R. Boudet: Quantum Mechanics in the Geometry of Space-Time Elementary Theory


Rouger Boudet
Quantum Mechanics in the Geometry of Space-Time Elementary Theory
Springer; 1st Edition. edition (June 21, 2011), paperback,
ISBN-10: 3642191983, ISBN-13: 978-3642191985
Price: 49.95 USD.

Preface

The aim of the work we propose is a contribution to the expression of the present particles theories in terms entirely relevant to the elements of the geometry of the Minkowski space–time M = R^{1;3}, that is those of the Grassmann algebra R^4, scalars, vectors, bivectors, pseudo-vectors, pseudo-scalars of R^4 associated with the signature (1, 3) which defines M, and, at the same time, the elimination of the complex language of the Pauli and Dirac matrices and spinors which is used in quantum mechanics.

The reasons for this change of language lie, in the first place, in the fact that this real language is the same as the one in which the results of experiments are written, which are necessarily real.

But there is another reason certainly more important. Experiments are generally achieved in a laboratory frame which is a galilean frame, and the fundamental laws of Nature are in fact independent of all galilean frame. So the theories must be expressed in an invariant form. Then geometrical objects appear, whose properties give in particular a clear interpretation of what we call energy. Also gauges are geometrically interpreted as rings of rotations of sub-spaces of local orthonormal moving frames. The energy–momentum tensors correspond to the product of a suitable physical constant by the infinitesimal rotation of these sub-spaces into themselves.

The passage of the expression of a theory from its form in a galilean frame to the one independent of all galilean frame, is difficult to obtain with the use of complex matrices and spinors language. The Dirac spinor which expresses the wave function W associated with a particle is nothing else by itself but a column of four complex numbers. The definition of its properties requires actions on this column of the Dirac complex matrices.

An immense step in clarity was achieved by the real form w given in 1967 by David Hestenes (Oersted Medal 2002) to the Dirac psi. In this form, the Lorentz rotation which allows the direct passage to the invariant entities appears explicitly. In particular the geometrical meaning of the gauges defined by the complex Lie rings U(1) and SU(2) becomes evident.

It should be emphasized, like an indisputable confirmation of the independent work of Hestenes, that a geometrical interpretation of the Dirac psi had been implicitly given, probably during the years 1930, by Arnold Sommerfeld in a calculation related to hydrogenic atoms, and more generally and explicitly by Georges Lochak in 1956. In these works psi is expressed by means of Dirac matrices, these last ones being implicitly identified with the vectors of the galilean frame in which the Dirac equation of the electron is written.

But the use of a tool, the Clifford algebra Cl(1,3)associated with the space M = R^{1;3}, introduced by D. Hestenes, brings considerable simplifications. Pages of calculations giving tensorial equations deduced from the complex language may be replaced by few lines. urthermore ambiguities associated with the use of the imaginary number i = sqrt(-1) are eliminated. The striking point lies in the fact that the ‘‘number i’’ which lies in the Dirac theory of the electron is a bivector of the Minkowski space–time M, a real object, which allows to define, after the above Lorentz rotation and the multiplication by (hbar c)/2, the proper angular momentum, or spin, of the electron.

In the same aim, to avoid the ambiguousness of the complex Quantum Field Theory, due to the unseasonable association i hbar of hbar and i in the expression of the electromagnetic potentials “in quite analogy with the ordinary quantum theory” (in fact the Dirac theory of the electron), we give a presentation of quantum electrodynamics entirely real. It is only based on the use of the Grassmann algebra of M and the inner product in M.

The more the theories of the particles become complicated, the more the links which can unify these theories in an identical vision of the laws of Nature have to be made explicit. When these laws are placed in the frame of the Minkowski space–time, the complete translation of these theories in the geometry of space–time appears as a necessity. Such is the reason for the writing of the present volume.

However, if this book contains a critique, sometimes severe, of the language based on the use of the complex matrices, spinors and Lie rings, this critique does not concern in any way the authors of works obtained by means of this language, which remain the foundations of Quantum Mechanics. The more this language is abstract with respect to the reality of the laws of Nature, the more these works appear to be admirable.

