User:YohanN7/Finite dimensional representations

Finite dimensional representations

For terminology, conventions and notation, please see the Notation section at the bottom of the article.

History

The development of the finite-dimensional representation theory of the Lorentz group mostly follows that of the subject in general. Lie groups originated with Sophus Lie in 1873. By 1888 the classification of simple Lie algebras was essentially completed by Wilhelm Killing. In 1913 the theorem of highest weight for representations of simple Lie algebras, the path that will be followed here, was completed by Elie Cartan. Richard Brauer was 1935-1938 largely responsible for the development of the connection between spin representations and Clifford algebras. The Lorentz group has also historically received special attention in representation theory, see infinite-dimensional unitary representations#history below, due to its exceptional importance in physics. Hermann Weyl, a mathematician who also made major contributions to the general theory, and the physicist E.P Wigner made substantial contributions over the years. Physicist P.A.M Dirac was perhaps the first to manifestly knit everything together in a practical application of major lasting importance with the Dirac equation in 1928.

The standard representation

The Lie algebra of so(3;1) is in the standard representation given by

${\begin{aligned}M^{23}&=-M^{32}=J_{1}=i{\biggl (}{\begin{smallmatrix}0&0&0&0\\0&0&0&0\\0&0&0&-1\\0&0&1&0\\\end{smallmatrix}}{\biggr )},M^{31}=-M^{13}=J_{2}=i{\biggl (}{\begin{smallmatrix}0&0&0&0\\0&0&0&1\\0&0&0&0\\0&-1&0&0\\\end{smallmatrix}}{\biggr )},M^{12}=-M^{21}=J_{3}=i{\biggl (}{\begin{smallmatrix}0&0&0&0\\0&0&-1&0\\0&1&0&0\\0&0&0&0\\\end{smallmatrix}}{\biggr )},\\M^{01}&=-M^{10}=K_{1}=i{\biggl (}{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0\\\end{smallmatrix}}{\biggr )},M^{02}=-M^{02}=K_{2}=i{\biggl (}{\begin{smallmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0\\\end{smallmatrix}}{\biggr )},M^{03}=-M^{30}=K_{3}=i{\biggl (}{\begin{smallmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0\\\end{smallmatrix}}{\biggr )}.\end{aligned}}$

They satisfy

$[M_{\mu \nu },M_{\rho \sigma }]=i(M_{\rho \nu }\eta _{\sigma \mu }+M_{\mu \rho }\eta _{\nu \sigma }-M_{\sigma \nu }\eta _{\rho \mu }-M_{\mu \sigma }\eta _{\nu \rho }),$

the commutation relations of so(3;1).

If X is a linear combinations of the generators with real coefficients,

$X=\mathbf {\theta \cdot J} +\mathbf {\xi \cdot K} =\theta _{1}J_{1}+\theta _{2}J_{1}2+\theta _{3}J_{4}+\xi _{1}K_{1}+\xi _{2}K_{2}+\xi _{3}K_{3}$

then the matrix exponential of iX,

$\Lambda =e^{iX}\equiv \sum _{n=0}^{\infty }{\frac {(iX)^{n}}{n!}}$

is a Lorentz transformation. In the standard representation, Lorentz transformations act on R⁴ and C⁴ by matrix multiplication,

$x\rightarrow Xx,\quad X\in so(3;1)$

$x\rightarrow \Lambda x,\quad \Lambda \in SO(3;1)^{+},x\in \mathbf {C} ^{4}(\mathbf {R} ^{4}).$

In some representation the is an expression defining the representation like like conjucation (X→AXA^-1 or some other linear operation. It these cases there always is a corresponding matrix G in Env(V) achieving the same thing by matrix multiplication fro the left, X→G'X.

