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- Herausgeber: Wiley-VCH GmbH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Introduction to Continuous Symmetries
Powerful and practical symmetry-based approaches to quantum phenomena
In Introduction to Continuous Symmetries, distinguished researcher Franck Laloë delivers an insightful and thought-provoking work demonstrating that the underlying equations of quantum mechanics emerge from very general symmetry considerations without the need to resort to artificial or ambiguous quantization rules. Starting at an elementary level, this book explains the computational techniques such as rotation invariance, irreducible tensor operators, the Wigner—Eckart theorem, and Lie groups that are necessary to understand nuclear physics, quantum optics, and advanced solid-state physics.
The author offers complementary resources that expand and elaborate on the fundamental concepts discussed in the book’s ten accessible chapters. Extensively explained examples and discussions accompany the step-by-step physical and mathematical reasoning. Readers will also find:
- A thorough introduction to symmetry transformations, including fundamental symmetries, symmetries in classical mechanics, and symmetries in quantum mechanics
- Comprehensive explorations of group theory, including the general properties and linear representations of groups
- Practical discussions of continuous groups and Lie groups, in particular SU(2) and SU(3)
- In-depth treatments of representations induced in the state space, including discussions of Wigner’s Theorem and the transformation of observables
Perfect for students of physics, mathematics, and theoretical chemistry, Introduction to Continuous Symmetries will also benefit theoretical physicists and applied mathematicians.
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Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright
Preface
Introduction
Chapter I: Symmetry transformations
A. Basic symmetries
B. Symmetries in classical mechanics
C. Symmetries in quantum mechanics
Notes
Complement A
I
: Eulerian and Lagrangian points of view in classical mechanics
1. Eulerian point of view
2. Lagrangian point of view
Complement B
I
: Noether's theorem for a classical field:
1. Lagrangian density and Lagrange equations for continuous variables
2. Symmetry transformations and current conservation
3. Generalization, relativistic notation
4. Local conservation of energy
Notes
Chapter II: Some ideas about group theory
A. General properties of groups
B. Linear representations of a group
Notes
Complement A
II
: Left coset of a subgroup; quotient group
1. Left cosets
2. Quotient group
Chapter III: Introduction to continuous groups and Lie groups
A. General properties
B. Examples
C. Galilean and Poincaré groups
Notes
Complement A
III
: Adjoint representation, Killing form, Casimir operator
1. Adjoint representation of a Lie algebra
2. Killing form ; scalaire product and change of basis in
3. Completely antisymmetric structure constants
4. Casimir operator
Notes
Chapter IV: Induced representations in the state space
A. Conditions imposed on the transformations in the state space
B. Wigner's theorem
C. Transformations of observables
D. Linear representations in the state space
E. Phase factors and projective representations
Notes
Complement A
IV
: Unitary projective representations, with finite dimension, of connected Lie groups. Bargmann's theorem.
1. Case where is simply connected
2. Case where is ‐connected
Notes
Complement B
IV
: Uhlhorn–Wigner theorem
1. Real space
2. Complex space
Notes
Chapter V: Representations of Galilean and Poincaré groups: mass, spin, and energy
A. Representations in the state space
B. Galilean group
C. Poincaré group
Notes
Complement A
V
: Proper Lorentz group and SL group
1. Link to the SL group
2. Little group associated with a four‐vector
3. operator
Notes
Complement B
V
: Commutation relations of the spin components, Pauli–Lubanski four‐vector
1. Operator
2. Pauli–Lubanski pseudovector
3. Energy‐momentum eigensubspace with any eigenvalues.
Note
Complement C
V
: Group of geometric displacements
1. Brief review: classical properties of displacements
2. Associated operators in the state space
Notes
Complement D
V
: Space reflection (parity)
1. Action in real space
2. Associated operator in the state space
3. Parity conservation
Notes
Chapter VI: Construction of state spaces and wave equations
A. Galilean group, the Schrödinger equation
B. Poincaré group, Klein–Gordon, Dirac, and Weyl equations
Notes
Complement A
VI
: Relativistic invariance of Dirac equation and non‐relativistic limit
1. Relativistic invariance
2. Non‐relativistic limit of the Dirac equation
Notes
Complement B
VI
: Finite Poincaré transformations and Dirac state space
1. Displacement group
2. Lorentz transformations
3. State space and Dirac operators
Complement C
VI
: Lagrangians and conservation laws for wave equations
1. Complex fields
2. Schrödinger equation
3. Klein–Gordon equation
4. Dirac equation
Notes
Chapter VII: Rotation group, angular momenta, spinors
A. General properties of rotation operators
B. Spin 1/2 particule; spinors
C. Addition of angular momenta
Notes
Complement A
VII
: Rotation of a spin 1/2 and SU matrices
1. Modification of a spin polarization induced by an SU matrix
2. The transformation is a rotation
3. Homomorphism
4. Link with the chapter VII discussion
5. Link with double‐valued representations
Notes
Complement B
VII
: Addition of more than two angular momenta
1. Zero total angular momentum; 3‐j coefficients
2. 6‐j Wigner coefficients
Note
Chapter VIII: Transformation of observables under rotation
Introduction
A. Scalar and vector operators
B. Tensor operators
C. Wigner–Eckart theorem
D. Applications and examples
Notes
Complement A
VIII
: Short review of classical tensors
1. Vectors
2. Tensors
