Domenico Fiorenza The standard model

Originally based on Giuseppe Malavolta master’s thesis. It is an ongoing open project, feel free to contribute. A pdf copy of the Giuseppe’s thesis is available here

The standard model: a dictionary for the mathematician

The spirit of the wiki is described in this short Introduction. Also all the Lie groups technology needed for representations is contained in Lie groups and algebras representations. It is convenient to read the Physics notation too.



Mass, helicity and spin


In Bourbaki’s parlance, given any mathematical struture 𝒮\mathcal{S} its symmetries are the structure-preserving invertible self-transformations of 𝒮\mathcal{S}. In the more modern categorical language this is expressed by saying that the symmetries of an object XX are the elements of its automorphism group.

Concretely, the situation one is customary faced with is the following: one has a basic strcuture 𝒮 basic\mathcal{S}_{basic} which is enriched with an additional structure, becoming 𝒮 rich\mathcal{S}_{rich}. Symmetries of 𝒮 rich\mathcal{S}_{rich} are then symmetries of 𝒮 basic\mathcal{S}_{basic} preserving the additional structure.


In linear algebra one can consider a real vector space VV and endow it with a metric gg; the symmetries of (V,g)(V,g) are then the isometries, i.e., the linear automorphisms φ:VV\varphi:V\to V such that φ *g=g\varphi^{*}g=g. Note that Aut(V,g)\mathrm{Aut}(V,g) is the stabilizer of gg for the action of Aut(V)\mathrm{Aut}(V) on the space of symmetric bilinear forms on VV. Similarly, if VV is even dimensional, one can endow it with a symplectic form ω\omega, and consider the group of symplectic transformations of (V,ω)(V,\omega). This example immediately generalizes from vector spaces to differential manifolds: one considers isometries of (pseudo-)Riemannian manifolds and symplectomorphisms of symplectic manifolds.


In classical mechanics one considers a differential manifold MM endowed with a Lagrangian, i.e., a smooth function L:TML:T M\to\mathbb{R}. Symmetries of such a Lagrangian system are diffeomorphisms φ:MM\varphi:M\to M such that (dφ) *L=L(d\varphi)^{*}L=L, where dφ:TMTMd\varphi:T M\to T M is the differential of φ\varphi.


The Lagrangian example above has a natural generalization in Hamiltonian mechanics. An Hamiltonian system is a triple (M,ω,H)(M,\omega,H) where (M,ω)(M,\omega) is a symplectic manifold, and H:MH:M\to\mathbb{R} is a smooth function, called the Hamiltonian. A symmetry of an Hamiltonian system is a symplectomorphism φ\varphi of (M,ω)(M,\omega) such that φ *H=H\varphi^{*}H=H.

In all of the above examples, symmetries are not just a group: they are a Lie group (eventually an infinite-dimensional one). So it is meaningful to talk of infinitesimal symmetries. Mathematically speaking, these are elements of the Lie algebra of the Lie group of symmetries. It is interesting to remark that in infinite-dimensional situations, the Lie algebra structure of the vector space of infinitesimal symmetries is a perfectly well define object, even when a rigorous infinite-dimensional Lie group structure on the group od symmetries is not defined.


In example \ref{ex-hamiltonian}, infinitesimal symmetries are vector fields XX on MM such that Xω=0\mathcal{L}_{X}\omega=0 and XH=0\mathcal{L}_{X}H=0, where X\mathcal{L}_{X} denotes the Lie derivative along XX. Among these symmetries, of particular interest are the Hamiltonian ones, i.e., vector fields of the form X f={f,}X_{f}=\{ f,-\}, where ff is a smooth function on MM and {,}\{-,-\} is the Poisson bracket induced by the symplectic struture of MM. Of the two conditions an Hamiltonian vector field has to satisfy in order to eb an infinitesimal symmetry of the system, the first one, X fω=0\mathcal{L}_{X_{f}}\omega=0 is always satisfied (this is Liouville’s theorem), whereas the second one, i.e., X fH=0\mathcal{L}_{X_{f}}H=0 is equivalent to {f,H}=0\{ f,H\}=0. In other terms, the Lie algebra of infinitesimal Hamiltonian symmetries is identified with the Lie algebra of smooth functions on MM Poisson-commuting with HH (modulo the constants, i.e., the kernel of fX ff\mapsto X_{f}). In terms of classical mechanics, each infinitesimal Hamiltonian symmetry ff is a constant of motion, i.e., if x:Mx:\mathbb{R}\to M is the time evolution of the Hamiltonian system with initial datum x(0)=x 0x(0)=x_{0}, then f(x(t))=f(x 0)f(x(t))=f(x_{0}) for every tt\in\mathbb{R}.


In quantum mechanics one considers a noncommutative version of example \ref{ex-infinitesimal-hamiltonian}, where smooth functions are replaced by self-adjoint operators and Poisson brackets are replaced by commutators. More precisely, quantum states are norm 1 vectors in an Hilbert space \mathcal{H} and the probability of a transition of the system from a state ϕ\phi to a state ψ\psi is

P(ϕψ)=|ϕ|ψ| 2. P(\phi\to\psi)=|\langle\phi|\psi\rangle|^{2}.

Symmetries of the system will have to preserve all these transition probabilities, so they will be unitary operators U:U:\mathcal{H}\to\mathcal{H}. Moreover, the system is endowed with a distinguished self-adjoint operator, the Hamiltonian H:H:\mathcal{H}\to\mathcal{H}, and the Poisson-commutation relation {f,H}=0\{ f,H\}=0 of Hamiltonain mechanics is translated in the commutation relation [U,H]=0[U,H]=0. This in particular implies that any symmetry UU of (,H)(\mathcal{H},H) preserves the HH-eigenspaces decomposition of \mathcal{H}. In the quantum mechanics parlance, the scalars in the spectrum of the operator HH are called the energy levels of the Hamiltonian.


In the above example we restricted our attention to unitary operators by the principle of “symmetries of the rich structure are a subgroup of symmetries of the basic structure”, and assuming that the basic structure was that of an Hilbert space, so that its symmetries were the automorphisms of \mathcal{H} as a Banach space. One could however consider more general symmetries, by looking at \mathcal{H} just as a set endowed with the function

P:× (ϕ,ψ) |ϕ|ψ| 2. \begin{aligned} P:\mathcal{H}\times\mathcal{H}&\to\mathbb{R}\\ (\phi,\psi)&\mapsto|\langle\phi|\psi\rangle|^{2}. \end{aligned}

and then consider the set of all invertible self-transformations of the set \mathcal{H} preserving PP. But this actually brings in nothing new: a PP-preserving self-transformation of \mathcal{H} is either a unitary transformation of \mathcal{H} or a unitary transformation of ¯\overline{\mathcal{H}} (Wigner’s theorem, wigner-unitary).

Groups of symmetries

For a given object XX in some category 𝒞\mathcal{C}, the complete description of the whole group of symmetries Aut(X)\mathrm{Aut}(X) may be an extremely difficult problem. In concrete situations one is often more interested in distinguished (in some sense) and well-behaved subgroups of Aut(X)\mathrm{Aut}(X), or more in general in group representation with values in Aut(X)\mathrm{Aut}(X). For instance, if GG is a Lie group, a realization of GG as a symmetry group for a quantum mechanical system is a group homomorphism

ρ:GU() \rho:G\to U(\mathcal{H})

which is smooth in the sense that for any two states ϕ\phi and ψ\psi in \mathcal{H}, the “matrix coefficient” ϕ|ρ(g)ψ\langle\phi|\rho(g)\cdot\psi\rangle is a smooth complex-valued function on GG. Here we are denoting by U()U(\mathcal{H}) the group of unitary operators on \mathcal{H}. We will also occasionally meet linear representations

ρ:GAut(), \rho:G\to\mathrm{Aut}(\mathcal{H}),

where Aut()\mathrm{Aut}(\mathcal{H}) is the group of automorphisms of \mathcal{H} in the category of Hilbert spaces. Since the category of Hilbert spaces is a full subcategory of Banach spaces, Aut()\mathrm{Aut}(\mathcal{H}) is the group of automorphisms of \mathcal{H} as a Banach space, i.e., the group of continuous invertible linear endomorphisms of \mathcal{H} with a continuous inverse.

Let us now focus on unitary representations. If 0\mathcal{H}_{0} is a subspace of \mathcal{H} which is stable under the unitary action of the symmetry group GG, then also 0 \mathcal{H}_{0}^{\perp} is GG-stable. This immediately implies the following result.


All unitary representations of a group GG are completely reducible, i.e., are direct sums of irreducible representations.

When GG is compact, each linear representation GAut()G\to\mathrm{Aut}(\mathcal{H}) is actually an unitary representation (up to conjugation). Indeed, let μ G\mu_{G} be the Haar measure of GG, i.e., the unique normalized biinvariant measure on GG and set

ϕ|ψ= Gρ(g)ϕ|ρ(g)ψdμ G(g). \langle\!\langle\phi|\psi\rangle\!\rangle=\int_{G}\langle\rho(g)\cdot\phi|\rho(g)\cdot\psi\rangle d\mu_{G}(g).

Then |\langle\!\langle-|-\rangle\!\rangle is an inner product on \mathcal{H} inducing an Hilbert space structure equivalent to the original one, and the representation ρ\rho is manifestly unitary with respect to this new inner product. So, by the above proposition, we obtain that all continuous linear representations of a compact group GG on an Hilbert space are completely reducible.

A less trivial result is the following.


Let GG be a compact Lie group (or, more in general, a compact topological group). Then, irreducible representations of GG are finite dimensional.

