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

## Symmetry as invariance

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

### 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 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 $\frac{1}{2}m(v_1^2 + v_2^2)$ and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the y-axis.

The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if v1 and v2 are interchanged.

## Local and global symmetries

Main articles: Global symmetry and Local symmetry

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

## Continuous symmetries

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.

### Spacetime symmetries

Main article: 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.

• Time translation: A physical system may have the same features over a certain interval of time δt; this is expressed mathematically as invariance under the transformation $t \, \rightarrow t + a$ for any real numbers t and a in the interval. In Mathematics, the real numbers may be described informally in several different ways For example, in classical mechanics, a particle solely acted upon by gravity will have gravitational potential energy $\, mgh$ when suspended from a height h above the Earth's surface. Potential energy can be thought of as Energy stored within a physical system Assuming no change in the height of the particle, this will be the total gravitational potential energy of the particle at all times. In other words, by considering the state of the particle at some time (in seconds) t0 and also at t0 + 3, say, the particle's total gravitational potential energy will be preserved.
• Spatial translation: These spatial symmetries are represented by transformations of the form $\vec{r} \, \rightarrow \vec{r} + \vec{a}$ 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 rotations and improper rotations. In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or In 3D Geometry, an improper rotation, also called rotoreflection or rotary reflection is depending on context a Linear transformation or The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit determinant. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n The latter are represented by square matrices with determinant -1 and consist of a proper rotation combined with a spatial reflection (inversion). For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article Rotation symmetry. Generally speaking an object with rotational symmetry is an object that looks the same after a certain amount of Rotation.
• Poincaré transformations: These are spatio-temporal symmetries which preserve distances in Minkowski spacetime, i. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity e. they are isometries of Minkowski space. They are studied primarily in special relativity. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Those isometries that leave the origin fixed are called Lorentz transformations and give rise to the symmetry known as Lorentz covariance. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally
• Projective symmetries: These are spatio-temporal symmetries which preserve the geodesic structure of spacetime. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces 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 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 Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter
• 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. Inversion transformations are a natural extension of Poincaré transformations to include all conformal One-to-one transformations on coordinate Space-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

## Discrete symmetries

Main article: Discrete symmetry

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.

• Time reversal: Many laws of physics describe real phenomena when the direction of time is reversed. T Symmetry is the symmetry of physical laws under a Time reversal transformation &mdash T t \mapsto -t Mathematically, this is represented by the transformation, $t \, \rightarrow - t$. For example, Newton's second law of motion still holds if, in the equation $F \, = m \ddot {r}$, t is replaced by t. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the This may be illustrated by describing the motion of a particle thrown up vertically (neglecting air resistance). For such a particle, position is symmetric with respect to the instant that the object is at its maximum height. Velocity at reversed time is reversed.
• Spatial inversion: These are represented by transformations of the form $\vec{r} \, \rightarrow - \vec{r}$ and indicate an invariance property of a system when the coordinates are 'inverted'. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate.
• Glide reflection: These are represented by a composition of a translation and a reflection. In Geometry, a glide reflection is a type of Isometry of the Euclidean plane: the combination of a reflection in a line and a translation These symmetries occur in some crystals and in some planar symmetries, known as wallpaper symmetries. In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern based on the

### C, P, and T symmetries

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:

• C-symmetry (charge symmetry) - every particle is replaced with its antiparticle. In Physics, C-symmetry means the symmetry of physical laws under a charge -conjugation transformation.
• P-symmetry (parity symmetry) - the universe is reflected as in a mirror. In Physics, a parity transformation (also called parity inversion) is the flip in the sign of one Spatial Coordinate.
• T-symmetry (time symmetry) - the direction of time is reversed. T Symmetry is the symmetry of physical laws under a Time reversal transformation &mdash T t \mapsto -t (This 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 properties like entropy. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy To properly time-reverse the universe, you would have to put the big bang and the resulting low-entropy conditions in the "future". Since our experience of time is related to entropy, the inhabitants of the resulting universe would then see that as the past. Entropy is the only quantity in the physical sciences that "picks" a particular direction for time sometimes called an Arrow of time. )

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

### Supersymmetry

Main article: Supersymmetry

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.

## Mathematics of physical symmetry

Main article: Symmetry group

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 $\, SO(3)$. 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 $\, SO(3)$. Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the Lorentz group (this may be generalised to the Poincaré group). 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 $\, S_3$. 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

### Conservation laws and symmetry

Main article: Noether's theorem

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.

 Class Invariance Conserved quantity Proper orthochronousLorentz symmetry translation in time  (homogeneity) energy translation in space  (homogeneity) linear momentum rotation in space  (isotropy) angular momentum Discrete symmetry P, coordinates' inversion spatial parity C, charge conjugation charge parity T, time reversal time parity CPT product of parities Internal symmetry(independent ofspacetime coordinates) 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 PxSU(2) G-parity SU(3) "winding number" baryon number SU(3) gauge transformation quark color SU(3) (approximate) quark flavor S((U2)xU(3))[U(1)xSU(2)xSU(3)] Standard Model

## References

• Brading, K. , and Castellani, E. , eds. , 2003. Symmetries in Physics: Philosophical Reflections. Cambridge Uni. Press.
• Rosen, Joe, 1995. Symmetry in Science: An Introduction to the General Theory. Springer-Verlag.
• Van Fraassen, B. C. , 1989. Laws and symmetry. Oxford Uni. Press.