Despite successes of modern physics, the existence of dark energy and matter is indicative that conventional mechanical accounting is lacking. The most basic of all mechanical principles is Newton’s second law, and conventionally, energy is just energy whether particle or wave energy. In this monograph, Louis de Broglie’s idea of simultaneous existence of both particle and associated wave is developed, with a novel proposal to account for mass and energy through a combined particle-wave theory. Newton’s second law of motion is replaced by a fully Lorentz invariant reformulation inclusive of both particles and waves.
The model springs from continuum mechanics and forms a natural extension of special relativistic mechanics. It involves the notion of “force in the direction of time” and every particle has both particle and wave energies, arising as characteristics of space and time respectively. Dark matter and energy then emerge as special or privileged states occurring for alignments of spatial forces with the force in the direction of time. Dark matter is essentially a backward wave and dark energy a forward wave, both propagating at the speed of light. The model includes special relativistic mechanics and Schrödinger’s quantum mechanics, and the major achievements of mechanics and quantum physics.
Our ideas of particles and waves are not yet properly formulated, and are bound up with the speed of light as an extreme limit and particle-wave demarcation. Sub-luminal particles have an associated superluminal wave, so if sub-luminal waves have an associated superluminal particle, then there emerges the prospect for faster than light travel with all the implications for future humanity. Carefully structured over special relativity and quantum mechanics, Mathematics of Particle-Wave Mechanical Systems is not a completed story, but perhaps the first mechanical model within which such exalted notions might be realistically and soberly examined. If ultimately the distant universe become accessible, this will necessitate thinking differently about particles, waves and the role imposed by the speed of light. The text constitutes a single proposal in that direction and a depository for mathematically related results. It will appeal to researchers and students of mathematical physics, applied mathematics and engineering mechanics.
Author(s): James M. Hill
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2022
Language: English
Pages: 378
Tags: de Broglie, Particle-Wave Mechanics, Newton's Second Law
Author's Foreword
Contents
1 Introduction
1.1 Introduction
1.2 General Introduction
1.3 Special Relativity
1.4 Quantum Mechanics
1.5 de Broglie Particle-Wave Mechanics
1.6 Plan of Text
1.7 Tables of Major Symbols and Basic Equations
2 Special Relativity
2.1 Introduction
2.2 Lorentz Transformations
2.3 Einstein Addition of Velocities Law
2.4 Lorentz Invariances
2.5 Lorentz Invariant Velocity Fields u(x, t)
2.6 General Framework for Lorentz Invariances
2.7 Integral Invariants of the Lorentz Group
2.8 Alternative Validation of Lorentz Invariants
2.9 Jacobians of the Lorentz Transformations
2.10 Space-Time Transformation x'= ct and t' = x/c
2.11 The de Broglie Wave Velocity u'= c2/u
2.12 Force and Physical Energy Arising from Work Done
2.13 Lorentz Invariant Energy-Momentum Relations
2.14 Force Invariance for Constant Velocity Frames
2.15 Example: Motion in an Invariant Potential Field
2.16 Alternative Energy-Mass Velocity Variation
3 General Formulation and Basic Equations
3.1 Introduction
3.2 Louis Victor de Broglie
3.3 James Clerk Maxwell
3.4 Four Types of Matter and Variable Rest Mass
3.5 Modified Newton's Laws of Motion
3.6 Identity for Spatial Physical Force ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (bold f) /StPNE pdfmark [/StBMC pdfmarkfps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
3.7 Assumed Existence of Work Done Function ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper W left parenthesis bold x comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkW(x, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
3.8 Forces ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (bold f) /StPNE pdfmark [/StBMC pdfmarkfps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and g Derivable from a Potential ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper V left parenthesis bold x comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkV(x, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
3.9 Correspondence with Maxwell's Equations
3.10 Centrally or Spherically Symmetric Systems
3.11 Newtonian Kinetic Energy and Momentum
3.12 Newtonian Wave-Like Solution
3.13 Newtonian Work Done ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (upper W left parenthesis u comma lamda right parenthesis) /StPNE pdfmark [/StBMC pdfmarkW(u, λ)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark from ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (partial differential f divided by partial differential t equals c squared partial differential g divided by partial differential x) /StPNE pdfmark [/StBMC pdfmark∂f/∂t = c2 ∂g/∂xps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
4 Special Results for One Space Dimension
4.1 Introduction
4.2 Basic Equations
4.3 General Reformulations of Basic Equations
4.4 Important Identity
4.5 Formulation in Terms of Lorentz Invariants
4.6 Differential Relations for Invariants ξ and η
4.7 de Broglie's Guidance Equation