This is proved by showing that if GG is compact then each unitary representation of GG is the direct sum of its finite-dimensional subrepresentations. A complete proof of this statement can be found, e.g., in brocker-tomdieck. Combining the two propositions above, one obtains the following principle: to understand compact Lie groups of symmetries in quantum mechanics, one has to study their finite dimensional irreducible representations.

From Lie groups to Lie algebras

Throughout this section we will assume the reader is familiar with the basics of the theory of Lie algebra and of the Lie groups/Lie algebra correspondence. So we will directly focus on the examples we will need in the sequel. Details on the general theory can be found, e.g., in fulton-harris.

If the Lie group GG is connected, then its group structure is entirely determined by an arbitrary small neighborhood of the identity element ee and so, ultimately, by the Lie algebra structure of its tangent space at the identity. The Lie algebra 𝔤=T eG\mathfrak{g}=T_{e}G is called the Lie algebra of the Lie group GG; it is canonically identified with the Lie algebra of left-invariant vector fields on GG by the Lie algebra homomorphism

{left invariant vector fields on $G$} 𝔤 X X e. \begin{aligned} \{\text{left invariant vector fields on $G$}\}&\xrightarrow{\sim} \mathfrak{g}\\ X&\mapsto X_{e}. \end{aligned}

If {e i} iI\{\mathbf{e}_{i}\}_{i\in I} is a linear basis of 𝔤\mathfrak{g}, then the Lie algebra structure of 𝔤\mathfrak{g} is completely encoded in the structure constants of the Lie bracket:

[e i,e j]=f ij ke k. [\mathbf{e}_{i},\mathbf{e}_{j}]=f^{k}_{ij}\mathbf{e}_{k}.

The Lie algebra 𝔰𝔬 3\mathfrak{so}_{3} of the Lie group SO(3)SO(3) of rotations in 3\mathbb{R}^{3} is the Lie algebra of 3×33\times3 real antisymmetric matrices. Its canonical basis is given by the generators of rotations around the xx-, yy-, and zz-axis, respectively:

X 1=(0 0 0 0 0 1 0 1 0);X 2=(0 0 1 0 0 0 1 0 0);X 3=(0 1 0 1 0 0 0 0 0). X_{1}=\left( \begin{matrix} 0&0&0\\ 0&0&-1\\ 0&1&0 \end{matrix} \right); \quad X_{2}=\left( \begin{matrix} 0&0&1\\ 0&0&0\\ -1&0&0 \end{matrix} \right); \quad X_{3}=\left( \begin{matrix} 0&-1&0\\ 1&0&0\\ 0&0&0 \end{matrix} \right).

These are colloquially called the infinitesimal rotations around the coordinate axes. It is starightforward to check the commutation relations

[X 1,X 2]=X 3;[X 2,X 3]=X 1;[X 3,X 1]=X 2, [X_{1},X_{2}]=X_{3};\qquad[X_{2},X_{3}]=X_{1};\qquad[X_{3},X_{1}]=X_{2},


[X i,X j]=ϵ ijkδ klX l. [X_{i},X_{j}]=\epsilon_{ijk}\delta^{kl}X_{l}.

In the physics literature it is customary to consider the complexified Lie algebra 𝔰𝔬 3;=𝔰𝔬 3\mathfrak{so}_{3;\mathbb{C}}=\mathfrak{so}_{3}\otimes\mathbb{C} with basis {J 1,J 2,J 3}\{ J_{1},J_{2},J_{3}\}, where J i=iX iJ_{i}=\mathbf{i}X_{i}. The commutation relations of the J iJ_{i} are

[J i,J j]=iϵ ijkδ klJ l. [J_{i},J_{j}]=\mathbf{i}\epsilon_{ijk}\delta^{kl}J_{l}.

The Lie algebra 𝔰𝔲 2\mathfrak{su}_{2} of the Lie group SU(2)SU(2) is the Lie algebra of 2×22\times2 complex anti-Hermitean matrices. The inner product (A,B)tr(A *B)(A,B)\mapsto\mathrm{tr}(A^{*}B) makes 𝔰𝔲 2\mathfrak{su}_{2} a 3-dimensional Euclidean space. The action of SU(2)SU(2) by conjugation on 𝔰𝔲 2\mathfrak{su}_{2} is isometric with respect to this inner product, thus giving a Lie group homomorphism SU(2)O(3)SU(2)\to O(3). Since SU(2)SU(2) is connected, the image of this homomorfism is necessarily contained in the connected component SO(3)SO(3). Moreover, one checks that SU(2)SO(3)SU(2)\to SO(3) is actually surjective and that its kernel is {Id,Id}\{\mathrm{Id},-\mathrm{Id}\}. Thus the homomorphism SU(2)SO(3)SU(2)\to SO(3) is a degree 2 covering map, and since SU(2)SU(2) is simply connected this exhibits SU(2)SU(2) as the universal cover of SO(3)SO(3):

SU(2)SU(2)/{±Id}SO(3). SU(2)\to SU(2)/\{\pm\mathrm{Id}\}\cong SO(3).

A covering map of Lie groups induces an isomorphism of the corresponding Lie algebras. So we have a natural isomorphism 𝔰𝔲 2𝔰𝔬 3\mathfrak{su}_{2}\cong\mathfrak{so}_{3}. Explicitly, a basis of 𝔰𝔲 2\mathfrak{su}_{2} is given by

Y 1=(i/2 0 0 i/2);Y 2=(0 1/2 1/2 0);Y 3=(0 i/2 i/2 0). Y_{1}=\left( \begin{matrix} \mathbf{i}/2&0\\ 0&-\mathbf{i}/2 \end{matrix} \right); \quad Y_{2}=\left( \begin{matrix} 0&1/2\\ -1/2&0 \end{matrix} \right); \quad Y_{3}=\left( \begin{matrix} 0&\mathbf{i}/2\\ \mathbf{i}/2&0 \end{matrix} \right).

One has

[Y 1,Y 2]=Y 3;[Y 2,Y 3]=Y 1;[Y 3,Y 1]=Y 2, [Y_{1},Y_{2}]=Y_{3};\qquad[Y_{2},Y_{3}]=Y_{1};\qquad[Y_{3},Y_{1}]=Y_{2},

and the isomorphism 𝔰𝔲 2𝔰𝔬 3\mathfrak{su}_{2}\xrightarrow{\sim}\mathfrak{so}_{3} is manifest. In physics literature, it is customary to consider the complexified Lie algebra 𝔰𝔲 2;=𝔰𝔲 2\mathfrak{su}_{2;\mathbb{C}}=\mathfrak{su}_{2}\otimes\mathbb{C} with basis {J 1,J 2,J 3}\{ J_{1},J_{2},J_{3}\} given by J i=iY iJ_{i}=\mathbf{i}Y_{i}. The commutation relations among the J iJ_{i} are, clearly,

[J i,J j]=iϵ ijkδ klJ l, [J_{i},J_{j}]=\mathbf{i}\epsilon_{ijk}\delta^{kl}J_{l},

as in the 𝔰𝔬 3;\mathfrak{so}_{3;\mathbb{C}} case. Note that

J 1=12σ 3;J 2=12σ 2;J 3=12σ 1 J_{1}=-\frac{1}{2}\sigma_{3}; \qquad J_{2}=-\frac{1}{2}\sigma_{2}; \qquad J_{3}=-\frac{1}{2}\sigma_{1}

where {σ 1,σ 2,σ 3}\{\sigma_{1},\sigma_{2},\sigma_{3}\} are the Pauli matrices.


The Lie group SO(3)SO(3) acts as a group of diffeomorphisms on 3\mathbb{R}^{3}. Hence we have an injective group homomorphisms SO(3)Diff( 3)SO(3)\to\mathrm{Diff}(\mathbb{R}^{3}) inducing an injective Lie algebra homomorphism

\mathfrak{so}_{3}\hookrightarrow\{\text{left $SO(3)$-invariant vector fields on $\mathbb{R}^{3}$}\}.

In particular, we can see 𝔰𝔬 3{left $SO(3)$-invariant vector fields on $\mathbb{R 3 \mathfrak{so}_{3}\hookrightarrow\{\text{left $SO(3)$-invariant vector fields on $\mathbb{R}^{3} as a SO(3)SO(3)-invariant vector field on 3\mathbb{R}^{3}. Explicitly,

X 1=x 2 3x 3 2;X 2=x 3 1x 1 3;X 3=x 1 2x 2 1, X_{1}=x^{2}\partial_{3}-x^{3}\partial_{2}; \qquad X_{2}=x^{3}\partial_{1}-x^{1}\partial_{3};\qquad X_{3}=x^{1}\partial_{2}-x^{2}\partial_{1},

which is best written in the compact form

X i=ϵ ijkx j k X_{i}=\epsilon_{ijk}x^{j}\partial^{k}

where we have used the metric in 3\mathbb{R}^{3} to raise the index in the derivation, i.e., i=δ ij j\partial^{i}=\delta^{ij}\partial_{j}. Complexifying this construction, we can identify the basis elements J iJ_{i} of 𝔰𝔬 3;𝕔\mathfrak{so}_{3;\mathbb{c}} with complex vector field on 3\mathbb{R}^{3}. It is customary to write

J ij=ϵ ijkJ k=i(x i jx j i). J^{ij}=\epsilon^{ijk}J_{k}=\mathbf{i}(x^{i}\partial^{j}-x^{j}\partial^{i}).

