Symmetries of Equations of Quantum Mechanics
Allerton Press Inc., New York, 1994, 480 pp., Format: Hardcover, ISBN:
0898640695
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Abstract
This book is devoted to the analysis of old (classical) and new (nonLie)
symmetries of the fundamental equations of quantum mechanics and classical
field theory, and to the classification and algebraictheoretical deduction
of equations of motion of arbitrary spin particles in both Poincaré
invariant approach. The authors present detailed information about the
representations of the Galilei and Poincaré groups and their possible
generalizations, and expound a new approach for investigating symmetries
of partial differential equations; this leads to finding previously unknown
algebras and groups of invariance of the Dirac, Maxwell and other equations.
Table of Contents
Chapter I. LOCAL SYMMETRY OF BASIS EQUATIONS OF RELATIVISTIC QUANTUM
THEORY
1. Local Symmetry of the KleinGordonFock Equation
1.1. Introduction
1.2. The Invariance Algebra of the KleinGordonFock Equation
1.3. Symmetry of the d'Alembert Equation
1.4. Lorentz Transformations
1.5. The Poincaré Group
1.6. The Conformal Transformations
1.7. The Discrete Symmetry Transformations
2. Local Symmetry of the Dirac Equation
2.1. The Dirac Equation
2.2. Various Formulations of the Dirac equation
2.3. Algebra of the Dirac Matrices
2.4. Symmetry Operators and Invariance Algebras
2.5. The Invariance Algebra of the Dirac Equation in the Class M_{1}
2.6. The Operators of Mass and Spin
2.7. Manifestly Hermitian Form of Poincaré Group Generators
2.8. Symmetries of the Massless Dirac Equation
2.9. Lorentz and Conformal Transformations of Solutions of the Dirac
Equation
2.10. P, T, and C Transformations
3. Maxwell's Equations
3.1. Introduction
3.2. Various Formulations of Maxwell's Equations
3.3. The Equation for the VectorPotential
3.4. The Invariance Algebra of Maxwell's Equations in the Class M
_{1}
3.5. Lorentz and Conformal Transformations
3.6. Symmetry Under the P, T, and C Transformations
3.7. Representations of the Conformal Algebra Corresponding to a Field
with Arbitrary Discrete Spin
3.8. Covariant Representations of the Algebras AP(1,3) and AC(1,3)
3.9. Conformal Transformations for Any Spin
Chapter II. REPRESENTATIONS OF THE POINCARÉ ALGEBRA AND WAVE
EQUATIONS FOR ARBITRARY SPIN
4. Irreducible Representation of the Poincaré Algebra
4.1. Introduction
4.2. Casimir Operators
4.3. Basis of an Irreducible Representation
4.4. The Explicit Form of the LubanskiPauli
Vector
4.5. Irreducible Representation of the Algebra A(c_{1},n)
4.6. Explicit Realizations of the Poincaré Algebra
4.7. Connections with the Canonical Realizations of ShirokovFoldyLomontMoses
4.8. Covariant Representations
5. Representations of the Discrete Symmetry Transformations
5.1. Introduction
5.2. Nonequivalent Multiplicators of the Group G_{8}
5.3. The General Form of the Discrete Symmetry Operators
5.4. The Operators P , T , and C for Representation
of Class I
5.5. Representations of Class II
5.6. Representations of Classes III IV
5.7. Representations of Class V
5.8. Concluding Remarks
6. PoincaréInvariant Equations of First Order
6.1. Introduction
6.2. The Poincaré Invariance Condition
6.3. The Explicit Form of the Matrices b_{m}
6.4. Additional Restristions for the Matrices b_{m}
6.5. The KemmerDuffinPetiau Equation
6.6. The DiracFierzPauli Equation for a Particle of Spin 3/2
6.7. Transition to the Schrödinger Form
7. PoincaréInvariant Equations
without Redundant Components
7.1. Preliminary Discussion .
7.2. Formulation of the Problem
7.3. The Explicit Form of Hamiltonians H^{I}_{s}
and H^{II}_{s}
