Mathematically, a group is a set G together with a pair-wise “group multiplication” on elements of G that satisfies the following axioms, (i) Multiplication is associative, meaning that xy(z)=x(yz) for x,y, and z any elements of G. (ii) G contains an identity element e for which ex=x=xe where x is any element of G. (iii) Each element of G has an inverse (x−1), satisfying x−1x=e=xx−1 (Armstrong, 1988; Birkhoff & MacLane, 1953). Groups for which all of the multiplications commute are said to be abelian.
Easily visualized examples of finite symmetry groups are provided by operations on familiar geometrical objects, such as squares, hexagons, cubes, and tetrahedrons, which carry vertices into themselves. In general, there will be a finite set of f linearly independent functions (in the space of the geometrical object) that are carried into linear combinations of themselves by the group operations. Thus, each group element corresponds to a matrix, and the corresponding set of f×f matrices is called a representation of the group (Landau & Lifschitz, 1958).
For a square that is lying on the plane with its center at the origin, group elements are rotations about the origin (by 90°, 180°, and 270°) and reflections (in horizontal and vertical axes and in the two diagonals). Group multiplication is defined as performing two operations in order, and the identity element leaves all vertices unchanged. The reader may find it helpful to cut out a square of cardboard, number the four corners, and construct a group multiplication table for these eight elements, noting that it is not abelian.
It has long been known that symmetries of a dynamic system influence the nature of possible characteristic solutions or natural modes of behavior (SeeSymmetry: equations vs. solutions). To appreciate this restriction, consider a system with reflection symmetry, and note that any solution can be resolved into two components that are respectively symmetric and antisymmetric about the plane of symmetry. In a linear system, the symmetric and antisymmetric components do not interact; thus, modes must be either symmetric or anti-symmetric about the plane. In a nonlinear system, however, there can also be natural modes of behavior that are neither symmetric nor antisymmetric. Such symmetry breaking is a fundamental feature of nonlinear dynamic systems, leading to the formation of solitons on optical fibers, impulses on nerve axons, and local modes in chemical molecules and molecular crystals.
Applications to Physical Chemistry
A typical chemical molecule is a simple geometric structure for which a symmetry group multiplication table has a finite number of entries (Herzberg, 1991; Wilson et al., 1980). Thus, the water molecule (H2O) has a single reflection plane; methane (CH4) is described by the same symmetry group as the regular tetrahedron, benzene (C6H6) by a planar hexagon, and so on. The physical chemist uses these finite symmetry groups to organize measurements of electronic and vibrational spectra, using the following notations: reflection group (σ), n-fold rotation symmetry (Cn), inversion symmetry (I), tetrahedral symmetry (T), and so on. A particularly lucid introduction to point symmetry notations is given by Landau & Lifschitz (1958).
Because quantum theory is linear, vibrational and electronic wave functions have the following property. If n successive applications of a symmetry operation return a molecule to its original orientation, the amplitude of the corresponding quantum wave function must change by an nth root of unity under the same operation. Under reflections, in other words, amplitudes change by factors of either +1 or −1 (symmetric or antisymmetric), whereas CH-stretching modes of benzene (C6) change under rotations of 60° by factors of exp(±imπ/3), where m=0, ±1, ±2, ±3, ±4, or ±5. Physical chemists label their spectral lines with notations that correspond to these factors (Landau & Lifschitz, 1958).
Although molecular vibrations can be quite nonlinear (especially those involving hydrogen atoms), this nonlinearity does not play a role in transitions from the ground state to first quantum levels of small molecules. (This is because the nonlinear operator of lowest order contains a product of two lowering operators, which annihilates first quantum states.) Thus, the excitations to the first quantum level are governed by linear symmetry considerations, and local modes in molecules are observed only for transitions to higher quantum levels.
The linear modes of a molecule with a center of inversion (I) appear either in infrared absorption or Raman scattering measurements, but not both. This “principle of mutual exclusion” helps to sort out the components of linear vibrational spectra. Local modes, which are nonlinear, can appear in both infrared and Raman measurements.
A periodic solid (or molecular crystal) with periodic boundary conditions can be viewed as a very large molecule, for which the number of elements of the corresponding symmetry group is also very large. Of particular interest are translations by lattice constants (a, b, and c) along the crystal axes, which bring the crystal back to its original configuration. The phase shift of an electronic or vibrational wave function under such a translation is where is called the crystal momentum. As the number of unit cells in the model approaches infinity, the three components of vary, respectively, from −π/a to π/a, −π/b to π/b, and −π/c to π/c. In vibrational modes of such systems, nonlinearity can arise from local lattice distortion, which allows symmetry breaking (local mode formation) to be observed at the first quantum level.
Applications to Field Theories
The above definition of a group does not require the number of group elements to be finite, and many partial differential equations provide examples of infinite-order symmetry groups.
The sine-Gordon (SG) equation, for example, is invariant under the independent variable transformation (x, t)→(ξ, τ), where and Taking this invariance as the property defining elements of the group, a symmetry group comprises all such transformations, parameterized by the continuous variable υ with |υ|<1. This is a one-dimensional version of the Lorentz transformation, which is shared by Maxwell’s equations. Interestingly, if u(x) is a time-independent solution of SG, then is also a solution that demonstrates Lorentz contraction (becomes smaller as υ→1).
In 1915, Emmy Noether established the following important result on the application of symmetry groups to field theories (José & Saletan, 1998).
Noether’s theorem.If a system is described by a Lagrangian that remains invariant under some continuous symmetry transformation, then there is a corresponding conservation law and constant of the motion.
This theorem has immediate implications. As fundamental descriptions of nature are assumed to be Lagrangian and independent of time displacements (the science of today is the same as it was yesterday), the corresponding conserved quantity is energy. In other words, the law of energy conservation stems from time invariance of scientific laws. Similarly, conservation of momentum and conservation of angular momentum arise, respectively, from the assumptions that scientific laws are independent of spatial displacements and angles of rotation.
Nowadays, it is widely assumed that the fundamental fields of nature are derived from a Lorentz invariant Lagrangian density or more generally based on the Poincaré group, which incorporates independence with respect to displacements in time and space (Kim & Noz, 1986). As physicists attempt to formulate a fundamental description of nature, they use Noether’s theorem to build in additional constants (charge, spin, rest mass, etc.) that have been empirically observed.
The pure soliton equations (Korteweg-de Vries, nonlinear Schrödinger, sine-Gordon, and so on) are Lagrangian systems that each have a countably infinite number of conservation laws and constants of the motion. It would be interesting to better understand the corresponding symmetries.
where 
is called the crystal momentum. As the number of unit cells in the model approaches infinity, the three components of 
vary, respectively, from −π/a to π/a, −π/b to π/b, and −π/c to π/c. In vibrational modes of such systems, nonlinearity can arise from local lattice distortion, which allows symmetry breaking (local mode formation) to be observed at the first quantum level.
and 
Taking this invariance as the property defining elements of the group, a symmetry group comprises all such transformations, parameterized by the continuous variable υ with |υ|<1. This is a one-dimensional version of the Lorentz transformation, which is shared by Maxwell’s equations. Interestingly, if u(x) is a time-independent solution of SG, then 
is also a solution that demonstrates Lorentz contraction (becomes smaller as υ→1).