Bassan, February 2011 Roger Boudet
(Preface: http://www.springerlink.com/content/978-3-642-19198-5/front-matter.pdf)

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Part I The Real Geometrical Algebra or Space–Time Algebra.
Comparison with the Language of the Complex Matrices
and Spinors
2 The Clifford Algebra Associated with the Minkowski Space–Time M . . . . 7
2.1 The Clifford Algebra Associated with an Euclidean Space . . . 7
2.2 The Clifford Algebras and the ‘‘Imaginary Number’’ sqrt(-1)  . . . . 9
2.3 The Field of the Hamilton Quaternions and the Ring
of the Biquaternion as Clþð3; 0Þ and Clð3; 0Þ ’ Clþð1; 3Þ. . . . . 10
3 Comparison Between the Real and the Complex Language . . . . . 13
3.1 The Space–Time Algebra and the Wave Function
Associated with a Particle: The Hestenes Spinor . . . . . . . . . . 13
3.2 The Takabayasi–Hestenes Moving Frame . . . . . . . . . . . . . . . 15
3.3 Equivalences Between the Hestenes and the Dirac Spinors . . . 15
3.4 Comparison Between the Dirac and the Hestenes Spinors . . . . 16
Part II The U(1) Gauge in Complex and Real Languages.
Geometrical Properties and Relation with the Spin and
the Energy of a Particle of Spin 1/2
4 Geometrical Properties of the U(1) Gauge . . . . . . . . . . . . . . . . . . 21
4.1 The Definition of the Gauge and the Invariance of a Change
of Gauge in the U(1) Gauge . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.1 The U(1) Gauge in Complex Language. . . . . . . . . . . 21
4.1.2 The U(1) Gauge Invariance in Complex
Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.3 A Paradox of the U(1) Gauge in Complex
Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2 The U(1) Gauge in Real Language . . . . . . . . . . . . . . . . . . . . 22
4.2.1 The Definition of the U(1) Gauge in Real
Language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2.2 The U(1) Gauge Invariance in Real Language . . . . . . 23
5 Relation Between the U(1) Gauge, the Spin and the Energy
of a Particle of Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.1 Relation Between the U(1) Gauge and the Bivector Spin . . . . 25
5.2 Relation Between the U(1) Gauge and the
Momentum–Energy Tensor Associated with the Particle . . . . . 25
5.3 Relation Between the U(1) Gauge and the Energy
of the Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Part III Geometrical Properties of the Dirac Theory
of the Electron
6 The Dirac Theory of the Electron in Real Language . . . . . . . . . . 29
6.1 The Hestenes Real form of the Dirac Equation . . . . . . . . . . . 29
6.2 The Probability Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.3 Conservation of the Probability Current. . . . . . . . . . . . . . . . . 30
6.4 The Proper (Bivector Spin) and the Total
Angular–Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
6.5 The Tetrode Energy–Momentum Tensor . . . . . . . . . . . . . . . . 31
6.6 Relation Between the Energy of the Electron and
the Infinitesimal Rotation of the ‘‘Spin Plane’’ . . . . . . . . . . . . 32
6.7 The Tetrode Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.8 The Lagrangian of the Dirac Electron . . . . . . . . . . . . . . . . . . 33
6.9 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 The Invariant Form of the Dirac Equation and Invariant
Properties of the Dirac Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.1 The Invariant Form of the Dirac Equation . . . . . . . . . . . . . . . 35
7.2 The Passage from the Equation of the Electron to the
One of the Positron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.3 The Free Dirac Electron, the Frequency and the Clock
of L. de Broglie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.4 The Dirac Electron, the Einstein Formula of the Photoeffect
and the L. de Broglie Frequency . . . . . . . . . . . . . . . . . . . . . 39
7.5 The Equation of the Lorentz Force Deduced from
the Dirac Theory of the Electron . . . . . . . . . . . . . . . . . . . . . 40
7.6 On the Passages of the Dirac Theory to the Classical Theory
of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Part IV The SU(2) Gauge and the Yang–Mills Theory in Complex
