Explorations in Mathematical Physics: The Concepts Behind an Elegant Language

Front Cover
Springer Science & Business Media, Sep 15, 2006 - Science - 544 pages
1 Review

Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature?

This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and three-dimensional rotations. You'll see how the accelerated frames of special relativity tell us about gravity. On the journey, you'll discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology.

The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for one-forms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods.

Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.

 

 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Selected pages

Contents

The Language of Physics
1
A Trip Down Linear Lane
6
21 Vector Spaces and Matrices
8
22 Inner Products
10
23 Crystallography and the Cobasis
12
The Use of Determinants
16
241 Definition and Properties of the Determinant
18
242 Determinants Handedness and the nDimensional Cross Product
21
Details of Setting Up Adams and Eves Coordinates
266
The Elegance and Power of Tensor Notation
270
811 Honing the Vector Idea
274
812 Two Types of Vectors
276
82 Vectors and Coordinate Changes
278
83 Generalising the Idea of Vector Length
281
831 Coordinate Transformation of the Metric
283
84 A Natural Basis for Covectors
285

243 Volume of a Parallelepiped in a HigherDimensional Space
24
244 The Cobasis and the Wedge Product
25
Changing Spaces
28
251 Diagonalising a Matrix
31
26 Diracs Bracket Notation
41
27 Brackets and Hermitian Operators
43
28 Frequency and Wavenumber
47
29 Deriving the Fourier Transform Using Brackets
52
210 Commutators and the Indeterminacy Principle
58
211 Evolving Wave Functions in Time
65
2111 Brackets and Wave Function Evolution
70
212 The Transition to Quantum Mechanics
71
The Natural Language of Random Processes
77
32 The Privileged Sum of Squares
82
321 Sums of Squares and the Random Walk
89
33 Least Squares Analysis Bayes Theorem and the Matrix Pseudo Inverse
93
331 Least Squares Analysis for Curve Fitting
95
34 Time Constants to Describe Growth and Decay
105
341 The Poisson Statistics of Radioactive Decay
106
342 The Mean Life of the Decaying Nuclei
110
343 The Notion of a Probability per Second
112
35 Logarithms and Exponentials in Statistical Mechanics
114
Chief Star of Statistical Mechanics
117
353 Logarithms and Decibels
119
36 Signal Processing and the zTransform
122
361 Deriving the Fibonacci Sequence from the zTransform
123
362 Convolving to Smoothen a Signal
124
37 The Discrete Fourier Transform
128
371 Sampling Using Nyquists Theorem
129
372 Discretising the Fourier Transform
131
373 Interpolating Real Data with the DFT
135
Presenting Solutions to Problems
140
381 Tailoring a Formula to a Given Set of Units
141
382 Calculating a Nuclear Scattering Rate
142
A Roundabout Route to Geometric Algebra
147
41 Matrix Representation of an Orientation
148
411 Describing an Orientation by a Rotation
150
42 Calculating the Matrix for an Arbitrary Rotation
151
421 Deriving the Rotation Matrix Rn9 via Diagonalisation
153
422 Are Rotations Vectors?
155
43 Combining Two Rotations
157
44 Rotations Lead to Complex Numbers and Quaternions
158
441 Tidying Up the Placeholders
163
45 Producing a Geometric Algebra
166
46 Rotations in Popular Usage
170
461 Describing an Orientation by Using Three Rotations
171
462 Confusing Euler Angle Orientation with Incremental Rotation
176
463 Quaternions Used in Computer Graphics
182
Special Relativity and the Lorentz Transform
185
52 The Postulates of Special Relativity
186
53 The Lorentz Transform
187
531 Paradoxes or Conundrums?
189
532 How Does Each Frame Measure the Other as Ageing Slowly?
192
54 The Symmetry of the Lorentz Transform
195
55 Using Radar to Derive Time Dilation
197
56 Space Time Becomes Spacetime
200
57 Spacetime Diagrams and Hyperbolic Geometry
202
58 The Lorentz Transform in an Arbitrary Direction
204
59 Energy and Momentum in Special Relativity
206
591 Einsteins Relation of Mass and Energy
210
FourVectors and the Road to Tensors
213
62 Running Nonrelativistically
216
63 Running Relativistically
218
632 The Length of the FourVelocity
223
65 Introducing Covectors and Fully Covariant Notation
231
Accelerated Frames Onward to the Principle of Covariance
233
71 The Clock Postulate
235
711 The Interval for Noninertial Observers
238
72 Coordinates for the Accelerated Frame
240
73 The Twin Conundrum
252
731 Making Eve Accelerate Uniformly
256
732 How the Twins Record Each Others Trips
258
74 A Glance Ahead to Gauge Theory
263
75 Covariant Notation and Generalising the Clock Postulate
264
841 Raising and Lowering Indices
290
85 Tensor Components with More than Two Indices
292
851 Bases for More General Tensors
294
852 The Metric Tensor Versus the Metric Matrix
296
86 The Gradient Operator and the Cobasis
297
861 The Gradient Operator in Fully Covariant Notation
301
862 Is a Metric Needed?
306
87 Normalised Basis Vectors
308
871 The Normalised Polar Basis in Celestial Mechanics
310
872 An Example of Using Vectors to Calculate an Effective Potential
312
873 Some Final Remarks on Vector Terminology
315
The Cross Product in General Coordinates
320
89 From Vector Calculus to Tensor Calculus
322
891 The Divergence in Tensor Notation
323
892 Christoffel Symbols for Cartesian Coordinates
326
893 Preparing to Make the Divergence Covariant
329
894 The Covariant Laplacian
332
895 The Covariant Curl
333
810 Exterior Calculus and the Theorems of Stokes and Gauss in Higher Dimensions
335
Curvature and Differential Geometry
349
911 Curves on Surfaces
354
Curves with No Geodesic Curvature
359
93 The Curvature of a Surface
362
931 The Method of Lagrange Multipliers
365
94 Gausss Extraordinary Theorem
369
95 Translating Vectors by Parallel Transport
373
96 Relating Parallel Transport to Curvature
378
The GaussBonnet Theorem in Euclidean 3Space
382
Variational Calculus and Field Theory
387
102 The Concept of a Field
389
103 The Lagrangian Formalism
391
1031 Lagranges Equation
393
1032 Other Variational Approaches
395
Hamiltons Principle
396
1034 Nothers Theorem and Lagrangian Invariances
398
First Steps to a Field Theory
399
1036 Nothers Theorem for a Scalar Field
402
104 Building a Lagrangian
404
1041 A Relativistic Lagrangian for a Charge in an EM Field
407
105 Producing the Schrodinger Equation
415
Fields Describe Particles Too
417
The Klein Gordon Equation
421
1062 A Route to the Dirac Equation
422
107 Gauge Theory and Quantum Electrodynamics
427
1072 A Gauge Transformation for the Dirac Lagrangian
429
108 The PathIntegral Approach to Quantum Mechanics
432
1081 Path Integrals Give the Schrodinger Equation
435
The Language of Decoherence
438
The Green Function Approach to Solving Field Equations
445
112 Deriving the Green Function for V2 via Fourier Theory
449
1121 The Other Way of Calculating the Integral 1120
456
113 Solving Maxwells Equations via the Green Function Approach
460
114 Variations on the Green Function Solution of Maxwells Equations
468
115 Fluctuation Dissipation and Times Arrow
470
Airliners Black Holes and Cosmology The ABC of General Relativity
471
122 The Pound Rebka Snider Experiments
474
123 A Space or Spacetime Description of Gravity?
476
1231 A Route to Curved Spacetime from Lagrangian Mechanics
478
1232 Free Particles Geodesics and Locally Inertial Frames
483
1233 Quantities That Are Conserved on Geodesics
489
124 A Path to Einsteins Equation
490
The Schwarzschild Metric
495
1251 Deriving Gravitational Redshift Again
498
126 The Schwarzschild Black Hole
500
1261 Tensor Components and Physical Measurements
506
Cartans Structural Equations
508
128 The Variational Approach to Einsteins Equation
511
1281 Adding Extra Field Terms to the Lagrangian Density
515
The Cosmological Constant
518
1283 Joining Electromagnetism to Gravity
519
1284 Path Integrals in General Relativity
523
Proper Distances in Cosmology
524
Index
531
Copyright

