
Contents
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3.1 Introduction 3.1 Introduction
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3.2 Special relativity and Lorentz transformations 3.2 Special relativity and Lorentz transformations
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3.2.1 Velocity addition 3.2.1 Velocity addition
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3.2.2 Lorentz vectors and scalars 3.2.2 Lorentz vectors and scalars
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3.2.3 Length contraction and time dilation 3.2.3 Length contraction and time dilation
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3.2.4 Twin puzzle 3.2.4 Twin puzzle
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3.3 Relativistic kinematics and the mass-energy equivalence 3.3 Relativistic kinematics and the mass-energy equivalence
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3.3.1 Relativistic Doppler effect 3.3.1 Relativistic Doppler effect
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3.3.2 Applications of relativistic kinematics 3.3.2 Applications of relativistic kinematics
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3.4 Quaternions 3.4 Quaternions
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3.4.4 Rotation quaternions 3.4.4 Rotation quaternions
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3.5 Dirac equation, spinors and matrices 3.5 Dirac equation, spinors and matrices
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3.5.1 More about Dirac matrices 3.5.1 More about Dirac matrices
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3.6 Symmetries of the Dirac equation∗ 3.6 Symmetries of the Dirac equation∗
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3.6.1 Parity 3.6.1 Parity
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3.6.6 Time reversal 3.6.6 Time reversal
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3.6.6 Charge conjugation 3.6.6 Charge conjugation
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3.7 Weyl and Majorana spinors, symmetry violations∗ 3.7 Weyl and Majorana spinors, symmetry violations∗
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3.7.7 Weyl spinors 3.7.7 Weyl spinors
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3.7.7 𝒫 and C𝒫 violations 3.7.7 𝒫 and C𝒫 violations
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3.7.3 Majorana spinors 3.7.3 Majorana spinors
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3.8 Lorentz group 3.8 Lorentz group
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3.8.1 SU(2) representations 3.8.1 SU(2) representations
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3.9 Cartan spinors and spin transformations in square-root space 3.9 Cartan spinors and spin transformations in square-root space
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3.9.1 Complex square-root coordinates on the light cone 3.9.1 Complex square-root coordinates on the light cone
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3.9.9 Spinor representation of spacetime coordinates and the Lorentz group 3.9.9 Spinor representation of spacetime coordinates and the Lorentz group
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3.10 Dyadics 3.10 Dyadics
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3.11 Cartesian tensors 3.11 Cartesian tensors
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3.11.1 3D Cartesian tensors 3.11.1 3D Cartesian tensors
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Tensor algebra Tensor algebra
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Permutation tensors Permutation tensors
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Parity transformation and pseudotensors Parity transformation and pseudotensors
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Parity and time-reversal symmetries Parity and time-reversal symmetries
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Tensor fields Tensor fields
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3.11.2 Tensor representations 3.11.2 Tensor representations
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Spherical harmonics Spherical harmonics
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3.12 Tensor analysis 3.12 Tensor analysis
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3.12.1 Tensor algebra 3.12.1 Tensor algebra
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Nonorthogonal basis in 3D space Nonorthogonal basis in 3D space
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Dual basis Dual basis
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Local coordinate transformations Local coordinate transformations
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Tensors as abstract objects Tensors as abstract objects
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New features in tensor algebra New features in tensor algebra
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3.12.2 Vector differential operations on tensor fields 3.12.2 Vector differential operations on tensor fields
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Appendix 3 Tables of mathematical formulas Appendix 3 Tables of mathematical formulas
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3 Relativistic square-root spaces∗
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Published:January 2013
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Abstract
Einstein's derivation of the Lorentz transformation between space and time coordinates is given. The resulting relativistic kinematics of £ Lorentz spacetime and momentum-energy vectors is readily applied to practical problems by using their scalar products that are Lorentz-invariant, the same in every Lorentz frame. Dirac went the other way by expanding the Lorentz quadratic invariant operator in the linear wave equation back into their original square-root vectors. The resulting Dirac spinor wave functions give full access to the rich spacetime properties of the wave motion in the square-root space. These properties include their spacetime symmetries, and the breaking of these symmetries by using other types of spinor wave functions. The connection is also made with Cartan's complexified square-root coordinates. Multidimensional arrays of spatial and spacetime vectors can be constructed and used. These objects include dyadics, Cartesian tensors and general tensors. Some of their useful properties are described.
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