7.4. Differential Equations of Motion for Spinning Particles
7.5. Connection with the ShirokovFoldy Representation
8. Equations in Dirac's Form for Arbitrary Spin Particles
8.1. Covariant Equations with Coefficients Forming the Clifford Algebra
8.2. Equations with the Minimal Number of Components
8.3. Connection with Equations without Superfluous Components
8.4. Lagrangian Formulation
8.5. DiracLike Wave Equations as a Universal Model of a Particle with
Arbitrary Spin
9. Equations for Massless Particles
9.1. Basic Definitions
9.2. A GroupTheoretical Derivation of Maxwell's Equations
9.3. ConformalInvariant Equations for Fields of Arbitrary Spin
9.4. Equations of Weyl's Type
9.5. Equations of Other Types for Massless Particles
10. Relativistic Particle of Arbitrary Spin in an External Electromagnetic
Field
10.1. The Principle of Minimal Interaction
10.2. Introduction of Minimal Interaction into First Order Wave Equations
10.3. Introduction of Interaction into Equations in Dirac's Form
10.4. A FourComponent Equation for Spinless Particles
10.5. Equations for Systems with Variable Spin
10.6. Introduction of Minimal Interaction into Equations without Superfluous
Components
10.7. Expansion in Power Series in 1/m
10.8. Causality Principle and Wave Equations for Particles of Arbitrary
Spin
10.9. The Causal Equation for SpinOne Particles with Positive Energies
Chapter III. REPRESENTATIONS OF THE GALILEI ALGEBRA AND GALILEIINVARIANT
WAVE EQUATIONS
11. Symmetries of the Schrödinger Equation
11.1. The Schrödinger Equation
11.2. Invariance Algebra of the Schrödinger Equation
11.3. The Galilei and Generalized Galilei Algebras
11.4. The Schrödinger Equation Group
11.5. The Galilei Group
11.6. The Transformations P and T
12. Representations of the Lie Algebra of the Galilei Group
12.1. The Galilei Relativity Principle and Equations of Quantum Mechanics
12.2. Classification of Irreducible Representations
12.3. The Explicit Form of Basis Elements of the Algebra AG(1,3)
12.4. Connections with Other Realizations
12.5. Covariant Representations
12.6. Representations of the Lie Algebra of the Homogeneous Galilei
Group
13. GalileiInvariant Wave Equations
13.1. Introduction
13.2. GalileiInvariance Conditions
13.3. Additional Restrictions for Matrices b_{m}
13.4. General Form of Matrices b_{m}
in the Basis  l;l ,m >
13.5. Equations of Minimal Dimension
13.6. Equations for Representations with Arbitrary Nilpotency Indices
14. GalileiInvariant Equations of the Schrödinger Type
14.1. Uniqueness of the Schrödinger equation
14.2. The Explicit Form of Hamiltonians of Arbitrary Spin Particles
14.3. Lagrangian Formulation
15. Galilean Particle of Arbitrary Spin in an External Electromagnetic
Field
15.1. Introduction of Minimal Interaction into FirstOrder Equations
15.2. Magnetic Moment of a Galilei Particle of Arbitrary Spin
15.3. Interaction with the Electric Field
15.4. Equations for a (2s +1)Component Wave Function
15.5. Introduction of the Minimal Interaction into SchrödingerType
Equations
15.6. Anomalous Interaction
Chapter IV. NONGEOMETRIC SYMMETRY
16. Higher Order Symmetry Operators of the KleinGordonFock and
Schrödinger Equations
16.1. The Generalized Approach to Studying of Symmetries of Partial
Differential Equations
16.2. Symmetry Operators of the KleinGordonFock Equation
16.3. Hidden Symmetries of the KleinGordonFock Equation
16.4. Higher Order Symmetry Operators of the d'Alembert Equation
16.5. Symmetry Operators of the Schrödinger Equation
16.6. Hidden Symmetries of the Schrödinger Equation
16.7. Symmetries of the QuasiRelativistic Evolution Equation
17. Nongeometric Symmetries of the Dirac Equation
