Fundamental Theories of Physics

Preface. Introduction. 1: Dirac Observables and psi do-s. 1.0 Introduction. 1.1 Some Special Distributions. 1.2. Strictly Classical Pseudodifferential Operators. 1.3. Ellipticity and Parametrix Construction. 1.4. L2-Boundedness and Weighted Sobolev Spaces 1.5. The Parametrix Method for Solving ODE-s 1.6. More on General psi do-Results. 2: Why Should Observables be Pseudodifferential? 2.0. Introduction. 2.1. Smoothness of Lie Group Action on psi do-s. 2.2. Rotation and Dilation Smoothness. 2.3. General Order and General H3-Spaces. 2.4. A Useful Result on L2-Inverses and Square Roots. 3: Decoupling with psi do-s. 3.0. Introduction. 3.1. The Foldy-Wouthuysen Transform. 3.2. Unitary Decoupling Modulo O (-infinity). 3.3. Relation to Smoothness of the Heisenberg Transform. 3.4. Some Comments Regarding Spectral Theory. 3.5. Complete Decoupling for V(x) not equivalent to 0. 3.6. Split and Decoupling are not Unique - Summary. 3.7. Decoupling for Time Dependent Potentials. 4: Smooth Pseudodifferential Heisenberg Representation. 4.0. Introduction. 4.1. Dirac Evolution with Time-Dependent Potentials. 4.2. Observables with Smooth Heisenberg Representation. 4.3. Dynamical Observables with Scalar Symbol. 4.4. Symbols Non-Scalar on S plusminus. 4.5. Spin and Current. 4.6. Classical Orbits for Particle and Spin. 5: The Algebra of Precisely Predictable Observables. 5.0. Introduction. 5.1. A Precise Result on psi do-Heisenberg Transforms. 5.2. Relations between the Algebras P(t). 5.3. About Prediction of Observables again. 5.4. Symbol Propagation along Flows. 5.5. The Particle Flows Components are Symbols. 5.6. A Secondary Correction for the Electrostatical Potential. 5.7. Smoothness and FW-Decoupling. 5.8. The Final Algebra of Precisely Predictables. 6: Lorentz Covariance of Precise Predictability. 6.0. Introduction. 6.1. A New Time Frame for a Dirac State. 6.2. Transformation of P and PX for Vanishing Fields. 6.3. Relating Hilbert Spaces; Evolution of the Spaces H' and H. 6.4. The General Time-Independent Case. 6.5. The Fourier Integral Operators around R. 6.6. Decoupling with Respect to H' and H(t). 6.7. A Complicated ODE with psi do-Coefficients. 6.8 Integral Kernels of e-functions. 7: Spectral Theory of Precisely Predictable Approximations. 7.0. Introduction. 7.1. A Second Order Model Program. 7.2. The Corrected Location Observable. 7.3. Electrostatic Potential and Relativistic Mass. 7.4. Separation of Variables in Spherical Coordinates. 7.5. Highlights of the Proof of Theorem 7.3.2. 7.6. The Regular Singularities. 7.7. The Singularity at infinity. 7.8. Final Arguments. 8: Dirac and Schr¶dinger Equations; a Comparison. 8.0. Introduction. 8.1. What is a C*-Algebra with Symbol? 8.2. Exponential Actions on A. 8.3. Strictly Classical Pseudodifferential Operators. 8.4. Characteristic Flow and Particle Flow. 8.5. The Harmonic Oscillator. References. General Notations. Index.