Symmetrier i Fysiken
Symmetries in Physics
5A1335 - 5A5367 for PhD students
Introductory Chapter to the lecture notes - under preparation!
PrefaceThis chapter gives a historic background and very brief introduction of the use of the symmetry concept and group theory in quantum physics. A list of books and web links is provided at the end of the chapter. There are several links in the text to sources of information on the web.
There exists a very large number of such web sources. A couple of mainly mathematical ones are
OrnamentsThe study of symmetry has a long history, much older than its applications in physics, making it one of the oldest of sciences. An introduction to the general aspects of symmetry is given by the following classic book:
The decorative aspect of symmetry is not the most important in the present context but it allows us to introduce some basic mathematics without worrying about the physics. In the figure below the 7 different symmetry types for 1-D ornaments are illustrated. The symmetry transformations of each ornament are those transformations of the plane which maps the ornament into itself. It is clear that these transformations can be combined, for every ordered pair of them there is a transformation obtained by first making one and then the other transformation. The order here is important, the resulting transformation will not be the same, in general, if the order is changed. Furthermore, there is for each transformation an inverse undoing it, and there is an identity map leaving each point in place. These three items are the defining properties of a group.
Group axiomsA group is a set G with a binary operation, the group 'multiplication', satisfying the following axioms:
(1) For every pair x,y of elements in G there is a product xy also in G.
A finite group has a finite number of elements.
An Abelian (commutative) group has a group product which is commutative: xy = yx for all pairs x,y in G. For Abelian groups the product is often written as addition, i.e. as x + y instead of xy, and the unit as a zero element.
This definition of a group is abstract; it does not give an interpretation of the group elements. We have seen in the examples above how each element of the group is identified with a transformation (permutation) of a geometrical objects, and we talk about groups of transformations or permutation groups. This is often the situation in applications, but in quantum mechanics we can also have symmetry groups without any obvious geometrical interpretation. An example is provided by the 'internal' symmetries of elementary particles: the transformed quantities are the quantum numbers which define elementary objects, e.g. the quarks.
A group acts as a permutation group on its own elements; and this can be done in several ways. The group element x can act on the left or on the right
y ---> xy, or y ---> y x-1 for all y in G,or as a conjugation
y ---> x y x-1 for all y in G.
For applications in quantum mechanics the idea of a matrix group is important. Matrix multiplication is associative, and the unit matrix I has the role of unit element. The matrix inverse exists for every matrix with a determinant different from zero. The determinant of a matrix product is the product of the determinant: Det(A B) = Det(A) Det(B), consequently we can build groups of matrices with non-zero determinants.
Polyhedra, molecules and crystalsWhen we go to 3-D symmetric structures, the classical source is the philosophy of Plato. In the Timaeus there is a construction of the elements, in which the cube, tetrahedron, octahedron, and icosahedron are given as the shapes of the atoms of earth, fire, air, and water. The fifth solid, the dodecahedron, is Plato's model for the whole universe. The mathematical work of Euclid resulted in the classification the regular polyhedra in 3-D (the Platonic solids).
The physical reality of the crystal lattices was finally confirmed using X-ray diffraction. The theory was worked out by Max von Laue (1912); it was confirmed by experiments by Friedrich and Knipping, and applied by William and Lawrence Bragg to the analysis of crystal structure.
MathematicsThe concept of a group is considered to have been introduced by Evariste Galois (1811-1832). His ideas were published in 1846, by Liouville. Some aspects of group theory had been studied even earlier: in number theory by Euler, Gauss, and others, and the theory of equations by Cauchy, Lagrange, and others.
The problem considered by Galois was the solubility of algebraic equations, and the Galois group is the permutation group of the roots of such an equation. Analysis of this group allows us to find out e.g. if a particular root can be constructed with ruler and compass.
The second half of the 19th century was a period of rapid development of group theory as a mathematical research field, involving many of the most famous mathematicians of those days, like Cauchy, Cayley, Hölder, Jordan and Klein. The continuous groups were introduced by Sophus Lie, this lead to many applications to geometry and differential equations, and to the classification of the simple finite-dimensional Lie groups by Killing and Cartan.