With this notation the commutation relations read

[J ij,J kl]=i(δ jkJ il+δ ilJ jkδ ikJ jlδ jlJ ik). [J^{ij},J^{kl}]=\mathbf{i}(\delta^{jk}J^{il}+\delta^{il}J^{jk}-\delta^{ik}J^{jl}-\delta^{jl}J^{ik}).

Universal enveloping algebras

Any associative algebra AA defines a Lie algebra A LieA_{Lie} simply by taking the commutator as the Lie bracket:

[a,b]=abba. [a,b]=ab-ba.

This construction is a functor

Associative algebrasLie algebras. \text{Associative algebras}\to\text{Lie algebras}.

Remarkably, this functor has a left adjoint

U:Lie algebrasAssociative algebras. U: \text{Lie algebras}\to\text{Associative algebras}.

That is, given a Lie algebra 𝔤\mathfrak{g} there exists an associative algebra U(𝔤)U(\mathfrak{g}) sucht that for any associative algebra AA,

Hom Lie(𝔤,A Lie)=Hom Assoc(U(𝔤),A). \mathrm{Hom}_{Lie}(\mathfrak{g},A_{Lie})=\mathrm{Hom}_{Assoc}(U(\mathfrak{g}),A).

In other words, any Lie algebra morphism 𝔤A Lie\mathfrak{g}\to A_{Lie} can be uniquely extended to an associative algebra morphism U(𝔤)AU(\mathfrak{g})\to A. Since U(𝔤)U(\mathfrak{g}) is characterized by an universal property, it is unique up to natural isomorphism. The algebra U(𝔤)U(\mathfrak{g}) is called the universal envelping algebra of 𝔤\mathfrak{g}. Note that, taking A=U(𝔤)A=U(\mathfrak{g}) one obtatins a canonical Lie algebra morphism ι 𝔤:𝔤U(𝔤) Lie\iota_{\mathfrak{g}}:\mathfrak{g}\to U(\mathfrak{g})_{Lie}, induced by the identity of U(𝔤)U(\mathfrak{g}). This morphism is actually injective and U(𝔤)U(\mathfrak{g}) is generated, as an associative algebra, by the image of ι 𝔤\iota_{\mathfrak{g}} (Poincaré-Birkhoff-Witt theorem).


A typical situation in which U(𝔤)U(\mathfrak{g}) occurs is the following. Consider a linear representation of a Lie group GG, i.e., a Lie graoup homomorphism GGL(V)G\to GL(V), where VV is some vector space. This induces a Lie algebra homomorphism 𝔤𝔤𝔩(V)=End(V) Lie\mathfrak{g}\to\mathfrak{gl}(V)=\mathrm{End}(V)_{Lie}, and so an associative algebra homomorphism

U(𝔤)End(V). U(\mathfrak{g})\to\mathrm{End}(V).

Let MM be a differential manifold, and let 𝒳(M)\mathcal{X}(M) be the Lie algebra of vector fields on MM. Finally, let 𝒟(M)\mathcal{D}(M) be the associative algebra of differential operators on MM. Then 𝒳(M)\mathcal{X}(M) is a sub-Lie algebra of 𝒟(M) Lie\mathcal{D}(M)_{Lie}. If a smooth action of a Lie group GG on MM is given, the above inclusion is refined to an inclusion (M) G𝒟(M) Lie G\mathcal{H}(M)^{G}\hookrightarrow\mathcal{D}(M)^{G}_{Lie} of left GG-invariant vector fields on MM into the Lie algebra of left GG-invariant differential operators on MM. In particular, if M=GM=G acting on itself by left multilication, we get a Lie algebra inclusion

𝔤𝒟(G) Lie G, \mathfrak{g}\hookrightarrow\mathcal{D}(G)^{G}_{Lie},

where we have used the identification of 𝔤\mathfrak{g} with left-invariant vector fields on GG. By the universal property of U(𝔤)U(\mathfrak{g}), this gives an associative algebra morphism from U(𝔤)U(\mathfrak{g}) to 𝒟(G) G \mathcal{D}(G)^{G}, which turns out to be an isomorphism:

U(𝔤){left-invariant differential operators on $G$}. U(\mathfrak{g})\xrightarrow{\sim}\{\text{left-invariant differential operators on $G$}\}.

More in general, if MM is a differential manifold with a smooth GG-action, we have a natural morphism of asociative algebras

U(𝔤){left $G$-invariant differential operators on $M$}. U(\mathfrak{g})\to\{\text{left $G$-invariant differential operators on $M$}\}.

Casimir elements and invariant differential operators

By definition, a Casimir element for a Lie algebra 𝔤\mathfrak{g} is an element in the center of its universal enveloping algebra, i.e. is an element CC in U(𝔤)U(\mathfrak{g}) such that [C,x]=0[C,x]=0 for any xx in U(𝔤)U(\mathfrak{g}). Since the algebra U(𝔤)U(\mathfrak{g}) is generated by the linear subspace 𝔤\mathfrak{g}, this is equivalent to [C,x]=0[C,x]=0 for any xx in 𝔤\mathfrak{g}. As remarked in the previous section, if 𝔤\mathfrak{g} is the Lie algebra of a Lie group GG, then U(𝔤)U(\mathfrak{g}) is identified with the algebra of left-invariant differential operators on GG. Therefore Casimir elements are identified with biinvariant differential operators on GG. More in general, if MM is a differential manifold endowed with a smooth GG-action, Casimir elements induce GG-biinvariant differential operators on MM. In what follows we will say GG-invariant differential operator to mean that a differential operator is biinvariant, whereas we will always say left GG-invariant or right GG-invariant when the operator is invariant only with respect to left- or right- translations.


For 𝔰𝔲 2𝔰𝔬 3\mathfrak{su}_{2}\cong\mathfrak{so}_{3} a Casimir operator is

J 2=δ ijJ iJ j. \| J\|^{2}=\delta^{ij}J_{i}J_{j}.


[J i,J 2] =δ jk[J i,J jJ k] =iδ jk([J i,J j]J k+J j[J i,J k]) =iδ jk(ϵ ijlδ lmJ mJ k+ϵ iklδ lmJ jJ m) =i j,l(ϵ ijlJ lJ j+ϵ ijlJ jJ l) =i j,l(ϵ ijl+ϵ ilj)J lJ j=0. \begin{aligned} [J_{i},\| J\|^{2}]&=\delta^{jk}[J_{i},J_{j}J_{k}]\\ &=\mathbf{i}\,\delta^{jk}([J_{i},J_{j}]J_{k}+J_{j}[J_{i},J_{k}])\\ &=\mathbf{i}\,\delta^{jk}(\epsilon_{ijl}\delta^{lm}J_{m}J_{k}+\epsilon_{ikl}\delta^{lm}J_{j}J_{m})\\ &=\mathbf{i}\sum_{j,l}(\epsilon_{ijl}J_{l}J_{j}+\epsilon_{ijl}J_{j}J_{l})\\ &=\mathbf{i}\sum_{j,l}(\epsilon_{ijl}+\epsilon_{ilj})J_{l}J_{j}=0. \end{aligned}

The operator J 2\| J\|^{2} is called angular momentum operator in physics. If ρ:SU(2)U()\rho:SU(2)\to U(\mathcal{H}) is a unitary representation of SU(2)SU(2), then the image of 𝔰𝔲 2\mathfrak{su}_{2} in End()\mathrm{End}(\mathcal{H}) consists of anti-Hermitean operators. Since multiplication by i\mathbf{i} turns an anti-Hermitean operator into an Hermitean one, the operators (corresponding to the) J iJ_{i} are Hermitean, and so is also

J 2: \| J\|^{2}:\mathcal{H}\to\mathcal{H}


(J 2) *=(δ ijJ iJ j) *=δ ijJ j *J i *=δ ijJ jJ i=J 2 (\| J\|^{2})^{*}=(\delta^{ij}J_{i}J_{j})^{*}=\delta_{ij}J_{j}^{*}J_{i}^{*}=\delta_{ij}J_{j}J_{i}=\| J\|^{2}

since δ\delta is real and symmetric. In particular J 2\| J\|^{2} is diagonalizable, its spectrum is real, and its eigenspaces are subrepresentations of ρ\rho. This means that J 2J^{2} acts as a scalar on irreducible unitary representations of SU(2)SU(2). This scalar is a numerical invariant attached to irreducible complex representations of 𝔰𝔲 2𝔰𝔬 3\mathfrak{su}_{2}\cong\mathfrak{so}_{3}. More precisely, we will see in Section \ref{su2} that for any nonnegative half-integer \ell there exist exactly one unitary irreducible representation ρ \rho_{\ell} of dimension 2+12\ell+1, and that these are all possible unitary irreducible representations of SU(2)SU(2). The operator J 2\| J\|^{2} acts as the multiplication by (+1)\ell(\ell+1) in the representation ρ \rho_{\ell}.


For the defining representation of SO(3)SO(3) on 3\mathbb{R}^{3}, the elements J iJ_{i} in End( 3)\mathrm{End}(\mathbb{R}^{3}) are described in Example \ref{ex-so3}. One therefore sees that

J 2=2Id. \| J\|^{2}=2\mathrm{Id}.

Complexifying this representation one obtains the =1\ell=1 representation of SU(2)SU(2).


The defining representation of SU(2)SU(2) on 2\mathbb{C}^{2} is the =1/2\ell=1/2 representation of SU(2)SU(2). For this representation, the elements J iJ_{i} in End( 2)\mathrm{End}(\mathbb{C}^{2}) are described in Example \ref{ex-su2}. One therefore sees that

J 2=34Id=12(12+1)Id \| J\|^{2}=\frac{3}{4}\mathrm{Id}=\frac{1}{2}\left(\frac{1}{2}+1\right)\mathrm{Id}

in this case.

