User:YohanN7/Representation theory of the Lorentz group

Properties of the finite dimensional representations

General properties

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

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

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

the commutation relations of so(3;1).

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

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

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→GX.

Dirac spinors

Let γ^μ denote the set of four 4-dimensional Gamma matrices, called the Dirac matrices. They will turn out to constitute a basis of a representation space V of a (½,½) representation. In this 4-vector representation, the elements of so(3;1) are defined to act by matrices σ^μν given by

$\sigma ^{\mu \nu }=-{\frac {i}{4}}[\gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu }],\quad \{\gamma ^{\mu },\gamma ^{\nu }\}=2\eta ^{\mu \nu }I_{4}$ (C1)

where I₄ is a 4×4 unit matrix, and η^μν is the spacetime metric according to

$\pi (M^{\mu \nu })(\gamma ^{\rho })\equiv \Sigma ^{\mu \nu }\gamma ^{\rho }\equiv ad_{\sigma ^{\mu \nu }}(\gamma ^{\rho })=[\sigma ^{\mu \nu },\gamma ^{\rho }]=-i\gamma ^{\mu }\eta ^{\nu \rho }+i\gamma ^{\nu }\eta ^{\mu \rho }.$ (C2)

In (C2), the far right equality follows from property (?) of the Clifford algebra. The second to last equality is (by definition) the mapping ad_σ^μν on a space of matrices. The other identities serve as definitions. The elements of Σ can be read off by using the far right side. The matrices π(M^μν) can, under this mapping, either be thought of as 4-dimensional matrices, Σ^μν, acting in the passive sense by matrix multiplication on the 4-dimensional subspace of M_n(C) spanned by the γ^μ, or they can be thought of as the σ^μν acting by commutation on the γ^μ.

The question arises whether the map π:so(3;1)→gl(V); M^μν → Σ^μν defines a representation. The map is linear, but explicit calculation using (??) shows that

$\pi ([M^{\mu \nu }),M^{\rho \sigma }])=-[\pi (M^{\mu \nu }),\pi (M^{\rho \sigma })].$ (C3)

so that the set Σ^μν does not constitute a representation of so(3;1). There is however a right action by π on the γ_μ yielding a representation.

Lie algebra embedding of so(3;1) in the Clifford algebra

Now define an action of so(3;1) on the σ^μν, and the linear space they span, given by

$\pi _{A}(M^{\mu \nu })(\sigma ^{\rho \sigma })\equiv \Sigma _{A}^{\mu \nu }\sigma ^{\rho \sigma }\equiv ad_{\sigma ^{\mu \nu }}(\sigma ^{\rho \sigma })=[\sigma ^{\mu \nu },\sigma ^{\rho \sigma }]=i(\eta ^{\sigma \mu }\sigma ^{\rho \nu }+\eta ^{\sigma \nu }\sigma ^{\rho \mu }-\eta ^{\mu \rho }\sigma ^{\nu \sigma }-\eta ^{\nu \rho }\sigma ^{\mu \sigma }),$. (C4)

The last equality in C4, which follow from the property ? of the gamma matrices, shows that the σ^μν constitute a representation of so(3;1). The Σ_A^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 elements of the Clifford algebra, Cl(3;1), generated by the 4-dimensional gamma matrices in 4 spacetime dimensions. The Lie algebra of so(3;1) is embedded in Cl(3;1) by π_A as the real subspace of M_n(C) spanned by the σ^μν.

Spinors introduced

Now introduce any 4-dimensional complex vector space U where the γ^μ act by matrix multiplication (U = C⁴ will do nicely). Let Λ = e^ω_μνMμν be a Lorentz transformation and define the action of the Lorentz group on U to be

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

Since the σ^μν constitute a representation of , earlier conclusions guarantee that the induced map

$\Pi :SO(3;1)^{+}\rightarrow End(U);\Lambda \rightarrow e^{i\pi _{A}(X)},e^{iX}=\Lambda ,X\in {\mathfrak {so}}(3;1)$ (C5)

either is a representation or a projective representation of SO(3;1)⁺.

The 4-Vector representation of SO(3;1)⁺ in the Clifforg algebra introduced

Earlier results also show that the induced action on End U, given explicitly by

$e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}\Gamma e^{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}=e^{ad_{i\sigma ^{mu\nu }}}(\Gamma ),\Gamma \in End(U)$ (C6)

is a representation of SO(3;1)⁺ on span{γ^μ}. This is a bona fide representation of SO(3;1)⁺, i.e., it is not projective. But the γ^μ form part of the basis for End(U). Therefore, the corresponding map for the γ^μ is

$\gamma ^{\rho }\rightarrow \Pi (\Lambda )\gamma ^{\rho }\Pi (\Lambda )^{-1}=e^{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}\gamma ^{\rho }e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}{\overset {?}{=}}\Lambda _{\mu }^{\rho }\gamma ^{\mu }=[e^{i\omega _{\mu \nu }M^{\mu \nu }}]_{\sigma }^{\rho }\gamma ^{\sigma }$ (C7)

Claim: The space span{γ^μ} is a 4-vector representation of SO(3;1)⁺. This holds if the equality with a question mark in C7 holds.

