Symmetry in physics refers to features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation. In Physics the word system has a technical meaning namely it is the portion of the physical Universe chosen for analysis Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In Mathematics a transform is an Operator applied to a function so that under the transform certain operations are simplified Observation is either an activity of a living being (such as a Human) which senses and assimilates the Knowledge of a Phenomenon, or the recording of data A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is "preserved" under some change.
The transformations may be continuous (such as rotation of a circle) or discrete (e. In Mathematics, a transformation could be any Function from a set X to itself A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Mathematics, a transformation could be any Function from a set X to itself g. , reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is Symmetries are frequently amenable to mathematical formulation and can be exploited to simplify many problems. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and
Invariance is specified mathematically by transformations that leave some quantity unchanged. This idea can apply to basic real-world observations. For example, temperature may be constant throughout a room. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature Since the temperature is independent of position within the room, the temperature is invariant under a shift in the measurer's position.
Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry. This article is about rotations in three-dimensional Euclidean space A rotation about any axis of the sphere will preserve how the sphere "looks". A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation
The above ideas lead to the useful idea of invariance when discussing observed physical symmetry; this can be applied to symmetries in forces as well.
For example, an electrical wire is said to exhibit cylindrical symmetry, because the electric field strength at a given distance r 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. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can 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 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.
Symmetries may be broadly classified as global or local. A global symmetry is a symmetry that holds for all points in the Spacetime under consideration as opposed to a Local symmetry that only holds for an In quantum field theory (QFT the forces between particles are mediated by other particles A global symmetry is one that holds at all points of spacetime, whereas a local symmetry is one that has a different symmetry transformation at different points of spacetime. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Local symmetries play an important role in physics as they form the basis for gauge theories. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations
The two examples of rotational symmetry described above - spherical and cylindrical - are each instances of continuous symmetry. In Mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some Symmetries as motions as opposed to e These are characterised by invariance following 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 will be the same on a given cylinder. Mathematically, continuous symmetries are described by continuous or smooth functions. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability An important subclass of continuous symmetries in physics are spacetime symmetries.
Continuous spacetime symmetries are symmetries involving transformations of space and time. Spacetime symmetries refers to aspects of Spacetime that can be described as exhibiting some form of Symmetry. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of These may be further classified as spatial symmetries, involving only the spatial geometry associated with a physical system; temporal symmetries, involving only changes in time; or spatio-temporal symmetries, involving changes in both space and time.
Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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, a local diffeomorphism is a Smooth map f: M &rarr N between Smooth manifolds such that for every point
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 Mathematics, a Killing vector field, named after Wilhelm Killing, is a Vector field on a Riemannian manifold (or Pseudo-Riemannian manifold In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries. For the Mechanical engineering and Architecture usage see Isometric projection. The article Isometries in physics discusses these symmetries in more detail. Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these
A discrete symmetry is a symmetry that describes non-continuous changes in a system. In Theoretical physics, a discrete symmetry is a Symmetry under the transformations of a Discrete group - i For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges.
The Standard model of particle physics has three related natural near-symmetries. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles These state that the universe is indistinguishable from one where:
Each of these symmetries is broken, but the Standard Model predicts that the combination of the three (that is, the three transformations at the same time) must be a symmetry, known as CPT symmetry. CPT symmetry is a fundamental symmetry of Physical laws under transformations that involve the inversions of charge, parity and CP violation, the violation of the combination of C and P symmetry, is a currently fruitful area of particle physics research, as well as being necessary for the presence of significant amounts of matter in the universe and thus the existence of life. In Particle physics, CP violation is a violation of the postulated CP symmetry of the laws of physics
A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the standard model. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the standard model, specifically a symmetry between bosons and fermions. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. If superpartners exist they must have masses greater than current particle accelerators can generate.
The transformations describing physical symmetries typically form a mathematical group. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Group theory is an important area of mathematics for physicists. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups.
Continuous symmetries are specified mathematically by continuous groups (called Lie groups). In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group 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 . In Mathematics, the orthogonal group of degree n over a field F (written as O( n, F) is the group of n (The 3 refers to the three-dimensional space of an ordinary sphere. ) Thus, the symmetry group of the sphere 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 Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime
Discrete symmetries are 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 S X or Sym( X) is the group whose underlying
An important type of physical theory based on local symmetries is called a gauge theory and the symmetries natural to such a theory are called gauge symmetries. Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations Gauge symmetries in the Standard model, used to describe three of the fundamental interactions, are based on the SU(3) × SU(2) × U(1) group. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms For a basic description see the article on the Standard Model. (Roughly speaking, the symmetries of the SU(3) group describe the strong force, the SU(2) group describes the weak interaction and the U(1) group describes the electromagnetic force. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature In Physics, the electromagnetic force is the force that the Electromagnetic field exerts on electrically charged particles )
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). In Physics, spontaneous symmetry breaking occurs when a system that is symmetric with respect to some Symmetry group goes into a Vacuum state Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its
The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves Noether's theorem gives a precise description of this relation. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has The 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. For example, the isometry of space gives rise to conservation of (linear) momentum, and isometry of time gives rise to conservation of energy.
A summary of some fundamental symmetries together with their conserved quantities is given in the table below.
|translation in time|
|translation in space|
|rotation in space|
|Discrete symmetry||P, coordinates' inversion||spatial parity|
|C, charge conjugation||charge parity|
|T, time reversal||time parity|
|CPT||product of parities|
|Internal symmetry(independent of|
|U(1) gauge transformation||electric charge|
|U(1) gauge transformation||lepton generation number|
|U(1) gauge transformation||hypercharge|
|U(1)Y gauge transformation||weak hypercharge|
|U(2) [U(1)xSU(2)]||electroweak force|
|SU(2) gauge transformation||isospin|
|SU(2)L gauge transformation||weak isospin|
|SU(3) "winding number"||baryon number|
|SU(3) gauge transformation||quark color|
|SU(3) (approximate)||quark flavor|