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. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

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. In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ...

Example 3

The electric field strength at a given distance r₀ 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 r₀. 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. In physics, an electric field or E-field is an effect produced by an electric charge that exerts a force on charged objects in its vicinity. ... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ...

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 v₁ and the other with speed v₂ 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 v₁ and v₂ 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

Main articles: global symmetry and local symmetry

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. In quantum field theory, a global symmetry is any symmetry of a model which is not a gauge symmetry. ... This article or section does not cite its references or sources. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ...

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 functions. An important subclass of continuous symmetries in physics are spacetime symmetries. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

Spacetime symmetries

Main article: 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 '). The role of symmetry in physics is important, for example, in simplifying solutions to many problems. ... Space has been an occupation for philosophers and scientists for much of human history, and hence it is difficult to provide an uncontroversial and clear definition outside of specific defined contexts (except scientific definition of space in physics and mathematics - see below). ... A pocket watch. ...

• Time translation: A physical system may have the same features over a certain interval of time δt ; this is expressed mathematically as invariance under the transformation 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 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) t₀ and also at t₀ + 3, say, the particle's total gravitational potential energy will be preserved.

• Spatial translation: These spatial symmetries are represented by transformations of the form 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.

Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms 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. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Potential energy (U, or Ep), a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ... Rotational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ... A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed. ... ‹The template below has been proposed for deletion. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. Definition of geodesic depends on the type of curved space. If the space carries a natural metric then geodesics are defined to be (locally) the shortest path between points on the space. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... // In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. ... Smooth could mean many things, including: Smooth function, a function that is infinitely differentiable, used in calculus and topology. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... Smooth could mean many things, including: Smooth function, a function that is infinitely differentiable, used in calculus and topology. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space. ... In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a diffeomorphism. ...

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. In mathematics, a Killing vector field is a vector field on a Riemannian manifold that preserves the metric. ... See: International System of Units, colloquially called the Metric System, and also metrication. ... In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Other continuous symmetries

Discrete symmetries

Main article: 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. In theoretical physics, a discrete symmetry is a symmetry under the transformations of a discrete group - i. ...

• Time reversal: Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, . For example, Newton's second law of motion still holds if, in the equation , 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 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.

T-symmetry is the symmetry of physical laws under a time-reversal transformation— The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*. In other words, the vector from X to P is the same as the vector from... Example of a glide reflection In geometry, a glide reflection is a type of isometry of the Euclidean plane. ... It has been suggested that crystallization processes be merged into this article or section. ...

Gauge symmetry

Many discrete symmetries are found in physics, especially particle physics. Particles erupt from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

Conservation laws and Noether's theorem

Main article: Noether's theorem

Symmetries of a physical system are intimately related to conservation laws for that system. This idea is encapsulated more precisely in Noether's theorem, which roughly states that each symmetry of a physical system gives rise to a conserved quantity for that system. Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between symmetries and conservation laws. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

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. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... Group theory is that branch of mathematics concerned with the study of groups. ...

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 . 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 Poincaré group). In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... The symmetry group of an object (e. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ...

Discrete symmetries tend to be described by discrete groups. For example, the symmetries of an equilateral triangle are described by the symmetric group . In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

In the Standard model of particle physics, the gauge group used to describe 3 of the fundamental forces 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). The Standard Model of Fundamental Particles and Interactions The Standard Model of particle physics is a theory which describes the strong, weak, and electromagnetic fundamental forces, as well as the fundamental particles that make up all matter. ... Particles erupt from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... This Lie group is the formulation of the Standard Model as a gauge theory with the gauge group SU(3) × SU(2) × U(1) or with a couple of fermion fields and a Higgs field, which is a and/or a . ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... Particles erupt from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn affected by the presence and motion of such particles. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) world + λογια (logia) discourse) is the study of the universe in its totality and by extension mans place in it. ...

Applications of symmetry

Physical problems can be simplified by noticing any symmetries that a system possesses.

See also

• Field (physics)
• Gauge theory
• Lagrangian
• Noether's theorem
• Particle physics and representation theory
• Principle of relativity
• Spacetime symmetries

In physics, a field is an assignment of a quantity to every point in space (or more generally, spacetime). ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between symmetries and conservation laws. ... There is a natural connection, first discovered by Eugene Wigner, between the properties of particles, the representation theory of Lie groups and Lie algebras, and the symmetries of the universe. ... Note: The principle of relativity should not be confused with the Theory of relativity. ... The role of symmetry in physics is important, for example, in simplifying solutions to many problems. ...

References

• Brading, K., and Castellani, E., eds., 2003. Symmetries in Physics: Philosophical Reflections. Cambridge Uni. Press.
• Van Fraassen, B. C., 1989. Laws and symmetry. Oxford Uni. Press.

External links

• Stanford Encyclopedia of Philosophy: Symmetry by Brading and Castellani.