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- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
"Problem Solving in Theoretical Physics" helps students mastering their theoretical physics courses by posing advanced problems and providing their solutions - along with discussions of their physical significance and possibilities for generalization and transfer to other fields.
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Seitenzahl: 732
Veröffentlichungsjahr: 2020
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Table of Contents
Cover
Preface to the English Edition
Preface
About the Companion Website
Part I: Problems
Chapter 1: Field Theory
Introduction
Problems
1.1 Vectors and tensors in Euclidean space
1.2 Vectors and tensors in Minkowski space
1.3 Relativistic kinematics
1.4 The Maxwell equations
1.5 The motion of a charged particle in an external field
1.6 Static electromagnetic field
1.7 Free electromagnetic field
1.8 The retarded potentials and radiation
1.9 Electromagnetic field of relativistic particles
1.10 The scattering of electromagnetic waves
Chapter 2: Quantum Mechanics
Introduction
2.1 Operators and states in quantum mechanics
2.2 One‐dimensional motion
2.3 Linear harmonic oscillator
2.4 Angular momentum and spin
2.5 Motion in a magnetic field
2.6 Motion in a centrally symmetric field
2.7 Semiclassical approximation
2.8 Perturbation theory
2.9 Relativistic quantum mechanics
2.10 Addition of angular momenta. The identity of particles
2.11 Theory of atoms and molecules
2.12 Theory of scattering
2.13 Theory of radiation
Chapter 3: Statistical Physics
Introduction
3.1 The Gibbs distribution: thermodynamic quantities and functions
3.2 Ideal quantum gases
3.2.1 The ideal Fermi gas
3.2.2 The ideal Bose gas
3.2.3 Ideal gases of elementary Bose excitations
3.3 Non‐ideal quantum systems (liquids): the fundamentals of condensed matter theory
3.3.1 Normal (nonsuperfluid) Fermi liquids
3.3.2 Superconductivity and the BCS theory
3.3.3 Weakly interacting Bose gases and the Gross‐Pitaevskii equation
3.3.4 Theory of superfluidity
3.4 Phase transitions and critical phenomena
3.4.1 The mean‐field approximation
3.4.2 The Ginzburg‐Landau functional
3.4.3 Fundamentals of the theory of critical phenomena
Part II: Solutions of Problems
Chapter 1: Field Theory
1.1 Vectors and tensors in Euclidean space
1.2 Vectors and tensors in Minkowski space
1.3 Relativistic kinematics
1.4 The Maxwell equations
1.5 Motion of a charged particle in an external field
1.6 Static electromagnetic field
1.7 Free electromagnetic field
1.8 The retarded potentials and radiation
1.9 Electromagnetic field of relativistic particles
1.10 The scattering of electromagnetic waves
Chapter 2a: Quantum Mechanics
2.1 Operators and states in quantum mechanics
2.2 One‐dimensional motion
2.3 Linear harmonic oscillator
2.4 Angular momentum and spin
2.5 Motion in a magnetic field
2.6 Motion in a centrally symmetric field
2.7 Semiclassical approximation
2.8 Perturbation theory
2.9 Relativistic quantum mechanics
2.10 Addition of angular momenta: The identity of particles
2.11 Theory of atoms and molecules
2.12 Theory of scattering
2.13 Theory of radiation
Chapter 3: Statistical Physics
3.1 The Gibbs distribution: the thermodynamic quantities and functions
3.2 Ideal quantum gases
3.2.1 The ideal Fermi gas
3.2.2 The ideal Bose gas
3.2.3 Ideal gases of elementary Bose excitations
3.3 Non‐ideal quantum systems (liquids): the fundamentals of condensed matter theory
3.3.1 Normal (nonsuperfluid) Fermi liquids
3.3.2 Superconductivity and the BCS theory
3.3.3 Weakly interacting Bose gases and the Gross‐Pitaevskii equation
3.3.4 Theory of superfluidity
3.4 Phase transitions and critical phenomena
3.4.1 The mean‐field approximation
3.4.2 The Ginzburg–Landau functional
3.4.3 Fundamentals of the theory of critical phenomena
Appendices
Bibliography
Theory of fields
Quantum mechanics
Statistical physics
General aspects
Index
End User License Agreement
List of Illustrations
Chapter 1a
Figure 1.1 The geometrical scheme of solving problem 1.3.6
Figure 1.2 The detection of light signals emitted by moving sources. The wor...