The Lorentz group

Let 1,3\mathbb{R}^{1,3} be the standard Minkowski space with metric η\eta of signature (+,,,)(+,-,-,-). The Lorentz group is the group O(1,3)O(1,3) of isometries of η\eta. In matrix form, an element Λ\Lambda of the Lorenz group is a 4×44\times4 real matrix Λ j i\Lambda^{i}_{j} such that

Λ k iη ijΛ l j=η kl \Lambda^{i}_{k}\eta_{ij}\Lambda^{j}_{l}=\eta_{kl}

The determinant map

det:O(1,3)O(1)={±1} \det:O(1,3)\to O(1)=\{\pm1\}

is surjective since

(1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1) \left( \begin{matrix} 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1 \end{matrix} \right)

is an element of O(1,3)O(1,3). The subgroup of Lorentz transformations of determinant 1 is called the subgroup of proper Lorentz transformations, and is denoted SO(1,3)SO(1,3). In contrast with what happens with O(4)O(4), where SO(4)SO(4) is the connected component of the identity, the group of proper Lorentz transformations is not connected. Indeed, since the columns of Λ j i\Lambda^{i}_{j} are an η\eta-orthonormal basis of 4\mathbb{R}^{4}, the first column Λ 0 i\Lambda^{i}_{0} satisfies η(Λ 0,Λ 0)=1\eta(\Lambda_{0},\Lambda_{0})=1, i.e.,

(Λ 0 0) 2(Λ 0 1) 2(Λ 0 2) 2(Λ 0 3) 2=1. ( \Lambda^{0}_{0})^{2}-( \Lambda^{1}_{0})^{2}-( \Lambda^{2}_{0})^{2}-( \Lambda^{3}_{0})^{2}=1.

This implies (Λ 0 0) 21(\Lambda^{0}_{0})^{2}\geq1 and so there are two disjoint possibilities: Λ 0 01\Lambda^{0}_{0}\geq1 or Λ 0 01\Lambda^{0}_{0}\leq-1. In the first case the Lorenz transformation is called orthocronous, in the seconda case anorthocronous. An example of anorthochronous Lorenz transformation is

(1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1) \left( \begin{matrix} -1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{matrix} \right)

The map O(1,3){±1}O(1,3)\to\{\pm1\} given by Λsgn(Λ 0 0)\Lambda\mapsto\mathrm{sgn}(\Lambda^{0}_{0}) is actually a group homomorphism, and so orthocronous Lorentz transformations are a subgroup of O(1,3)O(1,3). One can then show that the connected component of the identity in O(1,3)O(1,3) is precisely the subgroup SO +(1,3)SO^{+}(1,3) of proper orthocronous Lorentz transformations. In particular, O(1,3)O(1,3) has exactly four connected components.

The group SO +(1,3)SO^{+}(1,3) is a 6-dimensional connected Lie group. It is not simply connected, and its universal cover is the Lie group SL(2;)SL(2;\mathbb{C}). This is conveniently seen as follows. Let 𝔥𝔢𝔯 2\mathfrak{her}_{2} be 4-dimensional real vector space of 2×22\times2 Hermitean matrices. Since the determinant of an Hermitean matrix is a real number, we have a quadratic form

det:𝔥𝔢𝔯 2 \det:\mathfrak{her}_{2}\to\mathbb{R}

whose signature turns out to be (1,3)(1,3). This is seen by the linear isomorphism 4𝔥𝔢𝔯 2\mathbb{R}^{4}\to\mathfrak{her}_{2} given by

(x 0 x 1 x 2 x 3)(x 0+x 3 x 1ix 2 x 1+ix 2 x 0x 3). \left( \begin{matrix} x^{0}\\ x^{1}\\ x^{2}\\ x^{3} \end{matrix} \right)\mapsto\left(\begin{matrix} x^{0}+x^{3}& x^{1}-\mathbf{i}x^{2}\\ x^{1}+\mathbf{i}x^{2}& x^{0}-x^{3} \end{matrix} \right).


det(x 0+x 3 x 1ix 2 x 1+ix 2 x 0x 3)=(x 0) 2(x 1) 2(x 2) 2(x 3) 2. \det\left(\begin{matrix} x^{0}+x^{3}& x^{1}-\mathbf{i}x^{2}\\ x^{1}+\mathbf{i}x^{2}& x^{0}-x^{3} \end{matrix} \right)= (x^{0})^{2}-(x^{1})^{2}-(x^{2})^{2}-(x^{3})^{2}.

The group SL(2;)SL(2;\mathbb{C}) acts on 𝔥𝔢𝔯 2\mathfrak{her}_{2} by

APAP *, A\mapsto PAP^{*},

and this action clearly preserves the quadratic form det\det, so that we get a group homomorphism SL(2;)O(1,3)SL(2;\mathbb{C})\to O(1,3). Since SL(2;)SL(2;\mathbb{C}) is connected, the image of this homomorphism is contained in SO +(1,3)SO^{+}(1,3); moreover the morphism induced at the Lie algebra level is an isomorphism and so SL(2;)SO +(1,3)SL(2;\mathbb{C})\to SO^{+}(1,3) is a covering. The kernel of this map is {±Id}\{\pm\mathrm{Id}\} so SL(2;)SO +(1,3)SL(2;\mathbb{C})\to SO^{+}(1,3) is a two-fold covering; moreover, since SL(2;)SL(2;\mathbb{C}) is simply connected, this is the universal covering of SO +(1,3)SO^{+}(1,3):

SL(2,)SL(2,)/{±Id}SO +(1,3). SL(2,\mathbb{C})\to SL(2,\mathbb{C})/\{\pm\mathrm{Id}\}\cong SO^{+}(1,3).

{The Lorentz Lie algebra} The Lie algebra 𝔰𝔬 1,3\mathfrak{so}_{1,3} of the Lorentz group is a 6-dimensional real Lie algebra. As a matrix algebra, it is the algebra of 4×44\times4 real matrices A j iA^{i}_{j} such that

A k iη il+η kiA l i=0,for every $k,l$. A^{i}_{k}\eta_{il}+\eta_{ki}A^{i}_{l}=0, \quad\text{for every $k,l$}.

As above, we will be interested in the complexification 𝔰𝔬 1,3;=𝔰𝔬 1,3\mathfrak{so}_{1,3;\mathbb{C}}=\mathfrak{so}_{1,3}\otimes\mathbb{C}. A linear basis of 𝔰𝔬 1,3;\mathfrak{so}_{1,3;\mathbb{C}} is given by the six matrices

J ab=i(η acE c bE c aη cb),a<b, J^{ab}=\mathbf{i}\,(\eta^{ac}E^{b}_{c}-E^{a}_{c}\eta^{cb}),\qquad a\lt b,

where E j iE^{i}_{j} is the elementary matrix with 1 in position (i,j)(i,j) and 0 elsewhere. Indeed,

(J ab) k iη il+η ki(J ab) l i =i(η acδ biδ ckδ aiδ ckη cb)η il+iη ki(η acδ biδ clδ aiδ clη cb) =i(η akη blη kbη al+η kbη alη kaη lb)=0, \begin{aligned} (J^{ab})^{i}_{k}\eta_{il}+\eta_{ki}(J^{ab})^{i}_{l}&=\mathbf{i}\,(\eta^{ac}\delta^{bi}\delta_{ck}-\delta^{ai}\delta_{ck}\eta^{cb})\eta_{il}+\mathbf{i}\,\eta_{ki}(\eta^{ac}\delta^{bi}\delta_{cl}-\delta^{ai}\delta_{cl}\eta^{cb})\\ &= \mathbf{i}\,(\eta^{ak}\eta_{bl}-\eta^{kb}\eta_{al}+\eta_{kb}\eta^{al}-\eta_{ka}\eta^{lb})=0, \end{aligned}

where we used the identities η ij=η ji\eta_{ij}=\eta_{ji} and η ij+η ij\eta^{ij}+\eta_{ij} for ay i,ji,j. The commutation relations of the basis elements J abJ^{ab} are

[J ab,J cd]=i(η bcJ ad+η adJ bcη acJ bdη bdJ ac), [J^{ab},J^{cd}]=\mathbf{i}\,(\eta^{bc}J^{ad}+\eta^{ad}J^{bc}-\eta^{ac}J^{bd}-\eta^{bd}J^{ac}),

where we have set J aa=0J^{aa}=0 and J ba=J abJ^{ba}=-J^{ab} if a<ba\lt b. It is convenient to represent the basis elements J abJ^{ab} as SO +(1,3)SO^{+}(1,3) left-invariant vector fields on 4\mathbb{R}^{4}, as we did for the Lie algebra of SO(3)SO(3):

J ab=x a bx b a, J^{ab}=x^{a}\partial^{b}-x^{b}\partial^{a},

where i=η ij j\partial^{i}=\eta^{ij}\partial_{j}. The group SO(3)SO(3) embeds in SO +(1,3)SO^{+}(1,3) via

A(1 0 0 A) A\mapsto\left(\begin{matrix}1&0\\0&A \end{matrix}\right)

and so 𝔰𝔬 3\mathfrak{so}_{3} is a Lie subalgebra of 𝔰𝔬 1,3\mathfrak{so}_{1,3}. The generators of the copy of 𝔰𝔬 3\mathfrak{so}_{3} inside 𝔰𝔬 1,3\mathfrak{so}_{1,3} are clearly J 12J^{12}, J 23J^{23} and J 13J^{13}. For i,j,k{1,2,3}i,j,k\in\{1,2,3\} one writes

L i=ϵ ijkJ jk L_{i}=\epsilon^{ijk}J^{jk}

so that

L 1=J 23;J 2=J 13;L 3=J 12. L_{1}=J^{23}; \qquad J_{2}=-J^{13};\qquad L_{3}=J^{12}.

