This book explores the premise that a physical theory is an interpretation of the analytico–canonical formalism. Throughout the text, the investigation stresses that classical mechanics in its Lagrangian formulation is the formal backbone of theoretical physics. The authors start from a presentation of the analytico–canonical formalism for classical mechanics, and its applications in electromagnetism, Schrödinger's quantum mechanics, and field theories such as general relativity and gauge field theories, up to the Higgs mechanism.The analysis uses the main criterion used by physicists for a theory: to formulate a physical theory we write down a Lagrangian for it. A physical theory is a particular instance of the Lagrangian functional. So, there is already an unified physical theory. One only has to specify the corresponding Lagrangian (or Lagrangian density); the dynamical equations are the associated Euler–Lagrange equations. The theory of Suppes predicates as the main tool in the axiomatization and examples from the usual theories in physics. For applications, a whole plethora of results from logic that lead to interesting, and sometimes unexpected, consequences.This volume looks at where our physics happen and which mathematical universe we require for the description of our concrete physical events. It also explores if we use the constructive universe or if we need set–theoretically generic spacetimes.
Author(s): Newton C. A. da Costa, Francisco Antonio Doria
Series: Synthese Library, 441
Publisher: Springer
Year: 2022
Language: English
Pages: 204
City: Cham
Foreword
Contents
1 Preliminary
1.1 Hilbert's 6th Problem: The Mathematical Treatment of the Axioms of Physics
1.2 A Few Important Questions
1.3 A Theory of Time?
1.4 Physics as the Theory of Motion?
1.5 Possible Pitfalls
1.6 An Example
1.7 Historic Notes
1.8 Electromagnetism
1.9 General Relativity and Gravitation
1.10 Classical Mechanics
1.11 From Classical to Quantum Mechanics
1.12 To Sum It Up
Part I Physics: A Primer
2 Classical Mechanics
2.1 Motivation for Lagrange's Equations
2.2 Motivation for Hamilton's Equations
3 Variational Calculus
3.1 Variations
4 Lagrangian Formulation
4.1 Lagrange's Equations Out of a Variational Principle
4.2 Non-Potential Forces, Non-Holonomic Constraints
4.3 Lagrange Multipliers
5 Hamilton's Equations
5.1 Hamiltonian Formulation
5.2 Meaning of H
5.3 Phase Space
6 Hamilton–Jacobi Theory
6.1 Canonical Transformations
6.2 The Hamilton–Jacobi Equation
7 Where the Action Is
7.1 Höhepunkt
7.2 Geodesics as the Path of a System
7.3 The Harmonic Oscillator; Planetary Motion
7.4 Areal Velocity
7.5 The Equation for r
8 From Classical to Quantum
8.1 Wave Motion Associated to W
8.2 Schrödinger's Time-Independent Equation
8.3 Brief Comments
9 Field Theory
10 Electromagnetism
10.1 Covariant or 4-Dimensional Formulation, First Version
10.2 Einstein's Summation Convention, First Version
10.3 Covariant Current, Covariant Potential
10.4 The Electromagnetic Field Fμν
10.5 A Lagrangian Density for the Empty-Space MaxwellEquations
10.6 Dirac-like Formulation
11 Special Relativity
12 General Relativity
12.1 Tensor Calculus
12.2 Paths, Contravariant Vectors; Summation Convention, Second Version
12.3 Gradients, Covariant Vectors
12.4 Summation Convention
12.5 The Kronecker Delta δαβ
12.6 Tensors, Contravariant, Covariant and Mixed
12.7 The Covariant Derivative: Contravariant Vectors
12.8 The Covariant Derivative: Covariant Vectors
12.9 More Notational Conventions
12.10 Properties of the Affinity μνρ
12.11 The Curvature Tensor
12.12 Riemann's Tensor
12.13 Metric Tensor, Christoffel Symbols, Riemann–Christoffel Tensor
12.14 Geodesics
12.15 The Bianchi Identities
12.16 Ricci and Ricci–Einstein Tensors
12.17 The Einstein Field Equations
12.18 A Lagrangian Density for the Einstein Equations
13 Gauge Field Theories
13.1 A Lagrangian Density for the Empty-Space Gauge Field Equations
13.2 Gauge Transformations of the First and Second Kind
