Symmetry (physics)
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In
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some
transformation Transformation may refer to: Science and mathematics In biology and medicine * Metamorphosis, the biological process of changing physical form after birth or hatching * Malignant transformation, the process of cells becoming cancerous * Tran ...
. A family of particular transformations may be ''continuous'' (such as
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
of a circle) or ''
discrete Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory *Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a g ...
'' (e.g.,
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
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 group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s while discrete symmetries are described by
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
s (see '' Symmetry group''). These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as
group representations In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used t ...
and can, in addition, be exploited to simplify many problems. Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is described in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
by a group of transformations of the
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
known as the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
. Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
.


As a kind of invariance

Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example,
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room, we say that the temperature is ''invariant'' under a shift in an observer's position within the room. 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 In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or the ...
. A rotation about any
axis An axis (plural ''axes'') is an imaginary line around which an object rotates or is symmetrical. Axis may also refer to: Mathematics * Axis of rotation: see rotation around a fixed axis * Axis (mathematics), a designator for a Cartesian-coordinat ...
of the sphere will preserve how the sphere "looks".


Invariance in force

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 electric field due to an electrically charged wire of infinite length is said to exhibit
cylindrical symmetry In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane, or th ...
, because the
electric field strength An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field f ...
at a given distance ''r'' from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius ''r''. Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges. In Newton's theory of mechanics, given two bodies, each with mass ''m'', starting 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 In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
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 ''v''1 and ''v''2 are interchanged.


Local and global

Symmetries may be broadly classified as ''global'' or ''local''. A ''global symmetry'' is one that keeps a property invariant for a transformation that is applied simultaneously at all points of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
, whereas a ''local symmetry'' is one that keeps a property invariant when a possibly different symmetry transformation is applied at each point of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates, whereas a global symmetry is not. This implies that a global symmetry is also a local symmetry. Local symmetries play an important role in physics as they form the basis for
gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups ...
.


Continuous

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 discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to ano ...
. 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 transformations that change continuously as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.


Spacetime

Continuous ''spacetime symmetries'' are symmetries involving transformations of
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
and
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
. 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. * ''
Time translation Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ( ...
'': A physical system may have the same features over a certain interval of time Δ''t''; this is expressed mathematically as invariance under the transformation for any
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
parameters ''t'' and in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conver ...
''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 ''t'' and also at , 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 as
proper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
s and
improper rotation In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
s. The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
. The latter are represented by square matrices with determinant −1 and consist of a proper rotation combined with a spatial reflection (
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
). For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article ''
Rotation symmetry Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which i ...
''. * ''Poincaré transformations'': These are spatio-temporal symmetries which preserve distances in
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
, i.e. they are isometries of Minkowski space. They are studied primarily in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
. Those isometries that leave the origin fixed are called
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
s and give rise to the symmetry known as
Lorentz covariance In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same ...
. * ''Projective symmetries'': These are spatio-temporal symmetries which preserve the
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
structure of
spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
. They may be defined on any smooth manifold, but find many applications in the study of
exact solutions in general relativity In general relativity, an exact solution is a solution of the Einstein field equations whose derivation does not invoke simplifying assumptions, though the starting point for that derivation may be an idealized case like a perfectly spherical sh ...
. * ''Inversion transformations'': These are spatio-temporal symmetries which generalise Poincaré transformations to include other conformal one-to-one transformations on the space-time coordinates. Lengths are not invariant under
inversion transformations In mathematical physics, inversion transformations are a natural extension of Poincaré transformations to include all conformal one-to-one transformations on coordinate space-time. They are less studied in physics because unlike the rotations ...
but there is a cross-ratio on four points that is invariant. Mathematically, spacetime symmetries are usually described by
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
vector fields on a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. The underlying
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Form ...
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 fields which are those spacetime symmetries that preserve the underlying
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
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 In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
.