Among more recent results one remarkable feat is the classification of all finite simple groups, ca. 1950--1980, a massive collective effort. Especially important from the point of view of the physicist was the theory of group representations, started by Frobenius, Schur and Burnside around 1900 for finite groups. Later the theory was extended to Lie groups by Hermann Weyl and others. The full physical significance could be seen only after the development of quantum theory, but the key mathematical ideas were fully developed by the 1920's. This part of the mathematical theory is often called abstract harmonic analysis, a name which indicates that it is a generalization of the theory of Fourier series and integrals. It is still a thriving research field closely related to such subjects as the theory of special functions, orthogonal polynomials, and many others.
Important applications of group theory and the theory of representations are today found, beside physics and chemistry, in many fields of discrete mathematics. Examples include cryptography, error correcting codes and others relevant for computers and information technology.
MechanicsAnother important source for the symmetry concept was the study of space and time in the context of mechanics. The earliest ideas of invariance goes back (at least) to Galileo Galilei (1564-1642), who introduced the idea of relativity of motion to explain why we do not fall off an earth moving at high speed. (The second day of Dialogue Concerning the Two Chief World Systems).
The Newtonian equations of motion and their Lorentz covariant versions introduced by Einstein are seen to be inseparably connected to groups of equivalence transformations of space-time. As a result there is a number of groups named after some of the heroes from the history of mechanics: the Euclidean, Galilean, Lorentz and Poincaré groups.
The Euclidean group E(3) consists of those linear transformations of 3-space (R3) into itself, which leave invariant the Euclidean distance between any two points. It is built up from the group O(3) of rotations and reflections (orthogonal transformations) which leave invariant the Euclidean distance to the origin, and from the translations of R3, which just shift the origin of the coordinate system.
We can define O(3) as the group of 3x3 real orthogonal matrices: RT R = I. The subgroup of rotations SO(3), satisfying in addition Det(R) =1, is a Lie group of 3 parameters. So is the group of space translations; E(3) has 6 continuous real parameters.
If we add the time translations which change the zero point of the time coordinate, we can already formulate some of the most important invariance and conservation principles of physics.
Homogeneity of time (invariance under time translations) => conservation of the total energy of a closed system.
Homogeneity of space (invariance under space translations) => conservation of momentum in force-free motion.
Isotropy of space (invariance under space rotations) => conservation of angular momentum.
In 1918 Emmy Noether proved a general theorem connecting symmetries with conserved quantities, or integrals of the motion, as they are often called in classical mechanics.
If we remove the space-time translations from the Galilei group we are left with a 6-parameter homogeneous Galilei group consisting of orthogonal transformations and Galilean 'boosts', which are transformations to coordinate systems in uniform relative motion and which coincide at time 0.
In the Lorentz-Einstein form of relativity the homogeneous Galilei group is replaced by another 6-parameter group which mixes the space and time coordinates: the Lorentz group. This is the group of real linear transformations of space-time (represented by 4x4 real matrices) which leave invariant the pseudo-Euclidean form
c2t2 - x2 - y2 - z2
The Poincaré group or inhomogeneous Lorentz group combines the Lorentz group with the 4-parameter group of space-time translations.
When we use a group of space-time transformations like the Lorentz group, we find that the energy is not an invariant; instead the energy and momentum will transform as a four-vector, the energy-momentum vector. The rest mass of a particle remains invariant.
There are also special systems in non-homogeneous and non-isotropic environments which have a more restricted symmetry. For instance, there is the spherical symmetry (rotation invariance) of the Kepler problem of a light planet moving in the central force field of a heavy sun. The conservation of angular momentum is Kepler's second law. Particles moving in a periodic potential is another example,the symmetry group now contains a discrete group of translations.
Symmetry in quantum mechanicsSymmetry considerations are very important in quantum theory. The structure of the energy levels and eigenstates of quantum systems reflect the symmetries of the Hamiltonians. The probabilities for transitions between different states under a perturbation term in the Hamiltonian depend in a crucial way on the transformation properties of the perturbation.
We are going to take a close look at the consequences of symmetries for quantum systems in this course. Just now we will only recall some facts which should be known from the basic course in quantum physics and preview some general principles. We consider mainly physics where the relevant space-time transformations belong to the Galilei group, with some remarks on the consequences of Lorentz covariance. 'Relativistic' quantum theory demands a quantum field theory machinery for a fully consistent formalism.
