Mathematical foundations of New Physics (2019)
Bibliographic info
А. А. Ketsaris, Mathematical foundations of New Physics, Moscow, Green Print, 2019. Vol. 1 – 328 pages, Vol. 2 – 446 pages,
in Russian.
ISBN 978-5-6042069-5-9(v.1)
ISBN 978-5-6042069-7-3(v.2)
ISBN 978-5-6042069-6-6
Summary
This study concerns the field of fundamental generalizing concepts in present-day physics known as Unified Theory, Theory of Everything etc. Here we call it the New Physics.
Our key approach is to make algebraic generalization of two spaces: the space-time and the space of the action similar to the space-time.
We attribute the properties of the universal algebra to the space-time and the action space. This main concept allows us to explain quantum phenomena and give a new understanding of the wave function. Furthermore, it helps us to explain the hierarchy of fundamental elementary particles and make generalizations about them. A special case of universal algebra is Clifford's algebra assigned to leptons in our approach. Linear and bilinear transformations of universal algebra are set to match intermediate particles. These transformations make it possible to describe the interaction of fundamental and intermediate particles.
The book is intended for researchers in theoretical and mathematical physics, particle physics, gravitation theory and unified field theory, as well as for teachers, postgraduates and students of these specialties.
This work develops the concept outlined earlier in
The book contains two volumes and 6 parts.
The first volume includes the first three parts:
- Part 1. The frame of reference. Kinematics.
- Part 2. Curved space.
- Part 3. Dynamics. Action. Fundamental physical objects.
The second volume includes the next three parts:
- Part 4. Fundamental particles.
- Part 5. Intermediate physical objects.
- Part 6. Dynamic equations.
In the near future the monograph will go to the following Russian libraries:
- Национальное фондохранилище отечественных печатных изданий Информационного телеграфного агентства России (ИТАР-ТАСС);
- The Russian State Library (RSL);
- The National Library of Russia (NLR);
- Государственная публичная научно-техническая библиотека Сибирского отделения Российской академии наук (SPSTL SB RAS), Novosibirsk;
- Дальневосточная государственная научная библиотека, Khabarovsk;
- Библиотека Российской академии наук (БАН), St. Petersburg;
- Парламентская библиотека Российской Федерации, Moscow;
- Библиотека Администрации Президента Российской Федерации, Moscow;
- Крымская республиканская универсальная научная библиотека имени И. Я. Франко, Симферополь;
- Научная библиотека Московского государственного университета имени М. В. Ломоносова (МГУ), Moscow;
- Russian National Public
Library for Science and Technology, Moscow;
- Государственная публичная историческая библиотека России, Moscow;
- Всероссийская государственная библиотека иностранной литературы имени М. И. Рудомино, Moscow;
- Библиотека по естественным наукам Российской академии наук, Moscow.
Foundations of mathematical physics (1997)
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:
- Existence of new particles being symmetrical analog to leptons and quarks
is substantiated; this substantiation is created on a classification
of subalgebras of universal algebra of contravariant tensors.
- Dirac equation for eight-component complex wave function,
describing a system of two of particles such as electron and neutrino, is
generalizated.
- Dirac equation in the case of an arbitrary gauge field is
generalizated. Tensor of the field is taken also into account but not only
potential of the gauge field.
- The equations of a gravitational field are obtained. They consists from
three set of equations which contain a tensor of a curvature, tensor
of a torsion and matrix of linear transformation.
Among different methods worked out by the author the following is worth the
special attention:
- Deriving the quantum postulates from the structure equations for
action vector.
- Generalized variational principle, establishing correlation
between the dynamics equations and commutation relations for
operators; this principle permits to receive addends, responsible for a
self-operation of a gauge field.