3. Properties
4. Criterium for a tensor
5. Symmetric and antisymmetric tensors
6. Specific tensors
7. Irreducible tensors
Notes
Complement B
VIII
: Second‐order tensor operators
1. Tensor product of two vector operators
2. Cartesian components of the tensor in the general case
Complement C
VIII
: Multipole moments
1. Electric multipole moments
2. Magnetic multipole moments
3. Multipole moments of a quantum system with a given angular momentum
Notes
Complement D
VIII
: Density matrix expansion on irreducible tensor operators
1. Liouville space
2. Rotation transformation
3. Basis of the operators
4. Rotational invariance in a system's evolution
Notes
Chapter IX: Internal symmetries, SU(2) and SU(3) groups
Introduction
A. System of distinguishable but equivalent particles
B. SU group and isospin symmetry
C. SU symmetry
Notes
Complement A
IX
: The nature of a particle is equivalent to an internal quantum number
1. Partial or complete symmetrization, or antisymmetrization, of a state vector
2. Correspondence between the states of two physical systems
3. Physical consequences
Complement B
IX
: Operators changing the symmetry of a state vector by permutation
1. Fermions
2. Bosons
Chapter X: Symmetry breaking
A. Magnetism, breaking of rotational symmetry
B. A few other examples
Notes
Appendix: Time reversal
1. Time reversal in classical mechanics
2. Antilinear and antiunitary operators in quantum mechanics
3. Time reversal and antilinearity
4. Explicit form of the time reversal operator
5. Applications
Notes
Bibliography
SIMPLE INTRODUCTORY TEXTS
QUANTUM MECHANICS
GROUP THEORY AND SYMMETRIES
GALILEO AND POINCARÉ GROUPS
QUANTIZATION
NUCLEAR PHYSICS
EXPERIMENTS
ANGULAR MOMENTUM
CLASSICAL ELECTRODYNAMICS
BROKEN SYMMETRIES
Index
End User License Agreement
List of Illustrations
Chapter 1
Figure 1: Consider a transformation that changes an arbitrary physical syst...
Figure 2: The trajectories of a physical system are symbolized in this figur...
Figure 3: Two successive configurations of a physical system, and , are rep...
Figure 4 Two Galilean reference frames and have parallel axes, and their o...
Figure 5: Diagram expliciting the condition for the transformation , represe...
Chapter 2
Figure 1 The set of complex numbers () form the cyclic group ; as the multi...
Figure 2 The group is the set of the transformations in the plane that pres...
Figure 3 In three dimensions, a group includes three rotations around the a...
Chapter 3
Figure 1 Schematic representations of homeomorphic sets. Each set on the lef...
Figure 2 Schematic representations of non‐homeomorphic sets. Bicontinuitiy i...
Figure 3 Schematic representation of a “disconnected” set, composed of two p...
Figure 4 Two possible paths in a set, with endpoints and and followed by t...
Figure 5 Two paths having the same endpoints and are homotopic if a contin...
Figure 6 In the set of all the points inside a planar circle all the closed ...
Figure 7 In a set that includes “holes”, all the closed paths are not homoto...
Figure 8 A simple circumference is the variation domain of the parameter fo...
Figure 9 Variation domain of the parameters for the group , which has two “c...
Figure 10 The matrices of the SU group depend on three parameters, which can...
Figure 11 In each direction of the three‐dimensional space, the vector must...
Figure 12 Representation of a closed path in SU. It appears on the figure as...
Figure 13 Another representation of the SU group, different from the one sho...
Figure 14 Rotations are determined by a parameter whose extremity lies insi...
Figure 15 A path whose extremities are two diametrically opposed points on t...
Figure 16 A path that penetrates the sphere through two couples of opposite ...
Chapter 4
Figure 1 The same physical system is studied either in a first reference f...
Figure 2 In the complex plane, the vectors representing and have the same ...
Figure 3 The numbers and are represented, in the complex plane, by the vec...
Chapter 4a
Figure 1 In an infinitesimal domain, displayed in gray, around the identity ...
Figure 2 Two transformations and are the end points of two paths starting ...
Figure 3 Two different paths go from the identity transformation to the tra...
Chapter 5c
Figure 1: Euler angles in a rotation. A first rotation around the axis by a...
Figure 2: In an infinitesimal rotation by an angle around , point is displ...
Figure 3: In the space of parameters defining the translation operators, we...
Chapter 5d
Figure 1 In a parity operation, each point is tranformed into a point , sym...
Figure 2 When mobile rotates according to the right‐hand rule on a circular...
Figure 3 The product of a planar symmetry (mirror symmetry that transforms p...
Figure 4 If parity were conserved, two experiments symmetric with respect to...
Figure 5 Comparison of two symmetric experiments, not with respect to a poin...
Chapter 6
Figure 1 In one dimension, a function is translated by when subtracting th...
Chapter 7
Figure 1 The left sphere represents the rotation group , the one on the righ...
Figure 2 The continuous image in SU of a closed loop homotopic to zero in th...
Figure 3 The image of a closed path non‐homotopic to zero in the rotation gr...
Figure 4 The figure on the left‐hand side shows a closed path in the rotatio...
Figure 5 A source emits neutrons (spin particles) with polarized spins. Th...
Figure 6 Triangle law: for a Clebsch–Gordan coefficient to be different fro...
Chapter 7b
Figure 1 Two angular momenta with equal lengths but opposite directions add ...
Figure 2 Three angular momenta that add up to zero.
Figure 3 Different ways of adding together three angular momenta.
Figure 4 Tetrahedron associated with a coefficient.
Chapter 8
Figure 1 The left‐hand side of the figure schematizes a physical system des...
Figure 2 When the vector rotates fast enough around the vector , its averag...