A nontrivial example

(This section uses concepts introduced in later sections.= Let γ^{μ denote the set of four 4-dimensional Gamma matrices, called the Dirac matrices. They constitute the representation space V of a (½,0) ⊕ (0,½) representation. In this representation the elements of so(3;1) act through by matrices σμν defined by}

$\sigma ^{\mu \nu }=-{\frac {i}{4}}[\gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu }],\quad \gamma _{i}={\bigl (}{\begin{smallmatrix}0&\sigma ^{i}\\-\sigma ^{i}&0\\\end{smallmatrix}}{\bigr )},\gamma _{0}={\bigl (}{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}{\bigr )},$

where the σⁱ are the Pauli matrices, according to

$\pi (M^{\mu \nu })(\gamma ^{\rho })\equiv \Sigma ^{\mu \nu }\gamma ^{\rho }\equiv ad_{\sigma ^{\mu \nu }}(\lambda ^{\rho })=[\sigma ^{\mu \nu },\gamma ^{\rho }]=-i\gamma ^{\mu }\eta ^{\nu \rho }+i\gamma ^{\nu }\eta ^{\mu \rho }.$

It is not irreducible. The matrices π(M^μν) can, in this representation, be thought of as 4-dimensional matrices, Σ^μν, acting on the 4-dimensional subspace of M_{_n}(C) spanned by the γ^μ. There is also a representation of so(3;1) acting on the σ´s by

$\pi _{A}(M^{\mu \nu })(\sigma ^{\rho \sigma })\equiv \Sigma _{A}^{\mu \nu }\gamma ^{\rho }\equiv ad_{\sigma ^{\mu \nu }}(\sigma ^{\rho \sigma })=[\sigma ^{\mu \nu },\sigma ^{\rho \sigma }]=\eta ^{\nu \rho }\sigma ^{\mu \sigma }+\eta ^{\mu \rho }\sigma ^{\nu \sigma }-\eta ^{\sigma \mu }\sigma ^{\rho \nu }-\eta ^{\sigma \nu }\gamma ^{\rho \mu },$

as it must be, since in order for π to be a representation of so(3;1), the latter equation necessarily holds. The π_A(M^μν) are in this case 6-dimensional matrices, since the space in M_{_n}(C) spanned by the σ^μν is 6-dimensional.

The γ^μ and the σ^μν are both part of the Clifford algebra, Cl(3;1), generated by the 4-dimensional gamma matrices in 4 spacetime dimensions. The representation, Π of SO(3;1)⁺, corresponding to π is given by exponentiation

$e^{i\omega _{\mu \nu }\Sigma ^{\mu \nu }}\gamma ^{\rho }=e^{iad_{\omega _{\mu \nu }\sigma ^{\mu \nu }}}\gamma ^{\rho }=Ad_{e^{\omega _{\mu \nu }i\sigma ^{\mu \nu }}}\gamma ^{\rho }=e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}\gamma ^{\rho }e^{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}$ ( $=\Lambda _{\tau }^{\rho }\gamma ^{\tau }$ if the ω are infinitesimal),

where ω is antisymmetric in μ,ν. The relation between ad and Ad is a property of exponentiation of matrices. The parenthetical equality is the reason that the γ^μ are sometimes called 4-vectors. Now define a complex vector space U where the γ^μ act by matrix multiplication,

$u^{\beta }=[\gamma ^{\mu }]_{\alpha }^{\beta }u^{\alpha }.$

Define the action of the Lorentz group onUto be

$u\rightarrow \Pi (\Lambda )u$, in components, $u^{\alpha }=[e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}]_{\beta }^{\alpha }u^{\beta }.$

This representation is a a projective representation. The induced action on End U, given by AXA^-1, for the Lorentz group (with X = γ^μ, and A a matrix representation of Λ) is exactly the action found above. This is a bona fide representation of SO(3;1)⁺, i.e., it is not projective.

$\gamma ^{\rho }\rightarrow \Pi (\Lambda )\gamma ^{\rho }\Pi (\Lambda )^{-1}=e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}\gamma ^{\rho }e^{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}\equiv \Theta (\Lambda )\gamma ^{\rho }.$

The matrix Θ(Λ) effecting this transformation is a representative of SL(2;C), the double cover of SO(3;1), and its restriction to SO(3) is a representative of Spin(3), the double cover of SO(3). Both the elements of U and the elements of the Clifford algebra on the form a_μγ^μ are called spinors. By applying exactly the same reasoning to the π<subA> representation of so(3;1), one finds

$\sigma ^{\eta \zeta }\rightarrow \Pi (\Lambda )\sigma ^{\eta \zeta }\Pi (\Lambda )^{-1}=e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}\sigma ^{\eta \zeta }e^{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}$ ( $=\Lambda _{\rho }^{\eta }\Lambda _{\tau }^{\zeta }\sigma ^{\rho \tau }$ if the ω are infinitesimal)

The sigmas aren't called spinors however. The infinitesimal version of the transformation dictates the term antisymmetric tensor.

