## Bibliographic infoKetsaris A. A.,Foundations of mathematical physics, Moscow, Association of independent publishers, 1997. – 280 pages,
in Russian.
The second edition of the book under the new title:
## Algebraic foundations of physics## Space-time and action as universal algebraswas published in Editorial URSS Publishers. |

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.

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

- 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__

- 2.1
**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__

- 5.1
**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__

- 6.3.1

- 6.1
**Algebra LI(X). Space of leptino**__81__- 7.1
**Regular representation of basis vectors of algebra LI(X)**__83__

- 7.1
**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__

- 3.1
**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__

- 7.1
**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__

- 13.1
**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__

- 14.1
**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__

- 6.1
**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__