17.1. The Invariance Algebra of the Dirac Equation in the Class M
_{1}
17.2. Symmetries of the Dirac Equation in the Class of IntegroDifferential
Operators
17.3. Symmetries of the EightComponent Dirac Equation
17.4. Symmetry Under Linear and Antilinear Transformations
17.5. Hidden Symmetries of the Massless Dirac Equation
18. The Complete Set of Symmetry Operators of the Dirac Equation
18.1. Introduction and Definitions
18.2. The General Form of Symmetry Operators of Order n
18.3. Algebraic Properties of the FirstOrder Symmetry Operators
18.4. The Complete Set of Symmetry Operators of Arbitrary Order
18.5. Examples and Discussion
18.6. Symmetry Operators of the Massless Dirac Equation
19. Symmetries of Equations for Arbitrary Spin Particles
19.1. Symmetries of the KemmerDuffinPetiau Equation
19.2. Arbitrary Order Symmetry Operators of the KemmerDuffinPetiau
Equation
19.3. Symmetries of the DiracLike Equations for Arbitrary Spin Particles
19.4. Hidden Symmetries Admitted by Any PoincaréInvariant Wave
Equation
19.5. Symmetries of the LeviLeblond Equation
19.6. Symmetries of GalileiInvariant Equations for Arbitrary Spin
Particles
20. Nongeometric Symmetries of Maxwell's Equations
20.1. Invariance Under the Algebra AGL(2,C)
20.2. The Group of Nongeometric Symmetry of Maxwell's Equations
20.3. Symmetries of Maxwell's Equations in the Class M_{2}
20.4. Superalgebras of Symmetry Operators of Maxwell's Equations
20.5. Symmetries of Equations for the VectorPotential
21. Symmetries of the Schrödinger Equation with a Potential
21.1. Symmetries of the OneDimension Schrödinger Equation
21.2. The Potentials Admissing ThirdOrder Symmetries
21.3. TimeDependent Potentials
21.4. Algebraic Properties of Symmetry Operators
21.5. Complete Sets of Symmetry Operators for One and ThreeDimensional
Schrödinger equation
21.6. Symmetry Operators of the Supersymmetric Oscillator
22. Nongeometric Symmetries of Equations for Interacting Fields
22.1. The Dirac Equation for a Particle in an External Field
22.2. The Symmetry Operator of Dirac Type for Vector Particles
22.3. The Dirac Type Symmetry Operators for Particles of Any Spin
22.4. Other Symmetries of Equations for Arbitrary Spin Particles
22.5. Symmetries of a Galilei Particle of Arbitrary Spin in the Constant
Electromagnetic Field
22.6. Symmetries of Maxwell's Equations with Currents and Charges
22.7. Super and Parasupersymmetries
22.8. Symmetries in Elasticity
23. Conservation Laws and Constants of Motion
23.1. Introduction
23.2. Conservation Laws for the Dirac Field
23.3. Conservation Laws for the Massless Spinor Field
23.4. The Problem of Definition of Constants of Motion for the Electromagnetic
Field
23.5. Classical Conservation Laws for the Electromagnetic Field
23.6. The First Order Constants of Motion for the Electromagnetic Field
23.7. The Second Order Constants of Motion for the Electromagnetic
Field
23.8. Constants of Motion for the VectorPotential
Chapter V. GENERALIZED POINCARÉ GROUPS
24. The Group P(1,4)
24.1. Introduction
24.2. The Algebra AP(1,n)
24.3. Nonequivalent Realizations of the Tensor W _{ms}
24.4. The Basis of an Irreducible Representation
24.5. The Explicit Form of the Basis Elements of the algebra AP(1,4)
24.6. Connection with Other Realizations
25. Representations of the Algebra AP(1,4) in the PoincaréBasis
25.1. Subgroup Structure of the Group P(1,4)
25.2. PoincaréBasis
25.3. Reduction P(1,4) ® P(1,3)
of
Irreducible Representations of Class I
25.4. Reduction P(1,4) ® P(1,2)
25.5. Reduction of Irreducible Representations for the Case c _{1}=0
25.6. Reduction of Representations of Class IV
25.7. Reduction P(1,n) ® P(1.3)
26. Representations of the Algebra AP(1,4) in the G(1,3)