Figure 3 Polarization diagram for optical transitions between a lower level ...
Figure 4 Examples of polarization diagrams. Figure on the left: transition b...
Chapter 8c
Figure 1 A density of electric charges is localized inside a sphere of rad...
Figure 2 A first system of charges is concentrated around the origin, entire...
Figure 3 Typical pattern of the field lines created by a dipole placed at t...
Figure 4 Two systems of charges forming a pure quadrupole, with a zero total...
Figure 5 In the state with maximum polarization, the quadrupole is zero fo...
Chapter 9
Figure 1 Each vertical oval shape in the figure symbolizes a state space fo...
Figure 2 Diagram similar to figure 1, but for the case where the particles ,...
Figure 3 The Lie algebra of an angular momentum is of rank 1, so that an irr...
Figure 4 The values of , , and are indicated on three axes at 120 degrees f...
Figure 5 The action of the two operators results in a shift of the represen...
Figure 6 Starting from a point in the weight diagram, for which we assume ,...
Figure 7 To a point that is not located on any of the axes , , , correspond...
Figure 8 Starting from the points of figure 7 and by the action of the opera...
Figure 9 Figure (a) shows the diagram of the three point representation of ...
Figure 10 Three other possible diagrams for irreducible representations of t...
Figure 11 Weight diagram associated with representation , the product , obta...
Figure 12 Diagrams of irreducible representations derived from representatio...
Figure 13 Diagram of an irreducible representation derived from part (b) of ...
Figure 14 Weight diagram associated with the tensor product representation ,...
Figure 15 Irreducible representation, noted .
Figure 16 Octet of pseudoscalar mesons. The charges are indicated on the ax...
Figure 17 Octet of spin 1/2 baryons with, on the first line, the most usual ...
Figure 18 Decuplet of spin 3/2 baryons. The strangeness and charge axes are ...
Chapter 10
Figure 1: Graph of the free energy of a spin as a function of its polariza...
Figure 2: Free energy of a sample, composed of a large number of spins in ...
Figure 3: These graphs are similar to those in figure 2, but the magnetic fi...
Figure 4: Two condensates with and particles, occupying the individual sta...
Figure 5: Potential energy included in the field Lagrangian, as a function ...
Appendix
Figure 1 Figure on the left: initial motion (in a case where the particle sl...
Figure 2 Time‐reversal diagram in classical mechanics, in a representation s...
Figure 3 For a particle subjected to a potential that depends only on (ele...
Figure 4 For a charged particle subjected to a magnetic field , time reversa...
Figure 5 Diagram associated with the time reversal operator in quantum mech...
Figure 6 On the left‐hand side of the figure is shown a collision where two ...
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
Introduction
Begin Reading
Appendix: Time reversal
Bibliography
Index
End User License Agreement
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Introduction to Continuous Symmetries
From Space–Time to Quantum Mechanics
Franck LaloëTranslated by Nicole and Dan Ostrowsky
Author
Prof. Franck LaloëLaboratoire Kastler Brossel (ENS)24 rue Lhomond75231 Paris Cedex 05France
Cover: © agsandrew/Shutterstock
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.
Library of Congress Card No.: applied for
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© 2023 Wiley‐VCH GmbH, Boschstraße 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: 978‐3‐527‐41416‐1ePDF ISBN: 978‐3‐527‐84054‐0ePub ISBN: 978‐3‐527‐84053‐3
Instructor Material (= password protected) → Supplementary material for and instructors available for download www.wiley‐vch.de/9783527414161
Preface
The birth of Quantum Mechanics has often been compared to that of Relativity, initially introduced by Albert Einstein in 1905. The force of Einstein's reasoning was to start from physical ideas based on the invariance of light velocity, come up with a very general symmetry principle (the equivalence of the inertial reference frames), translate these ideas into equations, and then construct a theory that was the necessary consequence of these equations. This led to the theory of relativity in its so‐called “special” form. This deductive construction gave to the theories of relativity, both special and general (also based on a physical principle, the equivalence principle), a very convincing character. By contrast, Quantum Mechanics was not established as the consequence of such a superb abstract reasoning but rather as the result of a collection of experimental challenges: how can one explain black body radiation, atomic spectra, or the photoelectric effect?
After 25 years of efforts, from 1900 to 1925, the solution appeared almost as a magical algorithm, offering results explaining all the experiments. Within a few years, it was also shown that this formalism had a mathematical coherence, even though it could be written in different forms, either as wave equations or as matrix equations. It was a fantastic success for physics, subject however to one enormous reservation: whereas the equations were simple and coherent, and the predictions always verified, the physical objects themselves remained poorly defined. Numerous contradictory interpretations were proposed, desperately trying to reconstruct objects and properties based on these extraordinary equations. The lack of success of these attempts may have led to the conclusion that they were bound to fail and that there was nothing, or just something senseless, between the experimental data and the mathematical formalism.
Now that Quantum Mechanics is almost a century old, has the situation changed? Is it possible to identify some milestones along this rough road that might give a sense to this entire construction? Can we come back to the simple idea underlying physics that it does describe objects and their properties in the real world?