General properties

Isomorphisms

As a Lie algebra, the complexification of so(3;1), so(3;1)_C is isomorphic to sl(2;C) ⊕ sl(2;C) according to

so(3;1)_C = so(3;1) ⊕ i so(3;1) ≈ su(2)_C ⊕ su(2)_C ≈ sl(2;C) ⊕ sl(2;C) ≈ so(4;C).

The isomorphism so(3;1)_C = so(3;1) ⊕ i so(3;1) is, by definition, the complexification. The next one is shown in the previous section by making a complex change of basis, the one after that is a consequence of the well known su(2)_C ≈ sl(2;C).

The final isomorphism can be made plausible by switching to a new basis in the standard representation C⁴; let e₀ ↦ ie₀ where e₀ is the first basis vector. In this basis the original quadratic form −t² + x² + y² + z² defining O(3;1) on R⁴ becomes t² + x² + y² + z². Its symmetry group with unit determinant is SO(4;C).

Semisimplicity

The Lie algebra, so(3;1), of the Lorentz group is semisimple. All properties that are common to representations of semisimple Lie algebras are thus also properties of representations of so(3;1). An Analogous statement hold for representations of semisimple Lie groups and, in particular, the group SO(3;1). The Lie algebra so(3;1) is also simple. As a consequence, so(3;1) cannot be decomposed into a direct sum of two or more nonzero Lie algebras.

Since so(3;1)_C ≈ sl(2;C) ⊕ sl(2;C), it is not simple, but it is semisimple because sl(2;C) is simple. This decomposition makes it possible to express representations of so(3;1)_C and so(3;1) using known representations of sl(2;C). The representations of sl(2;C) will, in turn, follow from those of su(2) from the well known sl(2;C) ≈ su(2)_C.

The most useful fact from the semisimple representation theory is that a Lie algebra g is semisimple if and only if it has the complete reducibility property. This says that every representation of so(3;1) decomposes as a direct sum of the irreducible (m,n) representations. This statement too applies at the group level.

Building representations

In the other direction, one can from the irreducible representations form other representations by using standard constructions from general representation theory. These constructions include taking the complexification, direct sums, tensor products, and the dual of the representation space, and defining the action of the group or algebra appropriately. These constructs always yield representations from a given representation. Other constructs, like quotients, yield representations under certain hypotheses.

There are also representations that are inherent in the theory of Lie groups and Lie algebras.

• The standard representation of O(3;1) are the 4×4 matrix representations acting on R⁴ or C⁴ by matrix multiplication on column vectors. The matrices of O(3;1) are defined as those that preserve the quadratic form −t² + x² + y² + z² of R⁴. They are unique up to a similarity transformation corresponding to an orthogonal change of basis of R⁴.
• The standard representation of o(3;1) is the set of all matrices X such that the exponential map, given by e^itX, is in (the standard representation of) O(3;1) for all t∈R. A Lie algebra, g, is usually explicitly given by presenting a basis for g as a real vector space and the Lie brackets of the basis elements.
• Any Lie group G acts by conjugation on its Lie algebra, g, by the formula Ad_A(X) = AXA⁻¹. for A∈G and X∈g. This is the adjoint representation. (There is one representation Ad_A for each A.)
• A Lie algebra acts on itself according to ad_X(Y) = [X,Y]. This too is called the adjoint representation.
• The trivial representation simply takes any element of a group to the identity transformation. The corresponding representation for Lie algebras maps all elements the zero transformation. Finding trivial representation spaces given a general representation amounts to finding Lorentz scalars.

Most of the concepts above are used when building the (m,n) representations.