and E(4)  Basises
26.1. The G(1,3) Basis
26.2. Representations with P^{n}P_{n} >0
26.3. Representations of Classes IIIV
26.4. Covariant Representations
26.5. The E(4) Basis
26.6. Representations of the Poincaré Algebra in the G(1,2)
Basis
27. Wave Equations Invariant Under Generalized Poincaré Groups
27.1. Preliminary Notes
27.2. Generalized Dirac Equations
27.3. The Generalized KemmerDuffinPetiau
Equations
27.4. Covariant Systems of Equations
Chapter 6. EXACT SOLUTIONS OF LINEAR AND NONLINEAR EQUATIONS OF MOTION
28. Exact Solutions of Relativistic Wave Equations for Particles
of Arbitrary Spin
28.1. Introduction
28.2. Free Motion of Particles
28.3. Relativistic Particle of Arbitrary Spin in Homogeneous Magnetic
Field
28.4. A Particle of Arbitrary Spin in the Field of the Plane Electromagnetic
Wave
29. Relativistic Particles of Arbitrary Spin in the Coulomb Field
29.1. Separation of Variables in a Central Field
29.2. Solution of Equations for Radial Functions
29.3. Energy Levels of a Relativistic Particle of Arbitrary Spin in
the Coulomb Field
30. Exact Solutions of GalileiInvariant Wave Equations
30.1. Preliminary Notes
30.2. Nonrelativistic Particle in the Constant and Homogeneous Magnetic
Field
30.3. Nonrelativistic Particle of Arbitrary Spin in Crossed Electric
and Magnetic Fields
30.4. Nonrelativistic Particle of Arbitrary Spin in the Coulomb Field
31. Nonlinear Equations Invariant Under the Poincaré and Galilei
Groups
31.1. Introduction
31.2. Symmetry Analysis and Exact Solutions of the Scalar Nonlinear
Wave Equation
31.3. Symmetries and Exact Solutions of the Nonlinear Dirac Equation
31.4. Equations of Schrödinger type Invariant Under the Galilei
Group
31.5. Symmetries of Nonlinear Equations of Electrodynamics
31.6. Galilei Relativity Principle and the Nonlinear Heat Equation
31.7. Conditional Symmetry and Exact Solutions of the Boussinesq Equation
31.8. Exact Solutions of Linear and Nonlinear Schrödinger equation
Chapter 7. TWOPARTICLE EQUATIONS
32. TwoParticle Equations Invariant Under the Galilei Group
32.1. Preliminary Notes
32.2. Equations for Spinless Particles
32.3. Equations for Systems of Particles of Arbitrary Spin
32.4. TwoParticle Equations of First Order
32.5. Equations for Interacting Particles of Arbitrary Spin
33. QuasiRelativistic and PoincaréInvariant TwoParticle
Equations
33.1. Preliminary Notes
33.2. The Breit Equation
33.3. Transformation to the Quasidiagonal Form
33.4. The Breit Equation for Particles of Equal Masses
33.5. TwoParticle Equations Invariant Under the Group P(1,6)
33.6. Additional Constants of Motion for Two and ThreeParticle Equations
34. Exactly Solvable Models of TwoParticle Systems
34.1. The Nonrelativistic Model
34.2. The Relativistic TwoParticle Model
34.3. Solutions of TwoParticle Equations
34.4. Discussing of Spectra of the TwoParticle Models
Appendix 1. Lie Algebras, Superalgebras and Parasuperalgebras
Appendix 2. Generalized Killing Tensors
Appendix 3. Matrix Elements of Scalar Operators in the Basis
of Spherical Spinors
Preface to the English Edition
"In the beginning was the symmetry"
W. Heisenberg
"Hidden harmony is stronger then the explicit one"
Heraclitus
The English version of our book is published on the initiative of Dr.
Edward M. Michael, VicePresident of the Allerton Press Incorporated. It
is with great pleasure that we thank him for his interest in our work.
The present edition of this book is an improved version of the
Russian edition, and is greatly extended in some aspects. The main additions
occur in Chapter 4, where the new results concerning complete sets of symmetry
operators of arbitrary order for motion equations, symmetries in elasticity,
super and parasupersymmetry are presented. Moreover, Appendix II includes
the explicit description of generalized Killing tensors of arbitrary rank
and order: these play an important role in the study of higher order symmetries.
The main object of this book is symmetry. In contrast to Ovsiannikov's
term "group analysis" (of differential equations) [355]
we use the term "symmetry analysis" [123] in order to
emphasize the fact that it is not, in general, possible to formulate arbitrary
symmetry in the group theoretical language. We also use the term "nonLie
symmetry" when speaking about such symmetries which can not be found using
the classical Lie algorithm.
In order to deduce equations of motion we use the "nonLagrangian"
approach based on representations of the Poincaré and Galilei algebras.
That is, we use for this purpose the principles of Galilei and PoincaréEinstein
relativity formulated in algebraic terms. Sometimes we use the usual term
"relativistic equations" when speaking about Poincaréinvariant