Proof

In (C7) it is asserted, as a guess, that the representation on End(U) of SO(3;1)⁺ reduces to the 4-vector representation on the space spanned by the γ^μ. Now use the relationship between ad and Ad and assume that ω_μν are small.

$e^{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}\gamma ^{\rho }e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}{\overset {?}{=}}[e^{i\omega _{\mu \nu }M^{\mu \nu }}]_{\sigma }^{\rho }\gamma ^{\sigma }{\overset {Ad/ad}{\Leftrightarrow }}e^{ad_{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}}(\gamma ^{\rho }){\overset {?}{=}}[e^{i\omega _{\mu \nu }M^{\mu \nu }}]_{\sigma }^{\rho }\gamma ^{\sigma }{\overset {\omega ^{\mu \nu }small}{\Leftrightarrow }}$

$ad_{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}(\gamma ^{\rho }){\overset {?}{=}}[i\omega _{\mu \nu }M^{\mu \nu }]_{\sigma }^{\rho }\gamma ^{\sigma }{\overset {def}{\Leftrightarrow }}[-i\omega _{\mu \nu }\sigma ^{\mu \nu },\gamma ^{\rho }]{\overset {?}{=}}[i\omega _{\mu \nu }M^{\mu \nu }]_{\sigma }^{\rho }\gamma ^{\sigma }{\overset {linearity}{\Leftrightarrow }}$

$-[\sigma ^{\mu \nu },\gamma ^{\rho }]{\overset {?}{=}}[M^{\mu \nu }]_{\sigma }^{\rho }\gamma ^{\sigma }{\overset {(C2)}{\Leftrightarrow }}-i\gamma ^{\mu }\eta ^{\nu \rho }+i\gamma ^{\nu }\eta ^{\mu \rho }=[M^{\mu \nu }]_{\sigma }^{\rho }\gamma ^{\sigma }$

By inspection of ???,

$(M^{\mu \nu })_{\sigma }^{\rho }\gamma ^{\sigma }=i(\eta ^{\mu \rho }\delta _{\sigma }^{\nu }-\eta ^{\nu \rho }\delta _{\sigma }^{\mu })\gamma ^{\sigma }=i\eta ^{\mu \rho }\gamma ^{\nu }-i\eta ^{\nu \rho }\gamma ^{\mu },$

so the assertion is proved for ω^μν small.

The tensor representation of SO(3;1)+ in the Clifford algebra introduced

Let S = e^{iω_μνMμν} and consider how σ^μν transform under the induced action of S.

$S^{-1}\sigma ^{\mu \nu }S=S^{-1}[\gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu }]S={\frac {-i}{4}}S^{-1}[\gamma ^{\mu }SS^{-1}\gamma ^{\nu }-\gamma ^{\nu }SS^{-1}\gamma ^{\mu }]S=$

${\frac {-i}{4}}(\Lambda _{\mu }^{\rho }\gamma ^{\mu }\Lambda _{\nu }^{\sigma }\gamma ^{\nu }-\Lambda _{\nu }^{\sigma }\gamma ^{\nu }\Lambda _{\mu }^{\rho }\gamma ^{\mu })={\frac {-i}{4}}\Lambda _{\mu }^{\rho }\Lambda _{\nu }^{\sigma }(\gamma ^{\mu }\gamma ^{\nu }-\gamma ^{\nu }\gamma ^{\mu })=\Lambda _{\mu }^{\rho }\Lambda _{\nu }^{\sigma }\sigma ^{\mu \nu },$

where the known transformation rule of the γ^μ has been used. Thus the 6-dimensinal space span{σ^μν} is a tensor representation.

Stuff

Questionable

The complexification of this space is, as in then section on the (m,n) representations, isomorphic to sl(2,C) ⊕ sl(2,C) ≈ A_C ⊕ B_C. Now constrain attention to either of of A or B. The representation, Π of SO(3;1)⁺, corresponding to π_A is given by matrix exponentiation,

$e^{i\omega _{\mu \nu }\Sigma ^{\mu \nu }}\gamma ^{\rho }=e^{iad_{\omega _{\mu \nu }\sigma ^{\mu \nu }}}\gamma ^{\rho }=Ad_{e^{\omega _{\mu \nu }i\sigma ^{\mu \nu }}}\gamma ^{\rho }=e^{i\omega _{\mu \nu }\sigma ^{\mu \nu }}\gamma ^{\rho }e^{-i\omega _{\mu \nu }\sigma ^{\mu \nu }}\equiv \Theta (\Lambda )\gamma ^{\rho }$ ( $=\Lambda _{\tau }^{\rho }\gamma ^{\tau }$ 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. Since objects entirely in either A or B are considered, the representation thus obtained, is a representation of SL(2;C).

Superflous

This does indeed define a representation, at least for ω_μν small. This is rigorously seen by noting that the commutation relations among the σ^μν are the same as among the M^μν, and making use of the Baker-Campbell-Hausdorff formula, that says that group multiplication near the identity can be expressed in terms of commutators. For ω_μν not sufficiently small, one can still attempt to define the representation along a path from the origin in parameter space, and employing a trick using compactness of the path, again using the same formula to express the resulting matrix as a (finite) product of N factors, each of which use the original formula.

