A successful attempt to provide a Clean Observable Theory for Dirac´s Equation, free of contradictions like "Zitterbewegung"
The existence of an accurate split between physical states belonging to the electron and to the positron is discussed, as well as the fact that precisely predictable observables must preserve this split
This work presents a Clean Quantum Theory of the Electron, based on Dirac´s equation. "Clean" in the sense of a complete mathematical explanation of the well known paradoxes of Dirac´s theory and a connection to classical theory. It discusses the existence of an accurate split between physical states belonging to the electron and to the positron as well as the fact that precisely predictable observables must preserve this split.
|In this book we are attempting to o?er a modi?cation of Dirac´s theory of the electron we believe to be free of the usual paradoxa, so as perhaps to be acceptable as a clean quantum-mechanical treatment. While it seems to be a fact that the classical mechanics, from Newton to E- stein´s theory of gravitation, o?ers a very rigorous concept, free of contradictions and able to accurately predict motion of a mass point, quantum mechanics, even in its simplest cases, does not seem to have this kind of clarity. Almost it seems that everyone of its fathers had his own wave equation. For the quantum mechanical 1-body problem (with vanishing potentials) let 1 us focus on 3 di?erent wave equations : (I) The Klein-Gordon equation 3 2 2 2 2 (1) ? ?/?t +(1??)? =0 , ? = Laplacian = ? /?x . j 1 This equation may be written as ? ? (2) (?/?t?i 1??)(?/?t +i 1??)? =0 . Hereitmaybenotedthattheoperator1??hasawellde?nedpositive square root as unbounded self-adjoint positive operator of the Hilbert 2 3 spaceH = L (R ).
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 Schrödinger 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.
This work presents a "Clean Quantum Theory of the Electron", based on Dirac´s equation. "Clean" in the sense of a complete mathematical explanation of the well known paradoxes of Dirac´s theory, and a connection to classical theory, including the motion of a magnetic moment (spin) in the given field, all for a charged particle (of spin ½) moving in a given electromagnetic field.
This theory is relativistically covariant, and it may be regarded as a mathematically consistent quantum-mechanical generalization of the classical motion of such a particle, à la Newton and Einstein. Normally, our fields are time-independent, but also discussed is the time-dependent case, where slightly different features prevail. A "Schroedinger particle", such as a light quantum, experiences a very different (time-dependent) "Precise Predictablity of Observables". An attempt is made to compare both cases. There is not the Heisenberg uncertainty of location and momentum; rather, location alone possesses a built-in uncertainty of measurement.
Mathematically, our tools consist of the study of a pseudo-differential operator (i.e. an "observable") under conjugation with the Dirac propagator: such an operator has a "symbol" approximately propagating along classical orbits, while taking its "spin" along. This is correct only if the operator is "precisely predictable", that is, it must approximately commute with the Dirac Hamiltonian, and, in a sense, will preserve the subspaces of electronic and positronic states of the underlying Hilbert space.
Audience:
Theoretical Physicists, specifically in Quantum Mechanics.
Mathematicians, in the fields of Analysis, Spectral Theory of Self-adjoint differential operators, and Elementary Theory of Pseudo-Differential Operators
From the reviews:
"In this very interesting book, the author proposes a modification of Dirac´s theory of the electron, that he believes to be free of the systematic well-known difficulties that give rise to the usual paradoxes. ... the direction he describes in this book is promising, and will hopefully open the way to the construction of a more general framework." (Alberto Parmeggiani, Mathematical Reviews, Issue 2008 j)
"In this book Heinz Otto Cordes tries to make a contribution from the point of view of a mathematician, and it is certainly an interesting one. ... it is very pleasant reading for the more mathematically inclined person, and those with some interest in physics will enjoy the many insightful remarks about quantum mechanics immersed in the text. I warmly recommend this book to mathematicians and mathematical physicists interested in the Dirac equation." (Bernd Thaller, SIAM Review, Vol. 50 (2), 2008)
In this book we are attempting to o?er a modi?cation of Dirac's theory of the electron we believe to be free of the usual paradoxa, so as perhaps to be acceptable as a clean quantum-mechanical treatment. While it seems to be a fact that the classical mechanics, from Newton to E- stein's theory of gravitation, o?ers a very rigorous concept, free of contradictions and able to accurately predict motion of a mass point, quantum mechanics, even in its simplest cases, does not seem to have this kind of clarity. Almost it seems that everyone of its fathers had his own wave equation. For the quantum mechanical 1-body problem (with vanishing potentials) let 1 us focus on 3 di?erent wave equations : (I) The Klein-Gordon equation 3 2 2 2 2 (1) ? ?/?t +(1??)? =0 , ? = Laplacian = ? /?x . j 1 This equation may be written as ? ? (2) (?/?t?i 1??)(?/?t +i 1??)? =0 . Hereitmaybenotedthattheoperator1??hasawellde?nedpositive square root as unbounded self-adjoint positive operator of the Hilbert 2 3 spaceH = L (R ).
Dirac Observables and ?do-s.- Why Should Observables be Pseudodifferential?.- Decoupling with ?do-s.- Smooth Pseudodifferential Heisenberg Representation.- The Algebra of Precisely Predictable Observables.- Lorentz Covariance of Precise Predictability.- Spectral Theory of Precisely Predictable Approximations.- Dirac and Schrödinger Equations; a Comparison.
From the reviews:
"In this very interesting book, the author proposes a modification of Dirac's theory of the electron, that he believes to be free of the systematic well-known difficulties that give rise to the usual paradoxes. ... the direction he describes in this book is promising, and will hopefully open the way to the construction of a more general framework." (Alberto Parmeggiani, Mathematical Reviews, Issue 2008 j)
"In this book Heinz Otto Cordes tries to make a contribution from the point of view of a mathematician, and it is certainly an interesting one. ... it is very pleasant reading for the more mathematically inclined person, and those with some interest in physics will enjoy the many insightful remarks about quantum mechanics immersed in the text. I warmly recommend this book to mathematicians and mathematical physicists interested in the Dirac equation." (Bernd Thaller, SIAM Review, Vol. 50 (2), 2008)
Inhaltsverzeichnis
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 Schrödinger 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.
Klappentext
In this book we are attempting to o?er a modi?cation of Dirac¿s theory of the electron we believe to be free of the usual paradoxa, so as perhaps to be acceptable as a clean quantum-mechanical treatment. While it seems to be a fact that the classical mechanics, from Newton to E- stein¿s theory of gravitation, o?ers a very rigorous concept, free of contradictions and able to accurately predict motion of a mass point, quantum mechanics, even in its simplest cases, does not seem to have this kind of clarity. Almost it seems that everyone of its fathers had his own wave equation. For the quantum mechanical 1-body problem (with vanishing potentials) let 1 us focus on 3 di?erent wave equations : (I) The Klein-Gordon equation 3 2 2 2 2 (1) ? ?/?t +(1??)? =0 , ? = Laplacian = ? /?x . j 1 This equation may be written as ? ? (2) (?/?t?i 1??)(?/?t +i 1??)? =0 . Hereitmaybenotedthattheoperator1??hasawellde?nedpositive square root as unbounded self-adjoint positive operator of the Hilbert 2 3 spaceH = L (R ).
