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The Quantum Nature of Things: How Counting Leads to the Quantum World

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This book offers readers an entirely original and unconventional view of quantum mechanics. It is a view that accepts quantum mechanics as the natural way to think about the way nature works, rather than the view commonly expressed, especially in books on quantum physics, that quantum theory is weird and counterintuitive. It is based on the concept of itemization.

From this simple premise, quantities like energy and momentum, both linear and angular emerge naturally, as do configuration space, potentials, the electromagnetic field, many-body dynamics, special relativity and relativistic wave mechanics. The many-body dynamics, because it is not tied to physics from the outset, can be applied to population dynamics outside physics as well as the usual physical situations.

From this emerges much of the basic physics that describes, mathematically, how the natural world behaves.

This accessible introduction does not require exotic maths, and is aimed at inquisitive physics students and professionals who are interested in exploring unconventional approaches to physics. It may also be of interest to anyone studying quantum information theory or quantum computing.

Key Features

    • Provides a unique, new approach to understanding quantum mechanics.

    • Uses basic concepts and mathematical methods accessible at the undergraduate level.

    • Presents applications outside physics, including a newly devised and original model of cell division that shows how cancer-cell population explosions occur.

    Author(s): T. R. Robinson
    Publisher: CRC Press
    Year: 2023

    Language: English
    Pages: 281
    City: Boca Raton

    Cover
    Half Title
    Title Page
    Copyright Page
    Dedication
    Contents
    Preface
    Acknowledgments
    Prologue
    Chapter 1: The universal quantum hypothesis: Just items
    Chapter 2: An introduction to operators
    2.1. Linear operators
    2.2. Eigenvalues and eigenfunctions
    2.3. Commutation properties
    2.4. Adjoint and Hermitian operators
    2.5. Uncertainty
    2.6. Operator functions
    2.7. System evolution
    2.8. Evolution of expectation values
    2.9. Commutators and derivations
    Chapter 3: Natural number dynamics I: The basic formulation
    3.1. Non-negative operators and their factorization
    3.2. The natural number operator
    3.3. Uniqueness of N = A†A
    3.4. Representation of eigenstates
    3.4.1. Fock space representation
    3.4.2. Phasor operator representation
    3.4.3. Analytic representation
    3.4.4. N and u-v duality
    3.5. Eigenfunctions of A, U and V
    3.5.1. Eigenfunctions of A
    3.5.2. Eigenfunctions of u and v
    3.6. Expectation values involving A, U and V
    3.6.1. Fock space representation
    3.6.2. u-representation
    3.6.3. u-v Fourier duality
    3.6.4. U-V uncertainty
    3.7. Is there a N-ξ duality?
    3.8. Generalization of raising and lowering operators
    Chapter 4: Multi-category systems: Bosons and fermions
    4.1. A sum over categories
    4.2. Why we need time
    4.3. The emergence of fermions
    4.4. Fermionic Hermitian operators
    4.5. Linearization of square roots
    Chapter 5: The single-category system: The emergence of quantum mechanics
    5.1. The Schrödinger equation
    5.2. The emergence of configuration space
    5.3. Generalizing the Hamiltonian
    5.4. Eigenfunctions, expectation values and x-p duality
    5.5. The double-slit experiment
    5.6. Heisenberg's uncertainty principle
    5.6.1. Uncertainty in Fock space
    5.6.2. Uncertainty in analytic representation
    5.7. H-t duality?
    Chapter 6: Two-category systems
    6.1. A first look at interactions
    6.1.1. The bosonic case
    6.1.2. The fermionic case
    6.2. Time dependence
    6.2.1. Eigenfrequencies for two-category interactions
    6.2.2. Calculating the time dependent population numbers
    6.2.3. Quantum noise
    6.2.4. Number operator approach
    6.3. Complex rotation transformation
    6.4. Coupled creation and annihilation operators
    6.4.1. Bosonic case
    6.4.2. Fermionic case
    Chapter 7: Degenerate Two-category Systems
    7.1. Degenerate bosonic two-category systems
    7.2. Symmetry properties of the two-category bosonic system
    7.3. The emergence of two-dimensional configuration space
    7.4. The emergence of angular momentum
    7.5. A degenerate fermionic case
    7.6. Comment
    Chapter 8: Degenerate three-category systems
    8.1. Operators in degenerate three-category systems
    8.2. Three-dimensional configuration space and angular momentum
    8.2.1. Cartesian form
    8.2.2. Spherical polar form
    8.2.3. Separation of variables
    8.3. Central potentials
    8.4. Scalar and vector potentials
    8.5. The Maxwell equations
    8.6. Electromagnetic waves and photons
    8.7. Goldilocks and the three dimensions
    Chapter 9: Interactions in multi-category systems
    9.1. Multiple-category linear transformations
    9.2. Fermion interactions via boson exchange
    9.3. BCS theory
    9.3.1. The Fermi sea
    9.3.2. Fermion-fermion attraction
    9.3.3. The enegy gap
    Chapter 10: Field itemization
    10.1. Configuration space for fields
    10.2. Items in a field: The quantum field concept
    10.3. The Hamiltonian of an itemized field
    10.4. The particle/anti-particle Hamiltonian
    Chapter 11: Phase invariance: The emergence of space-time and wave mechanics
    11.1. Representing phase
    11.1.1. Phase as a scalar product
    11.1.2. Linear transformation
    11.1.3. Phase invariance for Euclidean geometry
    11.1.4. Phase invariance in pseudo-Euclidean geometry
    11.2. Special relativity
    11.3. Relativistic wave mechanics: The Dirac equation
    11.4. Phase kinematics
    11.5. 3+1 dimensions
    11.6. The origin of fermionic mass
    11.7. Angular momentum revisited: The emergence of spin
    11.8. Inhomogeneous propagation and the meaning of ξ
    11.9. Non-relativistic limit
    Chapter 12: Natural number dynamics II: Time-dependent population models
    12.1. One-for-one exchange
    12.2. The Lotka-Volterra equations
    12.3. Two-for-one exchange
    12.4. Parametric amplification in quantum optics
    12.5. Higher order interactions
    12.6. Cell division and cell population dynamics
    12.6.1. A simplified picture of cell division
    12.6.2. Normal cell division (NCD) model: Stable population
    12.6.3. Oncogenic cell division (OCD) model: Explosive population growth
    12.7. Vacuum fluctuations: A fermion/anti-fermion system
    Chapter 13: Epilogue
    13.1. Emergent physics
    13.2. What next?
    13.3. Forms and shadows: De-quantization and classical physics
    Appendix A: Peano axioms
    Appendix B: Virial theorem and discrete energy levels
    Appendix C: Wave kinematics
    Appendix D: Lie groups
    Appendix E: Differential operators
    Appendix F: Calculus without limits
    Appendix G: Mean-field approximation
    Appendix H: Legendre transformation
    Bibliography
    Index