Figure 1.3 The light reflection from a moving mirror
Figure 1.4 The energy (a) and angular (b) distributions of the neutrinos eme...
Figure 1.5 The orientation of the magnetic
H
and electric
E
fields in the Ca...
Figure 1.6 The hyperbolic motion
Figure 1.7 The guiding center approximation for the motion of a charged part...
Figure 1.8 The natural basis is as follows:
is the unit vector tangential ...
Figure 1.9 On the problem of charged particles moving in the Earth's magneti...
Figure 1.10 The plot of the force lines
Figure 1.11 The plot for the force lines of the magnetic dipole field
Figure 1.12 The schematic classification of the three zones in the field ind...
Figure 1.13 The fall of one charged particle on another during classical dip...
Figure 1.14 The radiating dipole above the conducting plane
Figure 1.15 The geometry for the relation between the field strengths at the...
Figure 1.16 The radiation pattern of a relativistic charged particle: (a) ve...
Figure 1.17 The electron radiation in a periodic electric field
Chapter 2a
Figure 2.1 The determination of the energy levels for the even states
Figure 2.2 The determination of the energy levels for the odd states
Figure 2.3 The plot for analyzing Eq. (3)
Figure 2.4 The allowed energy bands
Figure 2.5 The stationary states for the system of two
‐potential wells
Figure 2.6 The transmission coefficient for the potential barrier
Figure 2.7 The oblique incidence of the de Broglie wave onto the potential s...
Figure 2.8 The plot of the potential related to Pr. 2.7.1 (case (b))
Figure 2.9 The plot of the potential used in the
‐decay theory
Figure 2.10 The average value of the muon spin in the slowly rotating magnet...
Figure 2.11 The variation of the energy levels of a Dirac particle in a unif...
Figure 2.12 The Zeeman effect for the hyperfine energy levels of the ground ...
Figure 2.13 The Kratzer potential
Figure 2.14 The scattering cross sections of slow‐moving particles at a squa...
Figure 2.15 The differential cross section, given in units of classical valu...
Figure 2.16 The scattering of identical particles
Figure 2.17 The angular distribution of the
‐state radiation of the hydroge...
Chapter 3a
Figure 3.1 The plot of the heat capacity of a gas per one particle as a func...
Figure 3.2 The plot for the temperature behavior of the specific heat of par...
Figure 3.3 The behavior of sound velocity as a function of the magnetic fiel...
Figure 3.4 The plot of the oscillations of the magnetization versus the inve...
Figure 3.5 The behavior of surface tension as a function of the Wigner–Seitz...
Figure 3.6 The plot of the work function versus the Wigner–Seitz radius in t...
Figure 3.7 The energy of the gas versus the variational parameter
for some...
Figure 3.8 The diagram of the phase states. The solid line is the binodal. T...
Figure 3.9 The diagram of phase states in a ferroelectric. The second‐order ...
Figure 3.10 The phase diagram. The second‐order phase transitions are depict...
Figure 3.11 The phase diagram of a ferromagnetic superconductor. The second‐...
Figure 3.12 The superconducting order parameter in the granule as a function...
Guide
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Problem Solving in Theoretical Physics
Yury M. BelousovSerguei N. BurmistrovAlexei I. Ternov
Copyright
Authors
Dr. Yury M. Belousov
Moscow Institute of Physics and Technology
9 Institutskiy per.
141701 Dolgoprudny
Russia
Dr. Serguei N. Burmistrov
Kurchatov Institute
1, Akademika Kurchatova pl.
123182 Moscow
Russia
Dr. Alexei I. Ternov
Moscow Institute of Physics and Technology
9 Institutskiy per.
141701 Dolgoprudny
Russia
All books published by Wiley‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.
© 2020 Wiley‐VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany
All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.