Complexification

If g is a real Lie algebra, then its complexification is g_C = g ⊕ ig. A complex Lie algebra is its own complexification. Real linear representations (π,V) of g are in one-to-one correspondence with complex linear representations (π_C, V_C) of g_C. The action of π_C is given by

$\pi _{\mathbf {C} }(X)v=(\pi (X)+i\pi (X))v=\pi (X)v+i\pi (X)v,\qquad X\in g_{\mathbf {C} },v\in V.$

Direct sums

If (π_U,U) and (π_V,V) are representations of some Lie algebra g, then so is the direct sum (π_W = π_U ⊕ π_V, W = V ⊕ W). The action of π_W on this new space is given by

$\pi _{W}(X)(u,v)=(\pi _{U}(u),\pi _{V}(v)).$

A similar formula applies in the group case.

Tensor products

If G,H are Lie groups, then if Π_U, Π_V are representations of G and H respectively, the tensor product Π_W = Π_U⊗Π_V is a representation of G×G acting on W = U⊗V given by

$\Pi _{W}(g,h)(u\otimes v)=\Pi _{U}(g)\otimes \Pi _{V}(h)(u\otimes v)=\Pi _{U}(g)u\otimes \Pi _{V}(h)v.$

If H = G, then, by restricting the first representation to the diagonal, {(g,g)∈G×G}, Π_W may also be seen as a representation of G acting on U⊗V according to

$\Pi _{W}^{1}(g)(u\otimes v)=\Pi _{U}(g)\otimes \Pi _{V}(g)(u\otimes v)=\Pi _{U}(g)u\otimes \Pi _{V}(g)v.$

If g, h are Lie algebras and π_U, π_V are representations of g and h respectively, then the tensor product π_W = π_U ⊗ π_V, is a representation of g ⊕ h acting on W = U ⊗ V. It is given by

$\pi _{W}(X_{1},X_{2})(u\otimes v)=[\pi _{U}(X_{1})\otimes Id_{V}+Id_{U}\otimes \pi _{V}(X_{2})](u\otimes v)=\pi _{U}(X_{1})u\otimes v+u\otimes \pi _{V}(X_{2})v.$

If g = h, then π¹_W given by

$\pi _{W}^{1}(X)(u\otimes v)=[\pi _{U}(X)\otimes Id_{V}+Id_{U}\otimes \pi _{V}(X)](u\otimes v)=\pi _{U}(X)u\otimes v+u\otimes \pi _{V}(X)v.$

is also a representation of g acting on W = U ⊗ V.

The expressions use the identity (A ⊗ B)(u ⊗ v) = Au ⊗ Bv, which, in a basis for U and V, and hence also for U ⊗ V, defines the Kronecker product A ⊗ B of matrices A and B.

The Lie algebra and Lie group representations are related. If Π is a Lie group representation, then there is a corresponding representation π given by Π_*, the pushforward of Π. Conversely, if π is a Lie algebra representation and if the group whose Lie algebra is g is simply connected, then there is a Lie group representation Π given by the exponential mapping. If G is not simply connected, there may still be a corresponding group representation. In particular, the Lorentz group is not simply connected, so not all so(3;1) representations lift to representations of SO(3;1).

Dual representations

If V is a vector space, then V* is its dual space, the set of linear functionals on V. Let φ ∈ V* and v ∈ V. Any linear map A: V → V induces a dual map A^*: V* → V* given by

$(A^{*}(\phi ))(v)=\phi (A(v)).$

Given a representation, (π,V), there is a dual representation, (π*,V*) on V*. The action of the dual representation π* on V* is given by

$\pi ^{*}(X)(\phi )=-[\pi (X)]^{*}(\phi ).$

The corresponding expression at the level of groups is

$\Pi ^{*}(g)(\phi )=[\Pi (g^{-1})]^{*}(\phi ).$

When a basis for V is given and V* has the dual basis, then the dual map of A, A^* is the matrix transpose A^tr of A. The triple role of "*" should be observed.

Quotient representations

For any linear subspace H ⊂ V and any representation (π, V) of g, if H is invariant under the action of π, then there is a representation on the quotient V/H given by

$\pi _{V/H}(X)[v]=[\pi (X)v],\quad v\in V,X\in g$

where [v] ∈ V/H denotes the equivalence class of v. The same expression applies to group representations.