In the physics lieterature, the generators L 1,L 2,L 3L_{1},L_{2},L_{3} are called the infinitesimal rotations aroud the spatial axes of 1,3\mathbb{R}^{1,3}. Note that these precisely corresponds to the elements od 𝔰𝔬 3\mathfrak{so}_{3} we denoted by J 1,J 2,J 3J_{1},J_{2},J_{3} in \ref{ex-so3}.

The reamining three generators of 𝔰𝔬 1,3\mathfrak{so}_{1,3} are the elements

K i=J 0i=i(x 0 ix i 0)=i(x 0 i+x i 0). K_{i}=J^{0i}=\mathbf{i}(x^{0}\partial^{i}-x^{i}\partial^{0})=-\mathbf{i}(x^{0}\partial_{i}+x^{i}\partial_{0}).

These are called the boosts in the physics literature. The commutation relations of the generators J abJ^{ab} are conveniently written in terms of the L iL_{i} and the K jK_{j}:

[L i,L j]=iϵ ijkL k;[K i,K j]=iϵ ijkL k;[L i,K j]=iϵ ijkK k. [L_{i},L_{j}]=\mathbf{i}\epsilon_{ijk}L_{k}; \qquad[K_{i},K_{j}]=-\mathbf{i}\epsilon_{ijk}L_{k};\qquad[L_{i},K_{j}]=\mathbf{i}\epsilon_{ijk}K_{k}.

Therefore, if we consider the complex basis

J i +=12(L i+iK i);J i =12(L iiK i),i=1,2,3, J^{+}_{i}=\frac{1}{2}(L_{i}+\mathbf{i}K_{i}); \qquad J^{-}_{i}=\frac{1}{2}(L_{i}-\mathbf{i}K_{i}), \qquad i=1,2,3,

the commutation relations become

[J i +,J j +]=iϵ ijkJ k +;[J i ,J j ]=iϵ ijkJ k ;[J i +,J j ]=0. [J^{+}_{i},J^{+}_{j}]=\mathbf{i}\epsilon_{ijk}J^{+}_{k}; \qquad[J^{-}_{i},J^{-}_{j}]=\mathbf{i}\epsilon_{ijk}J^{-}_{k};\qquad[J^{+}_{i},J^{-}_{j}]=0.

So we have an isomorphism of complex Lie algebras

𝔰𝔬 1,3;𝔰𝔲 2;𝔰𝔲 2;. \mathfrak{so}_{1,3;\mathbb{C}}\cong\mathfrak{su}_{2;\mathbb{C}}\oplus\mathfrak{su}_{2;\mathbb{C}}.

It follows that irreducible complex representations of the Lorentz Lie algebra are tensor products of an irreducible representation of the left'' $\mathfrak{su}_{2}$ subalgebra and of an irreducible representation of theright’‘ 𝔰𝔲 2\mathfrak{su}_{2} subalgebra. In particular, as we will show in Section \ref{su2}, complex irreducible representations of 𝔰𝔬 1,3\mathfrak{so}_{1,3} are indexed by a pair of nonnegative half integers (j +,j )(j^{+},j^{-}), and we will write

(j +,j )=(j +,0)(0,j ) (j^{+},j^{-})=(j^{+},0)\otimes(0,j^{-})

to mean that the 𝔰𝔬 1,3\mathfrak{so}_{1,3} representation indexed by (j +,j )(j^{+},j^{-}) is the tensor product of the representation of the left copy of 𝔰𝔲 2\mathfrak{su}_{2} indexed by j +j^{+} and of the representation of the right copy of 𝔰𝔲 2\mathfrak{su}_{2} indexed by j j^{-}.

The Poincaré group and its Lie algebra

We now extend the Lorentz group by adding to it translations of 4\mathbb{R}^{4}. More precisely, we are considering the semidirect product

𝒫= 4O(1,3), \mathcal{P}=\mathbb{R}^{4}\rtimes O(1,3),

where the Lorentz group acts on 4\mathbb{R}^{4} by its defining representation. In particular we have a shoert exact sequence

0 4𝒫O(1,3)1. 0\to\mathbb{R}^{4}\to\mathcal{P}\to O(1,3)\to1.

This semidirect product is the group of isometries of R 1,3\mathbf{R}^{1,3} as a Minkowskian manifold, and is called the Poincaré group. It is a (noncompact) 10-dimensional Lie group. Its Lie algebra 𝔭\mathfrak{p} is obtained by adding to the Lorentz Lie algebra 𝔰𝔬 1,3\mathfrak{so}_{1,3} four new generators, corresponding to a basis of the abelian Lie algebra 4\mathbb{R}^{4} of infinitesimal translations of 1,3\mathbb{R}^{1,3}. It is customary to take as basis of 4\mathbb{R}^{4} the infinitesimal translations along the coodinate axis. In the identification of elements of 𝔭\mathfrak{p} with vector fields on 1,3\mathbb{R}^{1,3} these are just the coordinate vector fields i\partial_{i}. In the complexified Poincaré Lie algebra we set

P i=i i=iη ij j. P^{i}=\mathbf{i}\partial^{i}=\mathbf{i}\eta^{ij}\partial_{j}.

The commutation relations involving the new generators P iP^{i} are then easily seen to be

[P a,P b]=0;[P a,J bc]=i(η abP cη acP b). [P^{a},P^{b}]=0; \qquad[P^{a},J^{bc}]=\mathbf{i}(\eta^{ab}P^{c}-\eta^{ac}P^{b}).

A Casimir element for 𝔭\mathfrak{p} is

P 2=P aP a=η abP aP b. \| P\|^{2}=P_{a}P^{a}=\eta_{ab}P^{a}P^{b}.

This is an immediate consequence of the fact that the Poincaré group acts isometrically on 1,4\mathbb{R}^{1,4}. Indeed, one trivially has [P 2,P a]=0[\| P\|^{2},P^{a}]=0 and, if Λ=(Λ b a)\Lambda=(\Lambda^{a}_{b}) is an element in the Lorentz subgroup, then

Λ 1(P a)Λ=Λ b aP b, \Lambda^{-1}(P^{a})\Lambda=\Lambda^{a}_{b} P^{b},

where we have used that, since Λ\Lambda is an element of the Lorentz group,

(Λ 1) b a=η bcΛ d cη da. (\Lambda^{-1})^{a}_{b}=\eta_{bc}\Lambda^{c}_{d}\eta^{da}.


Λ 1(P 2)Λ=η abΛ c aη abΛ d bP cP d=η cdP cP d=P 2. \Lambda^{-1}(\| P\|^{2})\Lambda=\eta_{ab}\Lambda^{a}_{c}\eta_{ab}\Lambda^{b}_{d}P^{c}P^{d}=\eta_{cd}P^{c}P^{d}=\| P\|^{2}.

And the invariance of P\| P\| under the conjugacy action of the Lorentz group immediately gives the invariance of P\| P\| under the adjoint action of the Lorentz Lie algebra.

If one prefers a direct Lie algebra computation, then we trivially have [P 2,P a]=0[\| P\|^{2},P^{a}]=0, and

[P 2,J bc] =η daP d[P a,J bc]+η da[P d,J bc]P a =i(η daP d(η abP cη acP b)+η da(η dbP cη dcP b)P a) =i(δ d bP dP cδ d cP dP b+δ a bP cP aδ a cP bP a) =0. \begin{aligned} [\| P\|^{2},J^{bc}]&=\eta_{da}P^{d}[P^{a},J^{bc}]+\eta_{da}[P^{d},J^{bc}]P^{a}\\ &=\mathbf{i}(\eta_{da}P^{d}(\eta^{ab}P^{c}-\eta^{ac}P^{b})+\eta_{da}(\eta^{db}P^{c}-\eta^{dc}P^{b})P^{a})\\ &=\mathbf{i}(\delta_{d}^{b}P^{d}P^{c}-\delta_{d}^{c}P^{d}P^{b}+\delta_{a}^{b}P^{c}P^{a}-\delta_{a}^{c}P^{b}P^{a})\\ &=0. \end{aligned}

Therefore, if

𝒫U() \mathcal{P}\to U(\mathcal{H})

is a unitary representation, the Casimir element P 2\| P\|^{2} will act as an Hermitean operator

P 2: \| P\|^{2}:\mathcal{H}\to\mathcal{H}

and so P 2\| P\|^{2}-eigenspaces will be subrepresentations of \mathcal{H}. In particular, P 2\| P\|^{2} will act as a real scalar on every irreducible unitary representation of 𝒫\mathcal{P}. This scalar is called the mass of the irreducible unitary representation. This is the first ingredient for Wigner’s classification of irreducible unitary representations of the Poincaré group, we will describe in Section \ref{wigner}.

As remarked in Section \ref{casimir}, the Casimir operator P 2\| P\|^{2} corresponds to a (second-order) Poincaré biinvariant differential operator on 1,3\mathbb{R}^{1,3}. Explicitly, this second order operator is

= a a=η ab a b. -\square=-\partial_{a}\partial^{a}=-\eta_{ab}\partial^{a}\partial^{b}.

So P 2\| P\|^{2} corresponds to the opposite of the D’Alembert operator on 1,3\mathbb{R}^{1,3} and the eigenstates equation P 2ϕ=m 2ϕ\| P\|^{2}\phi=m^{2}\phi becomes the Klein-Gordon equation

(+m 2)ϕ=0. (\square+m^{2})\phi=0.

