Symmetry

Symmetry

In the most general sense, symmetry can be defined as a property that an entity has whereby it preserves some of its aspects under certain actual or possible transformations. A sphere is symmetrical because a rotation about its axis preserves its shape. A crystal structure is symmetrical with respect to certain translations in space. The existence of symmetries in natural phenomena and in human artifacts is pervasive. However, nature also displays important violations of symmetry: Some organic molecules come only or predominantly in left-handed varieties; the bilateral symmetry of most organisms is at best only approximate.

The general concept of symmetry applies not only to objects and their collections, but also to properties of objects, to processes they may undergo, as well as to more abstract entities such as mathematical structures, scientific laws, and symbolic and conceptual systems, including mythology and religion. Symmetry symbols pervade ancient cosmologies. Thus the concept of axis mundi (the world axis) is a famous mytho-poetic archetype expressing the idea of centrality in the arrangement of the Cosmos. Whether axis mundi is represented as a sacred mountain, tree, or ladder, it invariably signifies a possibility for humans to connect with heaven. The central image of Christianity, the cross, belongs in the same broad category, as far as its symbolic connotations are concerned. The concept of triadicity so essential to many religions is closely linked to symmetry considerations.

The abstract notion of symmetry also lies at the very foundation of natural science. The fundamental significance of symmetries for physics came to the fore early in the twentieth century. Prior developments in mathematics contributed to this. Thus, in his Erlangen Program (1872), the German mathematician Felix Klein (1849–1925) proposed interpreting geometry as the study of spatial properties that are invariant under certain groups of transformations (translations, rigid rotations, reflections, scaling, etc.). Emmy Noether (1882–1935) applied Klein's approach to theoretical physics to establish in 1915 a famous theorem relating physical conservation laws (of energy, momentum, and angular momentum) to symmetries of space and time (homogeneity and isotropy). By that time, Albert Einstein's (1879–1955) Theory of Relativity had engendered the notion of relativistic invariance, the kind of symmetry all genuine physical laws were expected to possess with respect to a group of coordinate transformations known as the Lorentz-Poincaré group. With this came the realization that symmetry (invariance) is a clue to reality: Only those physical properties that "survive" unchanged under appropriate transformations are real; those that do not are merely perspectival manifestations of the underlying reality.

With the development of particle physics the concept of symmetry was extended to internal degrees of freedom (quantum numbers), such as C (charge conjugation, the replacement of a particle by its antiparticle) and isospin (initially the quantum number distinguishing the proton from the neutron). Along with P (parity, roughly a mirror reflection of particle processes) and T (time-reversal operation), these were long believed to be exact symmetries, until the discovery in 1956 of C - and P -symmetry violations in certain weak interactions, and the discovery in 1964 of the violation of the combined CP -symmetry. However, theoretical considerations preclude violation of the more complex CPT -symmetry.

The emergence of quantum electrodynamics (QED), the first successful quantum relativistic theory describing the interaction of electrically charged spin-1/2 particles with the electromagnetic field, made the notion of gauge symmetry central to particle physics. The exact form of interaction turns out to be a consequence of imposing a local gauge invariance on a free-particle Lagrangian with respect to a particular group (U(1) in the case of QED) of transformations of its quantum state. Extending this principle to other interactions led to the unification of electromagnetic and weak forces in the Weinberg-Salam-Glashow theory on the basis of the symmetry group SU(2) × U(1) and to quantum chromodynamics (a theory of strong quark interactions based on the group SU(3)), and eventually paved the way for the ongoing search for a theory unifying all physical forces.

See also Laws of Nature

Bibliography

mainzer, klaus. symmetries of nature: a handbook for philosophy of nature and science. berlin and new york: walter de gruyter, 1996.

rosen, joe. symmetry in science. an introduction to the general theory. new york: springer-verlag, 1995.

yuri v. balashov