The symmetries of a quantum system can be represented by a group of unitary transformations (matrices) U in the Hilbert space; there is a group G and a map U: x ---> U(x), which is representation of the group: the group multiplication is represented as a operator (matrix) product
U(x)U(y) = U(xy) all x, y in G
The observables A of the quantum model will then transform as
A ---> U+ A U,
where + denotes the Hermitian conjugate (transpose - complex conjugate). For unitary transformations this is the inverse matrix:
U+ = U-1.
When does such a group of transformations represent a symmetry group?The first important point is that the transformations must transform the basic observables of the quantum model in a way which allows a physical interpretation. Consider a Schrödinger particle in 3-D. The basic observables are the vector operators Q (the position) and P (the momentum). We can always define a rotation of the coordinate system, or equivalently, a rotation of the observables. If R is an element in SO(3) then this rotation will be represented by a unitary operator U(R), and this transformation of the Hilbert should have the property of rotating the components of a vector operator like Q or P:
U+ P U = R[P].
We can then check that scalar operators like P2, Q . P (scalar product) are rotation invariant in the sense that the following two equivalent conditions hold
U+ P2 U = P2 <===> P2 U = U P2
Invariant operators by definition commute with the unitary operators which represent the transformations.
The transformations are symmetries of the quantum model if the Hamiltonian is an invariant operator.
Observables as generators of transformationsThe vector operators P, Q for a Schrödinger particle are themselves generators of space-time transformations. From the standard commutation relations one can show that (with hbar = 1)
U(a) = exp( - i a.P), a in R3
is a group of unitary operators in the Hilbert space of a Schrödinger particle, and that they perform a space translation
U(a)+ Q U(a) = Q + aI,
(where I is the identity operator). The model has the translations in R3 as an invariance group if and only if the following two equivalent conditions hold
U(a) H = H U(a) for all a in R3 <===> P H = H P
One can show that for Hamiltonians H(P,Q) which are functions of position and momentum, this means that in fact it is independent of position: H = H(P).
In the same way the group of unitary transformations
V(b) = exp( - i b.Q), b in R3
perform translations in momentum space, so this kind of transformation will turn up in the Galilei group. The angular momentum operators generate rotations in SO(3); they rotate all vector operators
L = Q x P,
for any R in SO(3) we can find a 3-vector w such that all vector operators are rotated by R
exp(i w.L) Q exp(-i w.L) = R[Q]
The time translations are, by definition, generated by the Hamiltonian; the transformations are of the form
U(t) = exp(-i t H).
Every observable which commutes with the Hamiltonian is a constant of the motion (invariant under time translations)
H A = A H <===> exp(i t H) A exp(-i t H) = A, for all t
Much of the quantum physics on the level of atomic physics, quantum optics, and quantum chemistry, is based on these observables which come from the space-time transformations. Note that the geometric interpretation of these transformation is precisely the same in the framework of classical mechanics. The similarities between classical and quantum mechanics come from the fact that there are classical and quantum models which share the same geometry and symmetry. But in quantum models the symmetry transformations are represented differently, as unitary transformations of the Hilbert space.
In nuclear physics and elementary particle physics new observables come in (e.g. the isospin quantum numbers and the other quark charges in the standard model). They generate symmetry groups which lack a classical counterpart, and they do not have any obvious relation with space-time transformations. These symmetries are often called internal symmetries in order to underline this fact.
Consequences of symmetries: multipletsSymmetries show up in the multiplet structure of discrete eigenvalues. Consider a single unitary transformation U in the Hilbert space, and an observable A which commutes with U
[U, A] = UA - AU = 0
If A has an eigenvector | a >, then U| a > will be an eigenvector with the same eigenvalue:
A | a > = a | a > ===> A U| a > = a U| a >
This means that either
(1) | a > is an eigenvector for both A and U
This mathematical argument leads to the conclusion: given a group of unitary operators in a Hilbert space representing a group G: U(x), x in G, any observable which is invariant under these transformations, i.e.
[U(x), A] = 0 for all x in G
the discrete eigenvalues and eigenvectors will show a characteristic multiplet structure: there will be a degeneracy due to the symmetry such that the eigenvectors belonging to each eigenvalue form an invariant subspace under the group of transformations.
The possible multiplet structures which can appear due to such a symmetry depends only on the structure of the group G!
