Symmetry in physics
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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 [r_0] 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 [r_0]. 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 [m] starting from rest at the origin and moving along the x-axis in opposite directions, one with speed [v_1] and the other with speed [v_2] the total kinetic energy of the system (as calculated from an observer at the origin) is [\frac(v_1^2 + v_2^2)] 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 [v_1] and [v_2] 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. Local symmetries tend to play an important role in physics, as measurements are performed in a limited region of space (or 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 its axis and the field strength magnitude will be the same on any given cylinder. Mathematically, continuous symmetries are usually described by continuous or smooth functions. An important subclass of continuous symmetries in physics are spacetime symmetries.
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 ').
- Time translation: A physical system may have the same features over a certain interval of time [\delta t] ; this is expressed mathematically as invariance under the transformation [t \, \rightarrow t + a ] for any real numbers t and t + a in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have gravitational potential energy [\, mgh] when suspended from a height [h] above the Earth's surface. Assuming no change in the height of the particle, this will be the total gravitational potential energy of the particle at all times. In other words, by considering the state of the particle at some time (in seconds) [t_0] and also at [t_0 + 3], say, the particle's total gravitational potential energy will be preserved.
- Spatial translation: These spatial symmetries are represented by transformations of the form [\vec \, \rightarrow \vec + \vec] and describe those situations where a property of the system does not change with a continuous change in location. For example, the temperature in a room may be independent of where the thermometer is located in the room.
- Spatial rotation: These spatial symmetries are classified into two types, namely, proper rotations and improper rotations. The former are just the 'ordinary' rotations; mathematically, they are square matrices with unit determinant. The latter are represented by square matrices with determinant -1 and consist of a proper rotation composed with a spatial reflection (inversion). For example, a ping-pong ball has rotational symmetry where the rotations are proper. Other types of spatial rotations are described in the article rotation symmetry.
- Poincaré transformations: These are spatio-temporal symmetries which preserve distances in Minkowski spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed are called Lorentz transformations and give rise to the symmetry known as Lorentz covariance.
- Projective symmetries: These are spatio-temporal symmetries which preserve the geodesic structure of spacetime. They may be defined on any smooth manifold, but find many applications in the study of exact solutions in general relativity.
Some of the most important vector fields are Killing vector fields 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
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.
- Time reversal: Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, [t \, \rightarrow - t ]. For example, Newton's second law of motion still holds if, in the equation [F \, = m \ddot ], [t] is replaced by [-t]. This may be illustrated by describing the motion of a particle thrown up vertically (neglecting air resistance). For such a particle, position is symmetric with respect to the instant that the object is at its maximum height. Velocity at reversed time is reversed.
- Spatial inversion: These are represented by transformations of the form [\vec \, \rightarrow - \vec] and indicate an invariance property of a system when the coordinates are 'inverted'. Example 4 above illustrates this spatial symmetry.
- Glide reflection: These are represented by a composition of a translation and a reflection. These symmetries occur in certain crystals.
Gauge symmetry
Many discrete symmetries are found in physics, especially particle physics.
Conservation laws and 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 implies that some physical property of that system is conserved, and conversely that each conserved quantity has a corresponding symmetry.
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 groups. Many physical symmetries are isometries and are specified by symmetry groups. 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 [\, SO(3)]. Thus, the symmetry group of the ping-pong ball with proper rotations is [\, SO(3)]. 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 Poincaré group).
Discrete symmetries tend to be described by discrete groups. For example, the symmetries of an equilateral triangle are described by the symmetric group [\, S_3].
In the Standard model of particle physics, the gauge group used to describe 3 of the fundamental interactions 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 in physical cosmology).
Applications of symmetry
Physical problems can be simplified by noticing any symmetries that a system possesses.
Supersymmetry
Extensions of symmetry to the concept of supersymmetry have been used to try to make theoretical advances in the standard model. Roughly speaking, supersymmetry is based on the idea that there is one remaining physical symmetry beyond those that are well-understood, a symmetry between bosons and fermions, so that each boson would have a symmetry partner fermion, called a superpartner, and vice versa. There are significant unsolved problems with the theory of supersymmetry, including that no known particle has the correct properties to be a superpartner of any other known particle, so that if superpartners exist, they apparently all must have greater mass than existing particle accelerators have been capable of generating.
See also
- Field (physics)
- Gauge theory
- Lagrangian
- Noether's theorem
- Particle physics and representation theory
- Principle of relativity
- Spacetime symmetries
References
- Brading, K., and Castellani, E., eds., 2003. Symmetries in Physics: Philosophical Reflections. Cambridge Uni. Press.
- Rosen, Joe, 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag.
- Van Fraassen, B. C., 1989. Laws and symmetry. Oxford Uni. Press.
External links
- Stanford Encyclopedia of Philosophy: [Symmetry] by Brading and Castellani.
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