求推荐讲解不定正交群的专业书籍（含定理证明与物理关联）

Hey there! I've got a few solid book recommendations for diving into indefinite orthogonal groups that should check all your boxes—covering foundational theorems, their proofs, and even connections to physics. Let's break them down:

• 《Lie Groups, Lie Algebras, and Representations: An Elementary Introduction》by Brian Hall

  • This is a fantastic entry point for learning about Lie groups, and it dedicates a section specifically to indefinite orthogonal groups. You’ll find rigorous, easy-to-follow proofs for foundational results like $O(p,q)\cong O(q,p)$ for all $p,q\in \mathbb{N}$.
  • It also places heavy emphasis on the Lorentz group ($O(3,1)$), the most physically relevant indefinite orthogonal group, walking through its central role in special relativity. This makes it perfect for linking the math to real-world physics applications.

• 《Symmetry, Representations, and Invariants》by Goodman and Wallach

  • If you want a deeper algebraic perspective, this book delivers. It systematically covers the structure of indefinite orthogonal groups, including detailed proofs of their isomorphism properties and representation theory fundamentals.
  • Throughout the text, it connects these groups to mathematical physics—touching on applications like symmetry analysis in quantum mechanics—so you’ll gain insight into why these groups matter beyond pure math.

• 《Orthogonal Lie Algebras and Their Representations》by Anthony Knapp

  • Knapp’s work is known for its precision, and this book focuses squarely on orthogonal Lie groups and their algebras, with indefinite cases as a core topic. You’ll find thorough proofs of $O(p,q)\cong O(q,p)$ and other key theorems, plus deep dives into root systems and advanced structure.
  • It also includes discussions of how indefinite orthogonal groups appear in particle physics (e.g., in gauge field theories), making it a great pick if you want to balance mathematical rigor with physical context.

• 《Mathematical Methods for Physicists》by Arfken, Weber, Harris

  • A staple for physics students, this book doesn’t cover all $O(p,q)$ cases comprehensively, but it uses the Lorentz group as a gateway to indefinite orthogonal groups. It explains their basic properties and ties everything directly to special relativity’s spacetime transformations, giving you an intuitive grasp of their physical importance.
  • If your primary goal is to understand how these groups apply to physics, this practical, example-driven resource will serve you well.

内容的提问来源于stack exchange，提问作者Sito