equations in spite of the fact that Galileiinvariant subjects are "relativistic"
also in the sense that they satisfy Galilei relativity principle.
Our book continues the series of monographs [[127],
[157], [171], [10^{*}],
[11^{*}] devoted to symmetries in mathematical
physics. Moreover, we will edit "Journal of Nonlinear Mathematical Physics"
which also will related to these problems.
We hope that our book will be useful for mathematicians and physicists
in the Englishspeaking world, and that it will stimulate the development
of new symmetry approaches in mathematical and theoretical physics.
Only finishing the contemplated work one
understands how it was necessary to begin it
B. Pascal
Preface
Over a period of more than a hundred years, starting from Fedorov's
works on symmetry of crystals, there has been a continuous and accelerating
growth in the number of researchers using methods of discrete and continuous
groups, algebras and superalgebras in different branches of modern natural
sciences. These methods have a universal nature and can serve as a basis
for a deep understanding of the relativity principles of Galilei and PoincaréEinstein,
of Mendeleev's periodic law, of principles of classification of elementary
particles and biological structures, of conservation laws in classical
and quantum mechanics etc.
The foundations of the theory of continuous groups were laid a
century ago by Sofus Lie, who proposed effective algorithms to calculate
symmetry groups for linear and nonlinear partial differential equations.
Today the classical Lie methods (completed by theory of representations
of Lie groups and algebras) are widely used in theoretical and mathematical
physics.
Our book is devoted to the analysis of old (classical) and new
(nonLie) symmetries of the basic equations of quantum mechanics and classical
field theory, classification and algebraic theoretical deduction of equations
of motion of arbitrary spin particles in both Poincaré and Galileiinvariant
approaches. We present detailed information about representations of the
Galilei and Poincaré groups and their possible generalizations,
and expound a new approach to investigation of symmetries of partial differential
equations, which enables to find unknown before algebras and groups of
invariance of the Dirac, Maxwell and other equations. We give solutions
of a number of problems of motion of arbitrary spin particles in an external
electromagnetic field. Most of the results are published for the first
time in a monographic literature.
The book is based mainly on the author's original works. The list
of references does not have any pretensions to completeness and contains
as a rule the papers immediately used by us.
We take this opportunity to express our deep gratitude to academicians
N.N. Bogoliubov, Yu.A. Mitropolskii, our teacher O.S. Parasiuk, correspondent
member of Russian Academy of Sciences V.G. Kadyshevskii, professors A.A.
Borgardt and M.K. Polivanov for essential and constant support of our researches
in developing the algebraictheoretical methods in theoretical and mathematical
physics. We are indebted to doctors L.F. Barannik, I.A. Egorchenko, N.I.
Serov, Z.I. Simenoh, V.V. Tretynyk, R.Z. Zhdanov and A.S. Zhukovski for
their help in the preparation of the manuscript.
Introduction
The symmetry principle plays an increasingly important role in modern
researches in mathematical and theoretical physics. This is connected with
the fact that the basis physical laws, mathematical models and equations
of motion possess explicit or unexplicit, geometric or nongeometric, local
or nonlocal symmetries. All the basic equations of mathematical physics,
i.e. the equations of Newton, Laplace, d'Alembert, EulerLagrange, Lame,
HamiltonJacobi, Maxwell, Schrodinger etc., have a very high symmetry.
It is a high symmetry which is a property distinguishing these equations
from other ones considered by mathematicians.
To construct a mathematical approach making it possible to distinguish
various symmetries is one of the main problems of mathematical physics.
There is a problem which is in some sense inverse to the one mentioned
above but is no less important. We say about the problem of describing
of mathematical models (equations) which have the given symmetry. Two such
problems are discussed in detail in this book.
We believe that the symmetry principle has to play the role of
a selection rule distinguishing such mathematical models which have certain
invariance properties. This principle is used (in the explicit or implicit
form) in a construction of modern physical theories, but unfortunately
is not much used in applied mathematics.
The requirement of invariance of an equation under a group enables
