Roger Penrose, one of the most accomplished scientists of our time, presents the only comprehensive and comprehensible account of the physics of the universe. From the very first attempts by the Greeks to grapple with the complexities of our known world to the latest application of infinity in physics, The Road to Reality carefully explores the movement of the smallest atomic particles and reaches into the vastness of intergalactic space. Here, Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing.

Preface

Acknowledgements

Notation

Prologue

1 The roots of science

　1.1 The cluest for the forces that shape the world

　1.2 Mathematical truth

　1.3 Is Plato's mathematical world 'real'?

　1.4 Three worlds and three deep mysteries

　1.5 The Good, the True, and the Beautiful

2 An ancient theorem and a modern question

　2.1 The Pythagorean theorem

　2.2 Euclid's postulates

　2.3 Similar-areas proof of the Pythagorean theorem

　2.4 Hyperbolic geometry: conformal picture

　2.5 Other representations of hyperbolic geometry

　2.6 Historical aspects of hyperbolic geometry

　2.7 Relation to physical space

3 Kinds of number in the physical world

　3.1 A Pythagorean catastrophe?

　3.2 The real-number system

　3.3 Real numbers in the physical world

　3.4 Do natural numbers need the physical world?

　3.5 Discrete numbers in the physical world

4 Magical complex numbers

　4.1 The magic number 'i'

　4.2 Solving equations with complex numbers

　4.3 Convergence of power series

　4.4 Caspar Wessel's complex plane

　4.5 How to construct the Mandelbrot set

5 Geometry of logarithms, powers, and roots

　5.1 Geometry of complex algebra

　5.2 The idea of the complex logarithm

　5.3 Multiple valuedness, natural logarithms

　5.4 Complex powers

　5.5 Some relations to modern particle physics

6 Real-number calculus

　6.1 What makes an honest function?

　6.2 Slopes of functions

　6.3 Higher derivatives; C∞-smooth functions

　6.4 The 'Eulerian' notion of a function?

　6.5 The rules of differentiation

　6.6 Integration

7 Complex-number calculus

　7.1 Complex smoothness; holomorphic functions

　7.2 Contour integration

　7.3 Power series from complex smoothness

　7.4 Analytic continuation

8 Riemann surfaces and complex mappings

　8.1 The idea of a Riemann surface

　8.2 Conformal mappings

　8.3 The Riemann sphere

　8.4 The genus of a compact Riemann surface

　8.5 The Riemann mapping theorem

9 Fourier decomposition and hyperfunctions

　9.1 Fourier series

　9.2 Functions on a circle

　9.3 Frequency splitting on the Riemann sphere

　9.4 The Fourier transform

　9.5 Frequency splitting from the Fourier transform

　9.6 What kind of function is appropriate?

　9.7 Hyperfunctions

10 Surfaces

11 Hypercomplex numbers

12 Manifolds of n dimensions

13 Symmetry groups

14 Calculus on manifolds

15 Fibre bundles and gauge connections

16 The ladder of infinity

17 Spacetime

18 Minkowskian geometry

19 The classical fields of maxwell and Einstein

20 Lagrangians and Hamiltonians

21 The quantum particle

22 Quantum algebra, geometry, and spin

23 The entangled quantum world

24 Dirac's electron and antiparticles

25 The standard model of particle physics

26 Quantum field theory

27 The Big Bang and its thermodynamic legacy

28 Speculative theories of the early universe

29 The measurement paradox

30 Gravity's rode in quantum state reduction

31 Supersymmetry, surpra-cimensionality,and strings

32 Einstein's narrower path;loop variables

33 More radical perspectives; twistor theory

34 Where lies the road to reality?

Epilogue

Bibliography

Index