13.3 A Brief Overview of the Higgs Mechanism
13.4 Time as a Field
13.5 Conclusion
Part II Axiomatics
14 Axiomatizations in ZFC
14.1 Suppes Predicates
14.2 A Close View of Suppes Predicates
14.3 Maxwell's Electromagnetic Theory Revisited
14.4 Relative Consistency of the Added Axioms
14.5 Hamiltonian Mechanics
14.6 General Relativity
14.7 Classical Gauge Fields
14.8 Quantum Theory of the Electron
14.9 General Field Theory
14.10 Other Domains of Science
14.11 Summing It Up
Part III Technicalities
15 Hierarchies
15.1 Species of Structures: Alternative View
15.2 Species of Mathematical Structures, Again
15.3 Suppes Predicates: Another View
15.4 Main Traits
15.5 What Is an Empirical Theory?
15.6 Suppes Predicates for Classical Field Theories in Physics, Revisited
15.7 Why Dirac–Like Equations?
15.8 Topology as the Cradle of Physical Data
15.9 Generalized Incompleteness: Preliminary
15.10 Generalized Incompleteness, a Simple Example
15.11 The Halting Function
15.12 Expressions for the Halting Function
15.13 Chaitin's Ω Number
15.14 Undecidability and Incompleteness
15.15 Main Undecidability and Incompleteness Result
15.16 A Generalized Rice Theorem
15.17 Higher Arithmetical Degrees
15.18 Expressions for Higher Degrees
15.19 From Turing Machines to Oracle Turing Machines
15.20 Incompleteness Revisited
15.21 The θ Function and the Arithmetical Hierarchy
15.22 Beyond Arithmetic
15.23 Applications to Mechanics and Chaos Theory
15.24 Undecidability and Incompleteness in Classical Mechanics
15.25 Chaos Theory is Undecidable and Incomplete
Part IV More Applications
16 Arnol'd's 1974 Problems
16.1 Main Tools
16.2 The Example: An Undecidable Hopf Bifurcation
16.3 The Intuitive Level
16.4 The Metamathematical Level
16.5 Fermat's Conjecture and the Hopf Bifurcation
16.6 Related Incompleteness Phenomena
16.7 Fermat's Conjecture Revisited
16.8 Incompleteness Implies Undecidability
16.9 Summing It Up
16.10 What Pertains to Dynamical Systems?
16.11 The Canard and a Free Particle that Looks Chaotic
17 Forcing and Gravitation
17.1 Main Tools: An Overview
17.2 Zermelo–Fraenkel Set Theory
17.3 Models
17.4 Boolean-Valued Models
17.5 Forcing and Boolean-Valued Models
17.6 Suppes Predicates
17.7 Axioms for General Relativity, Revisited
17.8 Applications
17.9 Geometries of Spacetimes
17.10 Einstein and Gödel
17.11 The Meaning of `Generic' Here
17.12 Preliminary Concepts and Results
17.13 Axiomatics
17.14 Spacetimes with Cosmic Time
17.15 The ZFC Set of All Spacetimes
17.16 Exoticisms
17.17 A Brief Introduction
17.18 Conjectures, Speculations, More Counterintuitive Results
17.19 Set Theory with Martin's Axiom
17.20 Category and Measure
17.21 Results About the Nongenericity of Global Time
17.22 Martin's Axiom Again
17.23 Can We Decide Whether an Arbitrary Spacetime Has a Global Time Coordinate?
17.24 Conclusion
17.25 Summing It Up
17.26 Envoi
17.27 Final Remarks, Riemann's Hypothesis
18 Economics and Ecology
18.1 Markets in Equilibrium May Have Noncomputable Prices
18.2 Oscillating Populations or a Chaotic Demography?
18.3 Will the Middle Class Survive?
Part V Computer Science
19 Fast-Growing Functions
19.1 Technical Constructions
19.2 The Monster Shows Its Frightening Face
19.3 Quasi-Trivial Machines
19.4 A Busy Beaver Like Function
19.5 BGS-Like Sets
19.6 Exotic BGSF Machines
19.7 A Few More Properties of the Counterexample Function
19.8 A Few Intuitions
Part VI Hypercomputation
20 Hypercomputation
20.1 Exploring Extensions
20.2 Expressions for the Halting Function and Beyond: An Informal Discussion
20.3 A Turing Machine with an Analog Oracle
20.4 Recall Richardson's Map
20.5 Richardson's Map; This Time We Require the Multidimensional Version
20.6 A Key Theorem
20.7 The Halting Function
20.8 Undecidability and Incompleteness, Again
20.9 Recall: Characteristic Functions for Higher ArithmeticDegrees
20.10 A Conjecture
20.11 Further Remarks
20.12 The Turing–Feferman Theorem: Very Brief Remarks
20.13 What Do We Lose with Hypercomputation?
References