I believe this book is such a milestone along that path. I discovered these ideas while being a graduate student in the “DEA Brossel” at the beginning of the 1980’s, and taking Franck's course that gave rise to this book. I was immediately fascinated by discovering that, instead of using mysterious and almost magical canonical quantization rules, most of quantum mechanics’ formalism could be obtained from much more down‐to‐earth considerations. You simply had to start from the rules governing classical physics and incorporate them into symmetry principles. These rules include geometric transformations, space and time translations, rotations, and may be extended to changes of reference frame in Galilean or Einstein relativity. Taking for granted that the description of quantum states requires a Hilbert space (this point will be discussed below), it can be shown that all the continuous symmetry transformations must be represented, in the mathematical sense, by unitary (or anti‐unitary) transformations acting in the Hilbert space. Starting from the structure itself of these generalized geometrical transformations, and making a small (but crucial) detour via Lie algebra to write the commutation relations between the infinitesimal generators of the symmetry operations, numerous essential characteristics of quantum mechanics “spontaneously” appear: the Schrödinger equation in the case of Galilean relativity, the Klein–Gordon and Dirac equations in the case of Einstein relativity, and (surprisingly!) the spin in both cases. These reconstructions are very clearly and convincingly presented in this work.
In the last part of this introduction, I would like to come back to the previous statement that “the description of a quantum system requires a Hilbert space”: why is this necessary? This question puzzled me for many years, but I believe I now have an answer, even though it is not (yet?) accepted by Franck. What is needed before this book is some form of a non‐classical probability theory based on projections (in a Hilbert space) rather than partitions (in a Borel set) as would be the case in classical physics. The basic idea of this new probability theory is to associate mutually orthogonal projectors to mutually exclusive measurement results. This is in agreement with a basic idea of the quantum world: mutually exclusive events corresponding to measurement results cannot be subdivided, and when performing feasible measurements on a given system there will never be more than N mutually exclusive results. Using Hilbert basis and distributions, this contextual quantization can be extended – with some mathematical precautions – to cases where N goes to infinity while remaining countable.
Accepting this fundamentally quantum idea, and a few simple other arguments, there is no longer a choice: very powerful mathematical theorems show that quantum mechanics is the only possible theory. More precisely, Uhlhorn's theorem, discussed in one of the complements of the book, shows that, in each context, unitary transformations between projectors are necessary to conserve the mutually exclusive character of the events. Gleason's theorem also shows that Born's law is necessary to ensure the general structure of a probability law. Once this probabilistic framework is established, one has reached the starting point of this book, and real physics – the continuous symmetry transformations – may come into play.
I am not sure Franck is willing to embark on this adventurous path, though it appeared to me as a consequence of the developments presented here. In any case, this book presents very exciting ideas for the reader to discover (or rediscover).
Philippe GrangierCNRS ‐ Institut d'Optique Graduate School ‐ Ecole Polytechnique,Palaiseau, March 2021
Introduction
This book started as a course given to students for several years, with its corresponding teaching notes. There were also a number of presentations given by the students to illustrate some specific points. At the very beginning, it was part of a master's course in “Quantum Physics” given at the end of the 1970’s at the Physic's Laboratory of the ENS in Paris. The main objective was to familiarize the graduate students with the computation techniques based on rotational invariance, irreducible tensor operators, the Wigner–Eckart theorem, etc. These techniques, often imported from nuclear physics, were becoming a basic tool in quantum optics, optical pumping theory, relaxation, etc. With that goal in mind, I still thought it useful to expand the context of the course and teach Wigner's ideas on the essential role of the Poincaré group generators. This aspect turned out to be particularly interesting for the students (and the teacher!), and the initial two‐hour course rapidly expanded. After a few years, this more fundamental aspect became a good half of the course. The present book naturally reflects this duality in the teaching objectives.
A reader may wish for example to rapidly master the technical tools of the course. After a general introduction, the reader can directly go to chapters VII and VIII for the principal results on rotational symmetries, or to chapter IX for particle exchange symmetries. With the same objective of offering practical useful tools, complement DV as well as the appendix, include forays into the domain of discrete symmetries: space parity and time reversal. Instead of considering, as is often the case, this latter as a symmetry in itself, we shall view it in the general frame of the space–time symmetries.
On the other hand, another reader may wish to explore the more fundamental aspects. The reader should then go to the chapters V and VI where we present the approach of Wigner [4] in the context of Einstein's special relativity, and that of Levy‐Leblond [32,33] for the Galilean case. These approaches show how “the quantum world arises from the classical one ” using very general hypotheses. We simply assume that classical space–time remains, in quantum mechanics, the general framework for describing the evolution of physical systems, and that this description is done in a linear (and complex) state space. There is no need to use more or less artificial and sometimes ambiguous “quantization rules” to go from a classical to a quantum description of a physical system. Using the sole properties of classical space–time one can predict the existence of linear quantum operators acting in the space state, and obeying very precise commutation rules. One can then build various relatively simple state spaces. Without adding any other hypothesis, one predicts the existence of several operators, one (diagonal) for the mass, one for the momentum, another one for the angular momentum, etc. In addition there appears, even for a point particle, a spin operator associated to an internal rotation, unimaginable in classical physics (a point object cannot rotate around itself). In a way, we can say that physical considerations (space–time has a classical structure imposed by special relativity) lead to mathematical results on possible quantum descriptions for the simplest objects (irreducible representations). These considerations enable us to construct various quantum wave equations: Schrödinger, Klein–Gordon, or Dirac equations.
Note that smaller characters are used for passages that can be skipped in a first reading. We also kept, whenever possible, the notations of reference [10]. After some hesitation, we did not use, in general, the relativity covariant notations. These notations are practically indispensable in field theory, and the reader should be familiar with them. They are also handy when treating, for example, the six generators of the Lorentz group as the components of a single second‐order tensor, or when building the Pauli–Lubanski generator, etc. On the other hand, they prevent us from retaining our goal: conduct parallel lines of reasoning for the Galilean and Poincaré groups, identify the additional 1/c terms appearing in the second case, and highlight the origin of the relativistic effects. We thus kept the more elementary notation, which, in the end, does not lead to longer computations when each steps of the reasoning is detailed.
