Heller Michael
Some mathematical Physics for Philosophers
Pontifical Council for Culture and Pontifical Gregorian University, 14 febbraio 2006, Libreria Editrice Vaticana
Paperback, pp. 128
Language: English
ISBN-13: 978-88-2097-724-5
Price: EUR 18,00
Description:
The STOQ Project Research Series - 1. The volume has been developed from an introductory course on Mathematical Physics delivered at the Gregorian University, Rome 2004. It is an original and unique attempt to explain mathematical methods in physics to students of phylosophy and theology whose last contact with mathematics and physics was during their secondary education. The book does not intend to offer a full course on mathematical physics, but to give some feeling of the beauty and effectiveness of modern mathematical methods in physics based on its intuitions and convey some appreciation of the philosophical significance of the mathematical modeling of the world. Even though this book is not geared to training professionals in the field of mathematical physics, mathematical rigor is not abandoned completely. The emphasis is placed on understanding concepts rather than completing exercises or solving problems.
The course presents schematically 13 lessons and requires substantial input from the lecturer or a tutor.
Table of Contents
Foreword
Presentation by Card. Paul Poupard
Introductory Remarks by the Author
1. The World of Structure
1.1 Nature of Mathematics
1.2 Basic Ideas of Category Theory
1.3 The Role of Mathematics in Physics
Part One: Mathematics of Relativity
2. The Manifold Model of Space-Time
2.1 Topology and Continuity
2.2 Smooth Manifold
2.3 Tangent Space
2.4 Physical Interpretation
3. Metric Model of Space-Time
3.1 Riemannian Metric
3.2 Lorentz Manifold
3.3 Relativistic Model of Space-Time
4. Geometry of Special Relativity
4.1 Space-Time Interval
4.2 Geometry of Motion
4.3 Lorentz Transformations
4.4 Relativistic Effects
5. Lorentz Frame Bundle
5.1 Introductory Remarks
5.2 Bundles and Fibre Bundles
5.3 Frame Bundle over Space-Time
6. The Newtonian Limit
6.1 When the Speed of Light Goes to Infinity
6.2 Galilean Relativity
7. Elements of General Relativity
7.1 From Inertial Frames to Gravity
7.2 The Galilean Principle
7.3 Einstein’s Field Equations
7.4 From General Relativity to Special Relativity
8. Elements of Friedman-Lemaître Cosmology
8.1 The Cosmological Problem
8.2 The Cosmological Principle
8.3 The Robertson-Walker Metric
8.4 The “Matter Content” of the Universe
8.5 Friedman’s Equation and Its Solutions
8.6 Critical Density
8.7 Other Cosmological Models
9. From Aristotle to Einstein
9.1 Dynamics and Geometry
9.2 Aristotelian Space-Time
9.3 Space-Time of Classical Mechanics without Gravity
9.4 Space-Time of Classical Mechanics with Gravity
9.5 Space-Time of Special and General Theories of Relativity
9.6 Comment
Part Two: Mathematics of Quanta
10. Quantum States and Hilbert Spaces
10.1 States of Quantum Systems
10.2 Topological Vector Space
10.3 Banach Space
10.4 Hilbert Space
10.5 Comment
10.6 The State Space of Quantum Mechanics
10.7 Superposition of States
11. Observables and Measurement
11.1 Observables
11.2 Measurement Results
11.3 Reduction of the State Vector
11.4 Penrose’s Comment
12. Quantum Dynamics
12.1 Schrödinger’s Equation
12.2 Heisenberg’s Equation
12.3 Indeterminacy Relations
12.4 Comments
13. Quantum Logic
13.1 The Dual Hilbert Space
13.2 Logical Structure of Quantum Mechanics
13.3 New Axioms for Quantum Mechanics
13.4 A Non-Aristotelian Logic
14. Algebraic Formulation of Quantum Mechanics
14.1 C*-algebras
14.2 New Formalism for Quantum Mechanics
14.3 Algebra of Observables
14.4 Comment
Bibliographical Notes (by the Author) - Lectures
Lecture 4- Geometry of Special Relativity
More advanced, but still accessible, introductions to special relativity can be found in the following books:
1. G.F.R. Ellis and R. Williams, Flat and Curved Space-Times, Clarendon Press, Oxford, 1988, chapters 1-4.
2. B.F. Schuttz, A First Course in General Relativity, Cambridge University Press, Cambridge - London - New York, 1985, chapters 1-4.