us in some cases to select this equation from a wide set of other admissible
ones. Thus, for example, there is the only system of Poincaréinvariant
partial differential equations of first order for two real vectors E
and H, and this is the system which reduces to Maxwell's equations.
It is possible to "deduce" the Dirac, Schrödinger and other equations
in an analogous way.
The main subject of the present book is the symmetry analysis of the
basic equations of quantum physics and deduction of equations for particles
of arbitrary spin, admitting different symmetry groups. Moreover we consider
twoparticle equations for any spin particles and exactly solvable problems
of such particles interaction with an external field.
The local invariance groups of the basic equations of quantum mechanics
(equations of Schrodinger, of Dirac etc.) are well known, but the proofs
that these groups are maximal (in the sense of Lie) are present only in
specific journals due to their complexity. Our opinion is that these proofs
have to be expounded in form easier to understand for a wide circle of
readers. These results are undoubtedly useful for a deeper understanding
of mathematical nature of the symmetry of the equations mentioned. We consider
local symmetries mainly in Chapter 1.
It is well known that the classical Lie symmetries do not exhaust
the invariance properties of an equation, so we find it is necessary to
expound the main results obtained in recent years in the study of nonLie
symmetries, super and parasupersymmetries. Moreover we present new constants
of motion of the basic equations of quantum physics, obtained by nonLie
methods. Of course it is interesting to demonstrate various applications
of symmetry methods to solving concrete physical problems, so we present
here a collection of examples of exactly solvable equations describing
interacting particles of arbitrary spins.
The existence of the corresponding exact solutions is caused by
the high symmetry of the models considered.
In accordance with the above, the main aims of the present book
are:
1. To give a good description of symmetry properties of the basic
equations of quantum mechanics. This description includes the classical
Lie symmetry (we give simple proofs that the known invariance groups of
the equations considered are maximally extensive) as well as the additional
(nonLie) symmetry.
2. To describe wide classes of equations having the same symmetry
as the basic equations of quantum mechanics. In this way we find the Poincaréinvariant
equations which do not lead to known contradictions with causality violation
by describing of higher spin particles in an external field, and the Galileiinvariant
wave equations for particles of any spin which give a correct description
of these particle interactions with the electromagnetic field. The last
equations describe the spinorbit coupling which is usually interpreted
as a purely relativistic effect.
3. To represent hidden (nonLie) symmetries (including super
and parasupersymmetries) of the main equations of quantum and classical
physics and to demonstrate existence of new constants of motion which can
not be found using the classical Lie method.
4. To demonstrate the effectiveness of the symmetry methods in
solving the problems of interaction of arbitrary spin particles with an
external field and in solving of nonlinear equations.
Besides that we expound in details the theory of irreducible representations
of the Lie algebras of the main groups of motion of fourdimensional spacetime
(i.e. groups of Poincaré and Galilei) and of generalized Poincaré
groups P(1,n) . We find different realizations of these representations
in the basises available to physical applications. We consider representations
of the discrete symmetry operators P , C and T , and
find nonequivalent realizations of them in the spaces of representations
of the Poincaré group.
The detailed list of contents gives a rather complete information about
subject of the book so we restrict ourselves by the preliminary notes given
above.
The main part of the book is based on the original papers of the
authors. Moreover we elucidate (as much as we are able) contributions of
other investigators in the branch considered.
We hope our book can serve as a kind of grouptheoretical introduction
to quantum mechanics and will be interesting for mathematicians and physicists
which use the grouptheoretical approach and other symmetry methods in
analysis and solution of partial differential equations.
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Footnotes:
^{1} Fushchych = Fushchich
(the first version is closer to the Ukrainian transcription)
Subject Index