Discrete

A ''discrete symmetry'' is a symmetry that describes non-continuous changes in a system. 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''. * '' 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 Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
still holds if, in the equation F \, = m \ddot , t is replaced by -t. This may be illustrated by recording the motion of an object thrown up vertically (neglecting air resistance) and then playing it back. The object will follow the same parabolic trajectory through the air, whether the recording is played normally or in reverse. Thus, position is symmetric with respect to the instant that the object is at its maximum height. * '' 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'. Stated another way, these are symmetries between a certain object and its mirror image. *''
Glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflection ...
'': These are represented by a composition of a translation and a reflection. These symmetries occur in some
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...
s and in some planar symmetries, known as wallpaper symmetries.


C, P, and T

The
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
of
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced. *
C-symmetry In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-symm ...
(charge symmetry), a universe where every particle is replaced with its
antiparticle In particle physics, every type of particle is associated with an antiparticle with the same mass but with opposite physical charges (such as electric charge). For example, the antiparticle of the electron is the positron (also known as an antie ...
*
P-symmetry In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
(parity symmetry), a universe where everything is mirrored along the three physical axes. This excludes weak interactions as demonstrated by
Chien-Shiung Wu ) , spouse = , residence = , nationality = ChineseAmerican , field = Physics , work_institutions = Institute of Physics, Academia SinicaUniversity of California at BerkeleySmith CollegePrinceton UniversityColumbia UniversityZhejiang Unive ...
. *
T-symmetry T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal, : T: t \mapsto -t. Since the second law of thermodynamics states that entropy increases as time flows toward the future ...
(time reversal symmetry), a universe where the direction of time is reversed. T-symmetry is counterintuitive (the future and the past are not symmetrical) but explained by the fact that the Standard Model describes local properties, not global ones like
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. To properly reverse the direction of time, one would have to put the Big Bang and the resulting low-entropy state in the "future". Since we perceive the "past" ("future") as having lower (higher) entropy than the present, the inhabitants of this hypothetical time-reversed universe would perceive the future in the same way as we perceive the past, and vice versa. These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called
CPT symmetry Charge, parity, and time reversal symmetry is a fundamental symmetry of physical laws under the simultaneous transformations of charge conjugation (C), parity transformation (P), and time reversal (T). CPT is the only combination of C, P, and T ...
.
CP violation In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics should be the ...
, the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of
baryonic matter In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classifie ...
in the universe. CP violation is a fruitful area of current research in
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
.


Supersymmetry

A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the Standard Model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the Standard Model, specifically a symmetry between
boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s and
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s. 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. Currently LHC is preparing for a run which tests supersymmetry.


Mathematics of physical symmetry

The transformations describing physical symmetries typically form a mathematical
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
.
Group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
is an important area of mathematics for physicists. Continuous symmetries are specified mathematically by ''continuous groups'' (called
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s). 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). (The '3' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere 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 In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
(this may be generalised to the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
). Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
S. A 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 symmetries in the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions - excluding gravity) in the universe and classifying a ...
, used to describe three of the
fundamental interaction In physics, the fundamental interactions, also known as fundamental forces, are the interactions that do not appear to be reducible to more basic interactions. There are four fundamental interactions known to exist: the gravitational and electro ...
s, are based on the SU(3) × SU(2) × U(1) group. (Roughly speaking, the symmetries of the SU(3) group describe the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the n ...
, the SU(2) group describes the
weak interaction In nuclear physics and particle physics, the weak interaction, which is also often called the weak force or weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, ...
and the U(1) group describes the
electromagnetic force In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
.) Also, the reduction by symmetry of the energy functional under the action by a group and
spontaneous symmetry breaking Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
of transformations of symmetric groups appear to elucidate topics in
particle physics Particle physics or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
(for example, the unification of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
and the
weak force Weak may refer to: Songs * Weak (AJR song), "Weak" (AJR song), 2016 * Weak (Melanie C song), "Weak" (Melanie C song), 2011 * Weak (SWV song), "Weak" (SWV song), 1993 * Weak (Skunk Anansie song), "Weak" (Skunk Anansie song), 1995 * "Weak", a song ...
in
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
).


Conservation laws and symmetry

The symmetry properties of a physical system are intimately related to the
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 over time. Exact conservation laws include conservation of energy, conservation of linear momentum, ...
characterizing that system.
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e. homogeneity of space) gives rise to conservation of (linear) momentum, and temporal translation symmetry (i.e. homogeneity of time) gives rise to
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means th ...
. The following table summarizes some fundamental symmetries and the associated conserved quantity.