Many colleagues and students of this course have given useful advice on the various versions of the teaching notes and the present manuscript. I cannot thank them all, and will only cite two, who later became well‐known researchers (and friends): Dominique Delande, one of the first who took the time to read in detail the initial teaching notes and proposed very useful corrections; Philippe Grangier, who has always been an enthusiastic supporter of the methods using symmetries for the “construction” of quantum mechanics, and has used them in his own research. The appendix on time reversal has greatly benefited from Guy Fishman's very pertinent remarks. Very special thanks are due to Michel Le Bellac who, after carefully reading several chapters, made several very interesting suggestions and encouraged me to add a few developments that were effectively lacking.
One of the best things that can happen to a book is to benefit from outstanding translators. They can bring out and correct weaknesses and missing elements of the texts ... as well as eliminate typographic errors. This book largely benefited from questions and remarks of its translators. Nicole and Dan Ostrowsky have, as they previously did for the three volumes of Quantum Mechanics with Claude Cohen‐Tannoudji and Bernard Diu, based their translation on an attentive reading, which led to numerous improvements. As for Carsten Henkel, the German translator, he never hesitated to deeply analyze the subjects treated in the book, which led to several important additions. I wish to express both my friendship and gratitude to all the translators for their valuable contributions.
Chapter ISymmetry transformations
A. Basic symmetries
A-1. Definition
A-2. Examples
A-3. Active and passive points of views
B. Symmetries in classical mechanics
B-1. Newton's equations
B-2. Lagrange's equations
B-3. Hamilton's equations
C. Symmetries in quantum mechanics
C-1. Quantization standard procedure
C-2. Symmetry transformations
C-3. General consequences
A. Basic symmetries
A-1. Definition
Consider a physical system which, at time , is in the state . For a classical system containing particles for example, could be defined by the values of the vectors giving their positions and velocities at time . Once it has evolved, the system is at time in the state .
Figure 1:Consider a transformation that changes an arbitrary physical system state into another state . If the sequence of the states and that of the states follow the same evolution laws, is said to be a symmetry transformation.
Imagine now a transformation that changes a system in a given state into another system in a state (figure 1). Obviously, the kind of transformations one can think of are numerous: translation of the positions, rotation through a constant or time‐dependent angle, multiplication by a factor 2 of the inter‐particle distances, changing the sign of the electric charges, etc.
Applying this transformation to a sequence of states that describe a possible motion of the system yields another sequence of states . By definition, is said to be a symmetry transformation if that latter sequence of states also describes a possible motion of the system, following the same evolution laws. This condition must be satisfied for any initial state . Therefore:
A transformation is said to be a symmetry transformation if it transforms any possible motion into another possible motion.
This means that in figure 1 one can, at any instant and for any motion, “finish closing the square ” with a transformation .
The definition of a symmetry transformation not only concerns a particular state of the system at a given instant (as, for example, when a given geometrical figure is said to be symmetrical), but all its history as well, i.e. the whole set of states the system successively goes through as time passes.
Comment:
One can also consider transformations that are not instantaneous, such as translations or expansions of the time scale, or Lorentz transformations of a field that is extended in space (its different points then undergo different time transformations), etc. In the left and right parts of figure 1, different time scales should then be used. In fact, for an extended system, a Lorentz transformation is not a succession of transformations, each acting at a single time , but a global transformation acting on the entire time evolution of the system.
A-2. Examples
A transformation is not necesssarily a symmetry transformation. For instance, space dilation by a factor 2 is not in general a symmetry transformation in classical mechanics, when the forces between particles depend on their distance. Neither is the one associated with the rotation of the system by an angle proportional to time (change to a non Galilean reference frame, where inertia effects are different). On the other hand, and this will be explained in detail, the translation or rotation by a fixed quantity of an isolated system is a symmetry operation (space homogeneity and isotropy).
Here are a number of so‐called fundamental symmetries:
space translations ;
space rotations ;
time translations ;
Lorentz (or Galilean) “relativistic” transformations;
(parity, meaning space symmetry with respect to the origin), (charge conjugation), and (time reversal);
exchange between identical particles.
Among these transformations, all are at the present considered as symmetry transformations for the whole set of physical laws governing isolated systems1, with the exception of , , and . These latter are symmetry transformations if the interactions within the system are electromagnetic (or strong), but no longer if weak interactions play a role (the product nevertheless remains a symmetry operation).
Note that the translation invariance of the evolution of an isolated system is a concept that would be hard to abandon completely as it is almost the definition of a so‐called “isolated physical system ”. As for the time translations, the basis of physics or even of the scientific method would be destroyed if they were not, at least approximately2, symmetry transformations for isolated systems: the same experiment performed today or tomorrow would yield different results.
As it is considered at present that all the physical laws known or to be discovered must satisfy all these symmetries, these latter can be viewed as fundamental “superlaws” (Wigner), well worth studying.
Comments:
If both and are symmetry transformations, so is their product (transformation obtained by applying first, and then ): a group structure is thus expected for the set of symmetry transformations.
In the case of the time‐reversal symmetry, one must actually change in
figure 1
into and invert the sense of the vertical arrow on the right, which represents the system evolution (
cf
. appendix and its figure 2).
