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Symmetry

equations mathematical symmetries physics

Symmetry is a property of some images, objects, and mathematical equations whereby reflections, rotations, or substitutions cause no change in properties or appearance. For example, the letter M is symmetrical across a line drawn down its center, a ball is symmetrical under all possible rotations, and the equation y = x2 (a parabola) is symmetrical under the substitution of -x for x. This equation's mathematical symmetry is equivalent to its graph's physical symmetry. The ability of mathematical symmetries to reflect the physical symmetries of the real world is of great importance in physics, especially particle physics.

Many real objects and forces at all size scales—subatomic particles, atoms, crystals, organisms, stars, and galaxies—exhibit symmetry, of which there are many kinds. Line or bilateral symmetry, the simplest and most familiar, is the symmetry of by any figure or object that can be divided along a central line and then restored (geometrically) to wholeness by reflecting its remaining half in a mirror.

Symmetries are not only defined in terms of reflection across a line. A sphere, for example, can be rotated through any angle without changing its appearance, and in mathematics is said to possess O(3) symmetry. The quantum field equations whose solutions describe the electron, which is, like a sphere, the same viewed from any direction, also have O(3) symmetry.

In particle physics, the mathematics of symmetry is an essential tool for producing an organized account of the confusing plethora of particles and forces observed in Nature and for making predictions based on that account. An extension of the parabola example shows how it is possible for mathematical symmetry to lead to the prediction of new phenomena. Consider a system of two equations, y = x2 and y = 4. There are two values of x that allow both equations to be true at once, x = 2 and x = -2. The two (x, y) pairs (2, 4) and (-2, 4) are termed the solutions to this system of two equations, because both both equations are simultaneously true if and only if x and y have these values. (The two solutions correspond to the points where a horizontal line, y = 4, would intersect the two rising arms of the parabola.) If this system two equations constitued an extremely simple theory of matter, and if one of its two solutions corresponded to a known particle, say with "spin" = x = 2 and "mass" = y = 4, then one might predict, based on the symmetry of the two solutions, that a particle with "spin" = -2 and "mass" = 4 should also exist. An analogous (though more complex) process has actually led physicists to predict, seek, and find certain fundamental particles, including the Ω baryon and the Η0 muon).

Symmetry, however, not only is a useful tool in mathematical physics, but has a profound connection to the laws of Nature. In 1915, German mathematician Emmy Noether (1882–1835) proved that every conservation law corresponds to a mathematical symmetry. A conservation law is a statement that says that the total amount of some quantity remains unchanged (i.e., is conserved) in any physical process.

Momentum, for example, is conserved when objects exert force on each other; electric charge is also conserved. The laws (mathematical equations) that describe momentum and charge must, therefore, display certain symmetries.

Noether's theorem works both ways: in the 1960s, a conserved quantum-mechanical quantity (unitary spin) was newly defined based on symmetries observed in the equations describing a class of fundamental particles termed hadrons, and has since become an accepted aspect of particle physics. As physicists struggle today to determine whether the potentially all-embracing theory of "strings" can truly account for all known physical phenomena, from quarks to gravity and the Big Bang, string theory's designers actively manipulate its symmetries in seeking to explore its implications.

Resources

Books

Barnett, R. Michael, Henry Mühry, and Helen R. Quinn. The Charm of Strange Quarks. New York: Springer-Verlag, 2000.

Elliot, J.P., and P.G. Dawber. Symmetry in Physics. New York: Oxford University Press, 1979.

Silverman, Mark. Probing the Atom Princeton, NJ: Princeton University Press, 2000.

Other

Cambridge University. "Cambridge Cosmology." [cited February 14, 2003]. <http://www.damtp.cam.ac.uk/user/gr/public/cos_home.html>.


Larry Gilman

KEY TERMS

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Noether's theorem

—Mathematical theorem stating that every conservation law in Nature must have a symmetrical mathematical description.

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