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| {{other uses2|Symmetry}}
| | Greetings! I am Marvella and I feel comfy when individuals use the full title. Hiring is his profession. Years in the past we moved to North Dakota. What I adore doing is playing baseball but I haven't produced a dime with it.<br><br>Also visit my blog post; [http://xrambo.com/blog/191590 home std test] |
| [[File:Brillouin_Zone_(1st,_FCC).svg|thumb|right|200px|First [[Brillouin zone]] of [[FCC lattice]] showing symmetry labels]]
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| In [[physics]], '''symmetry''' includes all features of a [[physical system]] that exhibit the property of [[symmetry]]—that is, under certain [[transformation (function)|transformations]], aspects of these systems are otherwise "unchanged", according to a particular [[observation]]. A '''symmetry of a physical system''' is a physical or mathematical feature of the system (observed or intrinsic) that is "preserved" under some change.
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| A family of particular transformations may be ''continuous'' (such as [[rotation]] of a circle) or ''[[discrete space|discrete]]'' (e.g., [[Reflection (physics)|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 group]]s while discrete symmetries are described by finite groups (see [[Symmetry group]]). Symmetries are frequently amenable to mathematical formulations such as [[representation of a Lie group|group representations]] and can be exploited to simplify many problems.
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| An important example of such symmetry is the [[General covariance|invariance]] of the form of physical laws under arbitrary differentiable coordinate transformations.
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| == Symmetry as invariance ==
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| 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. Since the temperature is independent of position within the room, the temperature is ''invariant'' under a shift in the measurer's position.
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| 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]]. A rotation about any [[Axis of rotation|axis]] of the sphere will preserve how the sphere "looks".
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| === Invariance in force ===
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| 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.
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| For example, an electric field due to a wire is said to exhibit [[rotational symmetry#Rotational symmetry with respect to any angle|cylindrical symmetry]], because the [[electric field strength]] at a given distance ''r'' from the 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''. 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. Suppose some configuration of charges (may be non-stationary) produce an electric field in some direction, then rotating the configuration of the charges (without disturbing the internal dynamics that produces the particular field) will lead to a net rotation of the direction of the electric field.These two properties are interconnected through the more general property that rotating ''any'' system of charges causes a corresponding rotation of the electric field.
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| In Newton's theory of mechanics, given two bodies, each with mass ''m'', starting from rest at the origin and moving along the ''x''-axis in opposite directions, one with speed ''v''<sub>1</sub> and the other with speed ''v''<sub>2</sub> the total [[kinetic energy]] of the system (as calculated from an observer at the origin) is {{nowrap|{{frac|1|2}}''m''(''v''<sub>1</sub><sup>2</sup> + ''v''<sub>2</sub><sup>2</sup>)}} and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the ''y''-axis.
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| 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''<sub>1</sub> and ''v''<sub>2</sub> are interchanged.
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| == Local and global symmetries ==
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| {{main|Global symmetry|Local symmetry}}
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| Symmetries may be broadly classified as ''global'' or ''local''. 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]]; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates. Local symmetries play an important role in physics as they form the basis for [[Gauge theory|gauge theories]].
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| ==Continuous symmetries==
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| The two examples of rotational symmetry described above - spherical and cylindrical - are each instances of [[continuous symmetry]]. 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 function|continuous]] or [[smooth function]]s. An important subclass of continuous symmetries in physics are spacetime symmetries.
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| ===Spacetime symmetries===
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| {{main|Spacetime symmetries}}
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| Continuous ''spacetime symmetries'' are symmetries involving transformations of [[space]] and [[time]]. 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.
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| * '''''Time translation''''': A physical system may have the same features over a certain interval of time <math>\delta t</math>; this is expressed mathematically as invariance under the transformation <math>t \, \rightarrow t + a </math> for any [[real number]]s ''t'' and ''a'' in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have [[gravitational potential energy]] <math>\, mgh</math> when suspended from a height <math>h</math> 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) <math>t_0</math> and also at <math>t_0 + 3</math>, say, the particle's total gravitational potential energy will be preserved.
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| * '''''Spatial translation''''': These spatial symmetries are represented by transformations of the form <math>\vec{r} \, \rightarrow \vec{r} + \vec{a}</math> 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.