Abelian algebra

Algebraictheoretical approach

Angular momentum

BornInfeld equation

Boussinesq equation

Breit equation

Casimir operator

Causality

Caushy problem

Center of mass

Change of variables

Charge conjugation

Commutator

Conditional symmetry

Conformal

algebra

invariance

transformations

Classical conservation laws

Conserved currents

Constants of motion

first order

for the Dirac field

for the electromagnetic field

for the vectorpotential

for two and threeparticle equations

in elasticity

second order

Coulomb

Covariant

representation

system of equations

massless field

Differential consequence

Dilatation

Dirac

constant of motion

form equation for arbitrary spin

equation

equation with positive energies

Elasticity

Energy levels

Energymomentum tensor

Equation

Boussinesq

continuity

covariant

D'Alembert

DiracFierzPauli

evolution

HamiltonJacobi

heat

Helmholtz

KemmerDuffinPetiau

KleinGordonFock

Laplace

LeviLeblond

LHG massless field

nonlinear Dirac

nonlinear Schrödinger

Schrödinger

Stuekelberg

TammSakataTaketani

wave

equations

EulerLagrange

Galileiinvariant

LeviLeblondHagenHurley

Maxwell's

nonlinear heat

reduced

twoparticle

Euclidean group

EulerLagrange equations

Exact solutions

Field

Coulomb

electric

external

magnetic

magnetic monopole

plane wave

Redmond

FoldyShirokov representation

FoldyWouthuysen operator

Formula

CampbellHausdorf

Zommerfeld

Function

Weierstrasse

Hermit

Galilei

algebra

group

invariant wave equations

relativity principle

transformations

Group

conformal

Euclidean

finite

Galilei

generalized Poincaré

Lorentz

oneparameter

orthogonal

Poincaré

HamiltonJacobi equation

Harmonic oscillator

Heat equation

HeavisideLarmorRainich transformation

Helicity

Helmholtz equation

Hermit

Hermitizing matrix

Hilbert space

Hydrogen atom

Integral transformation

Integrodifferential operator

Interaction

anomalous

electromagnetic

minimal

twoparticle

Invariance algebra

Invariant

Inversion

Isomorphism

Jacobi identity

KemmerDuffinPetiau

algebra

equation

matrices

Kepler problem

Killing tensor

Killing vector

LeviLeblond equation

Lie

Lie algebra

of the conformal group

of the Galilei group

LieBacklund symmetry

Lie algebra of the Poincaré group

Lie equations

Liouville equation

Local symmetry group

LubanskiPauli vector

Mass

center of

operator

reduced

rest

Matrix

Dirac

hermitizing

orthogonal

Pauli

spin

Maxwell's equations

Metric

Momentum

Multiplicator

Nilpotency index

Noether theorem

Nongeometric symmetry

NonLie symmetry

Nonlocal transformations

Normalization condition

Operator

Casimir

internal energy

mass

meanposition meanspin

symmetry

Overdetermined system

Parasuperalgebra

Parasupersymmetry

Pauli interaction

Pauli matrices

Poincaré

Potential

Projective representation

Projector

Prolongation

PseudoEuclidean space Ps

Quasiparticle

Reduced equations

Relativity principle

Representation

Scalar product

Schrödinger equation

Galilei invariant

two particle

space inversion

Spin

arbitrary

discrete

variable

Spinor

Superalgebra

Symmetry

conditional

hidden

nongeometric

Taylor series

Tensor

basic

energymomentum

generalized Killing

irreducible

Killing

zilch

Time reflection

Transformation

antilinear

charge conjugation

conformal

discrete symmetry

inversion

Galilei

Lorentz

scale

Transformation group

Variables

angular

internal

invariant

Vector

Wave

wave equation

Galileiinvariant

nonlinear

Poincaréinvariant

Weyl equation

Zilch tensor