Restrictions of representations

The restriction of a representation to a subalgebra or a subgroup will always yield a representation in the natural way. In particular, if g_C = g ⊕ ig, then if π_C is a complex linear representation of g_C, then π obtained by restricting π_C to a real subspace of the Lie algebra is a real linear representation of g. That is π(X) = π_C(X + i0).

The restriction of an irrep may or may not be irreducible. This is rather subtle for Lie algebras in terms of terminology. If a real linear representation has no complex nontrivial invariant subspaces, then it's complexification will certainly be irreducible too. The converse is also true. If π_C has no complex invariant subspaces, then if W is a complex invariant subspace for π, then it will be invariant under iπ. It follows that all X,Y ∈ g, π(X) + iπ(Y) ∈ W. There may however be nontrivial real invariant subspaces for π.

Subrepresentations

If V is a representation, then if U ⊂ V is a linear subspace that is stable under the action of a group or Lie algebra representation, the restriction of the domain, π_U = π|_U is a representation on U.

Induced representations

If (π₁,V) is a representation of a Lie algebra g, then there is an associated representation on End(V) given by

$\pi (X)A=[\pi (X),A],\quad A\in End(V),X\in g.$

Likewise, a representation (Π,V) of a group G yields a representation Π on End(V) given by

$\Pi (g)A=\Pi (g)A\Pi (g)^{-1},\quad A\in End(V),g\in G.$

If (Π,V) is a projective representation, then direct calculation shows that the induced representation on End(V) is, in fact, a representation.

The (m,n) representations

The (m,n) representations are in practice obtained in several steps. One may begine with the general form of so(3;1) given by

$[M_{\mu \nu },M_{\rho \sigma }]=i(M_{\rho \nu }\eta _{\sigma \mu }+M_{\mu \rho }\eta _{\nu \sigma }-M_{\sigma \nu }\eta _{\rho \mu }-M_{\mu \sigma }\eta _{\mu \rho }),$

where η is the Lorentz metric in flat spacetime with signature (−1,1,1,1), the M_μν are, for μ,ν ∈ {0,1,2,3} , objects of any kind from some real or complex vector space W endowed with a Lie bracket [·,·], and the quantities η are elements of η (0,1, or −1). The M are antisymmetric in μ and ν, or can be made so by M^{μν → (Mμν - Mνμ)/2} without affecting commutation relations. . Whether W is real or complex vector space, the M span a 6-dimensional real Lie algebra. This is the most compact way of writing down the so(3;1) algebra.

First rename according to

$J_{1}=M^{23},J_{2}=M^{31},J_{3}=J^{12},K_{1}=M^{10},K_{2}=M^{20}$ and $K_{3}=M^{30}$.

By direct computation it is found that

$[J_{i},J_{j}]=i\epsilon _{ijk}J_{k},[J_{i},K_{j}]=i\epsilon _{ijk}K_{k}$ and $[K_{i},K_{j}]=-i\epsilon _{ijk}K_{k}$

for i,j,k ∈ {1,2,3} . By antisymmetry in μ and ν, the renamed quantities span so(3;1).

Then complexify the vector space in which the J_i and K_i reside, W → W_C. The Lie algebra is complexified accordingly, sl(3;1) → sl(3;1)_C

Now define new objects in W_C by

${\textbf {A}}={\tfrac {1}{2}}({\textbf {J}}+i{\textbf {K}}),{\textbf {B}}={\tfrac {1}{2}}({\textbf {J}}-i{\textbf {K}})).$

These objects define two 3-dimensional real subspaces in W_C. They are found to satisfy

$[A_{i},A_{j}]=i\epsilon _{ijk}A_{k},[B_{i},B_{j}]=i\epsilon _{ijk}B_{k}$ and $[A_{i},B_{j}]=0.$

Thus A and B separately satisfy the commutation relations of the real Lie algebra su(2). Hence A ≈ B ≈ su(2). Since [A_i,B_j] = 0, A and B are ideals in the real algebra C generated by A and B, so C ≈ A ⊕ B ≈ su(2) ⊕ su(2). It's worth noting at this point that su(3;1) ≠ su(2) ⊕ su(2), in agreement with that su(3;1) is not semisimple.