Print ISBN: 978‐3‐527‐41396‐6
ePDF ISBN: 978‐3‐527‐82891‐3
ePub ISBN: 978‐3‐527‐82890‐6
Cover Design: SCHULZ Grafik‐Design, Fußgönheim, Germany
Preface to the English Edition
In the Moscow Institute of Physics and Technology (MIPT) the department of theoretical physics is one of the basic departments providing fundamental education for students. The department instructs second‐to‐fifth year students of all departments. The course of theoretical physics completes the fundamental education on physics.
Academician and Nobel Prize winner L. D. Landau was the first head of the department from 1946 to 1951. Many of his notable students (L. P. Gor'kov, I. M. Khalatnikov, E. M. Lifshitz, A. I. Larkin) taught at the department for various time. Some of the Landau doctoral students are still teaching. One of them, Professor S. S. Gershtein, is also one of first members of the Landau Theoretical Division in the Kapitza Institute for Physical Problems. The Landau scientific school formulated the fundamentals of instruction on theoretical physics in MIPT. It is no fortuity that the ten‐volume Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz, known widely in the world, is the main manual for our students. Our department offered its mite to perfecting this fundamental classical work. Professor Landau was succeeded by Professor V. B. Berestetskii, one of Landau's notable students, who headed the department until 1977. That time the structure of instructing theoretical physics was completed on the whole and the basics have remained unchanged up to the present. Academician S. T. Belyaev, known for his pioneer works on non‐ideal Bose‐Einstein condensed gases, headed the department from 1978 to 1991. He belonged to the first MIPT graduates.
The staff of the department of theoretical physics consists of highly qualified theorists, most of them are well‐known in science and technology as well as in such scientific research organizations as Kapitza Institute for Physical Problems, Institute for Theoretical and Experimental Physics, Landau Institute for Theoretical Physics, Lebedev Institute of Physics, Kurchatov Institute, Institute for High Energy Physics, and Institute for High Pressure Physics. The scientific interests and expertise of the Department's members embrace in essence all modern trends in theoretical physics.
Preparing the English edition, we have made a number of minor changes and also corrected misprints revealed since the first issue of the Russian edition. We would like to express our gratitude to all our colleagues and students for their critical comments and suggestions aimed at improving our book.
Moscow region
Dolgoprudny
December 2018
Yury M. Belousov
Serguei N. Burmistrov
Alexei I. Ternov
Preface
It is impossible to imagine the study of theoretical physics without mastering the technique of problem solution. The point is that theoretical physics does not belong to the humanities. In fact, both mastering and understanding only emerge in the course of solving problems while studying any section of theoretical physics. Here, we present the collected problems on the Course of Theoretical Physics together with detailed solutions and explanations. The problems and their variations over a long period have been posed for undergraduates of Moscow Institute of Physics and Technology on a so‐called basic level of complexity. The latter concept which has recently appeared reflects suitably the requirements of students specializing in physics but not necessarily as future theorists.
So, this book provides the problems for the three main Courses of Theoretical Physics: the classical theory of fields, quantum mechanics and statistical physics in accordance with the educational courses studied by MIPT undergraduates. This choice simply stems from the days of establishing the MIPT as a University and does not mean that other fields of theoretical physics are less important. Between the different parts of the Course of Theoretical Physics there exists an inseparable relationship. We have attempted to preserve this and thus combined all problems in one book.
Accordingly, the book consists of three sections. Each section is preceded with a brief introduction recalling basic notions necessary for the reader to analyze these problems later. If one wishes this is a sort of theoretical minimum which should be familar to undergraduates after completing the Course. This book can also be considered as a manual. The brief introduction does not imply step‐by‐step derivations of the formulae given. Should the derivation of a certain formula in itself be a useful problem we have tried to formulate its derivation as a separate problem. We suppose that such an approach helps the student to understand and learn better the terminal programme. At the same time we did not intend to increase the amount of problems as much as possible but to propose and discuss such an amount which we think is sufficient for learning and comprehending the main Courses of Theoretical Physics.