and Real Languages
8 Geometrical Properties of the SU(2) Gauge and the
Associated Momentum–Energy Tensor . . . . . . . . . . . . . . . . . . . . 45
8.1 The SU(2) Gauge in the General Yang–Mills Field Theory
in Complex Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2 The SU(2) Gauge and the Y.M. Theory in STA. . . . . . . . . . . 46
8.2.1 The SU(2) Gauge and the Gauge Invariance
in STA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8.2.2 A Momentum–Energy Tensor Associated with
the Y.M. Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8.2.3 The STA Form of the Y.M. Theory Lagrangian . . . . . 49
8.3 Conclusions About the SU(2) Gauge and the Y.M. Theory . . . 49
Part V The SU(2) 3 U(1) Gauge in Complex and Real Languages
9 Geometrical Properties of the SU(2) 3 U(1) Gauge . . . . . . . . . . . 53
9.1 Left and Right Parts of a Wave Function . . . . . . . . . . . . . . . 53
9.2 Left and Right Doublets Associated with
Two Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9.3 The Part SU(2) of the SU(2) 9 U(1) Gauge. . . . . . . . . . . . . . 56
9.4 The Part U(1) of the SU(2) 9 U(1) Gauge . . . . . . . . . . . . . . 56
9.5 Geometrical Interpretation of the SU(2) 9 U(1) Gauge
of a Left or Right Doublet . . . . . . . . . . . . . . . . . . . . . . . . . . 56
9.6 The Lagrangian in the SU(2) 9 U(1) Gauge. . . . . . . . . . . . . 57
Part VI The Glashow–Salam–Weinberg Electroweak Theory
10 The Electroweak Theory in STA: Global Presentation. . . . . . . . . 61
10.1 General Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10.2. The Particles and Their Wave Functions . . . . . . . . . . . . . . . . 62
10.2.1 The Right and Left Parts of the Wave Functions
of the Neutrino and the Electron . . . . . . . . . . . . . . . 62
10.2.2 A Left Doublet and Two Singlets . . . . . . . . . . . . . . . 62
10.3 The Currents Associated with the Wave Functions . . . . . . . . . 62
10.3.1 The Current Associated with the Right and Left
Parts of the Electron and Neutrino . . . . . . . . . . . . . . 63
10.3.2 The Currents Associated with the Left Doublet . . . . . 63
10.3.3 The Charge Currents . . . . . . . . . . . . . . . . . . . . . . . . 64
10.4 The Bosons and the Physical Constants. . . . . . . . . . . . . . . . . 65
10.4.1 The Physical Constants . . . . . . . . . . . . . . . . . . . . . . 65
10.4.2 The Bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
10.5 The Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
11 The Electroweak Theory in STA: Local Presentation. . . . . . . . . . 67
11.1 The Two Equivalent Decompositions of the Part LI
of the Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
11.2 The Decomposition of the Part LII of the Lagrangian into
a Charged and a Neutral Contribution . . . . . . . . . . . . . . . . . . 68
11.2.1 The Charged Contribution . . . . . . . . . . . . . . . . . . . . 69
11.2.2 The Neutral Contribution. . . . . . . . . . . . . . . . . . . . . 69
11.3 The Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
11.3.1 The Part U(1) of the SU(2) 9 U(1) Gauge . . . . . . . . 70
11.3.2 The Part SU(2) of the SU(2) 9 U(1) Gauge . . . . . . . 71
11.3.3 Zitterbewegung and Electroweak Currents
in Dirac Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Part VII On a Change of SU(3) into Three SU(2) 3 U(1)
12 On a Change of SU(3) into Three SU(2) 3 U(1) . . . . . . . . . . . . . 75
12.1 The Lie Group SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
12.1.1 The Gell–Mann Matrices ka. . . . . . . . . . . . . . . . . . . 75
12.1.2 The Column W on which the Gell–Mann
Matrices Act . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.1.3 Eight Vectors Ga . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.1.4 A Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
12.1.5 On the Algebraic Nature of the Wk . . . . . . . . . . . . . . 76
12.1.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
12.2 A passage From SU(3) to Three SU(2) 9 U(1) . . . . . . . . . . . 77
12.3 An Alternative to the Use of SU(3) in Quantum