For an isolated physical system, all the translations and rotations are symmetry transformations. If the physical system is subjected to an external potential (and hence no longer isolated), some of these transformations may still keep their symmetry character. This happens, for example, for the rotations around the origin O of a system subjected to a central potential around O.
A-3. Active and passive points of views
A transformation can be defined from two points of views. The first one concerns a single observer in a given reference frame. To any motion of the physical system, the observer associates another motion obtained by a transformation that can be a translation, a rotation, a time delay, etc. As we saw above, is a symmetry transformation if the observer can describe both motions with the same dynamical equations. The two motions only differ by their initial conditions. This is the so‐called “active” point of view that gives the observer the role of applying the transformation.
One can also adopt another “passive” point of view where a single motion of the system is described by two observers, each using its own reference frame deduced from one another by the transformation . Each observer will give the system, for example, a different position, or orientation, or velocity, etc. and hence a different mathematical description. is a symmetry transformation if the evolutions of the coordinates in the two reference frames are solutions of the same dynamical equations.
In short, the system changes in the active point of view, whereas the reference frame (the axes) changes in the passive point of view. Depending on the case, one or the other point of view will seem more natural. For example, for a time translation it is easy to imagine two different motions delayed in time as in the active point of view. On the other hand, when studying relativity where several Galilean reference frames are used by different observers, the passive point of view is often preferred.
Comments:
(i) As the definition of a translation, rotation, etc. amounts to a change of the space (or time) coordinates of the physical system, and as these coordinates define the relative position of the physical system with respect to the reference frame, it is clear that the active and passive points of view are in fact equivalent. Mathematically, the operations that must be performed on the equations to account for the effects of the transformation are identical in both cases. From a physics point of view, one can describe the transformation in the passive point of view with a single motion of the system, but seen from two different reference frames; but one can also switch to the active point of view and introduce a new motion that is seen, in the first reference frame, as the initial motion in the second.
(ii) The physical distinction between these two points of views becomes meaningful if one of the reference frames is prominent with respect to the other. This happens, for example, if there implicitly exists a third reference frame, independent both from and from the system under study, like the laboratory reference frame ( could then be the reference frame of the measuring devices, which, like the system, are mobile with respect to the laboratory). The difference between the two points of views then becomes clear: one moves either the system or the measuring devices.
Another example where the two points of view are not equivalent is when is an inertial reference frame, but not its transformed since the transformation is time‐dependent (for instance, a rotation at constant angular velocity). In such a case, the passive point of view is to be preferred in quantum mechanics 3, and we shall use it most of the time.
B. Symmetries in classical mechanics
Let us show that, in classical mechanics, symmetries enforce certain forms for the physical laws, and, in addition, impose constants of motion. We begin with a particularly simple example using Newton's equation, that is, .
B-1. Newton's equations
Consider two particles with mass and , positions and , and interacting through a potential . The equations of motion read:
where represents the second derivative of and the gradient with respect to the coordinates .
Translational invariance
Let be an arbitrary constant vector. Consider the transformation of the positions at every time :
(the two velocities are then unchanged). It transforms a possible motion into another possible motion (with the same potential ). Since the transformation does not change the accelerations, we have:
This means that, when the two variables increase by a quantity , the gradients of the potential function with respect to these two variables remains constant. It follows that, in this change of the two variables, only varies by a constant. This constant could be time‐dependent, but has no consequence on the particle's motion since it does not depend on their positions. If we furthermore require to go to zero at infinity, this constant is necessarily equal to zero. This means that the potential is invariant in the translation of the two variables, and is therefore only a function of :
This restricts the possible potentials, thus imposing a constraint on the form of the physical laws:
In addition, it is easy to show that , which leads to:
Accordingly, when translations are symmetry transformations, the total momentum is a constant of the motion.
Rotational invariance
If, in addition, rotations are symmetry transformations, other properties appear. Following the same line of reasoning as above, we see that the field of forces (or ), considered as a function of , is invariant under any rotation of the vector . It is thus a field such that and are parallel to and have a modulus that only depends on . It follows:
and:
The conservation over time of the total angular momentum thus comes from the rotational invariance of all the possible motions.
Time translation
The correspondence between the motions is given by:
where is an arbitrary constant (the new motion has a temporal advance of compared to the initial motion). This time translation is a symmetry transformation if:
The forces depend explicitly on the position but not on the time. It follows that:
The function does not have any physical meaning since it does not depend on the positions, and does not change the accelerations. We shall ignore it and, since is arbitrary in (I‐10b), choose a time‐independent potential energy that describes the possible motions:
Figure 2:The trajectories of a physical system are symbolized in this figure by the trajectory of a single position as a function of time. Two space–time reference systems and are used to describe this motion; they have a space offset and a time offset . The figure shows a motion described in the second reference system as being in advance, in both time and space, with respect to the description in the first reference system. Nevertheless, the origin has been moved to by a positive amount along the time axis, but by a negative amount along the position axis.
We now compute:
which leads to:
The total energy (kinetic and potential) is thus a constant of motion.
Comment:
Space and time translations belong to the same category of transformations, even though they show some differences. In the preceding equations, time was not considered as a system's dynamical variable like the position , but as a parameter the dynamical variables depend on. In relativity, however, time also plays the role of a coordinate that is needed, in addition to the three components of , to define an event in space–time. One should then be careful about signs, which may differ in both cases.