The solution of problems gives a student an opportunity to check his or her own real knowledge which should not theoretically be a bare set of memorized information. For a tutor, the problem solved by the student himself is the most trustworthy indicator of the comprehension depth of the subject studied. The independent solution of problems educates and perfects analytical and creative scientific thinking. The answers and methods of solving the problems are given in Part II of this book in the same sequence order as the problem statements in Part I. In the explanations of the problems and their solutions we attempt as far as possible to avoid the use of complex mathematical procedures or special methods of theoretical physics. The aim is to make the narrative comprehensible not only for students of the theoretical physics but also for a possibly much wider circle of students of various physical faculties. As a reference source, in the appendices we give some useful data about special functions of mathematical physics used frequently in solving various problems.
Now a few words about the problems. The specifics of stating the problems is that it is difficult to find the original authors as the problems, as a rule, have a popular and ubiquitous character. In fact, a portion of the problems can be found in remarkable collections of problems on field theory [6], quantum mechanics [35, 37, 38], and on statistical physics [48], and, naturally, in the corresponding volumes of the Course of Theoretical Physics by L.D. Landau and E.M. Lifshitz. The latter Course represents the fundamentals of the course of theoretical physics taught at Moscow Institute of Physics and Technology. As a rule, year by year until now these problems have been included in the students' homeworks. Some of these problems have entered our book since these problems became classical and a course of theoretical physics can hardly be imagined without such classical problems. However, in addition to such well‐known classical problems the reader will find plenty of original problems taking account of the specific features of the lectures delivered at various physical faculties. Thus, it should be implied that this book of collected problems is a result of the joint efforts of the members of the MIPT department of theoretical physics, both still working and deceased. We appreciate deeply Professor V. P. Smilga and Professor V. P. Kuznetsov who worked at the department from the very beginning and were the first to select and compile problems. A portion of the problems had already been included into Catecheism [65] and a booklet titled Applied Mathematics [66].
The authors are also grateful to the whole team of the MIPT Department of theoretical physics. In addition, we express our sincere thanks to Professor S. P. Alliluev, Professor S. T. Belyaev, Professor S. S. Gershtein, Professor L. A. Maksimov, and Professor R. O. Zaitsev, as well as to the deceased Professor V. B. Berestetskii, Professor B. T. Geilikman, Professor V. N. Gorelkin, and Professor I. A. Malkin who all contributed much to upgrading the Course of Theoretical Physics and compiling the homeworks. Special thanks are indebted to Dr. M. G. Ivanov for his help in preparing the section on aspects of field theory.
About the Companion Website
This book is accompanied by a companion website:
www.wiley-vch.de/ISBN9783527413966
The website includes figures available in the book.
Part IProblems
Chapter 1Field Theory
Introduction
1 Concept of a tensor
In theoretical physics all physical phenomena are described with the aid of various mathematical models in which the physical objects are associated with the mathematical ones. The mathematical objects describe the physical ones in space, specifying them in the different reference frames. Here, each point in space is associated with a set of coordinates, the number of them depending on the dimensionality of the space. The coordinates are usually denoted as where the index takes all possible values in accordance with the enumeration of the reference frame axes and is called free index. A set of coordinates pertaining to one point is referred to as radius‐vector and is usually, in the three‐dimensional case in particular, denoted with .
The properties of physical objects must not depend on the choice of reference frame and this determines the properties of mathematical objects corresponding to the physical ones. For example, the physical laws of conservation must be described with the aid of mathematical objects having the same form in various reference frames. These are called invariants. Various reference frames can be related with the aid of a certain coordinate transformation representing, from the formal mathematical viewpoint, a replacement written usually as . The prime is commonly ascribed to the radius‐vector in the reference frame transformed with respect to the initial frame. Since for describing physical objects it is also necessary to apply the inverse transformations, the continuous and non‐degenerate transformations alone should be used for replacing the coordinates for which , being the determinant of the Jacobi matrix, i. e. Jacobian of the given transformation.
The mathematical objects that remain unchangeable under all transformations (replacements) of the coordinates correspond to the invariants and these objects are referred to as scalars. A scalar has no free indices and thus can be denoted as . If in place of a simple scalar one specifies a scalar field, i. e. scalar as a function of the spatial point with coordinates , this function varies with the replacement of coordinates so that the previous values would correspond to new coordinates of the previous points:
If the coordinate transformation is linear, the elements of the Jacobi matrix are simply numbers and the expression for the radius‐vector of a point after transformation of the reference frame can be written in the following form:
It is customary always to perform a summation over the doubly repeated so‐called dummy index and thus the summation sign is usually omitted (rule of summation).
