Matrices

    6. Spectral Decomposition

    7. Hilbert Spaces

    8. Classical Orthogonal Polynomials

    9. Fourier Analysis

    10. Complex Calculus

    11. Calculus of Residues

    12. Advanced Topics

    13. Separation of Variables in Spherical Coordinates

    14. Second-Order Linear Differential Equations

    15. Complex Analysis of SOLDEs

    16. Integral Transforms and Differential Equations

    17. Introductory Operator Theory

    18. Integral Equations

    19. Sturm-Liouville Systems

    20. Green’s Functions in One Dimension

    21. Multidimensional Green’s Functions: Formalism

    22. Multidimensional Green’s Functions: Applications

    23. Group Theory

    24. Representation of Groups

    25. Representations of the Symmetric Group

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