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发表于 2013-11-14 23:09
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landao的我读了，毕竟那是几十年前的东西，不可能要求他对量子场论和弦论有很深刻的了解。
Zeidler原本在莱比锡马普所工作，是一位严肃的数学家和物理学家。尽管弦论还没有实验做支撑，但是，我相信，如果他肯花大力气研究量子场论和弦论，那就必然有他的道理。记得N年前，读他4卷《非线性泛函分析》就觉得写的深入浅出，公式很干净，不会上下指标打架，或是一大堆的花体字母。有很详细的推导过程，前后照应也得挺好，在书中会告诉你，这个公式，哪一章哪一节哪一页提到过，详细推导在情哪一章哪一页，哪一章有应用什么的。不用我到处翻
根据他的描述，弦论几乎囊括了所有数学分支，这使得我觉得很兴奋。我以前在硕士阶段攻读了分析 概率 几何3个数学方向（嘿嘿，那时候德理解错了studiumsordnung），数学方面的积累算是很厚。现在读了他的这套丛书后，我开始对以前觉得满枯燥数论的也开始产生了强烈的学习动机，想看看如何把分析概率几何数论代数等这些数学工具一起运用，来描述这个物理世界，嘿嘿
大致上来说有：
(a) algebra, algebraic geometry, and number theory,
(b) analysis and functional analysis,
(c) geometry and topology,
(d) information theory, theory of probability, and stochastic processes,
(e) scientific computing.
更细化
Lie groups and symmetry, Lie algebras, Kac–Moody algebras (gauge groups,
permutation groups, the Poincar´e group in relativistic physics, conformal
symmetry),
• graded Lie algebras (supersymmetry between bosons and fermions),
• calculus of variations and partial differential equations (the principle of
critical action),
• distributions (also called generalized functions) and partial differential
equations (Green’s functions, correlation functions, propagator kernels, or
resolvent kernels),
• distributions and renormalization (the Epstein–Glaser approach to quantum
field theory via the S-matrix),
• geometric optics and Huygens’ principle (symplectic geometry, contact
transformations, Poisson structures, Finsler geometry),
• Einstein’s Brownian motion, diffusion, stochastic processes and the Wiener
integral, Feynman’s functional integrals, Gaussian integrals in the theory of
probability, Fresnel integrals in geometric optics, the method of stationary
phase
non-Euclidean geometry, covariant derivatives and connections on fiber
bundles (Einstein’s theory of general relativity for the universe, and the
Standard Model in elementary particle physics),
• the geometrization of physics (Minkowski space geometry and Einstein’s
theory of special relativity, pseudo-Riemannian geometry and Einstein’s
theory of general relativity, Hilbert space geometry and quantum states,
projective geometry and quantum states, K¨ahler geometry and strings,
conformal geometry and strings),
• spectral theory for operators in Hilbert spaces and quantum systems,
• operator algebras and many-particle systems (states and observables),
• quantization of classical systems (method of operator algebras, Feynman’s
functional integrals, Weyl quantization, geometric quantization, deformation
quantization, stochastic quantization, the Riemann–Hilbert problem,
Hopf algebras and renormalization),
• combinatorics (Feynman diagrams, Hopf algebras),
• quantum information, quantum computers, and operator algebras,
• conformal quantum field theory and operator algebras,
• noncommutative geometry and operator algebras,
• vertex algebras (sporadic groups, monster and moonshine),
• Grassmann algebras and differential forms (de Rham cohomology),
• cohomology, Hilbert’s theory of syzygies, and BRST quantization of gauge
field theories,
• number theory and statistical physics,
• topology (mapping degree, Hopf bundle, Morse theory, Lyusternik–Schnirelman
theory, homology, cohomology, homotopy, characteristic classes, homological
algebra, K-theory),
• topological quantum numbers (e.g., the Gauss–Bonnet theorem, Chern
classes, and Chern numbers, Morse numbers, Floer homology),
• the Riemann–Roch–Hirzebruch theorem and the Atiyah–Singer index theorem,
• analytic continuation, functions of several complex variables (sheaf theory),
• string theory, conformal symmetry, moduli spaces of Riemann surfaces,
and K¨ahler manifolds. |
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发表于 2013-11-13 11:15