Mathematics

Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
s. The commutator of two of these infinitesimal transformations are equivalent to a third infinitesimal transformation of the same kind hence they form a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
. A general coordinate transformation described as the general field h(x) (also known as a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
) has the infinitesimal effect on a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
\phi(x),
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
\psi(x) or vector field A(x) that can be expressed (using the
Einstein summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
): : \delta\phi(x) = h^(x)\partial_\phi(x) : \delta\psi^\alpha(x) = h^(x)\partial_\psi^\alpha(x) + \partial_\mu h_\nu(x) \sigma_^ \psi^(x) : \delta A_\mu(x) = h^(x)\partial_A_\mu(x) + A_\nu(x)\partial_\mu h^(x) Without gravity only the Poincaré symmetries are preserved which restricts h(x) to be of the form: : h^(x) = M^x_\nu + P^\mu where M is an antisymmetric
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
(giving the Lorentz and rotational symmetries) and P is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field: : \delta\psi^\alpha(x) = \lambda(x).\tau^\psi^\beta(x) : \delta A_\mu(x) = \partial_\mu \lambda(x) , where \tau are generators of a particular
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
. So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of ''different'' types. Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind: : \delta \phi(x) = \Omega(x) \phi(x) If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form: : h^(x) = M^x_\nu + P^\mu + D x_\mu + K^ , x, ^2 - 2 K^\nu x_\nu x_\mu , with D generating scale transformations and K generating special conformal transformations. For example, super- Yang–Mills theory has this symmetry while
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
doesn't although other theories of gravity such as
conformal gravity Conformal gravity refers to gravity theories that are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations g_\rightarrow\Omega^2(x)g_ where g_ is the metric te ...
do. The 'action' of a field theory is an invariant under all the symmetries of the theory. Much of modern theoretical physics is to do with speculating on the various symmetries the Universe may have and finding the invariants to construct field theories as models. In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields.


See also

*
Conserved current In physics a conserved current is a current, j^\mu, that satisfies the continuity equation \partial_\mu j^\mu=0. The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume V, l ...
&
Charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
*
Coordinate-free A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system. Benefits Coordinate-free treatments generally all ...
* Covariance and contravariance * Fictitious force *
Galilean invariance Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames of reference. Galileo Galilei first described this principle in 1632 in his ''Dialogue Concerning the Two Chief World Systems'' using th ...
*
Principle of covariance In physics, the principle of covariance emphasizes the formulation of physical laws using only those physical quantities the measurements of which the observers in different frames of reference could unambiguously correlate. Mathematically, the ph ...
*
General covariance In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea is ...
*
Harmonic coordinate condition The harmonic coordinate condition is one of several coordinate conditions in general relativity, which make it possible to solve the Einstein field equations. A coordinate system is said to satisfy the harmonic coordinate condition if each of the co ...
*
Inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
*
List of mathematical topics in relativity {{Short description, none This is a list of mathematical topics in relativity (physics), relativity, by Wikipedia page. Special relativity *''Foundational issues'' **principle of relativity **speed of light **faster-than-light *biquaternion *conju ...
* Standard Model (mathematical formulation) *
Wheeler–Feynman absorber theory The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the assu ...


References


General readers

* * * Chapter 12 is a gentle introduction to symmetry, invariance, and conservation laws. *


Technical readers

* * * * Address to the 2002 meeting of the
Philosophy of Science Association The Philosophy of Science Association (PSA) is an international academic organization founded in 1933 that promotes research, teaching, and free discussion of issues in the philosophy of science from diverse standpoints. The PSA engages in activit ...
. * * * * *


External links


The Feynman Lectures on Physics Vol. I Ch. 52: Symmetry in Physical Laws
*
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...
:
Symmetry
—by K. Brading and E. Castellani.
Pedagogic Aids to Quantum Field Theory
Click on link to Chapter 6: Symmetry, Invariance, and Conservation for a simplified, step-by-step introduction to symmetry in physics. {{DEFAULTSORT:Symmetry In Physics Concepts in physics Conservation laws Diffeomorphisms Differential geometry Symmetry