Explorations in Mathematical Physics: The Concepts Behind an Elegant Language
Explorations in Mathematical Physics: The Concepts Behind an Elegant Language
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DOI：
10.5860/choice.44-4509
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发表时间：
2006-09
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影响因子：
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通讯作者：
Don Koks
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文献类型：
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作者：
Don Koks
序言。物理学的语言-线性车道之旅：向量空间和矩阵。内在产品晶体学和Cobasis。发现领域和领域：决定因素的使用。对角化和相似矩阵：变化的空间。狄拉克的符号括号和埃尔米特算子。频率和波数。使用括号推导傅立叶变换。换位子和不确定性原理。波函数在时间上的演化向量子力学的过渡。随机过程的自然语言：从条形图到直方图特权平方和。-最小二乘分析、贝叶斯定理和矩阵伪逆。描述增长和衰减的时间常数。统计力学中的对数和指数。信号处理和z变换。离散傅立叶变换正确和令人信服的：提出问题的解决方案。几何代数的迂回路线：方向的矩阵表示。计算任意旋转的矩阵。结合两个旋转。旋转导致复数和四元数。生成一个“几何”代数。常用的旋转。狭义相对论和洛伦兹变换：从不变性导出多普勒频移。狭义相对论的基本原理洛伦兹变换洛伦兹变换的对称性利用雷达推导时间膨胀。时空变成时空。时空图与双曲几何任意方向的洛伦兹变换狭义相对论中的能量和动量四向量和张量之路：数密度和通量密度。非相对论运行。运行相对论。其他四个向量的例子。介绍协向量和完全协变记法。加速帧：前进到协方差原理：时钟假设。加速帧的坐标。双胞胎之谜一个关于规范理论（Gauge Theory）的问题。协变记法和时钟公设的推广附录：设置亚当和夏娃坐标的详细信息。-张量记法的优雅和力量：回到向量，以更一般的方式。向量和坐标变化。向量长度的概念。共向量的基础。具有两个以上指数的张量分量。梯度算子和Cobasis。归一化基向量。体积元素、行列式和叉积。从向量演算到张量演算。外部微积分和斯托克斯和高斯定理在更高的维度。曲率与微分几何：平面上的曲率测地线：没有测地线曲率的曲线。曲面的曲率。高斯的非凡定理通过并行传输转换矢量。平行运输与曲率的关系从几何到拓扑：欧几里得三维空间中的高斯-博内定理变分微积分与场论：苍蝇与火车的故事场的概念。拉格朗日形式主义建立一个Lagrangian。产生薛定谔方程。量子场论：场也描述粒子！规范理论与量子电动力学量子力学的路径积分方法密度矩阵：退相干的语言。解场方程的绿色函数方法：绿色函数的概念。利用傅立叶理论推导拉普拉斯算子的绿色函数。通过绿色函数方法求解麦克斯韦方程。麦克斯韦方程组的绿色函数解的变化涨落耗散与时间之箭客机，黑洞和宇宙学：广义相对论的ABC：等效原理。的
Preface.- The Language of Physics.- A Trip Down Linear Lane: Vector Spaces and Matrices.- Inner Products.- Crystallography and the Cobasis.- Finding Areas and Volumes: The Use of Determinants.- Diagonalisation and Similar Matrices: Changing Spaces.- Dirac's Bracket Notation.- Brackets and Hermitian Operators.- Frequency and Wavenumber.- Deriving the Fourier Transform Using Brackets.- Commutators and the Indeterminacy Principle.- Evolving Wave Functions in Time.- The Transition to Quantum Mechanics.- The Natural Language of Random Processes: From Bar Graphs to Histograms.- The Privileged Sum of Squares.- Least Squares Analysis, Bayes' Theorem, and the Matrix Pseudo Inverse.- Time Constants to Describe Growth and Decay.- Logarithms and Exponentials in Statistical Mechanics.- Signal Processing and the z-Transform.- The Discrete Fourier Transform.- Correct and Convincing: Presenting Solutions to Problems.- A Roundabout Route to Geometric Algebra: Matrix Representation of an Orientation.- Calculating the Matrix for an Arbitrary Rotation.- Combining Two Rotations.- Rotations Lead to Complex Numbers and Quaternions.- Producing a 'Geometric' Algebra.- Rotations in Popular Usage.- Special Relativity and the Lorentz Transform: Deriving the Doppler Shift from an Invariance.- The Postulates of Special Relativity.- The Lorentz Transform.- The Symmetry of the Lorentz Transform.- Using Radar to Derive Time Dilation.- Space-time Becomes Spacetime.- Spacetime Diagrams and Hyperbolic Geometry.- The Lorentz Transform in an Arbitrary Direction.- Energy and Momentum in Special Relativity.- Four-Vectors and the Road to Tensors: Number Density and Flux Density.- Running Nonrelativistically.- Running Relativistically.- Examples of Other Four-Vectors.- Introducing Covectors and Fully Covariant Notation.- Accelerated Frames: Onward to the Principle ofCovariance: The Clock Postulate.- Coordinates for the Accelerated Frame.- The Twin Conundrum.- A Glance Ahead to Gauge Theory.- Covariant Notation and Generalising the Clock Postulate.- Appendix: Details of Setting Up Adam's and Eve's Coordinates.- The Elegance and Power of Tensor Notation: Back to Vectors, in a More General Way.- Vectors and Coordinate Changes.- Generalising the Idea of Vector Length.- A Basis for Covectors.- Tensor Components With More Than Two Indices.- The Gradient Operator and the Cobasis.- Normalised Basis Vectors.- Volume Elements, Determinants, and Cross Products Again.- From Vector Calculus to Tensor Calculus.- Exterior Calculus and the Theorems of Stokes and Gauss in Higher Dimensions.- Curvature and Differential Geometry: Curvature in the Plane.- Geodesics: Curves with No Geodesic Curvature.- The Curvature of a Surface.- Gauss's Extraordinary Theorem.- Translating Vectors by Parallel Transport.- Relating Parallel Transport to Curvature.- From Geometry to Topology: the Gauss-Bonnet Theorem in Euclidean 3-Space.- Variational Calculus and Field Theory: The Story of the Fly and the Train.- The Concept of a Field.- The Lagrangian Formalism.- Building a Lagrangian.- Producing the Schrodinger Equation.- Quantising Field Theory: Fields Describe Particles Too!.- Gauge Theory and Quantum Electrodynamics.- The Path Integral Approach to Quantum Mechanics.- Density Matrices: the Language of Decoherence.- The Green Function Approach to Solving Field Equations: The Idea of a Green Function.- Deriving the Green Function for the Laplacian Operator via Fourier Theory.- Solving Maxwell's Equations via the Green Function Approach.- Variations on the Green Function Solution of Maxwell's Equations.- Fluctuation-Dissipation and Time's Arrow.- Airliners, Black Holes, and Cosmology: The ABC of General Relativity: The Equivalence Principle.- The