Consider Lie algebra representations of su(2) ⊕ su(2), given by σ_m,n = σ_m ⊗ σ_n where σ_i are the irreducible (i + 1)-dimensional representations of su(2). By using the isomorphisms A ≈ B≈ su(2), representations ρ_m,n = ρ_m ⊗ ρ_n of A ⊕ B can be obtained. Explicitly, let (a_i:i = 1,2,3) be a basis for su(2) satisfying the same commutation relations as the A_i, and let h_A:A->su(2);h_A(A_i) = a_i be the isomorphism between A and su(2). Let h_B be the corresponding map for B mapping to the same basis for su(2), but labeled with b´s.

For the complexified Lie algebra one obtains su(3;1)_C ≈ (A ⊕ B)_C ≈ A_C ⊕ B_C ≈ sl(2;C) ⊕ sl(2;C). The elements

${\textbf {J}}=({\textbf {A}}+{\textbf {B}})$ and ${\textbf {K}}={\frac {1}{i}}({\textbf {A}}-{\textbf {B}})$

each span real su(2) subalgebras of (A ⊕ B)_C. The linear representations σ_m,n of su(2) ⊕ su(2) extend uniquely to complex linear representations τ_m,n of sl(2;C) ⊕ sl(2;C) by τ_m,n = τ_m ⊗ τ_m, where τ_i is the complexification of σ_i. Via the established isomorphisms, representations π^C_m,n of so(3;1)_C ≈ (A + B)_C are obtained, given by (ρ_m,n)^C.

Finally, by restriction to the real subspace spanned by the J_i and K_i, a representation π of sl(3;1) is obtained. Somewhat explicitly, the representation is given by

${\begin{aligned}\pi _{m,n}(J_{i})&=\pi _{m,n}^{\mathbf {C} }(J_{i})=\pi _{m,n}^{\mathbf {C} }(A_{i}+B_{i})=\rho _{m,n}^{\mathbf {C} }(A_{i},B_{i})\\&=\rho _{m,n}(A_{i},B_{i})=\sigma _{m,n}(h_{A}(A_{i}),h_{B}(B_{i}))=(\sigma _{m}\otimes \sigma _{n})(a_{i},b_{i}),\\&=\sigma _{m}(a_{i})\otimes 1_{n}+1_{m}\otimes \sigma _{n}(b_{i})\\\pi _{m,n}(K_{i})&={\frac {1}{i}}(\sigma _{m}(a_{i})\otimes 1_{n}-1_{m}\otimes \sigma _{n}(b_{i}))\end{aligned}}$

In the last line, the σ_j are taken in concrete sense of actual matrices. Thus the operation ⊗ should be seen as the Kronecker product of matrices. The matrices σ_m(a_i) can be taken as standard (2j + 1)-dimensional spin matrices J^(m)_i. Componentwise, for -m ≤ a ≤ m, -n ≤ b ≤ n, the equations become

${\begin{aligned}(\pi _{m,n}(J_{i}))_{a'b'ab}&=\delta _{b'b}(J_{i}^{(m)})_{a'a}+\delta _{a'a}(J_{i}^{(n)})_{b'b},\\(\pi _{m,n}(K_{i}))_{a'b'ab}&=i(\delta _{a'a}(J_{i}^{(n)})_{b'b}-\delta _{b'b}(J_{i}^{(m)})_{a'a}),\end{aligned}}$

where

${\begin{aligned}(J_{3}^{(j)})_{a'a}&=a\delta _{a'a},\\(J_{1}^{(j)}\pm iJ_{2}^{(j)})_{a'a}&={\sqrt {(j\mp a)(j\pm a+1)}}\delta _{a',a\pm 1}.\end{aligned}}$

Properties of the (m,n) representations

The (m,n) representations (irreps) constructed above are irreducible. The are the only irreducible representations. This is seen from the way they are constructed by appeal to the uniqueness of the su(2) irreps. They have dimension (2n + 1)(2m + 1). This too follows from properties of su(2).

The associated irreps of the connected component, SO(3;1), of the Lorentz group are, when they exist, never unitary. This follows from the fact that SO(3;1) is a connected, noncompact and simple group. A group with these properties has no nontrivial finite-dimensional unitary irreducible representations. At the level of the Lie algebra, not all representative matrices can be Herimitean.