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
- Foreword 8
- Introduction 10
- Basic concepts and definitions 13
- Primary concepts and judgments 13
- Solids and processes 13
- Motion of solids and processes 14
- Interactions 15
- Space-time 17
- 4-dimensional vector space 18
- Shifts of space-time. Space of fundamental particles 21
- Shifts of a vector space 21
- Spaces of tensors 25
- 2.1 Linear map. Power function of the first order 25
- 2.2 Space of tensors of the second order. Power function of the second order 25
- 2.3 Space of tensors of the n-th order. Power function of the n-th order 26
- 2.4 Polynomial. Universal space of contravariant tensors C(X) 28
- Universal algebra C(X) is algebra of shifts 29
- Representations of algebra of shifts 32
- Subalgebras of algebra of shifts 35
- 5.1 Normalized algebra of shifts R(X) 38
- 5.2 Symmetrization of tensors. Young diagram 38
- 5.3 Young tree and space of fundamental particles 47
- 5.4 Space of spin. Space of inertia 49
- 5.5 Commutation relations 50
- Clifford algebra. Space of leptons 53
- 6.1 Regular representation of basis vectors of Clifford algebra. Matrixes of Pauli and Dirac 56
- 6.2 Subalgebras of Clifford algebra 67
- 6.3 Product of Clifford algebras. Space of leptons and their neutrino 73
- 6.3.1 Algebra CL_{3}(X) and space of leptons 76
- 6.3.2 Algebra CL_{4}(X) and space of leptons 78
- Algebra LI(X). Space of leptino 81
- 7.1 Regular representation of basis vectors of algebra LI(X) 83
- Spaces of quarks Q(X) and quarkino QI(X) 93
- Linear maps. Rotations 105
- Linear map of space-time 105
- Linear map of algebra of shifts C(X) 106
- Rotations of universal space C(X) 109
- 3.1 Scalar product. Length of a vector 109
- 3.2 Rotations 110
- Projection of universal space C(X) 112
- Kinematic algebra T(X) = C(X) + U 114
- Conjugate space-time. Antiparticles 117
- Algebra of shifts in antispace-time is covariant universal algebra C(X) 117
- Algebra of fundamental particles and antiparticles C(X,X) 120
- Linear maps of a conjugate space. Rotations in antispace-time 121
- Projection of universal space C(X) 123
- Common algebra of rotations T=U + U 124
- Kinematic algebra in antispace-time T(X) = C(X) + U 125
- Common kinematic algebra T(X,X) = C(X) + C(X) + U + U 126
- Derivation. A gauge field 129
- Derivation of algebra of shifts C(X) 129
- Derivation of algebra of rotations U 131
- Derivation of common algebra of rotations T=U + U 134
- Derivation of kinematic algebra T(X) = C(X) + U 135
- Derivation of conjugate kinematic algebra T(X) = C(X) + U 136
- Derivation of common kinematic algebra T(X,X) = C(X) + U + C(X) + U 137
- Operators of derivation 140
- 7.1 Commutation relations for operators of derivation of algebra of shifts C(X) 140
- 7.2 Commutation relations for operators of derivation of algebra of rotations U 141
- 7.3 Commutation relations for operators of derivation of common algebra of rotations T=U + U 142
- 7.4 Commutation relations for operators of derivation of kinematic algebra T(X) = C(X) + U 143
- Second kinematic algebra. Multiplication of basis vectors 145
- Structure equations of the second kinematic algebra 149
- Second kinematic algebra. Commutation relations for operators of derivation 151
- Gauge group. The gauge field 155
- Structure equations of kinematic algebra in a gauge field 156
- Operators of derivation of kinematic algebra in a gauge field 159
- Structure equations of second kinematic algebra in a gauge field 161
- Operators of derivation of the second kinematic algebra in a gauge field 166
- Principle of equivalence 170
- Bianchi identities 172
- Parametrical representation of linear maps 173
- Covariant derivation 179
- Transformation of space and transformation of coordinates 181
- Covariant derivation on subgroup of gauge group 182
- Generalized structure equations and commutation relations 184
- Action. Dynamics equations. The equations of quantization 187
- Action vector. Action invariant 187
- Dynamic variable 188
- Action. Lagrangian. Dynamics equations 189
- Conservation law and dynamics equations 193
- Generalized action vector 194
- Lagrangian of the second order. Dynamics equations of the second order 197
- Lagrangian of the third order. Dynamics equations of the third order 200
- Connection between dynamic and field variable 201
- Wave function. The equations of quantization. Quantum postulates 202
- Equations of quantization in the Dirac form 204
- Equations of quantization and variational principle 206
- Generalized equations of quantization 207
- Transformations dynamic variable 208
- 13.1 First approximation 210
- 13.2 Second approximation 210
- 13.3 Common case 214
- Dynamics equations of the first order 221
- 14.1 First approximation 221
- 14.2 Second approximation 222
- 14.3 Dynamics equations in a gauge field. The second approximation 223
- 14.4 Third approximation 225
- Dynamics equations of the second order 229
- Dynamics equations of the third order 231
- Equations of dynamics of shifts, rotations and accelerations 233
- Equations of a field with sources 235
- Free field equations 236
- Classical dynamics equations 237
- Compatibility condition for dynamics equations 239
- Invariancy equations for dynamic variable 244
- Equations of mathematical physics 249
- Postulates of a quantum mechanics 249
- Quantum mechanics equations for leptons 250
- Quantum mechanics equations for leptino 251
- Electromagnetic interaction. Dynamics equations. Field equations 252
- Transformation of a electromagnetic field tensor due to an accelerated frame of reference 254
- Equations of quantization for leptons in an electromagnetic field 255
- 6.1 Electromagnetic interaction and components of leptons 256
- Electroweak interaction of leptons 257
- Spin interaction of leptons 262
- Force interaction. Field equations 265
- Gravitational interaction. Dynamics equations. Field equations 267
- Conclusions 269
- Bibliography 273
- Subject Index 275