4.8 Vanishing of Force g in Direction of Time
4.9 Clairaut's Differential Equation with Parameter u
4.10 Hamiltonian for One Space Dimension
4.11 Lagrangian for One Space Dimension
5 Exact Wave-Like Solution
5.1 Introduction
5.2 Wave-Like Solution
5.3 Work Done W(u, λ) from ∂f/∂t = c2 ∂g/∂x
5.4 Simple Derivation of Wave-Like Solution
5.5 Relation to Solution of Special Relativity
5.6 Relation to Hubble Parameter
5.7 Derivation of Integral for Hubble Formula
5.8 Dark Matter and Dark Energy as de Broglie States
6 Derivations and Formulae
6.1 Introduction
6.2 Derivation of Wave-Like Solution
6.3 Expressions for de Broglie Wave Energy
6.4 de Broglie Wave Energy for Particular λ
6.5 Alternative Approach to Evaluation of Integrals
6.6 Alternative Derivation for Wave Energy
6.7 Alternative Derivation of Exact Solution
6.8 Yet Another Approach to Evaluation of Integrals
7 Lorentz and Other Invariances
7.1 Introduction
7.2 Force Invariance Under Lorentz Transformations
7.3 Lorentz Invariance of ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (d script upper E divided by d p) /StPNE pdfmark [/StBMC pdfmarkd E/dpps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark or ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (d script upper E divided by d xi) /StPNE pdfmark [/StBMC pdfmarkdE/dξps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
7.4 Lorentz Invariance of Forces
7.5 Functional Dependence of Forces
7.6 Transformation x'= ct and t' = x/c
7.7 Force Invariance Under Superluminal Lorentz Frames
7.8 Particle and Wave Energies and Momenta
8 Further Results for One Space Dimension
8.1 Introduction
8.2 Wave Equation General Solution
8.3 Trivial Solution Only for Zero Spatial Force
8.4 Nontrivial Solutions for Zero Spatial Force
8.5 Generalisation of Wave-Like Solution
8.6 Solutions with Non-constant Rest Mass
8.7 Formulation for Variable Rest Mass
8.8 Characteristics ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (alpha equals c t plus x) /StPNE pdfmark [/StBMC pdfmarkα= ct + xps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (beta equals c t minus x) /StPNE pdfmark [/StBMC pdfmarkβ= ct - xps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
8.9 p(x, t) and ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (script upper E left parenthesis x comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkE(x, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Assumed Independent Variables
9 Centrally Symmetric Mechanical Systems
9.1 Introduction
9.2 Basic Equations with Spherical Symmetry
9.3 General Solutions for ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (script upper E left parenthesis r comma t right parenthesis) /StPNE pdfmark [/StBMC pdfmarkE(r, t)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and p(r, t)
9.4 Conservation of Energy ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (e plus script upper E plus upper V equals) /StPNE pdfmark [/StBMC pdfmarke + E + V = ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Constant
9.5 Fundamental Identity for f and g
9.6 ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (f equals plus or minus c left parenthesis g minus 2 p divided by r right parenthesis) /StPNE pdfmark [/StBMC pdfmarkf = c(g -2p/r)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Implies e0 Is Zero
9.7 Newtonian Gravitation and Schwarzschild Radius
9.8 Pseudo-Newtonian Gravitational Potential
9.9 Dark Matter-Dark Energy and Four Types of Matter
9.10 Positive Energy (I) e = (e02 + (pc)2)1/2, ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (e 0 not equals 0) /StPNE pdfmark [/StBMC pdfmarke0 ≠0ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
9.11 Negative Energy (II) e = - (e02 + (pc)2)1/2, ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (e 0 not equals 0) /StPNE pdfmark [/StBMC pdfmarke0≠0ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
9.12 Positive Energy (III) e = pc, e0 = 0
9.13 Negative Energy (IV) e = -pc, e0 = 0
9.14 Similarity Stretching Solutions of Wave Equation
9.15 Some Examples Involving the Dirac Delta Function
9.16 Calculation Details for Similarity Solutions
9.17 de Broglie's Centrally Symmetric Guidance Formula
10 Relation with Quantum Mechanics
10.1 Introduction
10.2 Quantum Mechanics and Schrödinger Wave Equation
10.3 Group Velocity and de Broglie Waves
10.4 Lorentz Invariants ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (xi equals e x minus c squared p t) /StPNE pdfmark [/StBMC pdfmarkξ= ex - c2 ptps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and ps: [/EMC pdfmark [/objdef Equ /Subtype /Span /ActualText (eta equals p x minus e t) /StPNE pdfmark [/StBMC pdfmarkη= px - etps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
10.5 Klein–Gordon Partial Differential Equation
10.6 Alternative Klein–Gordon–Schrödinger Equation
10.7 General Wave Structure of Solutions of Wave Equation
10.8 Wave Solutions of Klein–Gordon Equation
10.9 Time-Dependent Dirac Equation for Free Particle
11 Coordinate Transformations, Tensors and General Relativity
11.1 Summation Convention and Cartesian Tensors
11.2 Alternative Derivation of Basic Identity
11.3 General Curvilinear Coordinates
11.4 Partial Covariant Differentiation
11.5 Illustration for Single Space Dimension
11.6 Formulae for Ricci and Einstein Tensors
11.7 Two Illustrative Line Elements
11.8 Spiral Gravitating Structures
12 Conclusions, Summary and Postscript
12.1 Introduction
12.2 Conclusions
12.3 Summary
12.4 Postscript
Bibliography
Index
Index