Broken symmetries and supermultipletsThe exact degeneracy predicted by symmetry arguments seldom holds good in experiments. There are symmetry breaking interactions which will introduce small corrections to the energy or mass spectrum. Clearly such terms should be weaker than the forces responsible for the main features of the spectrum.
Typically there is a hierarchy of interactions going from the stronger to the weaker, with a corresponding ordering of the spectrum: supermultiplets for the stronger interactions are split up into smaller multiplets where the degeneracy holds with better approximation.
An example is given by the mass spectrum of the baryon octet in the SU(3) symmetry scheme of the standard model.
Dynamical symmetriesAlready in classical mechanics there are symmetries for particular models which cannot be expressed as coordinate transformations, instead they mix canonical coordinates and momenta (they are not 'contact' transformations) in a way which makes little sense for general physical systems. These symmetries have quantum counterparts, we often call them 'dynamical' symmetries
The most familiar example is the harmonic oscillator, where there are a large class of transformations which act in a linear way on the canonical position and momentum operators. For n-D isotropic harmonic oscillators there is a symmetry group SU(n), giving a high degree of degeneracy for large values of n.
Spontaneously broken symmetryThere is a different type of broken symmetry of great importance in several areas of physics, from the evolution of the universe to the thermodynamics of condensed matter. The Hamiltonian may have a certain symmetry, but the physically relevant macroscopic equilibrium states need not have this symmetry. Because it happens without a symmetry-breaking interaction it is often called spontaneous symmetry breaking. A standard example is given by multiparticle spin systems with full rotational invariance, which show a transition from an isotropic non-magnetic state to a ferromagnetic state with a macroscopic magnetic moment for low temperatures. This is a consequence of having many ground states in the theory which are macroscopically different.
A phase transition of a macroscopic system may thus involve a change in symmetry. This change can be continuous or discontinuous. The continuous changes in symmetry and the restrictions which group theory poses on the allowed processes is a vital ingredient of the Landau theory of second-order phase transitions.
Statistics of many-particle statesThe permutations of n objects form a group of order n!, the permutation group or symmetric group of order n. This group will inevitably have an important role for the physics of many-body systems.
Experience tells us that elementary quantum objects with the same quantum numbers are indistinguishable. In our mathematical description of a multi-quantum state a permutation of the constituents must not lead to a new quantum state.
The straightforward solution to this problem is to postulate that the only states allowed are those where permutation gives back the same state, with a possible phase factor. In 3-D space there are only two possibilities:
Fermions = the phase is the signature of the permutation, 1 for even, -1 for odd permutations.
Bosons = the phase is always 1.
If we have a 2-fermion system where the two particles are in quantum states with all quantum numbers the same, then the 2-particle state is automatically symmetric (parity +1) under exchange of the particles. But this is forbidden, by definition. Two fermions must always occupy different quantum states; this is the Pauli exclusion principle.
In a multi-particle system the phase is just the product of the phases for the constituents; a 2-fermion system behaves like a boson. The spin-statistics theorem says that systems with integer spin are bosons, systems with half-integer spins are fermions. This theorem can be proved in quantum field theory.
This symmetry principle will influence all other symmetries defined for multi-particle states, for instance, the possible values of the total angular momentum. These effects of statistics can be rather subtle, one example is the influence of nuclear spins on the quantum numbers of molecules which is seen in the difference between ortho- and parahydrogen. In the ortho form the two proton spins are parallel, in the para form they are paired to a total spin 0. As a result of the fermion character of the proton the allowed rotational angular momenta of the two forms are different, giving different heat capacities. The changes in the proton spin states are normally very slow, making the two forms appear as two macroscopically different gases.
In 2-D it is possible to allow quanta with different statistics, anyons, and multi-valued wave functions.
Symmetries of elementary particlesIn high energy physics the relevant group of space-time transformations is the Poincaré or inhomogeneous Lorentz group of special theory. Recall that these transformations do not commute with the Hamiltonian, instead there is a covariance condition that the energy and momentum should transform as a 4-vector. The physically relevant irreducible representations of this group, described by Eugene Wigner in 1939, are defined by a rest mass and an intrinsic spin.
Thus spin and rest mass define the transformation properties o under the Poincaré group. The symmetry between particles and antiparticles can also be traced to the the space-time transformation properties, and so can the CPT-theorem and the spin-statistics theorem.