We will come back to this in Section \ref{klein-gordon}

Induced representations and equivariant bundles

In this section we present the induced representation method. Essentially it consists in building a representation of a Lie group GG from a representation of a closed Lie subgroup HGH \subseteq G.


Given a representation VV of HH, the induced representation of GG, usually written Ind H G(V)\mathrm{Ind}^{G}_{H}(V) is by definition the vector space

Ind H G(V)={smooth functions f:GV such that f(hg)=hf(g)hH} \mathrm{Ind}^{G}_{H}(V)= \{\text{smooth functions } f : G \to V \text{ such that } f(hg)=hf(g) \quad\forall h \in H \}

with the obvious GG-action on it.

This construction has a nice geometric interpretation: Ind H G(V)\mathrm{Ind}^{G}_{H}(V) can be naturally realized as the space of sections of a GG-equivariant vector bundle on a GG-homogeneous space. Recall that a GG-equivariant vector bundle 𝒱\mathcal{V} on a GG-manifold MM is the datum of a lifting of the GG-action on MM to a GG-action on the total space of 𝒱\mathcal{V} which is linear on the fibers. Given GG, HH and VV as before, define the vector bundle G× HVG \times_{H} V over G/HG/H by

G× HV=G×V/ G \times_{H} V = G \times V / \sim

where \sim is the equivalence relation

(gh,v)(g,hv)gG,hH,vV. (gh,v) \sim(g,hv) \quad\forall g \in G, \forall h \in H, \forall v \in V .

The projection map π:G× HVG/H\pi: G \times_{H} V \to G/H is given by π(g,v)=gH\pi(g,v) = gH and the GG-action is given by g (g,v)=(g g,v)g^{\prime}(g,v)= (g^{\prime}g,v), for g Gg^{\prime}\in G. Denote by Γ(G/H,G× HV)\Gamma(G/H, G \times_{H} V ) the vector space of sections of G× HV G \times_{H} V. This vector space carries a natural action of GG: if σ\sigma is a section of G× HVG/HG \times_{H} V\to G/H and gg is an element of GG, then

(gσ) x:=g(σ(g 1x)). (g\sigma)_{x}:=g(\sigma (g^{-1} x)).

There is a natural isomorphism of representations of GG between Γ(G/H,G× HV)\Gamma(G/H,G \times_{H} V) and Ind H G(V)\mathrm{Ind}^{G}_{H}(V). Indeed, given a section sΓ(G/H,G× HV)s \in\Gamma(G/H,G \times_{H} V), let f sInd H G(V)f_{s} \in\mathrm{Ind}^{G}_{H}(V) be the function f s:GVf_{s}:G\to V defined by

f s(g)=g 1(s(g)). f_{s}(g)= g^{-1} (s(g)).

Conversely, given fInd H G(V)f \in\mathrm{Ind}^{G}_{H}(V), let s fΓ(G/H,G× HV)s_{f} \in\Gamma(G/H,G \times_{H} V) be section

s f(g)=(g,f(g)). s_{f}(g)= (g,f(g)).

It is straightforward to check that the given construction define morphisms of GG-representations between Γ(G/H,G× HV)\Gamma(G/H,G \times_{H} V) and Ind H G(V)\mathrm{Ind}^{G}_{H}(V) which are inverse each other.

The induced representation contrucition also has an important functroial interpretation: it is the adjoint of the restriction functor

| H:G-representationH-representations, \vert_{H}:G\text{-representation}\to H\text{-representations},

i.e. if VV is a representation of HH and WW is a representation of GG, then there is a natural isomorphism

Hom G(W,Ind H G(V))=Hom H(W| H,V). \mathrm{Hom}_{G}(W, \mathrm{Ind}^{G}_{H}(V)) = \mathrm{Hom}_{H}(W|_{H}, V).

This is known as Frobenius reciprocity formula.

Let us remark that bigger the subgroup HH, smaller the induced representation. For instance if H={e}H= \{ e \} and V=V=\mathbb{C}, then Ind H G(V)=C (G;)\mathrm{Ind}^{G}_{H}(V) = C^{\infty}(G;\mathbb{C}), which is an enormous space.

It is also interesting to remark that in general, even if the representation VV of HH is irreducible, we can’t state anything on the irreducibility of Ind H G(V)\mathrm{Ind}^{G}_{H}(V). On the other hand, if Ind H G(V)\mathrm{Ind}^{G}_{H}(V) is irreducible, then clearly the HH-representation VV is irreducible, since a splitting of VV induces a splitting of Ind H G(V)\mathrm{Ind}^{G}_{H}(V).

Wigner’s theorem

Every unitary irreducible representation of the Poincaré group is induced by a representation of the stabiler subgroup O(1,3) pO(1,3)_{\vec{p}} of some point p\vec{p} for the standard action of the Lorentz group on 1,3\mathbb{R}^{1,3}, a result originally due to Wigner wigner-induced. Furthermore, in the next section we will classify all the possibilities for 𝒫 p\mathcal{P}_{\vec{p}}.


𝒫U() \mathcal{P}\to U(\mathcal{H})

be a unitary representation of the Poicaré group on an Hilbert space \mathcal{H}, and let

𝔭 End() \mathfrak{p}_{\mathbb{C}}\to\mathrm{End}(\mathcal{H})

be the induced Lie algebra represention of the complexified Poincaré Lie algebra. As in the previous sections, let P μP^{\mu} the generators of the translations or 4\mathbb{R}^{4}, multiplied by i\mathbf{i}. Since 𝒫\mathcal{P} acts on \mathcal{H} by unitary operators, the real Lie algebra 𝔭\mathfrak{p} acts on \mathcal{H} by anti-Hermitean operators, and so the P μP^{\mu} act as Hermitean operators. In particular, they are diagonalizable, with a real spectrum. Since they commute, they are simultaneously diagonalizable, and the Hilbert space \mathcal{H} can be decomposed as

= p p, \mathcal{H} = \bigoplus_{p} \mathcal{H}_{\vec{p}},

with p=(p 0,p 1,p 2,p 3)\vec{p}=(p^{0},p^{1},p^{2},p^{3}) ranging in 4\mathbb{R}^{4}. In the above orthogonal decomposition, p\mathcal{H}_{\vec{p}} denotes the p\vec{p}-eigenspace for the P μP^{\mu}: elements of p\mathcal{H}_{\vec{p}} are vectors |p|\vec{p}\rangle of \mathcal{H} such that

P μ|p=p μ|p,for $\mu=0,\dots,3$. P^{\mu}|\vec{p}\rangle=p^{\mu}|\vec{p}\rangle, \qquad\text{for $\mu=0,\dots,3$}.

The action of the translations subgroup of 𝒫\mathcal{P} is then recovered siimply by exponentiation:

(Id,a)|p=e ia μP μ|p=e ia μp μ|p=e i(a|p)|p, (\mathrm{Id},\vec{a})|\vec{p}\rangle=e^{-\mathbf{i}a_{\mu}P^{\mu}}|\vec{p}\rangle=e^{-\mathbf{i}a_{\mu}p^{\mu}}|\vec{p}\rangle= e^{-\mathbf{i}(\vec{a}|\vec{p})}|\vec{p}\rangle,

where (a|p)=η(a,p)(\vec{a}|\vec{p})=\eta(\vec{a},\vec{p}) is the Minkowski inner product on 1,3\mathbb{R}^{1,3}. So in particular each subspace p\mathcal{H}_{\vec{p}} is stable for the action of the translation subgroup of 𝒫\mathcal{P}, which acts by scalar multiplication on each p\mathcal{H}_{\vec{p}}.

We are thus left with the problem of describing the action of the Lorentz group O(3,1)O(3,1): given an eigenvector |p|\vec{p}\rangle, we want to see where it is mapped by an element ΛO(3,1)\Lambda\in O(3,1). The key to answer this question is to notice that the Lie subalgebra of infinitesimal translation is preserved by the conjugacy action of PP on 𝔭 \mathfrak{p}_{\mathbb{C}}. More precisely, recall that if Λ=(Λ ν μ)\Lambda=(\Lambda^{\mu}_{\nu}), we have

Λ 1(P μ)Λ=Λ ν μP ν, \Lambda^{-1}(P^{\mu})\Lambda=\Lambda^{\mu}_{\nu}P^{\nu},

and let us act on Λ|p\Lambda|\vec{p}\rangle with the operator P μP^{\mu}. We have

P μΛ|p=Λ(Λ 1P μΛ)|p=Λ(Λ ν μP ν|p)=Λ(Λ ν μp ν|p)=(Λ ν μp ν)Λ|p. P^{\mu}\Lambda|\vec{p}\rangle= \Lambda(\Lambda^{-1}P^{\mu}\Lambda) |\vec{p}\rangle= \Lambda(\Lambda^{\mu}_{\nu}P^{\nu}|\vec{p}\rangle) = \Lambda(\Lambda^{\mu}_{\nu}p^{\nu}|\vec{p}\rangle)= (\Lambda^{\mu}_{\nu}p^{\nu})\,\Lambda|\vec{p}\rangle.

Therefore Λ|p\Lambda|\vec{p}\rangle is a P μP^{\mu}-eigenvector, with eigenvalue (Λp) μ(\Lambda\vec{p})^{\mu}. We express this in compact form as

Λ|p=|Λp. \Lambda|\vec{p}\rangle=|\Lambda\vec{p}\rangle.

In other words, an element Λ\Lambda in the Lorentz group induces an isomorphism

Λ: p Λp. \Lambda:\mathcal{H}_{\vec{p}}\xrightarrow{\sim}\mathcal{H}_{\Lambda\vec{p}}.

