Book by N. Marchuk: Field theory equations


  • Author: Nikolay Marchuk
  • Title: Field theory equations
  • Paperback: 290 pages
  • Publisher: CreateSpace Independent Publishing Platform (September 19, 2012)
  • Language: English
  • ISBN-10: 1479328073
  • ISBN-13: 978-1479328079
  • List Price: 30 USD.
Abstract: 
In this book, the equations of relativistic field theory are examined and, in particular, the covariance and symmetry properties of the Dirac-Maxwell and Dirac-Yang-Mills equations are considered. Several new sets of equations, called the model equations of field theory, are dealt with. These systems of equations reproduce the basic properties of standard systems of field theory equations. At the same time, the model equations exhibit a number of differences from the standard equations of field theory and, in particular, they have a new internal symmetry with respect to the pseudo-unitary (or symplectic, or spinor) group. Development of the concept of local pseudo-unitary (symplectic, spinor) symmetry of the model equations of field theory leads to momentous consequences. In certain sections of the book, the mathematical tools of Clifford and Atyah-Kahler algebras are applied.

1

2

Dirac—Maxwell equations

1.1   Minkowski space and tensor fields .   .  .  . .  .  .  .  . . . .   .  .  .

1.2   Algebra of Dirac matrices     .  .  . .  .  .  .  . .  .  .  .  . . .  .  .  .  .

1.3   Dirac—Maxwell equations in Minkowski space        . . . . . . . .

1.4   Charge conjugation of Dirac spinors               . . . . . . . . . . . . .

Model Dirac—Maxwell equations

19 19 21 24 32

36

  2.1 Relationship between ry-matrices and the pseudo-unitary group 5U12, 2)
     

36

  2.2 Model Dirac—Maxwell set of equations           . . . . . . . . . . . .

37

  2.3 Model Dirac—Maxwell equations with gauge pseudo-unitary symmetry  
     

38

  2.4 Formula for C&    .  .  . .  .  .  .  .  . .  .  .  .  . .  .  .  .  . .  .  .  .  . .

39

  2.5 Spinorization of the model equations               . . . . . . . . . . . . .

41

3

Clifford algebras

44

  3.1 Groups, vector spaces, algebras   .  . .  .  .  .  . .  .  .  .  .  . .  .  .

44

  3.2 Grassmann algebras A1n)     .  .  . .  .  .  .  . .  .  .  .  . .  .  .  .  . .

46

  3.3 Clifford algebras Ce1p, q) .  .  . .  .  .  .  .  . .  .  .  .  . .  .  .  .  . .

48

  3.4 Clifford multiplication of Grassmann algebra elements        . . .

52

  3.5 Commutators and anticommutators                 . . . . . . . . . . . . .

53

  3.6 Theorem on the convolution of generators             . . . . . . . . . .

61

  3.7 Conjugation operators    .  .  .  .  . .  .  .  .  . .  .  .  .  . .  .  .  .  . .

62

  3.8 Complex Clifford algebras Cep1p, q)   . . . . .     . . . .    . . .    . .

66

  3.9 Structure of unitary (or Euclidean) space on the Clifford algebras
         . .  .  .  .  . .  .  .  .  . .  .  .  .  . .

67

3.10 Hermitian idempotents, left-hand ideals and related structures 72

3.11 Normal representations of Clifford algebra elements in the form of complex matrices . . . . . . . . . . . . . . . . 75

3.12 Matrix representations of the algebra Ce11, 3) . . . . . . . . 81

3.13 Other matrix representations of the algebra Ce11, 3) . . . . . 84

3.14 Secondary generators of the algebra Ce11, 3) . . . . . . . . . . . . . . . . . . . . . .                    86

3.15 Simplest operations on elements of the algebra Ce11, 3) . . . 88

3.16 Local generalized Pauli theorem . . . . . . . . . . . . . . . . 91

4 Lie groups and Lie algebras, related to the Clifford algebras                                                        94

4.1 Unitary group of the Clifford algebra . . . . . . . . . . . . . 94

4.2 The case of Clifford algebra ;e(1, 3) . . . . . . . . . . . . . . 96

4.3 Pseudo-unitary group of the Clifford algebra . . . . . . . . . . . . . . . . . . . . .                       99

4.4 Symplectic subgroup of the pseudo-unitary group . . . . . . . . . . . . . . . . . . 104

4.5  Spinor and orthogonal groups . . . . . . . . . . . . . . . . . 107

4.6 Exponent of elements of second rank                         . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.7 External exponential function of elements of second rank                         . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.8 Groups Pin(1, 3), Pin+(1, 3), Spin(1, 3), Spin+(1, 3) and Pin8(1, 3)                          . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.9 The set ;e#222(1, 3) and the amplitude . . . . . . . . . . . . 121

4.10 Unitary subgroups of the pseudo-unitary, symplectic, and spinor groups . . . . . . . . . . . . . . . . . . . . . . . . 125