The standard model of elementary particles introduces internal
symmetries, which we can understand (at present) only by
introducing new observables and quantum numbers which have no obvious
relation to space-time properties.
In particle physics the basic building blocks (quarks etc) are defined in this way on the basis of their transformation properties under symmetry groups, e.g. SU(3) and SU(4). These groups, and the corresponding observables which provide the quantum numbers, are defining parts of the model. These ``internal'' observables and quantum numbers do not have any classical counterpart (with the important exception of the electric charge), they can only be investigated through the particle mass spectra and reactions.
Symmetries in quantum chemistry and solid stateThe symmetry aspects are crucial for the theory of chemical bonds and chemical reactions. In many of these problems the crucial symmetry is that of the potential seen by electrons moving in the electric field of the nuclei. The Born-Oppenheimer approximation treats the nuclei as essentially static on the time scale of the electron dynamics.
A more difficult problem is that of predicting the shape of molecules on the basis of the atomic orbitals. In some molecules the relation is well understood.
For instance, the structures of ammonia, diamond, graphite, aromatics and Fullerenes are due to the geometry defined by the atomic structure of carbon and the 'hybridization' of its atomic orbitals. The structure of diamond is due to sp3 hybridized orbitals.
Not so long ago the icosahedral group was thought to have no physical application, today several examples are known. The C60 molecule has had most media coverage, and was the first to be discovered of the Fullerenes. The characteristic structure with 3 bonds in a nearly plane configuration for every C atom is due to sp2 hybridized orbitals. The same mechanism is at work in graphite and in C6H6.
Nanotubes are essentially rolled up graphite sheets, 1 atom thick. The restriction that the hexagonal structure of the sheet has to be respected in this process defines a discrete set of different symmetries for the nanotubes. They are presently investigated for their electrical and mechanical properties; applications to microelectronics and micro-mechanics are expected.
Computer algebraComputer programs for numerical calculations in solid state theory, quantum chemistry, atomic structure, etc, will often use the relevant group theory. There are also symbolic computer algebra programs which can handle groups. In Maple the group and combinat packages allow us to create and manipulate finite groups, permutation groups in particular. Mathematica also has interesting capabilities, in particular for the rotation group. There are several other programs of this type, only more specialized, like GAP and Magma.
Online lecture notes and books
BooksWe can only set down a small selection of books. A number of more specialized references are given in the text.
Some more general books on symmetries with numerous illustrations:
Heilbronner and Dunitz: Reflections on Symmetry, Wiley-VCH 1993
Hargittai and Hargittai: Symmetry through the Eyes of a Chemist, 2nd ed, Plenum 1995
Cromwell: Polyhedra, Cambridge UP 1997
Weyl: Symmetry, Princeton UP
Books on symmetry in quantum physics and chemistry:
Lederman and Hill: Symmetry and the Beautiful Universe , Prometheus Books 2004
Jones: Groups, Representations and Physics, 2nd edn, IOP Publishing 1998
Inui, Tanabe and Onodera: Group Theory and its Applications in Physics,
2nd corr printing, Springer-Verlag 1996
Ludwig and Falter: Symmetries in Physics, 2nd ed, Springer-Verlag 1996.
Messiah: Quantum Mechanics, vol 2, North Holland 1967,
Chapters 13-15, Appendices C & D
Greiner and Müller: Quantum Mechanics - Symmetries, 2nd ed, Springer-Verlag 1994.
Elliott and Dawber: Symmetry in Physics, 2 vols, Macmillan 1985
Cotton: Chemical Applications of Group Theory, 3rd ed, Wiley 1990.
Mathematical group theory
Armstrong: Groups and symmetry, 3rd printing, Springer-Verlag 1997
Fraleigh: A first course in abstract algebra, 6. ed. Addison-Wesley, 1999
Humphreys: A Course in Group Theory, Oxford UP 1996
Guides to the physical literature:
Park: Resource letter SP-1. Symmetry in physics, Am. J. Phys. 36(7), 577-584 (1968).
Rosen: Resource letter SP-2, Symmetry and group theory in physics, Am. J. Phys. 49(4), 304-319 (1981).
Cheng and Li: Resource letter GI-1, Gauge invariance, Am. J. Phys. 56(7), 586-599 (1988)
Commins: Resource letter ETDSTS-1, Experimental tests of the discrete
Last modified 5 Sept 2005