Grand Unified Theory

Русский

Foundations of mathematical physics

Bibliographic info

Ketsaris A. A., Foundations of mathematical physics, Moscow, Association of independent publishers, 1997. – 280 pages, in Russian.

Table of contents >>

The second edition of the book under the new title:

Algebraic foundations of physics

Space-time and action as universal algebras
was published in Editorial URSS Publishers.

Summary

A variant of the uniform theory of interactions is considered. It is based on a replacement of 4-dimensional space-time to the space of tensors of all ranks and on a multi-dimensional generalization of the Lagrange least action principle. The author uses methods of algebra and differential geometry including Cartan's method of differential forms. The most of the calculations is given in detail.

Among number of problems considered in this work, the following can be treated as providing some new results:

Among different methods worked out by the author the following is worth the special attention:

The book is intended for the experts and lecturers in theoretical physics and mathematicians as well as for students of these specialities.


Table of contents

Anti-spaces are marked by blue color (for example, X).

  1. Foreword 8
  2. Introduction 10
  3. Basic concepts and definitions 13
    1. Primary concepts and judgments 13
    2. Solids and processes 13
    3. Motion of solids and processes 14
    4. Interactions 15
    5. Space-time 17
    6. 4-dimensional vector space 18
  4. Shifts of space-time. Space of fundamental particles 21
    1. Shifts of a vector space 21
    2. Spaces of tensors 25
      1. 2.1 Linear map. Power function of the first order 25
      2. 2.2 Space of tensors of the second order. Power function of the second order 25
      3. 2.3 Space of tensors of the n-th order. Power function of the n-th order 26
      4. 2.4 Polynomial. Universal space of contravariant tensors C(X) 28
    3. Universal algebra C(X) is algebra of shifts 29
    4. Representations of algebra of shifts 32
    5. Subalgebras of algebra of shifts 35
      1. 5.1 Normalized algebra of shifts R(X) 38
      2. 5.2 Symmetrization of tensors. Young diagram 38
      3. 5.3 Young tree and space of fundamental particles 47
      4. 5.4 Space of spin. Space of inertia 49
      5. 5.5 Commutation relations 50
    6. Clifford algebra. Space of leptons 53
      1. 6.1 Regular representation of basis vectors of Clifford algebra. Matrixes of Pauli and Dirac 56
      2. 6.2 Subalgebras of Clifford algebra 67
      3. 6.3 Product of Clifford algebras. Space of leptons and their neutrino 73
        1. 6.3.1 Algebra CL3(X) and space of leptons 76
        2. 6.3.2 Algebra CL4(X) and space of leptons 78
    7. Algebra LI(X). Space of leptino 81
      1. 7.1 Regular representation of basis vectors of algebra LI(X) 83
    8. Spaces of quarks Q(X) and quarkino QI(X) 93
  5. Linear maps. Rotations 105
    1. Linear map of space-time 105
    2. Linear map of algebra of shifts C(X) 106
    3. Rotations of universal space C(X) 109
      1. 3.1 Scalar product. Length of a vector 109
      2. 3.2 Rotations 110
    4. Projection of universal space C(X) 112
    5. Kinematic algebra T(X) = C(X) + U 114
  6. Conjugate space-time. Antiparticles 117
    1. Algebra of shifts in antispace-time is covariant universal algebra C(X) 117
    2. Algebra of fundamental particles and antiparticles C(X,X) 120
    3. Linear maps of a conjugate space. Rotations in antispace-time 121
    4. Projection of universal space C(X) 123
    5. Common algebra of rotations T=U + U 124
    6. Kinematic algebra in antispace-time T(X) = C(X) + U 125
    7. Common kinematic algebra T(X,X) = C(X) + C(X) + U + U 126
  7. Derivation. A gauge field 129
    1. Derivation of algebra of shifts C(X) 129
    2. Derivation of algebra of rotations U 131
    3. Derivation of common algebra of rotations T=U + U 134
    4. Derivation of kinematic algebra T(X) = C(X) + U 135
    5. Derivation of conjugate kinematic algebra T(X) = C(X) + U 136
    6. Derivation of common kinematic algebra T(X,X) = C(X) + U + C(X) + U 137
    7. Operators of derivation 140
      1. 7.1 Commutation relations for operators of derivation of algebra of shifts C(X) 140
      2. 7.2 Commutation relations for operators of derivation of algebra of rotations U 141
      3. 7.3 Commutation relations for operators of derivation of common algebra of rotations T=U + U 142
      4. 7.4 Commutation relations for operators of derivation of kinematic algebra T(X) = C(X) + U 143
    8. Second kinematic algebra. Multiplication of basis vectors 145
    9. Structure equations of the second kinematic algebra 149
    10. Second kinematic algebra. Commutation relations for operators of derivation 151
    11. Gauge group. The gauge field 155
    12. Structure equations of kinematic algebra in a gauge field 156
    13. Operators of derivation of kinematic algebra in a gauge field 159
    14. Structure equations of second kinematic algebra in a gauge field 161
    15. Operators of derivation of the second kinematic algebra in a gauge field 166
    16. Principle of equivalence 170
    17. Bianchi identities 172
    18. Parametrical representation of linear maps 173
    19. Covariant derivation 179
    20. Transformation of space and transformation of coordinates 181
    21. Covariant derivation on subgroup of gauge group 182
    22. Generalized structure equations and commutation relations 184
  8. Action. Dynamics equations. The equations of quantization 187
    1. Action vector. Action invariant 187
    2. Dynamic variable 188
    3. Action. Lagrangian. Dynamics equations 189
    4. Conservation law and dynamics equations 193
    5. Generalized action vector 194
    6. Lagrangian of the second order. Dynamics equations of the second order 197
    7. Lagrangian of the third order. Dynamics equations of the third order 200
    8. Connection between dynamic and field variable 201
    9. Wave function. The equations of quantization. Quantum postulates 202
    10. Equations of quantization in the Dirac form 204
    11. Equations of quantization and variational principle 206
    12. Generalized equations of quantization 207
    13. Transformations dynamic variable 208
      1. 13.1 First approximation 210
      2. 13.2 Second approximation 210
      3. 13.3 Common case 214
    14. Dynamics equations of the first order 221
      1. 14.1 First approximation 221
      2. 14.2 Second approximation 222
      3. 14.3 Dynamics equations in a gauge field. The second approximation 223
      4. 14.4 Third approximation 225
    15. Dynamics equations of the second order 229
    16. Dynamics equations of the third order 231
    17. Equations of dynamics of shifts, rotations and accelerations 233
    18. Equations of a field with sources 235
    19. Free field equations 236
    20. Classical dynamics equations 237
    21. Compatibility condition for dynamics equations 239
    22. Invariancy equations for dynamic variable 244
  9. Equations of mathematical physics 249
    1. Postulates of a quantum mechanics 249
    2. Quantum mechanics equations for leptons 250
    3. Quantum mechanics equations for leptino 251
    4. Electromagnetic interaction. Dynamics equations. Field equations 252
    5. Transformation of a electromagnetic field tensor due to an accelerated frame of reference 254
    6. Equations of quantization for leptons in an electromagnetic field 255
      1. 6.1 Electromagnetic interaction and components of leptons 256
    7. Electroweak interaction of leptons 257
    8. Spin interaction of leptons 262
    9. Force interaction. Field equations 265
    10. Gravitational interaction. Dynamics equations. Field equations 267
  10. Conclusions 269
  11. Bibliography 273
  12. Subject Index 275