Provided that the coordinate transformation is nonlinear, the law of transformation is valid not for the radius‐vector but for its differential
A vector or contravariant vector is called a set of quantities varying under the transformation of a reference frame in the same way as the components of the radius‐vector:
The derivative of the scalar field with respect to the radius‐vector components, i.e. the gradient, is a multicomponent quantity as well. Under transformation of the reference frame its form will vary with the aid of the inverse transformation matrix. Such an object is referred to as a covariant vector (covector) and is written with a lower index:
For orthogonal transformations, the transformation laws for vectors and covectors coincide. In this case there is no necessity to distinguish between these two objects and use the upper and lower indices.
If the scalar field is doubly differentiated with respect to the radius‐vector components, there appears a set of quantities described by two free indices transforming in the same way as the replacement of coordinates, namely as a product of components of two covariant vectors:
Such mathematical objects are called covariant tensors of second rank. Accordingly, a (contravariant) tensor of second rank is a set of quantities with two upper indices which are transformed as a product of the corresponding components of two vectors. The rank of a tensor is determined by the number of its indices. In general, a tensor can be of any rank and have both upper and lower indices. The position of each index is important:
In particular, a scalar is a tensor without indices, a vector is a tensor with one single upper index, and a covariant vector is a tensor with one single lower index.
The following operations are defined for tensors.
Contraction is a summation over a pair of recurring indices, one of them being the upper and the other being the lower, . Contracting over a single pair of indices decreases the rank of a tensor by two. The contraction is also defined for two tensors and here the number of pairs of same indices can be arbitrary, e.g. . Note that, in general,
The contraction of two tensors of first rank is nothing but their scalar product .
The tensor or Kronecker product is an elementwise multiplication of tensor components with sets of different indices . As a result, one obtains a tensor whose rank equals the sum of the ranks of the tensors in the product.
The tensor equality implies that two tensors with the same set of the lower and upper indices are equal
if the difference of the corresponding components of these tensors vanishes in an arbitrary reference frame. Thus, the equality of two tensors in ‐dimensional space corresponds to a system of equations, being the tensor rank. It follows that vectors and tensors allow one to write physical relations in a form independent of the chosen reference frames. This is because both sides of the tensor equality transform according to the same rule under replacement of coordinates.
A combination of correctly constructed tensors results in another valid tensor provided the following rules of the index balance are observed:
Each summand and each term of the equality must have the same sets of free indices, namely, indices with the same attributes must be in the identical (lower or upper) positions and the attribute of each free index runs once in each term.
In each summand and in every part of the equality there may also be (or not) dummy indices, namely, every dummy index in each term either is absent or can be found exactly twice (once as an upper and once as a lower index).
If any index occurs three times or more in one term, there is an error in the formula.
If some index is found twice in the upper position or in the lower position, the formula has an error. (Otherwise, we deal with tensors corresponding to orthogonal transformations.)
We can rename any free index if we change its name equally in all summands. Here, one should keep in mind that the new index name must differ from the names of the other indices used in every summand. Analogously, we can replace the free index with its numerical value.
We can rename any dummy index in some summand, replacing simultaneously the name of both its positions. In this operation the new name of the index should not coincide with the names of other indices used in the summand.
One can obtain a tensor of second rank by differentiating the components of the radius‐vector with respect to its proper components. This results in a symmetric tensor whose components are invariant with respect to any coordinate transformations:
The tensor is usually referred to as the Kronecker symbol.
Among various invariants an especially important one is the element of length, its square being . Since this is a scalar, its value must remain unchanged by a replacement of coordinates:
According to the definition, the factor in front of is a mixed second‐rank tensor which can be denoted as . If both indices are lower ones, a covariant tensor of second rank can be introduced as
and thus the element of length can be written using the contravariant vector components alone:
The tensor is called the metric. It is easy to see that the contravariant tensor proves to be inverse to , i. e. . With replacement of variables the metric tensor transforms according to the definition
For a second‐rank tensor, one can calculate its determinant. Calculating the determinant for both sides we arrive at or .