The outlined procedure would yield a representation if SO(3;1) was simply connected. Now, since SO(3;1) is doubly connected, the matrix depends on the homotopy class of the chosen path so that for two Lorentz transformations Λ₁,Λ₂ D(Λ₁)D(Λ₂) = +/-1*D(Λ₁Λ₂) depending on the path classes chosen. This corresponds to a representation "up to a phase".

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_A 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.

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 mapping, 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.

Complex conjugate representations

If (π, V) is a representation of g, then

$({\overline {\pi }},V)$

given by

${\overline {\pi }}(X)={\overline {\pi (X)}},X\in g$

Adjoint representations

If (π, V) is a representation of g, then

$({\overline {\pi ^{*}}},V^{*})$

is the adjoint representation.

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.

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 → (M^μν - M^νμ)/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

${\begin{aligned}\pi _{m,n}(J_{i})&=\pi _{m,n}^{\mathbf {C} }(J_{i})=\pi _{m,n}^{\mathbf {C} }(A_{i}+B_{i})=\rho _{m,n}^{\mathbf {C} }(A_{i},B_{i})\\&=\rho _{m,n}(A_{i},B_{i})=\sigma _{m,n}(h_{A}(A_{i}),h_{B}(B_{i}))=(\sigma _{m}\otimes \sigma _{n})(a_{i},b_{i}),\\&=\sigma _{m}(a_{i})\otimes 1_{2n+1}+1_{2m+1}\otimes \sigma _{n}(b_{i})\\\pi _{m,n}(K_{i})&={\frac {1}{i}}(\sigma _{m}(a_{i})\otimes 1_{2n+1}-1_{2m+1}\otimes \sigma _{n}(b_{i}))\end{aligned}}$

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.

Explicit expressions

The matrices σ_m(a_i) can be taken as standard (2j + 1)-dimensional spin matrices J^(m)_i,

${\begin{aligned}\pi _{m,n}(J_{i})&=J_{i}^{(m)}\otimes 1_{(2n+1)}+1_{(2m+1)}\otimes J_{i}^{(n)}\\\pi _{m,n}(K_{i})&={\frac {1}{i}}(J_{i}^{(m)}\otimes 1_{(2n+1)}-1_{(2m+1)}\otimes J_{i}^{(n)}).\end{aligned}}$

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

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

Weyl spinors and bispinors

By taking , in turn, m = ⁠1/2⁠, n = 0 and m = 0, n = ⁠1/2⁠ and by setting

$J_{i}^{({\frac {1}{2}})}={\frac {1}{2}}\sigma _{i}$ (W1)

in the general expression (G1), and by using the trivial relations 1₁ = 1 and J⁽⁰⁾ = 0, one obtains

:${\begin{aligned}\pi _{({\frac {1}{2}},0)}(J_{i})&={\frac {1}{2}}(\sigma _{i}\otimes 1_{(1)}+1_{(2)}\otimes J_{i}^{(0)})={\frac {1}{2}}\sigma _{i}\quad \pi _{({\frac {1}{2}},0)}(K_{i})={\frac {i}{2}}(1_{(2)}\otimes J_{i}^{(0)}-\sigma _{i}\otimes 1_{(1)})=-{\frac {i}{2}}\sigma _{i},\\\pi _{(0,{\frac {1}{2}})}(J_{i})&={\frac {1}{2}}(J_{i}^{(0)}\otimes 1_{(2)}+1_{(1)}\otimes \sigma _{i})={\frac {1}{2}}\sigma _{i}\quad \pi _{(0,{\frac {1}{2}})}(K_{i})={\frac {i}{2}}(1_{(1)}\otimes \sigma _{i}-J_{i}^{(0)}\otimes 1_{(2)})=+{\frac {i}{2}}\sigma _{i}.\end{aligned}}$ (W2)

These are the left-handed and right-handed Weyl spinor representations. They act on 2-dimensional complex vector spaces (with a choice of basis) V_L and V_R, whose elements are left- and right-handed Weyl spinors, by matrix multiplication.

Given (π(⁠1/2⁠,0),V_L) and (π(0,⁠1/2⁠),V_R) one may form their direct sum as representations,

:${\begin{aligned}\pi _{({\frac {1}{2}},0)\oplus (0,{\frac {1}{2}})}(J_{i})&={\frac {1}{2}}{\biggl (}{\begin{matrix}\sigma _{i}&0\\0&\sigma _{i}\\\end{matrix}}{\biggr )},\\\pi _{({\frac {1}{2}},0)\oplus (0,{\frac {1}{2}})}(K_{i})&={\frac {i}{2}}{\biggl (}{\begin{matrix}\sigma _{i}&0\\0&-\sigma _{i}\\\end{matrix}}{\biggr )}\\\end{aligned}}.$ (W3)

This is, up to a similarity transformation, the Dirac spinor representation of so(3,1). It acts on the 4-component elements (Ψ_L, Ψ_R) of (V_L⊕V_R), called bispinors, by matrix multiplication. The representation may be obtained in a more general and basis independent way using Clifford algebras.