The non-unitarity of the (m,n) irreps is not a problem in the relativistic quantum theory, since the objects the representations act on are not required to have a Lorentz invariant positive definite norm, as is the case in nonrelativistic quantum mechanics with rotations (SO(3)) and wave functions.

The (m,n) representation, however, is unitary when restricted to the subgroup SO(3), but these representations are not irreducible as representations of SO(3). A Clebsch–Gordan decomposition can be applied showing that an (m,n) representation have SO(3)-invariant subspaces of dimension m+n, m+n−1, ..., |m−n| where each possible dimension occurs exactly once.

The (m,n) representation is the dual of the (n,m) representations. Generally, dual representations may or may not be isomorphic as representations.

Properties of general finite dimensional representations

Since so(3;1) is semisimple, and since the irreducible (m,n) representations are all known, it follows that every finite dimensional representation of the Lorentz group can be expressed as a direct sum of the (m,n). If π is any representation of so(3;1), then

$\pi =\oplus _{m,n\in \mathbb {N} }a_{mn}\pi _{m,n}.$

The (m,n) representations are known explicitly in terms of representative matrices π_m,n(X). The spaces V_m,n on which they act, the representation spaces, can be built up using Clebsch–Gordan decomposition. The building material is V_(½,½) and V_{(½,0) ⊕ (0,½)}. The rules for general reduction of tensor products can be deduced from the corresponding rules for sl(2;1) or, equivalently, those of su(2) or so(3).

In particular, tensor products of the (m,n) representations decompose as direct sums of (p,q) terms where p ≤ m, q ≤ n. For instance, (m,0) ⊗ (0,m) ≈ (m,m), where π has been dropped from the notation since focus is on the representation space.

Application of the rules to (½,½) ⊗ [(½,0) ⊕ (0,½)] yields

$({\tfrac {1}{2}},{\tfrac {1}{2}})\otimes [({\tfrac {1}{2}},0)\oplus (0,{\tfrac {1}{2}})]=(1,{\tfrac {1}{2}})\oplus ({\tfrac {1}{2}},1)\oplus (0,{\tfrac {1}{2}})\oplus ({\tfrac {1}{2}},0).$

This is a 16-dimensional spinor-vector representation. If ψ is a spinor-vector in this representation with components ψ^μ_α in a vector-spinor basis v_μ ⊗ γ^α, then the subspace defined by γ_μψ^μ_α = 0, α ∈ {0,1,2,3} transforms under (½,0) ⊕ (0,½). The 12-dimensional complement of this subspace (by irreducibility) transforms under (1,½) ⊕ (½,1). The equation ψ^μ_αγ_μ = 0 is a simple version of the Rarita–Schwinger equation.

In general, every representation is a direct sum of tensors (including the vector and the scalar irreps) for which m + n an integer, or spinor-tensors, for m + n half an odd integer. General tensors of rank N transform as a tensor product T of N (½,½) representations. Irreducible terms (m,n) with m = N/2, N/2 − 1, ..., and m = N/2, N/2 − 1, ... can be extracted by reduction of T. Every (m,n) representation with m + n an integer are found in this way. The (m,n) representations where m + n is half an odd integer are obtained by forming the tensor product of tensor representations and the (½,½) representation.

Group vs Lie algebra representations

A Lie algebra representation may or may not have a corresponding group representation. The correspondence at the level of compact Lie groups is that there always is a corresponding group representation of the connected component of the group if the group is simply connected.

The Lie algebra and Lie group representations are, when both exist, related. If Π is a Lie group representation, then there is a corresponding representation π given by Π_*, the pushforward of Π. Conversely, if π is a Lie algebra representation and if the group whose Lie algebra is g is simply connected, then there is a Lie group representation Π given by the exponential mapping. If G is not simply connected, there may still be a corresponding group representation.

If the group G corresponding to g is a matrix group (linear group), then the exponential mapping amounts to taking the matrix exponential of the representative elements of the Lie algebra;

$\Pi (e^{i\pi (X)})=e^{i\pi (X)},\quad X\in g.$

A proof that the above relation yields a representation of the group depends on simple connectedness of G and uses the qualitative statement of the Baker-Campbell-Hausdorff formula. In the other direction, given a representation Π of a matrix group, the formula

$\pi (X)={\frac {d}{dt}}\Pi (e^{itX}),\quad X\in g$

evsluated at t = 0 yields a representation of the Lie algebra.