Moreover, if π p: p \pi_{\vec{p}}:\mathcal{H}\to\mathcal{H}_{\vec{p}} denotes the projection on the p\vec{p}-eigenspace, then

π p=Λ 1π ΛpΛ, \pi_{\vec{p}}=\Lambda^{-1}\pi_{\Lambda\vec{p}}\Lambda,

for any Λ\Lambda in the Lorentz group. Indeed, both sides act as the zero operator on an eigenvector |q|\vec{q}\rangle with qp\vec{q}\neq\vec{p}, whereas, acting on |p|\vec{p}\rangle we have

Λ 1π ΛpΛ|p=Λ 1Λ|p=π p|p, \Lambda^{-1}\pi_{\Lambda\vec{p}}\Lambda|\vec{p}\rangle= \Lambda^{-1}\Lambda|\vec{p}\rangle=\pi_{\vec{p}}|\vec{p}\rangle,

since Λ|p\Lambda|\vec{p}\rangle is an element in the eigenspace Λp\mathcal{H}_{\Lambda\vec{p}}. Summing up, for a fixed p 0\vec{p}_{0}, the direct sum

pO(1,3)p 0 p, \bigoplus_{\vec{p}\in O(1,3)\vec{p}_{0}}\mathcal{H}_{\vec{p}},

where p\vec{p} ranges in the Lorentz orbit of p 0\vec{p}_{0}, is a subrepresentation of the representation of 𝒫\mathcal{P} we started with. So if the original representatin was irreducible and p\vec{p} is in the spectrum of the P μP^{\mu}, then

= qO(1,3)p 0 p. \mathcal{H}=\bigoplus_{\vec{q}\in O(1,3)\vec{p}_{0}}\mathcal{H}_{\vec{p}}.

Let m 2=p 0 2=p 0μp 0 μm^{2}=\|\vec{p}_{0}\|^{2}=p_{0\mu} {p_{0}}^{\mu}. The Lorentz group preserves the Minkowski norm, and, if m 20m^{2}\neq0, acts transitively on the set on norm m 2m^{2} vectors, so that p\vec{p} is in the Lorentz orbit of p 0\vec{p}_{0} if and only if p 2=m 2\| p\|^{2}=m^{2}. Therefore, if m 20m^{2}\neq0 we have found

= p 2=m 2 p. \mathcal{H}=\bigoplus_{\|\vec{p}\|^{2}=m^{2}}\mathcal{H}_{\vec{p}}.

Note that from this, in particular we recover that the Casimir element P 2\| P\|^{2} acts as the scalar m 2m^{2} on \mathcal{H}, so that m 2m^{2} is the mass of the irreducible representation \mathcal{H}, in the notations of Section \ref{poincare}. Indeed, by the above decomposition, for any eigenvector |p|\vec{p}\rangle in \mathcal{H} we have

P 2|p=η μνP μP ν|p=η μνp μp ν|p=m 2|p. \| P\|^{2}|\vec{p}\rangle=\eta_{\mu\nu}P^{\mu}P^{\nu}|\vec{p}\rangle=\eta_{\mu\nu}p^{\mu}p^{\nu}|\vec{p}\rangle= m^{2}|\vec{p}\rangle.

Denote by X m 2X_{m^{2}} the mass m 2m^{2} hyperboloid, i.e., the set

X m 2={p 1,3 such that p 2=m 2} X_{m^{2}}=\{\vec{p}\in\mathbb{R}^{1,3}\text{ such that }\|\vec{p}\|^{2}=m^{2}\}

Then the collection of eigenspaces p\mathcal{H}_{\vec{p}} is a vector bundle m 2\mathcal{E}_{m^{2}} over X m 2X_{m^{2}}, and the Hilbert space \mathcal{H} is naturally identified with the Hilbert space of L 2L^{2}-sections of this bundle (with respect to the spectral measure on X m 2X_{m^{2}}). Indeed, each vector ψ\psi in \mathcal{H} is identified with the section σ ψ\sigma_{\psi} defined by σ ψ:pπ p(ψ)\sigma_{\psi}:\vec{p}\mapsto\pi_{\vec{p}}(\psi), where π p: p\pi_{\vec{p}}: \mathcal{H}\to\mathcal{H}_{\vec{p}} is the projection on the p\vec{p}-eigenspace. Moreover the vector bundle \mathcal{E} is clearly O(1,3)O(1,3)-equivariant, and the map

\sigma:\mathcal{H}\xrightarrow{\sim}\{\text{sections of $\mathcal{E}_{m^{2}}$ over $X_{m^{2}}$}\}

is an isomorphism of representations of σ:{sections of $\mathcal{E m 2 \sigma:\mathcal{H}\xrightarrow{\sim}\{\text{sections of $\mathcal{E}_{m^{2}}. This last statement is nothing but the identity π p=Λ 1π ΛpΛ\pi_{\vec{p}}=\Lambda^{-1}\pi_{\Lambda\vec{p}}\Lambda derived above. Therefore we see that we are precisely in the situation described in Section \ref{induced}: let 𝒫\mathcal{P} act on 1,3\mathbb{R}^{1,3} via if we denote by O(1,3) p 0O(1,3)_{\vec{p}_{0}} the stabilizer of p 0\vec{p}_{0} under the Lorentz action, then the Lorentz orbit of p\vec{p} is

X m 2=O(1,3)/O(1,3) p 0 X_{m^{2}}= O(1,3)/O(1,3)_{\vec{p}_{0}}

and the representation of the Lorentz subgroup of 𝒫\mathcal{P} on \mathcal{H} is the representation induced on the space of sections of an equivariant bundle on a Lorentz-homogeneous space by a representation of the stabilizer subgroup O(1,3) p 0O(1,3)_{\vec{p}_{0}} on the fiber p\mathcal{H}_{\vec{p}}. The description of the Poincaré action is completed by recalling that the subgroup of translations acts by scalar mutiplication by e i(a|p)e^{\mathbf{i}(\vec{a}|\vec{p})} on the fibers.

The situation for m 2=0m^{2}=0 is completely similar, but we have to distinguish two cases. Indeed the Lorentz action on the set of zero-norm vectors is not transitive but there are two orbits: one consisting of the 00 vector alone (the vacuum state of physics parlance), and the other consisting of all nonzero zero-norm vectors of 1,3\mathbb{R}^{1,3} (the light cone in the physics jargon). Also, almost nothing changes if instead of the Poincaré group 𝒫\mathcal{P} we consider the universal cover of the connected component of the identity:

𝒫˜= 4SO +(1,3)˜. \tilde{\mathcal{P}}=\widetilde{ \mathbb{R}^{4}\rtimes SO^{+}(1,3)}.

Indeed, nothing changes at the Lie algebra level, and at the group level the only difference is that the orbit space for the action of the universal cover SL(2;SL(2;\mathbb{C} of the proper orthochronous Lorentz group SO +(1,3)SO^{+}(1,3) on 1,3\mathbb{R}^{1,3} is more refined than the orbit space for the full Lorentz group.

Therefore we have finally proven the following.


[Wigner] Irreducible unitary representation of the Poincaré group 𝒫\mathcal{P} are classified by pairs (𝒪,s)(\mathcal{O},s), where 𝒪\mathcal{O} is a Lorentz orbit in 1,3\mathbb{R}^{1,3} and ss is (the isomorphism class of) an irreducible representation of the stabilizer O(1,3) pO(1,3)_{\vec{p}} of a point p\vec{p} in 𝒪\mathcal{O} under the Lorentz action. Moreover, the (𝒪,s)(\mathcal{O},s)-representation is induced by the ss-representation of O(1,3) pO(1,3)_{\vec{p}}. The analogous statement holds for the universal cover 4SL(2;)\mathbb{R}^{4}\rtimes SL(2;\mathbb{C}) of the identity element in the Poincaré group.

The stabilizer subgroups of SL(2;)SL(2;\mathbb{C}) in Wigner’s theorem are called little groups in the physics terminology.

SL(2;)SL(2;\mathbb{C})-orbits on 1,3\mathbb{R}^{1,3} and elemetary particles

We now exhibit the classification SL(2;)SL(2;\mathbb{C})-orbits on 1,3\mathbb{R}^{1,3} and the corresponding little groups. In view of Wigner’s theorem this gives a classification of the irreducible unitary representations of 𝒫\mathcal{P}. When the unitary irreducible representation of the little group involved is finite dimensional, the corresponding representation of 4SL(2;)\mathbb{R}^{4}\rtimes SL(2;\mathbb{C}) is called an elementary particle in physics.

  1. The origin {0} 4\{0 \} \in\mathbb{R}^{4} is a singleton orbit, stabilized by the whole SL(2;)SL(2;\mathbb{C}). As we are going to show below, the only finite dimensional unitary irreducible representation of SL(2;)SL(2;\mathbb{C}) is the trivial one. The corresponding representation of the Poincaré group is called the vacuum state in physics.

  2. {p 1,3|p μp μ=0,p 0>0}\{ \vec{p} \in\mathbb{R}^{1,3} | p_{\mu}p^{\mu}=0, p_{0} \gt 0 \} . Up to rescaling, a representative for this orbit is p 0=(1,0,0,1)\vec{p}_{0}=(1,0,0,1). Recalling the definition of the SL(2;)SL(2;\mathbb{C})-action on 1,3\mathbb{R}^{1,3}, the corresponding little group is

    H={ASL 2()|AM p 0A *=M p 0},M p 0=(1 0 0 0). H = \{ A \in SL_{2}(\mathbb{C}) | AM_{\vec{p}_{0}}A^{*}=M_{\vec{p}_{0}} \}, \quad M_{\vec{p}_{0}}= \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right).