To see why in a simple case, let us consider a one‐dimensional space. Figure 2 shows the motion of a particle seen by two different observers placed in reference frames and . These reference frames have a space offset and a time offset . The two offsets are counted positive if the second observer sees a motion that has progressed in space and time with respect to the first observer – this is the case shown in the figure. The same event is then described, either by the coordinates , or by the coordinates given by:
We notice an opposite sign between the space and time variations. The minus sign for the time is not surprising: if the event “the train arrives at the station” occurs at ten to twelve instead of twelve, the train is ten minutes early.
The sign of is also different in (I‐9) and (I‐14a). The reason lies in the difference between the two points of view mentioned above: in the first case, time was considered as a parameter the dynamical variables depend on, in the second case, as the coordinate of a space–time event.
Starting from equation (I-14a), we can recover the plus sign of (I-9) . The same motion is described in the first reference frame by a function of time , and in the second reference frame by a different function . If times and are such that , they relate to the same event, so that , and therefore:
Time now follows a plus sign. The conclusion is that, when time plays the role of a parameter the position depends on, one has to add (and not subtract) to obtain a time progression, in agreement with (I-9).
We shall go no further in the symmetry studies around Newton's equations as these are not a convenient starting point for the quantization of a physical system. It is best to use either the Lagrangian or the Hamiltonian formalism.
B-2. Lagrange's equations
B-2-a. General formalism
The system is described4 by a set of generalized coordinates , with , which define its configuration (distinct from its state , which also contains velocities); it can also depend on a number of parameters (particles’ masses, charges, etc.). One associates with that system a function called “Lagrangian”, which depends on all the coordinates and :
and such that the equations of motion are given by the Lagrangian equations:
In these equations, stands for the time derivative of . Remember that the total time derivative of a function is given by:
Like Newton's equations, the Lagrangian equations are of second‐order with respect to the time.
They are equivalent to a principle of least action that yields, in a global way (rather than local in time), the possible motions of the physical system. This principle states that, among all the possible motions leading the system at time from the configuration (symbolizing the set of all ) to the configuration at time , the only realizable motion (that which satisfies the equations of motion) must have a stationary action :
Figure 3, where the set of coordinates is symbolized by a single ordinate axis , shows with dashed lines several a priori possible paths, and with a solid line the path actually followed by the system, as it minimizes .
The primary advantage of the Lagrangian point of view is its great generality, as, with a proper choice of the function , the equations of motion of a large number of physical systems can be put in the form (I‐16). Furthermore, whatever the variables chosen to describe the system (for example, taking spherical instead of Cartesian coordinates, etc. ), the system's equations of motion keep the same form (I-16), which is not the case for Newton's equations.
For a system of particles of mass , with an interaction potential , the Lagrangian can be written:
Here, the 3 components of the particles’ position vectors play the role of the , and is the kinetic energy:
Figure 3:Two successive configurations of a physical system, and , are represented in a symbolic way by the value of a single position variable on the vertical axis. These states being fixed, various “paths” (set of states at all the intermediary times) going from to are shown with dashed lines. None of these virtual paths will be followed by the system for real, except the one, drawn with a solid line, that makes action stationary.
In this example, the forces can be obtained by taking the derivative of a potential, but Lagrangian formalism can be used in more general cases. One can, for example, compute the evolution of a system of particles interacting with a given electromagnetic field described by a scalar potential and a vector potential . A possible Lagrangian is then (cf., for example, Appendix III, § 4.b, of [9]):
(where is the electric charge of the particle). As mentioned above, many other equations of motion (where the values of the fields at each point of space become dynamical variables, playing the role of the and ) can be obtained from Lagrange's equations and a variational principle. This is the case, for example, of Maxwell's equations.
Comments:
(i) It should not be assumed that a unique Lagrangian corresponds to the equations of motion of a given physical system. There are actually many “equivalent” Lagrangians. For example, if is an arbitrary function , one can, starting from a Lagrangian obtain another5 Lagrangian :
The variation of the Lagrangian is:
and:
This means that the contributions of to each side of equation (I-16) are identical and hence compensate each other. Lagrangians that lead to the same differential equations of motion are called “equivalent Lagrangians”.
Another way to check that and are equivalent is to note that the corresponding variation of the action is written:
The variation only depends on , and , , and not on the path followed by the system between and . It follows that and will always be stationary for the same paths.
(ii) Reciprocally, it is sometimes suggested that the difference between two equivalent Lagrangians must be a total derivative with respect to time. This is simply not true.
A first simple counter‐example is to multiply the Lagrangian by an arbitrary constant. Actually, the ensemble of the equivalent Lagrangians is in general much larger. For example, for a free particle, one can choose either or given by:
where , , and are arbitrary constants.
It might be interesting to establish a general procedure to find all the Lagrangians equivalent to a given Lagrangian. This would enable one to find the necessary and sufficient conditions for a given transformation to be a symmetry transformation. Furthermore, it would allow seeing if the quantization obtained from these Lagrangians leads to the same physical results. This general problem does not seem to have been solved at the present.
B-2-b. Constants of motion; Noether's theorem
α. Simple cases
The invariance properties of can lead to the existence of constants of motion. For example, imagine the Lagrangian (I‐19) is invariant under the translation of the whole set of particles by a given value (substitution ). In that case:
leads to, according to (I-16):
where is defined as:
Conservation of the total momentum is thus a consequence of space translation invariance of the Lagrangian6.
Similarly, if is invariant under a time translation:
Defining the function as:
it is easy to show, using (I-16) and (I‐28) that, as the system evolves, remains constant:
Time translation invariance thus leads to the energy being a constant of motion.