The metric tensor can be used for raising or lowering the indices:
There exists one more useful invariant tensor which has the same rank as the space in which a physical object is treated. This is a completely antisymmetric tensor whose components change sign under permutation of any pair of indices and whose nonzero components are . For definiteness, we consider the three‐dimensional case. Then, the completely antisymmetric tensor is a third‐rank tensor, . Due to the antisymmetric property all six components differ only in sign. It is agreed to put +1 for the component having the right ordering of indices, i. e. in Cartesian coordinates . Let us write the relation of the tensor components obtained after a coordinate transformation with the initial one:
A summation over twice recurring indices taking account of the sign interchange gives
where is the Jacobian of the transformation. If the Jacobian of the transformation equals unity, e.g. in the case of pure rotations, the tensor components remain unchanged1 . In the three‐dimensional case the tensor is usually called the Levi‐Civita symbol.
2 Vectors and tensors in Euclidean space
In Euclidean space there exist a preferred systems of coordinates called Cartesian coordinates in which the components of the metric tensor () are given by the unit matrix. In this case the raising or lowering of indeces does not change the values of tensor components. The Jacobi matrices for coordinate transformations are always orthogonal, , and vectors with upper and lower indices transform identically.
With the exception of specified cases we will use Cartesian coordinates for describing Euclidean space:, allowing us to disregard the difference in the upper and lower indices. We will use Latin characters for denoting tensor indices in Euclidean space and write the indices as subscripts.
Usually, in Euclidean space a vector is denoted either by a bold character or by an arrow over its symbol.
Transformations of the Cartesian reference frame for which the reference origin remains invariant reduce to rotations around some axes, reflections in planes, and inversion. For rotations, the Jacobian is , and for reflections and inversion one has . Therefore, the tensor behaves as a tensor for rotational transformations but its components change sign for reflections. In other words, this tensor has an attribute different from that of a genuine tensor. Thus, the tensor is often referred to as a pseudotensor.
The scalar, cross (vector) and scalar triple (mixed) products in tensor notation take the form
As one sees, the cross product is a contraction of a pseudotensor and two vectors over two pairs of indices, and thus the resultant vector is not genuine and is usually referred to as pseudovector or axial vector.
The differential operator or del, presented by the nabla symbol, is in Cartesian coordinates a vector with the following components:
In three‐dimensional Euclidean space the basic functions, having invariant meaning and thus often used, read in tensor notation as follows:
The integral Gauss theorem in invariant form is given by
The surface element or the normal is implied to be directed outside.
The Gauss theorem in tensor form reads as
The integral Stokes theorem in invariant form is given by
The Stokes theorem in tensor form reads as
3 Vectors and tensors in Minkowski space
In the four‐dimensional Minkowski space there also exists a preferred system of coordinates (Lorentz coordinates) usually referred to as reference frames. In Lorentz coordinates the components of the metric tensor (Minkowski metric) are written as (). In this case, raising or lowering an index may change the sign of the tensor component, resulting in the necessity to distinguish between upper and lower indices.
With the aid of the metric and the inverse metric one can raise or lower the indices in the following way:
Greek characters will stand for the tensor indices in Minkowski space.
With the exception of a few specified cases we use Lorentz coordinates for describing Minkowski space, , being the speed of light. A point in Minkowski space is called an event. It is obvious that
A vector in Minkowski space is often denoted by an underlined character, .
The four‐dimensional distance between two events , the length of a four‐dimensional radius‐vector (or, for short, a 4‐radius‐vector), is called an interval, its square being equal to
In general the square of an interval may be positive, negative, or zero. The case is referred to as a timelike one (two events may occur at the same point in space but at different time points). The case , this is said to be spacelike. For the lightlike line one finds , this represents the light or zero interval.