These expressions for bispinors and Weyl spinors all extend by linearity of Lie algebras and representations to all of so(3,1). Expressions for the group representations are obtained by exponentiation.

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;

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

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.

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.

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)^-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

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.

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.

The Rarita-Schwinger field

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 quantities defined by γ_μψ^μ_α, α ∈ {0,1,2,3} transform under the (½,0) ⊕ (0,½) representation.

The equation

$\gamma _{\mu }\Psi _{\alpha }^{\mu }=0$

then isolates the Rarita–Schwinger field. This 12-dimensional subspace, by irreducibility, transforms under the (1,½) ⊕ (½,1) representation. Further breaking down the representation, when viewed as a representation of SO(3), there are two spin 3/2 terms, and two spin 1/2 terms,

$(1,{\tfrac {1}{2}})\oplus ({\tfrac {1}{2}},1)=[{\tfrac {3}{2}}]\oplus [{\tfrac {1}{2}}]\oplus [{\tfrac {1}{2}}]\oplus [{\tfrac {3}{2}}]\qquad [6+6=4+2+2+4],$

where the l h s refers to so(3;1) representations and the r h s to so(3) representation spaces and the far right indicates the dimensionality of each term.

Now impose the Dirac equation

$[\gamma ^{\nu }\partial _{\nu }+m]\Psi _{\alpha }^{\mu }=0$

on each spinor in the vector. This removes the doubling so that 3/2 ⊕ 1/2 remains as far as spin content goes. Finally, demand that

$\partial _{\mu }\Psi _{\alpha }^{\mu }=0.$

With these conditions, the field describes a single particle of spin 3/2. Note that the three equations each has a spinor index, meaning that there are altogether 3*4 = 12 linear conditions leaving 4 degrees of freedom for the spin 3/2 particle. This fits with the fact that the spin z-component should take on values -3/2, -1/2, 1/2, 3/2. Antiparticles are not accommodated though.

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 mapping becomes X ↦ exp(itX).
• The basis elements of a Lie algebra are sometimes called infinitesimal generators of the group, or merely generators.

Terminology

• A Lie algebra is complex if it equals its complexification.
• 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 a complex representation.
• 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.
• Representations of a complex Lie algebra are of course always real linear, but need, in fact, not be complex linear. They may be conjugate linear.
• If π (Π) is a complex representation of a Lie algebra (Lie group), then π (Π) is irreducible if and only if there are no nontrivial complex subspaces.

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

Consequences of Lorentz invariance

The decomposition of the representations under rotations will, when combined with other requirements in applications than well behaved Lorentz transformation properties, lead to restrictions on which vectors in a representation can actually represent states and operators. For instance, a physical elementary particle should have well defined spin that does not change under Lorentz transformations. If the number of independent components of the quantities is less than the dimension (m + 1)(n + 1) of the irrep, then constraints must be imposed on states and the operators operating on the states.

Further constraints may be brought in by demanding a prescribed behavior under parity transformations, i.e. invariance under space inversion represented by the matrix in the full Lorentz group having (1,−1,−1,−1) on the diagonal and 0 elsewhere. Likewise for time reversal transformations represented by (−1,1,1,1). In the physics terminology, a vector quantity in a representation that transforms into minus itself under the parity transormation is a called a pseudo-vector. If it goes into itself it is just a vector. Analogous terminology exists for scalars quantities and tensor quantities. Theories with pseudo-type objects are (perhaps confusingly) also considered invariant in a certain sense. Parity is said to be conserved. An example of a theory lacking parity invariance is the weak interaction. Similar remarks apply to time reversal invariance and the combination of the two inversions, (−1,−1,−1,−1).

Quantum Mechanics

According to standard quantum mechanical rules, a particle with spin j will need a (2j + 1)-dimensional space so that its spin z-component can take on the values j, j−1, ..., −j. A particle with spin j that transforms under the (m,n) representation must therefore be represented by a state vector that remains in one of the rotationally invariant subspaces.

These subspaces do not mix under rotations but they do mix under boosts. An example is given by the vector representation (½,½), which splits into spin j = ½ − ½ = 0 (1-dimensional, e.g. the time component of the electromagnetic vector potential A) and spin j = ½ + ½ = 1 (3-dimensional, e.g. space components A of A) representations. These subspaces don't mix under rotations.

In the application of the theory to quantum mechanics, there are frequently symmetries due to exchange symmetry of identical particles. These result in equivalence relations on the vector space of quantum states. The corresponding quotient space has a natural vector space structure. Any representation of a group or Lie algebra on the original space, for which the kernel of the quotient map is a stable subspace, will descend to the quotient space.