3. E.F. Taylor and J.A. Wheeler, Spacetime Physics, Freeman and Co., San Francisco - London, 1966.
Lecture 7 - Elements of General Relativity
The books listed at the end of Lecture 4 contain also chapters devoted to general relativity. As a less “technical” reading I recommend:
• R. Geroch, General Relativity from A to B, The University of Chicago Press, Chicago - London, 1978.
Lecture 8 - Friedman-Lemaître Cosmology
There are many books on cosmology with varying difficulty levels worth reading. A good introduction (with some mathematics) is:
1. A. Liddle, An Introduction to Modern Cosmology, Wiley and Sons, Chichester, 1999.
For reading about cosmology, I recommend:
2. B. Greene, The Elegant Universe, A.W. Norton and Company, 1999 (my favourite!).
Or, from a more observational side:
3. M. Rees, Before the Beginning, Simon and Schuster, London, 1998.
There are also chapters devoted to cosmology in the books listed at the end of Lecture 4.
Lecture 9 - From Aristotle to Einstein
Reconstruction of former dynamical theories in terms of modern geometric structures can be found in the following publications:
1. Ehlers, J., “The Nature and Structure of Space-Time”, in: The Physicist’s Conception of Nature, ed.: J. Mehra, Raidel, 1973, 51-91.
2. Heller, M., Theoretical Foundations of Cosmology, World Scientific, 1992, Appendix, 112-125.
3. Penrose, R., “Structure of Space-Time”, in: Battelle Rencontres, eds.: C.M. De Witt, J.A. Wheeler, Benjamin, 1968, 121-235.
4. Raine, D.J. and Heller, M., The Science of Space-Time, Pachart, 1981.
Lecture 10 – Quantum States and Hilbert Spaces
Usually, books on quantum mechanics are either popular with no mathematics, or highly technical. As a supplementary reading I would recommend:
Elementary introduction:
• J. C. Polkinghorn, The Quantum World, Penguin Books, London, 1990.
Giving a deeper insight:
• R. Penrose, Imperor’s New Mind, Oxford University Press, New York - Oxford, 1989, Chapter 6.
Requiring more advanced mathematical preparation but nicely discussing conceptual and structural aspects:
• C. J. Isham, Lectures on Quantum Theory, Imperial College Press, Singapore - London, 1995.
Bibliographical Notes (by the Author) - Years
• E.F. Taylor and J.A. Wheeler, Spacetime Physics, Freeman and Co., San Francisco - London, 1966.
• Penrose, R., “Structure of Space-Time”, in: Battelle Rencontres, eds.: C.M. De Witt, J.A. Wheeler, Benjamin, 1968, 121-235.
• Ehlers, J., “The Nature and Structure of Space-Time”, in: The Physicist’s Conception of Nature, ed.: J. Mehra, Raidel, 1973, 51-91.
• R. Geroch, General Relativity from A to B, The University of Chicago Press, Chicago - London, 1978.
• Raine, D.J. and Heller, M., The Science of Space-Time, Pachart, 1981.
• B.F. Schuttz, A First Course in General Relativity, Cambridge University Press, Cambridge - London - New York, 1985, chapters 1-4.
• G.F.R. Ellis and R. Williams, Flat and Curved Space-Times, Clarendon Press, Oxford, 1988, chapters 1-4.
• R. Penrose, Imperor’s New Mind, Oxford University Press, New York - Oxford, 1989, Chapter 6.
• J. C. Polkinghorn, The Quantum World, Penguin Books, London, 1990.
• Heller, M., Theoretical Foundations of Cosmology, World Scientific, 1992, Appendix, 112-125.
• C. J. Isham, Lectures on Quantum Theory, Imperial College Press, Singapore - London, 1995.
• M. Rees, Before the Beginning, Simon and Schuster, London, 1998.
• A. Liddle, An Introduction to Modern Cosmology, Wiley and Sons, Chichester, 1999.
• B. Greene, The Elegant Universe, A.W. Norton and Company, 1999.
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