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| * '''''Spatial rotation''''': These spatial symmetries are classified as [[proper rotation]]s and [[improper rotation]]s. The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit [[determinant]]. The latter are represented by square matrices with determinant −1 and consist of a proper rotation combined with a spatial reflection ([[Point reflection|inversion]])<!-- odd-dimensional? -->. For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article ''[[Rotation symmetry]]''.
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| * '''''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 transformation]]s and give rise to the symmetry known as [[Lorentz covariance]].
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| * '''''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]].
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| * '''''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]] but there is a cross-ratio on four points that is invariant.
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| Mathematically, spacetime symmetries are usually described by [[Smooth function|smooth]] [[vector field]]s on a [[smooth manifold]]. The underlying [[local diffeomorphism]]s associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.
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| Some of the most important vector fields are [[Killing vector field]]s which are those spacetime symmetries that preserve the underlying [[Metric tensor|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]].
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| ==Discrete symmetries==
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| {{main|Discrete symmetry}}
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| 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''.
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| * '''''[[T-symmetry|Time reversal]]''''': Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, <math>t \, \rightarrow - t </math>. For example, [[Newton's second law of motion]] still holds if, in the equation <math>F \, = m \ddot {r} </math>, <math>t</math> is replaced by <math>-t</math>. 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 [[Parabola|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.
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| * '''''[[Parity (physics)|Spatial inversion]]''''': These are represented by transformations of the form <math>\vec{r} \, \rightarrow - \vec{r}</math> and indicate an invariance property of a system when the coordinates are 'inverted'. Said another way, these are symmetries between a certain object and its [[mirror image]].
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| *'''''[[Glide reflection]]''''': These are represented by a composition of a translation and a reflection. These symmetries occur in some [[crystal]]s and in some planar symmetries, known as [[wallpaper group|wallpaper symmetries]].
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| === C, P, and T symmetries ===
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| The [[Standard model]] of [[particle physics]] has three related natural near-symmetries. These state that the actual universe about us is indistinguishable from one where:
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| *Every particle is replaced with its [[antiparticle]]. This is [[C-symmetry]] (charge symmetry);
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| *Everything appears as if reflected in a mirror. This is [[Parity (physics)|P-symmetry]] (parity symmetry);
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| *The [[entropy (arrow of time)|direction of time]] is reversed. This is [[T-symmetry]] (time symmetry).
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| T-symmetry is counterintuitive (surely 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]]. 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 (see [[Entropy (arrow of time)|perception of time]]), the inhabitants of this hypothetical time-reversed universe would perceive the future in the same way as we perceive the past.
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| 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]]. [[CP violation]], the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of [[baryonic matter]] in the universe. CP violation is a fruitful area of current research in [[particle physics]].
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| === Supersymmetry ===
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| {{main|Supersymmetry}}
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| 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]]s and [[fermion]]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. If superpartners exist they must have masses greater than current [[particle accelerator]]s can generate.
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| ==Mathematics of physical symmetry==
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| {{main|Symmetry group}}
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| {{see also|Symmetry in quantum mechanics|Symmetries in general relativity}}
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| The transformations describing physical symmetries typically form a mathematical [[group (mathematics)|group]]. [[Group theory]] is an important area of mathematics for physicists.
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| Continuous symmetries are specified mathematically by ''continuous groups'' (called [[Lie group]]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]] <math>\, SO(3)</math>. (The ''3'' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is <math>\, SO(3)</math>. 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]]).
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| Discrete symmetries are described by discrete groups. For example, the symmetries of an equilateral triangle are described by the [[symmetric group]] <math>\, S_3</math>.
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| An important type of physical theory based on ''local'' symmetries is called a [[Gauge theory|''gauge'' theory]] and the symmetries natural to such a theory are called [[Gauge symmetry|gauge symmetries]]. Gauge symmetries in the [[Standard model]], used to describe three of the [[fundamental interaction]]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 SU(2) group describes the [[weak interaction]] and the U(1) group describes the [[electromagnetic force]].)
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| 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]]).
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| ===Conservation laws and symmetry===
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| {{main|Noether's theorem}}
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| The symmetry properties of a physical system are intimately related to the [[conservation laws]] characterizing that system. [[Noether's theorem]] 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, the [[isometry]] of [[space]] gives rise to [[conservation of momentum|conservation of (linear) momentum]], and [[isometry]] of [[time]] gives rise to [[conservation of energy]].
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| The following table summarizes some fundamental symmetries and the associated conserved quantity.