For example, if S is a space of single-particle states, then V = S ⊗ S is a space of 2-particle states. If the particles are bosons, i.e. if u ⊗ v ≈ u ⊗ v for all u,v ∈ S, then the linear subspace H ⊂ V spanned by expressions of the type c(u⊗v − u⊗v) can be seen as physically equivalent to the zero state (the null vector in V). Any tensor product, π_S1 ⊗ π_S2, of representations of S will respect the induced equivalence relation and pass to the quotient space. This exchange symmetry can be described by the permutation group S₂. It acts by permutation of the tensor factors. Equivalently, it acts by permuting indices on tensor components. More complicated exchange symmetries are described by S_n and its subgroups and representations.

Quantum field theory

A few physically reasonable assumptions will have far reaching consequences when combined with Lorentz invariance within quantum field theory (QFT). In ths section, a few basic assumtions of QFT are explicitly outlined.

The Hilbert space

The space of physical states in QFT is an infinite-dimensional Hilbert space that is built up from single-particle states using tensor products and direct sums. By the usual rules of tensor products, a basis for the one-particle states will yield a basis for any tensor product of 1-particle Hilbert spaces. A similar comment applies to taking direct sums. Every state in the Hilbert space is a superposition of multi-particle states. See the article on Fock space for details.

A typical element of Hilbert space will look like

$A|\alpha _{1}\rangle \otimes |\beta _{1}\rangle \otimes \cdots \otimes |\gamma _{1}\rangle \oplus \cdots \oplus B|\eta _{2}\rangle \otimes |\zeta _{2}\rangle \otimes \cdots \otimes |\theta _{n}\rangle$,

where α_i, β_i, etc. are complete sets of quantum numbers, |α_i〉, |β_i〉, etc. are single particle states, and A,B are constants. Not all of these states are particularly meaningful. The meaningful states will usually exhibit certain exchange symmetries and are subject to normalization in most computations. More general states are given by integrals, most commonly over momentum of multi-particle states with definite momenta in each factor. An example of this type is given in the following sections.

Linear operators Hilbert space

The construction allows a particularly useful basis for the set of linear operators on the space. The creation and annihilation operators are specified by defining their action on the multi-particle states. As their names indicate, they take n-particle states to (n + 1)-particle states and (n − 1)-particle states respectively. It might be noted that the existence of these operators has nothing to do with whether particles can actually be created or destroyed. The effect of the creation operator on a multi-particle state is defined by

$a(q)^{\dagger }|q_{1}\rangle \otimes |q_{2}\rangle \otimes \cdots \otimes |q_{n}\rangle =|q\rangle \otimes |q_{1}\rangle \otimes |q_{2}\rangle \otimes \cdots \otimes |q_{n}\rangle .$

In particlular, if |VAC> denotes the vacuum, then

$a^{\dagger }(q_{1})a^{\dagger }(q_{2})\cdots a^{\dagger }(q_{n})|VAC\rangle =|q_{1}\rangle \otimes |q_{2}\rangle \otimes \cdots \otimes |q_{n}\rangle .$

Here q denotes the complete set of quantum observables {p,σ,n} where n is the particle type, p 3-momentum, and σ is the spin z-component. If there are more discrete quantum numbers, they are assumed to be included in the σ-label.

A more intricate state is given by a possibly bound particle antiparticle pair, e.g. positronium, given by

$\Phi =\sum _{\sigma \sigma ^{\prime }}\int \int \chi ({\textbf {p}},\sigma ,{\textbf {p}}^{\prime },\sigma ^{\prime })a^{\dagger }({\textbf {p}},\sigma )a^{c\dagger }({\textbf {p}}^{\prime },\sigma ^{\prime })|VAC\rangle d^{3}{\textbf {p}}d^{3}{\textbf {p}}^{\prime }$

where a^† creates a particle and a^†c its antiparticle, and χ is the wave function.

The annihilation operator is defined to be the adjoint of a,

$a(q)=(a(q)^{\dagger })^{\dagger }.$

Its effect on an n-particle state is slightly more complicated due to the possible exchange symmetries described below. It is in any case a linear combination (n terms) of (n − 1)-particle state.

The creation and annihilation operators usually obey relations among themselves. This is typically expressed by commutator or anticommutator relations between them. Physically, these relations origin in various exchange symmetries between states. On Hilbert space they induce equivalence relations resulting in subspaces representing the same physical state. The resulting quotient space represents, up to normalization, the unique physical state. One consequence of this is that the effect of the creation operator a^†(q) of a fermionic particle on a state-vector |α〉 where the state q is occupied is to destroy the state, a^†(q)|α〉 = 0.

Any linear operator on Hilbert space can be expressed in terms of creation and annihilation operators. The expression is a polynomial the a, a^† with momentum-dependent coefficients integrated over all momenta.

As a consequence of the (anti-) commutation relations for bosonic and fermionic fields, the Hamiltonian takes the simple form

$H=\int a(q)^{\dagger }a(q)E(q)dq,E(\mathbf {p} ,\sigma ,n)={\sqrt {\mathbf {p} ^{2}+m_{n}^{2}}}$

Here, dq is a shorthand for summing over particle types and discrete labels, and integrating over the continuous labels (momenta).

Transformation of single-particle states

The single particle states are assumed to transform under some, not necessarily irreducible, representation of the Lorentz group. To say this again, a state representing a physical free particle is assumed to have definite Lorentz transformation properties.

