Symmetry in physics Totally Explained

Everything about Symmetry In Physics totally explained

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. A symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that's "preserved" under some change. The transformations may be continuous (such as rotation of a circle) or discrete (for example, 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). Symmetries are frequently amenable to mathematical formulation and can be exploited to simplify many problems.

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. 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. A rotation about any axis 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 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

. Rotating the wire about its own axis doesn't change its position, hence it'll 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

v_1

and the other with speed

v_2

the total kinetic energy of the system (as calculated from an observer at the origin) is

frac

and indicate an invariance property of a system when the coordinates are 'inverted'.

Glide reflection: These are represented by a composition of a translation and a reflection. These symmetries occur in some crystals and in some planar symmetries, known as wallpaper symmetries.

C, P, and T symmetries

The Standard model of particle physics has three related natural near-symmetries. These state that the universe is indistinguishable from one where:

C-symmetry (charge symmetry) - every particle is replaced with its antiparticle.

P-symmetry (parity symmetry) - the universe is reflected as in a mirror.

T-symmetry (time symmetry) - the direction of time is reversed. (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. To properly time-reverse the universe, you'd 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.) 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. 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.

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's another physical symmetry beyond those already developed in the standard model, specifically a symmetry between bosons and fermions. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry hasn't 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

The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists.
Continuous symmetries are specified mathematically by continuous groups (called Lie groups). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group

, 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 (this may be generalised to the Poincaré group).
Discrete symmetries are described by discrete groups. For example, the symmetries of an equilateral triangle are described by the symmetric group

, S_3

.
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 symmetries in the Standard model, used to describe three of the fundamental interactions, 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.)
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).

Conservation laws and symmetry

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 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.

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