Question

Significance of unit determinant in particle physics

Answer 1

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or. A. . Geometrically, it can be viewed as the scaling factor of the linear transformation described by the matrix.

Answer 2

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1.

(More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.)

The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on [text]{\displaystyle \mathbb {C} ^{n}}\ {C} ^{n}.[a] It is itself a subgroup of the general linear group, {\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n,\mathbb {C} )}{\displaystyle \operatorname {SU} (n)\subset \operatorname {U} (n)\subset \operatorname {GL} (n,\mathbb {C} )}.[/tex]

The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1]

The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.[b] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.