Free one-particle states can be characterized by a set of labels {p, σ, ...} where p is linear momentum, σ is the spin z-component or helicity for massless particles, and the ellipsis denote other discrete labels. Under a Lorentz transformation of the space–time variables (t,x,y,z) ↦ (t′,x′,y′,z′) a one particle state |p,σ,...〉 vector (bra-ket notation) will be affected by a unitary or antiunitary transformation |p,σ,...〉 ↦ |p′,σ_−j′,...〉 + |p′,σ_−j+1′,...〉 ... + |p′,σ_j′,...〉 of Hilbert space. Wigner's theorem asserts and proves the existence of such a transformation.

With the choice of parameters as above, p transforms under the 4-vector representation (½,½). Thus for a Lorentz transformation Λ in the standard 4-vector representation (½,½), p′ = Λp (matrix multiplication). The σ-label will transform under some finite-dimensional representation. Considered as a column vector σ transforms as σ = C(Λ,p)σ, where C is a matrix. The complete expression for a free massive single-particle state reads

$U(\Lambda )|p,\sigma \rangle ={\sqrt {\frac {(\Lambda p)^{0}}{p^{0}}}}\sum _{\sigma ^{\prime }}D_{\sigma ^{\prime }\sigma }^{(j)}(W(\Lambda ,p))|\Lambda p,\sigma ^{\prime }\rangle$,

where W(Λ,p) ⊂ SO(3) is the Wigner rotation corresponding to Λ and p. The Wigner rotation is a consistently chosen rotation for a Lorentz transormation taking a massive particle at rest to momentum p. The matrix D is the (2j + 1)-dimensional representation of the rotation group SO(3).

In a (only slightly) less abstract setting, the ket |p,σ〉 may be represented by functions of space-time (with p as a parameter) as entries in a (2j + 1)-dimensional column vector. In this case the functions will be e^ipx. Other sets of parameters are also possible. One can also, for instance, use the set {p_r,j(j+1),σ} where p_r is a continuous index representing "radial momentum", and j(j + 1) is total angular momentum. In this case, the corresponding functions are built up from spherical harmonics and spherical Bessel functions. These infinite sets of functions must transform among themselves under the infinite-dimensional representations of the Lorentz group.

The set {e^ipx} does constitute representation of the Lorentz group using the rule D(Λ)e^ip·x ↦ e^{iΛ−1p·x′} where D(Λ) is an infinite-dimensional representation on function space of Λ taking x to x′. It is not irreducible however.

Transformation of multi-particle states

The transformation properties of multi-particle states follow from the properties of the single-particle states under formation of direct sums and tensor products of representations. The properties of more complicated states (e.g. coherent states) follow by linearity.

Transformation of linear operators

The transformation properties the creation and annihilation operators follow too using the representations induced on End(H) and hence the transformation properties of all operators once they are expressed in terms of creation and annihilation operators. The transformation rule for the creation operator is

$U(\Lambda )a^{\dagger }(p,\sigma ,n)U^{-1}(\Lambda )=\sum _{\sigma ^{\prime }}D_{\sigma ^{\prime }\sigma }(W(\Lambda ,p))a^{\dagger }(p_{\Lambda },\sigma ^{\prime },n).$

The behavior of creation and annihilation operators under Lorentz transformations restricts the form both of the free quantum fields and their interactions. A few consequences for free fields will be outlined below.

The S-matrix

The S-matrix is unitary and assumed to be Lorentz invariant. The first condition follows from its (rigorous) definition. It is a "matrix" connecting two complete sets of basis vectors for Hilbert space, that of the "in states" and that of the "out states".

The unitarity simply says that probability amplitudes S_βα = 〈β|α〉 for processes α → β are the same as those for〈U(Λ)β|U(Λ)α〉. The U(Λ) are the unitary operators on Hilbert space corresponding to the Lorentz transformation Λ. When this is written out explicitly (observing that it holds for all in- and out-states) one obtains a definition of Lorentz invariance of the S-matrix. The precise equation expressing Lorentz invariance of the S-Matrix is rather involved.. In principle, this relation can be expressed in terms of one-particle states and creation and annahilation operators, and their respective known Lorentz transformation properties.

The S-matrix will be Lorentz invariant if the interaction V can be written as

$V=\int {\mathcal {H}}(\mathbf {x} ,t)d^{3}x,$

and the Hamiltonian density transforms as

$U_{0}(\Lambda ,a){\mathcal {H}}(x)U_{0}^{-1}(\Lambda ,a)={\mathcal {H}}(\Lambda x+a),$

and, in addition, the causality condition below is satisfied. The Hamiltonian density is in general a polynomial (with constant coefficients) in the creation and annihilation fields.

Quantum fields

Quantum fields are expressed as linear combinations,

$\psi =\kappa \psi ^{+}+\lambda \psi ^{-}$

of annihilation fields and creation fields,

$\psi _{l}^{+}=\sum \int u_{l}(x;p,\sigma ,n)a(p,\sigma ,n)d^{3}p,\qquad \psi _{l}^{-}=\sum \int v_{l}(x;p,\sigma ,n)a^{\dagger }(p,\sigma ,n)d^{3}p.$

Here, the a* is the creation operator, tacking on a single particle of type n with momentum p and spin z-component σ to any state (ignoring exchange symmetries). The annihilation operator a* is its adjoint. The index l runs over all considered particle types and also over all irreducible representations as well as components of these representations.