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| {| class="wikitable"
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| ||'''Class'''
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| ||'''[[Invariant (physics)|Invariance]]'''
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| ||'''[[Conservation law|Conserved quantity]]'''
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| |-
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| ||Proper orthochronous<br />[[Lorentz symmetry]]
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| ||translation in time<br />  <SMALL>([[homogeneity (physics)|homogeneity]])</SMALL>
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| ||[[conservation of energy|energy]]
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| |-
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| ||[[translational invariance|translation in space]]<br />  <SMALL>([[homogeneity (physics)|homogeneity]])</SMALL>
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| ||[[conservation of momentum|linear momentum]]
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| |-
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| ||
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| ||[[rotational invariance|rotation in space]]<br />  <SMALL>([[isotropy]])</SMALL>
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| ||[[conservation of angular momentum|angular momentum]]
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| |-
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| ||[[Discrete symmetry]]
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| ||P, coordinate inversion
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| ||[[CP-symmetry|spatial parity]]
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| |-
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| ||C, [[charge conjugation]]
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| ||[[CP-symmetry|charge parity]]
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| |-
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| ||T, time reversal
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| ||[[T-symmetry|time parity]]
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| |-
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| ||[[CPT symmetry|CPT]]
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| ||product of parities
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| |-
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| ||[[Internal symmetry]] (independent of<br />[[spacetime]] [[coordinate]]s)
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| ||[[U(1)]] [[gauge transformation]]
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| ||[[electric charge]]
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| |-
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| ||[[U(1)]] [[gauge transformation]]
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| ||[[lepton number|lepton generation number]]
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| |-
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| ||[[U(1)]] gauge transformation
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| ||[[hypercharge]]
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| |-
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| ||[[U(1)]]<SUB>Y</SUB> [[gauge transformation]]
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| ||[[weak hypercharge]]
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| |-
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| ||U(2) [U(1) × [[SU(2)]]]
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| ||[[electroweak force]]
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| |-
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| ||SU(2) gauge transformation
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| ||[[isospin]]
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| |-
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| ||[[SU(2)]]<SUB>L</SUB> gauge transformation
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| ||[[weak isospin]]
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| |-
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| ||P × SU(2)
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| ||[[G-parity]]
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| |-
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| ||SU(3) "winding number"
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| ||[[baryon number]]
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| |-
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| ||SU(3) gauge transformation
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| || [[quark color]]
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| |-
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| ||[[SU(3)]] (approximate)
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| ||[[flavor (physics)|quark flavor]]
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| |-
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| ||S(U(2) × U(3))<br />[[U(1)]] × [[SU(2)]] × [[SU(3)]]]
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| ||[[Standard Model]]
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| |-
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| |}
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| ==Mathematics==
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| Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various [[particle field]]s. The [[commutator]] of two of these infinitessimal transformations are equivalent to a third infinitessimal transformation of the same kind hence they form a [[Lie algebra]].
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| A general coordinate transformation (also known as a [[diffeomorphism]]) has the infinitessimal effect on a [[scalar field|scalar]], [[spinor field|spinor]] and [[vector field]] for example:
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| <math>
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| \delta\phi(x) = h^{\mu}(x)\partial_{\mu}\phi(x)
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| </math>
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| <math>
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| \delta\psi^\alpha(x) = h^{\mu}(x)\partial_{\mu}\psi^\alpha(x) + \partial_\mu h_\nu(x) \sigma_{\mu\nu}^{\alpha \beta} \psi^{\beta}(x)
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| </math>
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| <math>
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| \delta A_\mu(x) = h^{\nu}(x)\partial_{\nu}A_\mu(x) + A_\nu(x)\partial_\nu h_\mu(x)
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| </math>
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| for a general field, <math>h(x)</math>. Without gravity only the Poincaré symmetries are preserved which restricts <math>h(x)</math> to be of the form:
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| <math>
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| h^{\mu}(x) = M^{\mu \nu}x_\nu + P^\mu
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| </math>
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| where '''M''' is an antisymmetric [[Matrix (mathematics)|matrix]] (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:
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| <math>
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| \delta\psi^\alpha(x) = \lambda(x).\tau^{\alpha\beta}\psi^\beta(x)
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| </math>
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| <math>
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| \delta A_\mu(x) = \partial_\mu \lambda(x)
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| </math>
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| where <math>\tau</math> are generators of a particular [[Lie group]]. 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.