Other editions - View all

Common terms and phrases

About the author (2006)

"With enjoyable and sometimes surprising excursions along the way, the journey provides a fresh look at many familiar topics, as it takes us from basic linear mathematics to general relativity... look forward to having your geometric intuition nourished and expanded by the author’s intelligent commentaries."

Eugen Merzbacher, University of North Carolina, Chapel Hill

Have you ever wondered why the language of modern physics centres on geometry? Or how quantum operators and Dirac brackets work? What a convolution really is? What tensors are all about? Or what field theory and lagrangians are, and why gravity is described as curvature?

This book takes you on a tour of the main ideas forming the language of modern mathematical physics. Here you will meet novel approaches to concepts such as determinants and geometry, wave function evolution, statistics, signal processing, and three-dimensional rotations. You will see how the accelerated frames of special relativity tell us about gravity. On the journey, you will discover how tensor notation relates to vector calculus, how differential geometry is built on intuitive concepts, and how variational calculus leads to field theory. You will meet quantum measurement theory, along with Green functions and the art of complex integration, and finally general relativity and cosmology.

The book takes a fresh approach to tensor analysis built solely on the metric and vectors, with no need for one-forms. This gives a much more geometrical and intuitive insight into vector and tensor calculus, together with general relativity, than do traditional, more abstract methods.

Don Koks is a physicist at the Defence Science and Technology Organisation in Adelaide, Australia. His doctorate in quantum cosmology was obtained from the Department of Physics and Mathematical Physics at Adelaide University. Prior work at the University of Auckland specialised in applied accelerator physics, along with pure and applied mathematics.

Bibliographic information