Geometric Methods in Mathematical Physics I

Geometric Methods in Mathematical Physics I: Tensors, Special Relativity, Spinors provides a rigorous and pedagogically coherent introduction to the algebraic and geometric language used in modern mathematical physics.The book develops the foundations of multilinear algebra and tensor calculus from first principles, covering dual and conjugate spaces, multilinear maps, tensor products, the universal property of tensor products, tensor algebra, abstract index notation, exterior algebra, scalar products, metric tensors, pseudo-tensors, and tensor densities. Particular attention is given to the relationship between the abstract mathematical definition of tensors and the practical index notation commonly used in physics.The text then applies these tools to group theory, representation theory, and relativistic physics. It discusses tensor products of group representations, symmetry of tensors, Grassmann algebra, pseudo-orthogonal groups, and polar decomposition. These topics prepare the reader for a geometric presentation of Special Relativity, including Minkowski spacetime, Lorentz and Poincaré transformations, relativistic kinematics and dynamics, four-momentum, conservation laws, four-force, and the stress-energy tensor for macroscopic systems.The final chapters introduce the structure of the Lorentz group, its Lie algebra, boosts and rotations, the relation between SL(2,C) and SO(1,3), and the basic theory of Weyl and Dirac spinors, including the Dirac equation.

Designed for graduate students and advanced readers in mathematics, physics, and mathematical physics, the book offers a compact but rigorous bridge between multilinear algebra, tensor methods, Special Relativity, and spinorial techniques. It is suitable both as a course text and as a reference for readers seeking a solid mathematical foundation for the geometric methods of theoretical physics.