The requirement of Lorentz invariance of the S-matrix, when applied to the fields, using known properties of the creation and annihilation operators, leads to the equations

$\sum _{\bar {\sigma }}u_{\bar {l}}(\Lambda x+b;{\textbf {p}}_{\Lambda },{\bar {\sigma }},n)D_{{\bar {\sigma }}\sigma }^{(j_{n})}(W(\Lambda ,p))={\sqrt {\frac {p^{0}}{(\Lambda p)^{0}}}}\sum _{l}D_{{\bar {l}}l}(\Lambda )e^{(i(\Lambda p)\cdot b)}u_{l}(x;{\textbf {p}},\sigma ,n),$

$\sum _{\bar {\sigma }}v_{\bar {l}}(\Lambda x+b;{\textbf {p}}_{\Lambda },{\bar {\sigma }},n)D_{{\bar {\sigma }}\sigma }^{(j_{n})*}(W(\Lambda ,p))={\sqrt {\frac {p^{0}}{(\Lambda p)^{0}}}}\sum _{l}D_{{\bar {l}}l}(\Lambda )e^{(-i(\Lambda p)\cdot b)}v_{l}(x;{\textbf {p}},\sigma ,n).$

The u and v are referred to as coefficient functions. In the sequel it will be seen that these functions, and hence the field operator, will satisfy certain differential equations. In the parametrization using p it is seen by considering translations (the full Poincaré group is considered) that

$u_{l}(x;p,\sigma )=(2\pi )^{-3/2}e^{ipx}u_{l}(p,\sigma )$ and

$v_{l}(x;p,\sigma )=(2\pi )^{-3/2}e^{-ipx}v_{l}(p,\sigma )$,

where the species index n have been dropped.

For zero momemtum, by considering rotations and infinitesimal rotations in turn, one obtains the relations

${\begin{aligned}\sum _{\bar {\sigma }}u_{\bar {l}}(0,{\bar {\sigma }})\mathbf {J} _{{\bar {\sigma }}\sigma }^{(j)}&=\sum _{\bar {l}}\mathbf {\mathcal {J}} _{{\bar {l}}l}u_{l}(0,\sigma ),&\sum _{\bar {\sigma }}u_{\bar {l}}(0,{\bar {\sigma }})D_{{\bar {\sigma }}\sigma }^{(j)}(R)=\sum _{\bar {l}}D_{{\bar {l}}l}(R)u_{l}(0,\sigma )\\\sum _{\bar {\sigma }}v_{\bar {l}}(0,{\bar {\sigma }})\mathbf {J} _{{\bar {\sigma }}\sigma }^{(j)*}&=-\sum _{\bar {l}}\mathbf {\mathcal {J}} _{{\bar {l}}l}u_{l}(0,\sigma ),&\sum _{\bar {\sigma }}v_{\bar {l}}(0,{\bar {\sigma }})D_{{\bar {\sigma }}\sigma }^{(j)*}(R)=\sum _{\bar {l}}{D}_{{\bar {l}}l}(R)u_{l}(0,\sigma )\end{aligned}}$

for the Lie algebra representations (left) and the group. In these equation, the J are spin matrices for spin j, and the MATHCAL J is some, not necessarily irreducible, representation of so(3;1). The D are representations of the Lorentz group, while the D^j are representations of SO(3).

The behavior of u and v is governed strongly by which (m,n) irrep under which the fields transform. One first considers how the fields must appear at zero momentum, p = 0 (massive particles only). The coefficient functions have (m + 1)(n + 1) components, but only (2j + 1) of those can be independent (corresponding to the allowed values for σ). It is, in principle, easy to find u(0) and v(0) if (m + 1)(n + 1) = (2j + 1). Additional assumptions, like parity invariance are taken into account at this point. If (m + 1)(n + 1) ≠ (2j + 1), then further constraints must be imposed.

With knowledge of u_l(0,σ) and v_l(0,σ) the appearance at finite momenta p can be found by applying a standard (m,n) transformation corresponding to a specific Λ(p) taking (0,0,0) to p to (the vectors, spinors, tensors or spinor-tensors) u and v respectively. These standard are given by

$u_{l}({\textbf {p}},\sigma )=(m/q_{0})^{\tfrac {1}{2}}\sum _{\bar {l}}D_{l,{\bar {l}}}(L(p))u_{\bar {l}}({\textbf {0}},\sigma ),\qquad v_{l}({\textbf {p}},\sigma )=(m/q_{0})^{\tfrac {1}{2}}\sum _{\bar {l}}D_{l,{\bar {l}}}(L(p))v_{\bar {l}}({\textbf {0}},\sigma ),$

where L is a standard Lorentz boost taking zero momentum to q, and D is its representation.