The Lorentz group is not simply connected, and already at the level of the compact doubly connected subgroup SO(3) it is seen that not all (m,n) representations lift to the group. The (m,n) so(3;1) representations have corresponding representations of only if m and n are both integer.

Projective representations

Even if there is no representation of the group gorresponding to a particular representation of the Lie algebra, there may be a projective representation. If D(Λ) denotes the representative of a Lorentz transformation in a projective representation, then

$D(\Lambda _{1}\Lambda _{2})=e^{i\Phi (\Lambda _{1},\Lambda _{2};v)}D(\Lambda _{1})D(\Lambda _{2})v,\qquad v\in V,A_{1},A_{2}\in G.$

The possible dependence of the phase factor Φ on the vector v on which D is acting indicates the presence of central charges in the Lie algebra. This corresponds in quantum mehanics to superselection rules.

Extension to the full Lorentz group

The (m,n) representation can be extended to a (possibly projective) representation of all of O(3;1) if and only if m = n. This follows from considering the adjoint action Ad_P of P∈O(3;1) on so(3;1), where P is the standard representative of space inversion, diag(1,−1,−1,−1), given by

$Ad_{P}(J_{i})=PJ_{i}P^{-1}=J_{i},\qquad Ad_{P}(K_{i})=PK_{i}P^{-1}=-K_{i}.$

If π is any representation of so(3;1) and Π is an associated group representation, then Π acts on the representation space of π by the adjoint action, X→Π(g)XΠ(g)sup>-1 for X∈π(so(3;1). This is equivalent to

$\Pi (P)\pi (B_{i})\Pi (P)^{-1}=\pi (A_{i})$

which cannot hold if A_i and B_i have different dimensions. If m ≠ n then (m,n) ⊕ (n,m) can be extended to an irreducible (possibly projective) representation of O(3;1). It is ,by the above construction, not irreducible as a representation of so(3;1).

Time reversal, T, acts similarly on so(3;1) by

$Ad_{T}(J_{i})=TJ_{i}T^{-1}=-J_{i},\qquad Ad_{P}(K_{i})=TK_{i}T^{-1}=K_{i}$

The Dirac representation, (½,0) ⊕ (0,½), is usually taken to include space and time inversions. Without it, it is not irreducible

Notation

The terminology used is, in the interest of being brief, sometimes abused. This means that a term may be used for objects or concepts that don't correspond exactly to the objects or concepts referred to in the precise definition of the term.

There exists different conventions regarding Lie algebras and the signature of the Lorentz metric. These different conventions have their origin in practical utility. The physicists convention for the Lie algebra stems from the convention that physical quantities (eigenvalues of certain operators on Hilbert space) should take on real values. The mathematical convention would yield purely imaginary values.

The choice of spacetime metric has a less deep meaning, except that a particular signature and choice of basis for the Lie algebra is computationally convenient for the problem at hand.

Conventions

• This article uses the signature (−1,1,1,1) for the Lorentz metric.

• The Lorentz group is O(3;1).
• The subgroup with unit determinant is SO(3;1). This the proper Lorentz group. It excludes (pure) space inversions and time-reversals.
• The connected component of the Lorentz group is SO⁺(3;1) with Lie algebra so(3;1). These transformations are called proper and orthochronous. They preserve the sign of the 0-component of a 4-vector.

• A Lie algebra is closed under [X_i,Y_j] = iC_ij^{kX_k. This differs from the usual definition by a factor of i. Likewise, the exponential map becomes X ↦ exp(itX).}
• The basis elements of a Lie algebra are sometimes called infinitesimal generators of the group, or merely generators.

Terminology

• If (π,V) is a representation, then both π(V) and V are called representations. The vector space V is sometimes called the representation space.
• If V is complex, then π is said to be complex.
• A representation (π,V) is real (complex) linear if π is real (complex) linear. This means that there are complex representations of real Lie algebras.
• A Lie algebra is complex if it equals its complexification.

Clear explicit distinction between a Lie algebra and its complexification, and between real linear and complex linear representations is generally intended throughout the article.