5 Field theory model equations in the Clifford algebra formalism                                                                                             130

5.1 Tensors with values in Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . 130

5.2   Yang—Mills equations . . . . . . . . . . . . . . . . . . . . . . 132

5.3   Model Dirac—Yang—Mills equations . . . . . . . . . . . . . . 133

5.4 Object characterization in the model equations . . . . . . . . . . . . . . . . . . . . . 137

5.5 Covariant transformations and the symmetries of model equations . . . . . . . . . . . . . . 139

5.6   Properties of the model Dirac—Yang—Mills equations . . . . . . . . . . . . . . . . . . 143

5.7 Hamiltonian form of model Dirac—Maxwell equations . . . . . . . . . . . . . . . . . . . 146

5.8 The localization of pseudo-unitary symmetry . . . . . . . . . . . . . . . . . . 150

5.9 Model equations with two Yang—Mills fields . . . . . . . . . . . . . . . . . . . . . . 154

5.10 Semi-divergent form of the model Dirac equation . . . . . . 157

5.11 Model Dirac—Yang—Mills equations with local spinor symmetry . . . . . . . . . . . . . . . . . . 158

5.12 A hypothesis concerning the description of antiparticles and particles of opposite spins . . . . . . . . . . . . . . . . . . . 160

5.13 The operation of charge conjugation . . . . . . . . . . . . . 161

6 Model equations on the pseudo-Riemannian manifold                  165

6.1 The pseudo-Riemannian spinor manifold . . . . . . . . . . . 165

6.2 Model equations on the pseudo-Riemannian manifold . . . . . . . . . . . . . 174

6.3   Model equations with local spinor symmetry on the pseudo-Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . 178

7 Model equations in the Atyah-K,ahler algebra formalism 182

7.1 Differential forms and tetrad on the spinor manifold . . . . . . . . . . . . . . . . . . . . . 182

7.2 Tensors with values in the Atyah—Kfahler algebra . . . . . . . . . . . . . . . . . . . 185

7.3 Unitary, pseudo-unitary and spinor groups in the Atyah—Kfahler algebra formalism . . . . . . . . . . . . 187

7.4 Formal partial derivatives <& . . . . . . . . . . . . . . . . . 189

7.5 Operators ~, d, S                       . . . . . . . . . . . . . . . . . . . . . . . . 191

7.6 Relationship between the spinor manifold X’,) and Riemann—Cartan space . . . . . . . . . . . . . . . 193

7.7 Formal covariant derivatives . . . . . . . . . . . . . . . . . . 198
7.8 Model equations with the pseudo-unitary symmetry . . . . 199

7.9 Model equations with local spinor symmetry . . . . . . . . . 204

8 Model field theory equations within the matrix formalism                                                          209

8.1 Model Dirac—Maxwell equations . . . . . . . . . . . . . . . . 209

8.2 Relationship between the standard and the model Dirac— Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . 215

8.3 Model Dirac—Yang—Mills equations . . . . . . . . . . . . . . 216

8.4 Model Dirac—Yang—Mills equations with local pseudo-unitary symmetry . . . . . . . . . . . . . 218

8.5 Model equations with two Yang—Mills fields . . . . . . . . . 222

8.6 Model set of equations with spinor local symmetry . . . . . 224

9 Special model equations                                                                   230

9.1 The main idea. . . . . . . . . . . . . . . . . . . . . . . . . . 230

9.2 Lie algebras of anti-Hermitian differential forms. . . . . . . 236

9.3 Basic equations.. . . . . . . . . . . . . . . . . . . . . . . . . 238

9.4 Non—Abelian laws of charge conservation. . . . . . . . . . . 239

9.5 Unitary and spinor gauge symmetries. . . . . . . . . . . . . 241

10 Amplitude in relativistic field equations                                        243

10.1 Model Dirac—Maxwell equations with local spinor symmetry 243

10.2 Special model Dirac—Maxwell equations . . . . . . . . . . . 245

10.3 Solutions of the special model Dirac equation of the plane-wave type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

10.4 Fixation of the spinor gauge . . . . . . . . . . . . . . . . . . 249

10.5 The special case of a& 3 3 . . . . . . . . . . . . . . . . . . . 250

11 Supplements                                                                                    255

11.1 Formulae for commutators and anti-commutators . . . . . . . . . . . . . . . . . . . . . 255

11.2 Matrix representations for generators of the Clifford algebra 258

11.3 Determinant, spectrum and Hermitian conjugation of elements of the Clifford algebra . . . . . . . . . . . . . . . . . 266

11.4 Expression of tetrad components through the metric tensor components . . . . . . . . . . . . 269

11.5 Associated and external matrices . . . . . . . . . . . . . . . 273

11.6 Algebraic tensor operations . . . . . . . . . . . . . . . . . . 277

11.7 P.S.   .       . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Source: Emails by Nikolay Marchuk, nmarchuk2005_at_yandex.ru, 14+16 July 2013

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