Causality

The principle of causality is assumed to hold. The latter can be expressed more technically by assuming the slightly weaker cluster decomposition principle. In this setting one finds that the Hamiltonian density, and hence free field operators must commute at spacelike distances by using the known transformation properties of the creation and annihilation operators. If not, the cluster decomposition may be violated meaning, in principle, that experiments made at CERN can interfere with experiments at Fermilab or elsewhere in the universe. Mathematically the causality principle now reads

$[{\mathcal {H}}(x),{\mathcal {H}}(y)]=0\Rightarrow [\psi _{l}(x),\psi _{l^{\prime }}^{\dagger }(y)]=0$

for (x−y) spacelike, where † denotes the adjoint, and l, l′ are component indices of the field operator.

Free field equations and gauge principles

The commutator equation leads to free field equations for the field operators. The basic example is that all components of all massive quantum fields satisfy the free Klein–Gordon equation.

For a spin ½ particle with mass m in the (½,0) ⊕ (0,½) representation, the added assumption of parity invariance under the full Lorentz group the causality principle leads to the free field Dirac equation. Starting with the MASTER EQUATION, using the Pauli spin matrices for spin ½, an application of Schur's lemma leads to the (most) general ansatz

$u(0,{\tfrac {1}{2}})={\begin{bmatrix}c_{+}\\0\\c_{-}\\0\end{bmatrix}},u(0,-{\tfrac {1}{2}})={\begin{bmatrix}0\\c_{+}\\0\\c_{-}\end{bmatrix}},v(0,{\tfrac {1}{2}})={\begin{bmatrix}0\\d_{+}\\0\\d_{-}\end{bmatrix}},v(0,-{\tfrac {1}{2}})=-{\begin{bmatrix}d_{+}\\0\\d_{-}\\0\end{bmatrix}}$

for the coefficient functions at 0 momenta. The choice of an overall scale and parity invariance fixes two of the unknown parameters in the ansatz. The commutator equation explicitly reads

$[\psi _{l}(x),\psi _{\bar {l}}^{\dagger }]=(2\pi )^{-3}\int [|\kappa |^{2}N_{l{\bar {l}}}({\textbf {p}})e^{ip\cdot (x-y)}-|\lambda |^{2}M_{l{\bar {l}}}({\textbf {p}})e^{-ip\cdot (x-y)}]d^{3}p=0,$ where

$N_{l{\bar {l}}}({\textbf {p}})=\sum _{\sigma }u_{l}({\textbf {p}},\sigma )u_{\bar {l}}^{*}({\textbf {p}},\sigma ),M_{l{\bar {l}}}({\textbf {p}})=\sum _{\sigma }v_{l}({\textbf {p}},\sigma )v_{\bar {l}}^{*}({\textbf {p}},\sigma ).$

The equation fixes the last unknowns in the zero momentum coefficient functions and further implies that thy satisfy (ip^μγ_μ + m)u(p,σ)_l = 0 and (−ip^μγ_μ + m)u(p,σ)_l = 0 respectively. These are the momentum space versions of the Dirac equation and its adjoint in its original form. The index l runs over the 4 components of the Dirac field. The γ^μ are the Gamma matrices, also called the Dirac matrices, of dimension 4. For the field operator one obtains

$(\gamma ^{\mu }\partial _{\mu }-m)\psi =0.$

The appearance of the partial derivative is a consequence of properties (p ↔ id⁄dx) of the Fourier transform. This free field equations is obeyed by all free massive spin-½ particles having party invariance in the (½,0) ⊕ (0,½) representation.

The approach used here should be contrasted with the method of canonical quantization starting with a Lagrangian density for the free field and postulating canonical equal time commutation relations for the field operator. The equations obtained are precisely the Heisenberg equations of motion for the field operator. The corresponding Schrödinger picture equations can be obtained by standard means. These equations may or may not have formal similarity with the corresponding Heisenberg equations. In the case of the Klein–Gordon field, the Dirac field, the electoromagnetic field (below), and the Proca field, the equations of motion are formally identical.

Similar considerations and the (1,0) ⊕ (0,1) representation lead to the free field Maxwell equation. The Maxwell field tensor, f^μν resides in the (1,0) ⊕ (0,1) representation, which is 16-dimensional, but it has only 6 independent components. These are related by the free field Maxwell equation.

Principles of gauge invariance arise in this way too. When considering massless fields x^μ of spin 1 in the 4-vector representation (½,½), one finds that such fields will not be 4-vectors in general. There are degrees of freedom that does not correspond to physical degrees of freedom. These fields are nonetheless useful because Lorentz invariant quantities can be constructed from them. One example is the Maxwell field which is given by

$f_{\mu \nu }=\partial _{\mu }a_{\nu }-\partial _{\nu }a_{\mu }.$

Lorentz scalars (i.e. (0,0) representations) can be formed by contraction. The quantity A^μΨ_μ, where Ψ is the electron–positron field, is an ingredient in the Lagrangian in quantum electrodynamics (QED) representing the interaction between electrons and photons.

Other consequences

A couple more profound consequences of Lorentz invariance in QFT include the following.

• The existence of antiparticles
• The CPT theorem
• The spin-statistics theorem

References

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