Chromodynamics Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Part VIII Addendum
13 A Real Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 83
13.1 General Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
13.2 Electromagnetism: The Electromagnetic Potential. . . . . . . . . . 84
13.2.1 Principles on the Potential . . . . . . . . . . . . . . . . . . . . 84
13.2.2 The Potential Created by a Population of Charges . . . 85
13.2.3 Notion of Charge Current . . . . . . . . . . . . . . . . . . . . 86
13.2.4 The Lorentz Formula of the Retarded Potentials. . . . . 87
13.2.5 On the Invariances in the Formula of the
Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 88
13.3 Electrodynamics: The Electromagnetic Field,
the Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
13.3.1 General Definition . . . . . . . . . . . . . . . . . . . . . . . . . 89
13.3.2 Case of Two Punctual Charges: The Coulomb Law . . 89
13.3.3 Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . 90
13.3.4 Electric and Magnetic Fields Deduced from the
Lorentz Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 91
13.3.5 The Poynting Vector . . . . . . . . . . . . . . . . . . . . . . . . 93
13.4 Electrodynamics in the Dirac Theory of the Electron . . . . . . . 93
13.4.1 The Dirac Probability Currents. . . . . . . . . . . . . . . . . 94
13.4.2 Current Associated with a Level E of Energy . . . . . . 94
13.4.3 Emission of an Electromagnetic Field . . . . . . . . . . . . 95
13.4.4 Spontaneous Emission. . . . . . . . . . . . . . . . . . . . . . . 95
13.4.5 Interaction with a Plane Wave . . . . . . . . . . . . . . . . . 96
13.4.6 The Lamb Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Part IX Appendices
14 Real Algebras Associated with an Euclidean Space . . . . . . . . . . . 105
14.1 The Grassmann (or Exterior) Algebra of Rn . . . . . . . . . . . . . 105
14.2 The Inner Products of an Euclidean Space E ¼ Rq;nq . . . . . . 105
14.3 The Clifford Algebra CIðEÞ Associated with
an Euclidean Space E ¼ Rp;np . . . . . . . . . . . . . . . . . . . . . . 106
14.4 A Construction of the Clifford Algebra . . . . . . . . . . . . . . . . . 108
14.5 The Group OðEÞ in CIðEÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 109
15 Relation Between the Dirac Spinor and the Hestenes Spinor . . . . 111
15.1 The Pauli Spinor and Matrices . . . . . . . . . . . . . . . . . . . . . . . 111
15.2 The Dirac spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
15.3 The Quaternion as a Real Form of the Pauli spinor . . . . . . . . 113
15.4 The Biquaternion as a Real Form of the Dirac spinor . . . . . . . 114
16 The Movement in Space–Time of a Local
Orthonormal Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
16.1 C.1 The Group SOþðEÞ and the Infinitesimal
Rotations in ClðEÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
16.2 Study on Properties of Local Moving Frames . . . . . . . . . . . . 116
16.3 Infinitesimal Rotation of a Local Frame . . . . . . . . . . . . . . . . 116
16.4 Infinitesimal Rotation of Local Sub-Frames . . . . . . . . . . . . . . 117
16.5 Effect of a Local Finite Rotation of a Local Sub-Frame . . . . . 118
17 Incompatibilities in the Use of the Isospin Matrices . . . . . . . . . . . 121
17.1 W is an ‘‘Ordinary’’ Dirac Spinor . . . . . . . . . . . . . . . . . . . . . 121
17.2 W is a Couple (Wa; Wb) of Dirac Spinors . . . . . . . . . . . . . . . 121
17.3 W is a Right or a Left Doublet. . . . . . . . . . . . . . . . . . . . . . . 122
17.4 Questions about the Nature of the Wave Function . . . . . . . . . 122
18 A Proof of the Tetrode Theorem. . . . . . . . . . . . . . . . . . . . . . . . . 123
19 About the Quantum Fields Theory . . . . . . . . . . . . . . . . . . . . . . . 125
19.1 On the Construction of the QFT. . . . . . . . . . . . . . . . . . . . . . 125
19.2 Questions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
19.3 An Artifice in the Lamb Shift Calculation . . . . . . . . . . . . . . . 127

(Contents: http://www.springerlink.com/content/978-3-642-19198-5/front-matter.pdf)

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