The completely antisymmetric tensor is a tensor with respect to the transformations with unit Jacobian, i. e. for rotations and other transformations conserving the volume and orientation of the basis. In the four‐dimensional case it is usually defined by
With the aid of the completely antisymmetric tensor in Minkowski space one can introduce the objects dual to scalar, vector, and antisymmetric tensors of rank 2, 3 and 4:
The differential operator (4‐gradient operator) in linear coordinates represents a covariant vector and reads as
With the exception of specified cases we consider in the following only linear coordinate transformations conserving the Minkowski metric. A set of such transformations embraces translations, spatial reflections, reversion of time, spatial rotations, Lorentz transformations, and their possible combinations.
4 Relativistic kinematics
The Lorentz transformation determines the transfer from one inertial reference frame to another. The rotation‐free Lorentz transformation is also called a boost. Contravariant vectors and tensors in four‐dimensional space are simply referred to as four‐vectors and four‐tensors. The standard Lorentz transformation of an arbitrary 4‐vector , determined for the case when the velocity of the reference frame moving relative to the laboratory frame is directed along the axis, has the form2
Covariant vectors transform with the aid of the transposed inverse matrix which can be obtained by replacing with . The boosts in the or direction can be derived with the corresponding permutations of rows and columns in the transformation matrix.
A contravariant vector and its components are usually written as , and a covariant vector reads correspondingly as .
The scalar or dot product of two vectors equals
The interval between events, associated with the particle motion, is expressed in terms of the particle velocity as the invariant . Here, is the proper time of a particle, i.e. the time in the frame in which the particle is at rest. The proper time is often used for the definition of various kinematic quantities, differing from the interval along the timelike world line by a factor of light speed .
The 4‐velocity and 4‐acceleration have the forms
From the definition of 4‐velocity and 4‐acceleration we have
The 4‐momentum of a particle with mass reads as
where is the energy and is the three‐dimensional momentum.
The rest energy is unambiguously associated with the mass of a particle3 and is also an invariant.
The general definition of mass can be derived from the invariant
For a massless particle, the 4‐velocity is not a well‐defined quantity but the 4‐momentum can readily be determined, obeying the general relation in this case.
For a system of particles, the 4‐momentum conservation law holds:
where the sum is taken over all particles before and after any interaction among them, e.g. scattering, decay, reactions, etc. The total energy and spatial momentum are conserved. However, both the number and the type of particles can vary as well as the total mass of all particles. Thus, on the left‐hand and right‐hand sides of Eq. 1.4 the index of summation is denoted by different characters. The 4‐momentum conservation law, written in the form 1.4, is very convenient for solving various problems with the aid of four‐dimensional invariants obtained by separating the squared 4‐momentum of one or several particles from the above‐mentioned equation. For example, one can write for two particles of known masses
Application of the method of 4‐invariants depends on the process under consideration.
An elastic process is a process in which the amount of particles and their types remain unchanged and, in particular, the particle masses do not vary either.
An inelastic process is a process with varying numbers and/or types and, in particular, masses of particles.
Among kinematic problems associated with inelastic processes a very important problem is the determination of the threshold of reactions in the production of particles that differ from the initial ones. The energy threshold of the reaction is the minimum kinetic energy of the incident particle, necessary for the creation of a new particle. For the system of particles, one can determine the effective mass as an energy in the center‐of‐inertia frame, using the invariance of the squared 4‐momentum:
As follows from Eq. 1.5, the energy will be minimal if the particles are at rest in the center‐of‐inertia frame. Then, the effective mass equals the sum of the masses of all particles after the reaction:
5 The Maxwell equations
(1) The Maxwell equations in three‐dimensional form. The Maxwell equations are commonly written in the form of two pairs, one without sources and the other with sources. In three‐dimensional (in differential) form the equations read as
The continuity equation or charge conservation law results from the second pair of the Maxwell equations:
The densities of charge and current induced by a point particle are equal to
The first pair of the Maxwell equations allows us to parameterize the electromagnetic field by introducing the scalar and vector potentials:
(2) The Maxwell equations in four‐dimensional form. The scalar and vector potentials treated together form the components of a 4‐vector . The electric and magnetic fields can be expressed via the components of the electromagnetic field 4‐tensor. The latter is determined with the aid of derivatives of the 4‐potential components with respect to the components of the 4‐radius‐vector according to
The components of electric and magnetic fields are connected with the components of the antisymmetric field 4‐tensor as follows:
This can clearly be represented in the matrix form
Introducing the electromagnetic field tensor allows one to write the Maxwell equations in four‐dimensional form. Then, the first pair of the equations is determined by the 4‐divergence of the dual tensor :
The 4‐vector of the electric current density reads and the continuity equation is the divergence of the 4‐current density:
The potentials of the given electromagnetic field are ambiguously determined. In fact, due to gauge symmetry it is always possible to perform a gauge transformation of the form
where is an arbitrary function. The transformations considered leave the strengths of electromagnetic field and invariant.
