W.I. FUSHCHICH & A.G. NIKITIN

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 (non-Lie) symmetries of the fundamental equations of quantum mechanics and classical field theory, and to the classification and algebraic-theoretical 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 Klein-Gordon-Fock Equation

1.1. Introduction
1.2. The Invariance Algebra of the Klein-Gordon-Fock 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 M1
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 Vector-Potential
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 Lubanski-Pauli Vector
4.5. Irreducible Representation of the Algebra A(c1,n)
4.6. Explicit Realizations of the Poincaré Algebra
4.7. Connections with the Canonical Realizations of Shirokov-Foldy-Lomont-Moses
4.8. Covariant Representations

5. Representations of the Discrete Symmetry Transformations

5.1. Introduction
5.2. Nonequivalent Multiplicators of the Group G8
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. Poincaré-Invariant Equations of First Order

6.1. Introduction
6.2. The Poincaré Invariance Condition
6.3. The Explicit Form of the Matrices bm
6.4. Additional Restristions for the Matrices bm
6.5. The Kemmer-Duffin-Petiau Equation
6.6. The Dirac-Fierz-Pauli Equation for a Particle of Spin 3/2
6.7. Transition to the Schrödinger Form

7. Poincaré-Invariant Equations without Redundant Components

7.1. Preliminary Discussion .
7.2. Formulation of the Problem
7.3. The Explicit Form of Hamiltonians HIs and HIIs
7.4. Differential Equations of Motion for Spinning Particles
7.5. Connection with the Shirokov-Foldy 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. Dirac-Like Wave Equations as a Universal Model of a Particle with Arbitrary Spin

9. Equations for Massless Particles

9.1. Basic Definitions
9.2. A Group-Theoretical Derivation of Maxwell's Equations
9.3. Conformal-Invariant 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 Four-Component 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 Spin-One Particles with Positive Energies

Chapter III. REPRESENTATIONS OF THE GALILEI ALGEBRA AND GALILEI-INVARIANT 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. Galilei-Invariant Wave Equations

13.1. Introduction
13.2. Galilei-Invariance Conditions
13.3. Additional Restrictions for Matrices bm
13.4. General Form of Matrices bm in the Basis | l;l ,m >
13.5. Equations of Minimal Dimension
13.6. Equations for Representations with Arbitrary Nilpotency Indices

14. Galilei-Invariant 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 First-Order 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 Schrödinger-Type Equations
15.6. Anomalous Interaction

Chapter IV. NONGEOMETRIC SYMMETRY

16. Higher Order Symmetry Operators of the Klein-Gordon-Fock and Schrödinger Equations

16.1. The Generalized Approach to Studying of Symmetries of Partial Differential Equations
16.2. Symmetry Operators of the Klein-Gordon-Fock Equation
16.3. Hidden Symmetries of the Klein-Gordon-Fock 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 Quasi-Relativistic 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 Integro-Differential Operators
17.3. Symmetries of the Eight-Component 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 First-Order 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 Kemmer-Duffin-Petiau Equation
19.2. Arbitrary Order Symmetry Operators of the Kemmer-Duffin-Petiau Equation
19.3. Symmetries of the Dirac-Like Equations for Arbitrary Spin Particles
19.4. Hidden Symmetries Admitted by Any Poincaré-Invariant Wave Equation
19.5. Symmetries of the Levi-Leblond Equation
19.6. Symmetries of Galilei-Invariant 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 M2
20.4. Superalgebras of Symmetry Operators of Maxwell's Equations
20.5. Symmetries of Equations for the Vector-Potential

21. Symmetries of the Schrödinger Equation with a Potential

21.1. Symmetries of the One-Dimension Schrödinger Equation
21.2. The Potentials Admissing Third-Order Symmetries
21.3. Time-Dependent Potentials
21.4. Algebraic Properties of Symmetry Operators
21.5. Complete Sets of Symmetry Operators for One- and Three-Dimensional 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 Vector-Potential

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 Poincaré-Basis

25.1. Subgroup Structure of the Group P(1,4)
25.2. Poincaré-Basis
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 PnPn >0
26.3. Representations of Classes II-IV
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 Kemmer-Duffin-Petiau 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 Galilei-Invariant 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. TWO-PARTICLE EQUATIONS

32. Two-Particle 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. Two-Particle Equations of First Order
32.5. Equations for Interacting Particles of Arbitrary Spin

33. Quasi-Relativistic and Poincaré-Invariant Two-Particle 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. Two-Particle Equations Invariant Under the Group P(1,6)
33.6. Additional Constants of Motion for Two- and Three-Particle Equations

34. Exactly Solvable Models of Two-Particle Systems

34.1. The Nonrelativistic Model
34.2. The Relativistic Two-Particle Model
34.3. Solutions of Two-Particle Equations
34.4. Discussing of Spectra of the Two-Particle 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, Vice-President 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 "non-Lie 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 "non-Lagrangian" approach based on representations of the Poincaré and Galilei algebras. That is, we use for this purpose the principles of Galilei and Poincaré-Einstein relativity formulated in algebraic terms. Sometimes we use the usual term "relativistic equations" when speaking about Poincaré-invariant equations in spite of the fact that Galilei-invariant 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 English-speaking 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 Poincaré-Einstein, 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 (non-Lie) 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 Galilei-invariant 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 algebraic-theoretical 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 non-geometric, local or non-local symmetries. All the basic equations of mathematical physics, i.e. the equations of Newton, Laplace, d'Alembert, Euler-Lagrange, Lame, Hamilton-Jacobi, 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 Poincaré-invariant 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 two-particle 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 non-Lie symmetries, super- and parasupersymmetries. Moreover we present new constants of motion of the basic equations of quantum physics, obtained by non-Lie 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 (non-Lie) 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 Poincaré-invariant equations which do not lead to known contradictions with causality violation by describing of higher spin particles in an external field, and the Galilei-invariant 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 spin-orbit coupling which is usually interpreted as a purely relativistic effect.

 3. To represent hidden (non-Lie) 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 four-dimensional space-time (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 group-theoretical introduction to quantum mechanics and will be interesting for mathematicians and physicists which use the group-theoretical approach and other symmetry methods in analysis and solution of partial differential equations.
 
 

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  40. H.C. Corben and J. Schwinger, The Physical Review, vol. 58, p. 953
  41. W.I. Fushchych and R.Z. Zhdanov, "Conditional Symmetry and Reduction of Partial Differential Equations", Ukrainskii Matematicheskii Zhurnal, vol. 44, no. 7, pp. 970-982, 1992; [in English] Ukrainian Mathematical Journal, vol. 44, no. 7, pp. 875-886, 1992.

    42.  

     
     
     

Footnotes:

1 Fushchych = Fushchich (the first version is closer to the Ukrainian transcription)


 
 
 

Subject Index