Quantum mechanics in the geometry of space-time: Elementary theory

143 Pages·2011·0.67 MB·English

by  Boudet, Roger

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Eric Le Ru, Victoria University of Wellington, New Zealand
James Overduin, Towson University, USA
Vesselin Petkov, Concordia University, Canada
Charles H.-T. Wang, The University of Aberdeen, UK
Andrew Whitaker, Queen’s University Belfast, UK
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Roger Boudet
Quantum Mechanics in the
Geometry of Space–Time
Elementary Theory
123
Roger Boudet
Honorary Professor
Université de Provence
Av.de Servian 7
34290Bassan
France
e-mail: [email protected]
ISSN 2191-5423 e-ISSN2191-5431
ISBN 978-3-642-19198-5 e-ISBN978-3-642-19199-2
DOI 10.1007/978-3-642-19199-2
SpringerHeidelbergDordrechtLondonNewYork
(cid:2)RogerBoudet2011
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Preface
Theaimoftheworkweproposeisacontributiontotheexpressionofthepresent
particlestheoriesintermsentirelyrelevanttotheelementsofthegeometryofthe
Minkowski space–time M ¼R1;3, that is those of the Grassmann algebra ^R4,
scalars, vectors, bivectors, pseudo-vectors, pseudo-scalars of R4 associated with
thesignature(1,3)whichdefinesM,and,atthesametime,theeliminationofthe
complex language of the Pauli and Dirac matrices and spinors which is used in
quantum mechanics.
Thereasonsforthischangeoflanguagelie,inthefirstplace,inthefactthatthis
real language is the same as the one in which the results of experiments are
written, which are necessarily real.
Butthereisanotherreasoncertainlymoreimportant.Experimentsaregenerally
achievedinalaboratoryframewhichisagalileanframe,andthefundamentallaws
of Nature are in fact independent of all galilean frame. So the theories must be
expressedinaninvariantform.Thengeometricalobjectsappear,whoseproperties
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
wavefunctionWassociatedwithaparticleisnothingelsebyitselfbutacolumnof
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 W. In this form, the Lorentz
rotationwhichallowsthedirectpassagetotheinvariantentitiesappearsexplicitly.
In particular the geometrical meaning of the gauges defined by the complex Lie
rings Uð1ÞandSUð2Þ becomes evident.
v
vi Preface
It should be emphasized, like an indisputable confirmation of the independent
work of Hestenes, that a geometrical interpretation of the Dirac W 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 W is expressed by means of Dirac
matrices,theselastonesbeingimplicitlyidentifiedwiththevectorsofthegalilean
frame in which the Dirac equation of the electron is written.
But the use of a tool, the Clifford algebra Clð1;3Þ associated with the space
M ¼R1;3, introduced by D. Hestenes, brings considerable simplifications. Pages
ofcalculationsgivingtensorialequationsdeducedfromthecomplexlanguagemay
be replaced by few lines. Furthermore ambiguities associated with the use of the
p
ffiffiffiffiffiffiffi
imaginarynumberi¼ (cid:2)1areeliminated. Thestriking pointlies inthefact 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
Lorentzrotationandthemultiplicationby(cid:2)hc=2,theproperangularmomentum,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(cid:2)hof (cid:2)handi 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
electrodynamicsentirelyreal.ItisonlybasedontheuseoftheGrassmannalgebra
of M and the inner product in M.
The more the theories of the particles become complicated, the more the links
whichcanunifythesetheoriesinanidenticalvisionofthelawsofNaturehaveto
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
basedontheuseofthecomplexmatrices,spinorsandLierings,thiscritiquedoes
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
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
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
p
ffiffiffiffiffiffiffi
2.2 The Clifford Algebras and the ‘‘Imaginary Number’’ (cid:2)1. . . . 9
2.3 The Field of the Hamilton Quaternions and the Ring
of the Biquaternion as Clþð3;0ÞandClð3;0Þ’Clþð1;3Þ. . . . . 10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
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
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
vii
viii Contents
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
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
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
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Contents ix
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
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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