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| 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:
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| <math>
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| \delta \phi(x) = \Omega(x) \phi(x)
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| </math>
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| 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:
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| <math>
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| h^{\mu}(x) = M^{\mu \nu}x_\nu + P^\mu + D x_\mu + K^{\mu} |x|^2 - 2 K^\nu x_\nu x_\mu
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| </math>
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| with '''D''' generating scale transformations and '''K''' generating special conformal transformations. For example N=4 super-[[Yang-Mills]] theory has this symmetry while [[General Relativity]] doesn't although other theories of gravity such as [[conformal gravity]] do. The 'action' of a field theory is an [[invariant (physics)|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.
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| 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.
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| ==See also==
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| *[[Conservation law]]
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| *[[Conserved current]]
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| *[[Coordinate-free]]
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| *[[Covariance and contravariance of vectors|Covariance and contravariance]]
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| *[[Diffeomorphism]]
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| *[[Fictitious force]]
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| *[[Galilean invariance]]
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| *[[Gauge theory]]
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| *[[General covariance]]
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| *[[Harmonic coordinate condition]]
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| *[[Inertial frame of reference]]
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| *[[Lie group]]
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| *[[List of mathematical topics in relativity]]
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| *[[Lorentz covariance]]
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| *[[Noether's theorem]]
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| * [[Poincaré group]]
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| * [[Special relativity]]
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| * [[Spontaneous symmetry breaking]]
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| * [[Standard model]]
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| * [[Standard model (mathematical formulation)]]
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| * [[Symmetry breaking]]
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| * [[Wheeler–Feynman absorber theory|Wheeler–Feynman Time-Symmetric Theory]]
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| ==References==
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| ===General readers===
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| *[[Leon Lederman]] and [[Christopher T. Hill]] (2005) ''Symmetry and the Beautiful Universe''. Amherst NY: Prometheus Books.
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| * Schumm, Bruce (2004) ''Deep Down Things''. Johns Hopkins Univ. Press.
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| *[[Victor J. Stenger]] (2000) ''Timeless Reality: Symmetry, Simplicity, and Multiple Universes''. Buffalo NY: Prometheus Books. Chpt. 12 is a gentle introduction to symmetry, invariance, and conservation laws.
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| *[[Anthony Zee]] (2007) ''[http://press.princeton.edu/titles/8509.html Fearful Symmetry: The search for beauty in modern physics,]'' 2nd ed. Princeton University Press. ISBN 978-0-691-00946-9. 1986 1st ed. published by Macmillan.
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| ===Technical===
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| *Brading, K., and Castellani, E., eds. (2003) ''Symmetries in Physics: Philosophical Reflections''. Cambridge Univ. Press.
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| * -------- (2007) "Symmetries and Invariances in Classical Physics" in Butterfield, J., and [[John Earman]], eds., ''Philosophy of Physic Part B''. North Holland: 1331-68.
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| * Debs, T. and Redhead, M. (2007) ''Objectivity, Invariance, and Convention: Symmetry in Physical Science''. Harvard Univ. Press.
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| * [[John Earman]] (2002) "[http://philsci-archive.pitt.edu/archive/00000878/00/PSA2002.pdf Laws, Symmetry, and Symmetry Breaking: Invariance, Conservations Principles, and Objectivity.]" Address to the 2002 meeting of the [[Philosophy of Science Association]].
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| *Mainzer, K. (1996) ''Symmetries of nature''. Berlin: De Gruyter.
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| *Mouchet, A. "Reflections on the four facets of symmetry: how physics exemplifies rational thinking". European Physical Journal H 38 (2013) 661 [http://hal.archives-ouvertes.fr/hal-00637572 hal.archives-ouvertes.fr:hal-00637572]
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| *Thompson, William J. (1994) ''Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems''. Wiley. ISBN 0-471-55264.
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| *[[Bas Van Fraassen]] (1989) ''Laws and symmetry''. Oxford Univ. Press.
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| *[[Eugene Wigner]] (1967) ''Symmetries and Reflections''. Indiana Univ. Press.
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| ==External links==
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| * [[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/symmetry-breaking/ Symmetry]" -- by K. Brading and E. Castellani.
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| * [http://www.quantumfieldtheory.info 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.
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| {{DEFAULTSORT:Symmetry In Physics}}
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| [[Category:Concepts in physics]]
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| [[Category:Conservation laws]]
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| [[Category:Diffeomorphisms]]
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| [[Category:Differential geometry]]
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| [[Category:Symmetry]]
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