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Symmetry in physics refers to various features of a physical system that can be said to exhibit Symmetry . These symmetries are usually formulated mathematically and can be exploited to simplify many problems. SYMMETRY AS INVARIANCE A symmetry of a physical system is a (physical or mathematical) feature of the system that is preserved under some change. Some examples of symmetry are given below. Example 1 The temperature in a room may be constant. The temperature being independent of position within the room, it is said that the temperature is unchanged by a shift in position. Example 2 An unmarked ping-pong ball, when rotated about its centre, will look exactly as it did before the rotation. The ping-pong ball is said to exhibit Spherical Symmetry . A rotation about any axis of the ball will preserve how the ball looks. Example 3 The Electric Field Strength at a given distance from an electrically charged wire of infinite length will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius . The wire is said to exhibit Cylindrical Symmetry . Rotating the wire about its own axis does not change its position, hence it will preserve the field. The field strength at a rotated position is the same, but its direction is rotated accordingly. These two properties are interconnected through the more general property that rotating ''any'' system of charges causes a corresponding rotation of the electric field. Example 4 In Newton's theory of mechanics, given two equal masses starting from rest at the origin and moving along the x-axis in opposite directions, one with speed and the other with speed the total kinetic energy of the system (as calculated from an observer at the origin) is and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the y-axis. The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if and are interchanged. The above ideas lead to the useful idea of ''invariance'' when discussing symmetry. Invariance is usually specified mathematically by transformations that leave some quantity unchanged. These transformations may be continuous (such as rotations) or discrete (such as reflections) and lead to corresponding types of symmetries. LOCAL AND GLOBAL SYMMETRIES Symmetries may be broadly classified as global and local. A ''global symmetry'' is one that holds at all points of Spacetime , whereas a ''local symmetry'' is one that only holds on a certain Subset of the whole Spacetime . CONTINUOUS SYMMETRIES The two examples of rotational symmetry described above - spherical and cylindrical - are each instances of continuous symmetry. These are characterised by a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about it's axis and the field strength magnitude will be the same on any given cylinder. Mathematically, continuous symmetries are usually described by Continuous or Smooth Function s. An important subclass of continuous symmetries in physics are spacetime symmetries. Spacetime symmetries See Also: Spacetime symmetries ''Spacetime symmetries'' are those continuous symmetries that involve transformations of Space and Time . These may be further divided into 3 categories. Many symmetries in physics are described by continuous changes of the spatial geometry associated with a physical system (' ''spatial symmetries'' '), others only involve continuous changes in time (' ''temporal symmetries'' ') or continuous changes in both space and time (' ''spatio-temporal symmetries'' ').
Mathematically, spacetime symmetries are usually described by Smooth Vector Field s on a Smooth Manifold . The underlying Local Diffeomorphism s associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system. Some of the most important vector fields are Killing Vector Field s which are those spacetime symmetries that preserve the underlying Metric structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of Isometries . The article Isometries In Physics discusses examples of these symmetries in more detail. Other continuous symmetries DISCRETE SYMMETRIES See Also: Discrete symmetry A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete symmetry, as only rotations by integral multiples of 90 degrees will preserve the square's original outlook. Discrete symmetries often involve some type of 'swapping', these swaps usually being called ''reflections'' or ''interchanges''.
Gauge symmetry Many discrete symmetries are found in physics, especially Particle Physics . CONSERVATION LAWS AND NOETHER'S THEOREM See Also: Noether's theorem The symmetry properties of a physical system are intimately related to the Conservation Laws characterizing that system. Noether's Theorem makes this fact precise. Roughly, that theorem states that each symmetry of a physical system means that some quantitative property of that system is conserved. SYMMETRY GROUPS Many of the important transformations describing physical symmetries form a Group . This has led to Group Theory being one of the areas of mathematics most studied by physicists. Continuous symmetries are specified mathematically by 'continuous groups' called Lie Group s. Many physical symmetries are isometries and are specified by Symmetry Group s. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the Special Orthogonal Group . Thus, the symmetry group of the ping-pong ball with proper rotations is . Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the Lorentz Group (this may be generalised to the PoincarĂ© Group ). Discrete symmetries tend to be described by discrete groups. For example, the symmetries of an equilateral triangle are described by the Symmetric Group . In the Standard Model of Particle Physics , the gauge group used to describe 3 of the Fundamental Force s is SU(3) × SU(2) × U(1) . Also, the reduction by symmetry of the energy functional under the action by a group and Spontaneous Symmetry Breaking of transformations of symmetric groups appear to elucidate topics in Particle Physics (for example, the unification of Electromagnetism and the Weak Force and Cosmology ). APPLICATIONS OF SYMMETRY Physical problems can be simplified by noticing any symmetries that a system possesses. SEE ALSO
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