In the two examples above, we have assumed that is invariant, but the invariance of the equations of motion in a transformation does not forcibly imply that the Lagrangian itself is invariant under this transformation. Another possibility (cf. comment above) is that the transformation adds to a total derivative with respect to time. Emily Noether has shown in 1918 that, in this case as well, the invariance of the transformation also leads to the existence of a constant of motion.
β. General demonstration of the theorem
Consider a transformation of the generalized coordinates of an arbitrary system, that can be written as:
The transformation is supposed to be infinitesimal and:
where is an infinitely small quantity. The variations of the time derivatives of the are given by:
Note that , contrary to , may depend on , as can be seen from the definition of :
which leads to:
In the transformation , the infinitesimal variation of the Lagrangian is written:
If the transformation is chosen in such a way that is proportional to the total time derivative of a function :
Noether's theorem states that the function:
is a constant of motion; is zero along all the possible trajectories of the physical system.
Demonstration: the total time derivative of is:
(the second equality, obtained from the equations of motion, is valid along any possible trajectory of the system). It follows, using (I‐36) and (I‐37):
Inserting this result into the time derivative of (I‐38) leads to , which proves the theorem.
Comments:
If depends only on the coordinates and of the time , the variation of the Lagrangian depends on the , the , and of time; then provides a Lagrangian that is equivalent to . But, if depends on the , the function also depends on the , which is no longer compatible with the standard definition of a Lagrangian. The Noether theorem nevertheless remains valid in this case; see, for instance, the first example below.
One may have to use Lagrange's equations to go from (
I-36
) to (
I-37
). As an example, they can be used to express the appearing in general in (
I-36
), as a function of the and ; one can then look for a function independent of the (and whose total time derivative does not contain the ).
Obviously, Noether's theorem is only interesting if it yields a non‐trivial constant of motion , and not, for example, a zero or a constant independent of the generalized coordinates!
If one accepts functions such as , there is a greater chance of obtaining zero information. A trivial case is to take completely arbitrary functions , and write in the form (I-37) choosing the function (this is always possible since (I‐40) simply states that, along the trajectories, is always proportional to the total derivative of that function ). We then get the trivial result .
γ. Examples
Here are a few simple examples of the application of Noether's theorem; the theorem is also valid in field theory, where it has important applications – cf. complement BI.
Consider first an arbitrary system whose Lagrangian is not explicitly time‐dependent. As a transformation law, we choose:
where is a constant that plays the role of . Operation shifts the time evolution of the system; we are thus dealing with a time translation. As is infinitesimal, we can write:
so that:
On the other hand:
(since by hypothesis ). We find that the function is none other than the Lagrangian itself. According to (I-38), the constant of motion is:
which is simply the usual definition of the Hamiltonian (energy).
Let us now go back to the example discussed in § B‐2‐b above, and find which constants of motion derive from the invariance under space translation, or under a change of Galilean reference frame. The Lagrangian is written as:
where the interaction potential is invariant under the translation of all the positions (as usual, we have replaced the by the , or more precisely by the three components of these vectors). The invariance of under the transformation readily yields the conservation of the total momentum. The 3 components of the infinitesimal vector now play the role of 3 infinitely small ; setting , , , yield as constants of motion the 3 components of the vector:
We now introduce a change of Galilean reference frame by:
Relations (I-33) then become:
so that we have and for ; the 3 components of the vector now play the role of . In this case, is written:
If is translation invariant as in (I‐26), the sum of all its gradients is equal to zero, and:
where:
is the position of all the particles’ center of mass, multiplied by the sum of masses. Replacing by in (I-38) yields the (vectorial) constant of the motion:
Hence:
Dividing this equation by the sum of masses we check that, as expected, the particles’ center of mass moves at constant velocity.
Finally, imagine that is not translation invariant, but that the sum of the (outside) forces acting on the particles is a constant vector . The variation of is then:
with:
The constant of motion is now:
where, in the second equality, we have taken into account the linear time variation of the total momentum . As a result, we have to add a term to the right‐hand side of (I‐53), which shows that the center of mass now moves with a constant acceleration.
Exercise:
Consider a particle of mass , position , momentum , and angular momentum . This particle is subjected to a central potential ( is a positive integer). Introducing the transformation
show that:
In the case of a Coulomb or Newton potential , show that:
and that, consequently, the vector (Runge–Lenz vector):
is a constant. Physical interpretation: the points of the particle's (plane) trajectory where its velocity is perpendicular to are fixed. Instead of following a “rosette pattern”, the trajectory is a closed curve. It is actually an ellipse whose major axis is parallel to and whose eccentricity equals .
Comment:
In (I‐33), we assumed that the transformation only concerned the (and the ), but not the time. This restriction can be lifted by introducing, in addition to the variations and written in (I‐33), a time variation:
Relation (I‐35) is no longer valid in this case. Writing:
we readily obtain, to first‐order in :
meaning:
We also have:
Writing as:
it can be shown that the function:
is a constant of motion. Using Lagrange's equation to get the total time derivative of and (I‐65a) to get that of , one shows that:
Using relation (I‐64) to compute , and finally replacing by its value (I‐63), all the terms in the right‐hand side of this equality cancel each other two by two. is indeed a constant of motion.
Exercise: Setting , , show, as in (I‐44), that is a constant of the motion if .
B-3. Hamilton's equations
Before leaving classical mechanics, let us quickly review the Hamiltonian formalism, which is often used for the “quantization” of a physical system. Note that another quantization procedure starts from the Lagrangian formalism [10] (Feynman's postulates), and is often used, especially in field theory.
B-3-a. General formalism
To each generalized coordinate of the system we associate the conjugate momentum:
We assume that all the
