Editing Representation theory of SU(2) (section)

==Most important irreducible representations and their applications==
{{main|Rotation group SO(3)#A note on Lie algebras}}

Representations of SU(2) describe non-relativistic [[Spin (physics)|spin]], due to being a double covering of the [[rotation]] group of [[Euclidean space|Euclidean 3-space]]. [[special relativity|Relativistic]] spin is described by the [[representation theory of SL2(C)|representation theory of SL<sub>2</sub>('''C''')]], a supergroup of SU(2), which in a similar way covers [[Lorentz group|SO<sup>+</sup>(1;3)]], the relativistic version of the rotation group. SU(2) symmetry also supports concepts of [[isobaric spin]]<!-- BTW could this one be U(2)? --> and [[weak isospin]], collectively known as ''isospin''.

The representation with <math>m = 1</math> (i.e., <math>l = 1/2</math> in the physics convention) is the '''2''' representation, the [[fundamental representation]] of SU(2). When an element of SU(2) is written as a [[complex number|complex]] {{math|2 × 2}} [[matrix (mathematics)|matrix]], it is simply a [[matrix multiplication|multiplication]] of [[column vector|column 2-vectors]]. It is known in physics as the [[spin-1/2]] and, historically, as the multiplication of [[quaternion]]s (more precisely, multiplication by a [[unit vector|unit]] quaternion). This representation can also be viewed as a double-valued [[projective representation]] of the rotation group SO(3).

The representation with <math>m = 2</math> (i.e., <math>l = 1</math>) is the '''3''' representation, the [[adjoint representation]]. It describes 3-d [[rotation (mathematics)|rotations]], the standard representation of SO(3), so [[real number]]s are sufficient for it. Physicists use it for the description of [[rest mass|massive]] spin-1 particles, such as [[vector meson]]s, but its importance for spin theory is much higher because it anchors spin states to the [[geometry]] of the physical [[three-dimensional space|3-space]]. This representation emerged simultaneously with the '''2''' when [[William Rowan Hamilton]] introduced [[versor]]s, his term for elements of SU(2). Note that Hamilton did not use standard [[group theory]] terminology since his work preceded Lie group developments.

The <math>m = 3</math> (i.e. <math>l = 3/2</math>) representation is used in [[particle physics]] for certain [[baryon]]s, such as the [[delta baryon|Δ]].