In order to restrict in part the randomness associated with gauge symmetry, additional gauge conditions, simply called the gauge, are imposed on the potentials. The following gauge conditions are most common:
Substituting the explicit expression for tensor given in the 4‐potential components into the second pair of Eqs. 1.8, we have
For this case, it is convenient to choose the Lorentz gauge condition in order to obtain the wave equation for the 4‐potential components (the d'Alembert equation):
Here, the box stands for the d'Alembertian or d'Alembert operator which can be treated as the Laplace operator in Minkowski space.
After choosing the Lorentz gauge, it is still possible to subject the potentials to residual gauge transformations which also do not modify the fields and and do not disturb the Lorentz gauge:
The electric and magnetic fields, as 4‐tensor components, transform with the transition from one reference frame to another according to the Lorentz transformations for the components of a 4‐tensor. In the case of the usual Lorentz transformation we have
Note that it is simpler to derive the rule for the transformation of the electric field components from the transform of the dual tensor .
The electromagnetic field tensor together with the dual tensor allows one to determine two invariants of the electromagnetic field:
(3) The action function for the electromagnetic field. The equations for the electromagnetic field and for the motion of charged particles in the general form can be obtained starting from the principle of least action. In its most general form the action function is represented as a sum of terms describing the separate subsystems of the system in question and their interaction. Hence, the action for the electromagnetic field and electrically charged particles should consist of these three contributions:
i.e. actions for particles, field and particle‐field interactions. On account of the universal principle of least action, the action must be a scalar, i.e. a 4‐invariant in our case. Deriving specific expressions for the action, one should keep in mind additional conditions. In particular, the theory obtained should be gauge‐invariant and, in addition, provide a principle of correspondence, i.e. crossover to the known expressions in the nonrelativistic limit.
On derivation of the equations for a field interacting with charged particles it is necessary to employ the second and third terms in 1.11. It is convenient to write these terms in the form of the action for the electromagnetic field with sources,
being the Lagrangian density. The integrals along the world lines of particles are represented as integrals over space‐time, and the current density 4‐vector is introduced according to
Here, is the 4‐radius‐vector of the th particle, is its 4‐velocity 1.2, is the interval associated with the proper time of a particle, and is the four‐dimensional ‐function. Integrating in 1.13 over with the aid of Eq. (A.7), we arrive at the familiar formula 1.6. Note that the Lagrangian density is a scalar function since the element of four‐dimensional volume is a scalar as well.
The components of 4‐potential of the field are here the independent generalized coordinates in which the variation of action is calculated. Under the condition of vanishing variation of the action with respect to , i.e. , we obtain the Euler‐Lagrange equation for the electromagnetic field:
The second pair of the Maxwell equations 1.8 results from this equation. According to the definition of tensor the first pair of Eqs. 1.7 is satisfied identically.
(4) The energy‐momentum tensor. Based on the action for the electromagnetic field , see 1.11 or the first term in 1.12, one can find the so‐called canonical energy‐momentum tensor:
The definition of the energy‐momentum tensor 1.14, in general, is not unambiguous. In fact, any tensor with zero 4‐divergence can be added to the initial tensor, i.e.
The latter is commonly chosen so that the final energy‐momentum tensor would be symmetric, i.e. .
The symmetric energy‐momentum tensor for the electromagnetic field reads
In three‐dimensional notation, the tensor splits into blocks:
where the energy density and the energy flux density vector or Poynting vector for the electromagnetic field are equal to
The momentum density and the momentum flux density tensor4 of electromagnetic field are defined as
For a closed system consisting of an electromagnetic field and particles interacting with the field, the energy‐momentum conservation law holds:
