User:Maschen/irrep

"Irrep" redirects to here.

In mathematics, specifically in the representation theory of groups, an irreducible representation or irrep of a group is a representation of a group which cannot be decomposed further into other representations of the group. The decomposition could be into direct sums, or tensor products. Stated in other ways, an irrep of a group is a representation of the group:

• that cannot be brought into a diagonalized block matrix by a similarity transformation,
• which cannot be expressed in terms of a representation of lower dimension,
• which has no nontrivial invariant subspaces.[1]

Group theory is a powerful and natural mathematical language used to describe symmetries in physics and chemistry, as fundamental as spacetime symmetries and particle physics through molecular geometry and condensed matter physics, because symmetries always form mathematical groups.

The importance of irreducible representations in quantum mechanics is that the energy levels of the system are labelled by the irreducible representations of the symmetry group of the system, allowing the selection rules to be determined. [2] Irreducible representations of the Lorentz group in relativistic quantum mechanics are used to derive relativistic wave equations for particles with spin, due to their relation to the spin angular momentum matrices. Relativistic four vectors, tensor fields, and spinor fields transform according to reducible representations of the Lorentz group which are built out of certain irreps.

For simplicity, throughout this article "rep" will always refer to "representation of the group" and "irrep" for "irreducible representation of the group", also "element" means "element of the group", unless explicitly stated otherwise.

History

Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act over a field K of arbitrary characteristic, rather than a vector of real or complex numbers. The structure analogous to an irreducible representation in the resulting theory is a simple module.

Overview

Notation and terminology of group representations

Group elements can be represented by matrices, although the term "represented" has a specific and precise meaning in this context. A representation of a group is a mapping from the group elements to the general linear group of matrices. As notation, let a, b, c... denote elements of a group G with group product signified without any symbol, so ab is the group product of a and b and is also an element of G, and let representations be indicated by D. The representation of a is written

$\displaystyle D(a)={\begin{pmatrix}[D(a)]_{11}&[D(a)]_{12}&\cdots &[D(a)]_{1n}\\[D(a)]_{21}&[D(a)]_{22}&\cdots &[D(a)]_{2n}\\\vdots &\vdots &\ddots &\vdots \\[D(a)]_{n1}&[D(a)]_{n2}&\cdots &[D(a)]_{nn}\\\end{pmatrix}}$

in the same notation for functions as in mathematical analysis and linear algebra. By definition of group representations, the representation of a group product is translated into matrix multiplication of the representations:

${\displaystyle D(ab)=D(a)D(b)}$

If e is the identity element of the group (so that ae = ea = a, etc.), then D(e) is an identity matrix, or identically a block matrix of identity matrices, since we must have

${\displaystyle D(ea)=D(ae)=D(a)D(e)=D(e)D(a)=D(a)}$

and similarly for all other group elements.

Reducible and irreducible representations

A representation is reducible if a similar matrix P can be found for the similarity transformation:

${\displaystyle D(a)\rightarrow P^{-1}D(a)P}$

which diagonalizes every matrix in the representation into the same pattern of diagonal blocks - each of the blocks are representation of the group independent of each other. This means the representation can be decomposed into a direct sum of k matrices:

${\displaystyle D(a)={\begin{pmatrix}D^{(1)}(a)&0&\cdots &0\\0&D^{(2)}(a)&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &D^{(k)}(a)\\\end{pmatrix}}=D^{(1)}(a)\oplus D^{(2)}(a)\oplus \cdots \oplus D^{(k)}(a)}$

so D(a) is reducible, and it is customary to label the decomposed matrices by a superscript in brackets, as in D(n)(a) for n = 1, 2, ..., k.

The dimension of D(a) is the sum of the dimensions of the blocks:

${\displaystyle \mathrm {dim} [D(a)]=\mathrm {dim} [D^{(1)}(a)]+\mathrm {dim} [D^{(2)}(a)]+\ldots +\mathrm {dim} [D^{(k)}(a)]}$

If this is not possible, then the representation is irreducible - an "irrep".

Basis functions interrelated by continuous symmetries

Functions interrelated by a continuous symmetry transform into each other by means of a representation of the transformation. Let f1, f2, ..., fm be m functions, each a function of n variables x1, x2, ..., xn, and let ${\displaystyle {\widehat {\Omega }}}$ be a symmetry operator (say for spatial translations or rotations), parametrized by p parameters λ = (λ1, λ2, ..., λp), so that ${\displaystyle {\widehat {\Omega }}\equiv {\widehat {\Omega }}({\boldsymbol {\lambda }})}$. The operators themselves are elements of the underlying symmetry group (spatial translations or rotations). Collecting the variables into an n-dimensional column vector:

${\displaystyle \mathbf {x} ={\begin{pmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{pmatrix}}}$

and functions into an m-dimensional row vector:

${\displaystyle \mathbf {f} ={\begin{pmatrix}f_{1}&f_{2}&\cdots &f_{m}\end{pmatrix}}}$

and treating ${\displaystyle {\widehat {\Omega }}({\boldsymbol {\lambda }})}$ as an n × n matrix, then

${\displaystyle {\widehat {\Omega }}({\boldsymbol {\lambda }})\mathbf {f} (\mathbf {x} )=\mathbf {f} ({{\widehat {\Omega }}({\boldsymbol {\lambda }})}^{-1}\mathbf {x} )=\mathbf {f} (\mathbf {x} )D(\Omega )}$

where the representation of the transformation, D(Ω), is an m × m matrix. The functions in f are called basis functions.

For a physical example, f = (ψ1 ψ2 ... ψn) may be a collection of wavefunctions labelled by quantum numbers, for some quantum system, which are each functions of the position vector coordinates x = (x, y, z). If one considers what happens when the coordinate system is rotated (a passive transformation), then ${\displaystyle {\widehat {\Omega }}}$ would be the 3d rotation operator, parametrized by some angle turned around some axis.

<<Still to clarify/correct where needed and continue onto irreps>>

Applications in theoretical physics and chemistry

In quantum physics and quantum chemistry, each set of degenerate eigenstates of the Hamiltonian operator makes up a representation of the symmetry group of the Hamiltonian, that barring accidental degeneracies will correspond to an irreducible representation. Identifying the irreducible representations therefore allows one to label the states, predict how they will split under perturbations; and predict non-zero transition elements.

Lorentz group

The irreps of D(K) and D(J), where J is the generator of rotations and K the generator of boosts, can be used to build to spin representations of the Lorentz group, because they are related to the spin matrices of quantum mechanics. This allows them to derive relativistic wave equations.

Real irreducible representations and spin

Defining new generators:

${\displaystyle \mathbf {A} ={\frac {\mathbf {J} +i\mathbf {K} }{2}}\,,\quad \mathbf {B} ={\frac {\mathbf {J} -i\mathbf {K} }{2}}\,,}$

so A and B are simply complex conjugates of each other, it follows they satisfy the symmetrically formed commutators:

${\displaystyle \left[A_{i},A_{j}\right]=\varepsilon _{ijk}A_{k}\,,\quad \left[B_{i},B_{j}\right]=\varepsilon _{ijk}B_{k}\,,\quad \left[A_{i},B_{j}\right]=0\,,}$

and these are essentially the commutators the orbital and spin angular momentum operators satisfy. Therefore A and B form operator algebras analogous to angular momentum; same ladder operators, z-projections, etc., independently of each other as each of their components mutually commute. By the analogy to the spin quantum number, we can introduce positive integers or half integers, a, b, with corresponding sets of values m = a, a − 1, ... −a + 1, −a and n = b, b − 1, ... −b + 1, −b. The matrices satisfying the above commutation relations are the same as for spins a and b:

${\displaystyle \left(A\right)_{m'n',mn}=\delta _{n'n}\left(\mathbf {J} ^{(m)}\right)_{m'm}\,\quad \left(B\right)_{m'n',mn}=\delta _{m'm}\left(\mathbf {J} ^{(n)}\right)_{n'n}}$

where

${\displaystyle \left(J_{z}^{(m)}\right)_{m'm}=m\delta _{m'm}\,\quad \left(J_{x}^{(m)}\pm iJ_{y}^{(m)}\right)_{m'm}=m\delta _{a',a\pm 1}{\sqrt {(a\mp m)(a\pm m+1)}}}$

and similarly for n, in which J(m) is a (2m + 1)×(2m + 1) square matrix and J(n) a (2n + 1)×(2n + 1) square matrix. The integers or half-integers m and n numerate all the irreducible representations by, in two equivalent notations used by authors: D(m, n) ≡ (m, n), which are each [(2m + 1)(2n + 1)]×[(2m + 1)(2n + 1)] square matrices.

Applying this to particles with spin s;

• left-handed (2s + 1)-component spinors transform under the real irreps D(s, 0),
• right-handed (2s + 1)-component spinors transform under the real irreps D(0, s),
• taking direct sums symbolized by (see direct sum of matrices for the simpler matrix concept), one obtains the representations under which 2(2s + 1)-component spinors transform: D(m, n)D(n, m) where m + n = s. These are also real irreps, but as shown above, they split into complex conjugates.

In these cases the D refers to any of D(J), D(K), or a full Lorentz transformation D(Λ).