    A=(a b c d), A= \left( \begin{matrix} a & b \\ c & d \end{matrix}\right),

    the equation AM p 0A *=M p 0AM_{\vec{p}_{0}}A^{*}=M_{\vec{p}_{0}} becomes

    (|a| 2 ac¯ ca¯ |c| 2)=(1 0 0 0) \left( \begin{matrix} |a|^{2} & a\bar{c} \\ c\bar{a} & |c|^{2} \end{matrix}\right) = \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)

    and so we find |a|=1|a|=1, bb \in\mathbb{C} and c=0c=0, i.e.,

H={(a b 0 a¯),|a|=1}. H= \left\{ \left( \begin{matrix} a & b \\ 0 & \bar{a} \end{matrix} \right) , |a|=1 \right\} .

Consider now the bijection ×U(1)H\mathbb{C}\times U(1)\leftrightarrow H given by

(z,ζ)(z z¯ζ 0 z¯) (z,\zeta)\leftrightarrow\left( \begin{matrix} z & \bar{z}\zeta\\ 0 & \bar{z} \end{matrix} \right)

With this notation, the multiplication in HH reads

(z,ζ)(zζ)(z z¯ζ 0 z¯)(z z¯ζ 0 z¯)=(zz zz¯ζ+z¯z¯ζ 0 z¯z¯)(zz,z 2ζ+ζ). (z,\zeta)(z'\zeta')\leftrightarrow\left( \begin{matrix} z & \bar{z}\zeta\\ 0 & \bar{z} \end{matrix}\right) \left( \begin{matrix} z' & \bar{z}'\zeta' \\ 0 & \bar{z}' \end{matrix}\right) = \left( \begin{matrix} zz' & z\bar{z}'\zeta' +\bar{z}\bar{z}'\zeta\\ 0 & \bar{z}\bar{z}' \end{matrix}\right)\leftrightarrow(zz',z^{2}\zeta'+\zeta).

Hence, the group HH is identified the semidirect product U(1)\mathbb{C}\rtimes U(1) with U(1)U(1) acting on \mathbb{C} by (z,ζ)z 2ζ(z,\zeta)\mapsto z^{2}\zeta. This is the double cover of the semidirect product of \mathbb{C} with U(1)U(1) given by the standard U(1)U(1) action (z,ζ)zζ(z,\zeta)\mapsto z\zeta. Since the standard U(1)U(1)-action on \mathbb{C} is naturally identified with the SO(2)SO(2)-action on 2\mathbb{R}^{2}, the little group HH is the double cover of the group SE(2)SE(2) of orientation preserving (affine) isometries of the Euclidean plane. Its finite dimensional unitary irreducible representations are classified by an half-integer ε\varepsilon, called the helicity of the representation. The corresponding elementary particle is called a massless helicity ε\varepsilon particle.

  1. {p 4|p μp μ=0,p 0<0}\{ p \in\mathbb{R}^{4} | p_{\mu}p^{\mu}=0, p_{0} \lt 0 \}. The little group is the analogous of the precedent case. The corresponding elementary particle is called a massless helicity ε\varepsilon antiparticle.

  2. {p 4|p μp μ>0,p 0>0}\{ p \in\mathbb{R}^{4} | p_{\mu}p^{\mu}\gt0, p_{0}\gt0 \}. Up to rescaling, a representative for this orbit is p 0=(1,0,0,0)\vec{p}_{0}=(1,0,0,0), which corresponds to the Hermitean matrix

    M p 0=(1 0 0 1). M_{\vec{p}_{0}}=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right).

    Hence the little group is

    H={ASL(2;)|AIdA *=Id}=SU(2). H = \{ A \in SL(2;\mathbb{C}) | A\mathrm{Id}A^{*}=\mathrm{Id} \}=SU(2).

    We will show next that irreducible unitary representations of SU(2)SU(2) are classified by a nonnegative half-integer \ell called the spin. The corresponding elementary particle is called a mass mm spin \ell particle.

  3. {p 4|p μp μ>0,p 0<0}\{ p \in\mathbb{R}^{4} | p_{\mu}p^{\mu}\gt0, p_{0}\lt 0 \}. The little group is the analogous of the previous case. The corresponding elementary particle is called a mass mm spin \ell antiparticle.

  4. {p 4|p μp μ<0}\{ p \in\mathbb{R}^{4} | p_{\mu}p^{\mu}\lt 0 \} Up to rescaling, a representative for this orbit is p 0=(0,1,0,0)\vec{p}_{0}=(0,1,0,0). The little group for this orbit is then:

    H={ASL(2;)|AM p 0A *=M p 0},M p 0=(0 1 1 0) H = \{ A \in SL(2;\mathbb{C}) | AM_{\vec{p}_{0}}A^{*}=M_{\vec{p}_{0}} \},\qquad M_{\vec{p}_{0}}= \left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right)

    It is convenient to rewrite the defining relation for elements of HH as A=M p 0(A *) 1M p 0 1A=M_{\vec{p}_{0}}(A^{*})^{-1}M_{\vec{p}_{0}}^{-1}. The right-hand side of this expression is

    M p 0(A *) 1M p 0 1=(0 1 1 0)(d¯ c¯ b¯ a¯)(0 1 1 0)=(a¯ b¯ c¯ d¯)=A¯, M_{\vec{p}_{0}}(A^{*})^{-1}M_{\vec{p}_{0}}^{-1} = \left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right) \left( \begin{matrix} \bar{d} & -\bar{c} \\ -\bar{b} & \bar{a} \end{matrix}\right) \left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix}\right) = \left( \begin{matrix} \bar{a} & \bar{b} \\ \bar{c} & \bar{d} \end{matrix}\right) = \bar{A},

    so that H=SL(2;)H=SL(2;\mathbb{R}). Corresponding elementary particles are nonphysical since they would have negative mass-square.

Finite dimensional unitary representations of SL(2,)SL(2,\mathbb{C})

Here we prove that the only finite dimensional unitary representation of SL(2;)SL(2;\mathbb{C}) is the trivial one. We borrow the following argument from knapp-trapa, where the case of SL(2;)SL(2;\mathbb{R})-representations is treated. Let

ρ:SL(2;)U(n) \rho:SL(2;\mathbb{C})\to U(n)

be a continuous reperesentation, and consider the subset of SL(2;)SL(2;\mathbb{C}) consisting of the matrices of the form

A(z)=(1 z 0 1),z0. A(z)=\left( \begin{matrix} 1 & z\\ 0 &1 \end{matrix} \right), \qquad z\neq0.

All these matrices have the same Jordan normal form, so they are in the same SL(2;)SL(2;\mathbb{C})-conjugacy class. Therefore their images ρ(A(z))\rho(A(z)) are in the same U(n)U(n)-conjugacy class. Let us call this conjugacy class CC. Since U(n)U(n) is compact, conjugacy classes in U(n)U(n) are closed, so

Id U(n)=lim z0ρ(A(z)) \mathrm{Id}_{U(n)}=\lim_{z\to0}\rho(A(z))

is an element of CC (here we have used the continuity of ρ\rho). But this means that CC is the conjugacy class of the identity in U(n)U(n) and so consists of the identity alone. Since by construction ρ(A(z))\rho(A(z)) lies in CC for every z0z\neq0, we have thus shown that ρ(A(z))=Id U(n)\rho(A(z))=\mathrm{Id}_{U(n)} for every zz in \mathbb{C}. The identical argument applies to the matrices

B(w)=(1 0 w 1),w0. B(w)=\left( \begin{matrix} 1 & 0\\ w &1 \end{matrix} \right), \qquad w\neq0.

The matrices A(z)A(z) and B(w)B(w) generate SL(2;)SL(2;\mathbb{C}), as is easily seen by noticing that the two matrices

(0 1 0 0)and(0 0 1 0) \left( \begin{matrix} 0& 1\\ 0 &0 \end{matrix} \right) \qquad\text{and}\qquad\left( \begin{matrix} 0 & 0\\ 1 &0 \end{matrix} \right)

generate the Lie algebra 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C}). Therefore, ρ\rho is the trivial representation.

It is worth remarking that SL(2;)SL(2;\mathbb{C}) admits nontrivial infinite dimensional unitary representations, see, e.g. dao-nguyen.

Finite dimensional unitary representations of U(1)\mathbb{C}\rtimes U(1)

In this section we show that irreducible unitary finite-dimensional representations of the double cover SE˜(2)\widetilde{SE}(2) of the group SE(2)SE(2) of affine isometries of 2\mathbb{R}^{2} are classified by an half-integer ε\varepsilon; moreover ε\varepsilon is an integer precisely when the given reresentation SE˜(2)U(n)\widetilde{SE}(2)\to U(n) factors through SE˜(2)SE(2)\widetilde{SE}(2)\to SE(2). We thank Andrea Maffei for having shown us this proof.

We begin by recalling that SE˜(2)\widetilde{SE}(2) is the semidirect product U(1)\mathbb{C}\rtimes U(1), with U(1)U(1) acting on \mathbb{C} by (z,ζ)z 2ζ(z,\zeta)\mapsto z^{2}\zeta. Let now

ρ:U(1)U(n) \rho:\mathbb{C}\rtimes U(1)\to U(n)

be a finite dimensional representation. The restriction of ρ\rho to the normal subgroup \mathbb{C} gives a representation

ρ| :U(n) \rho\vert_{\mathbb{C}}:\mathbb{C}\to U(n)

of the real Lie group \mathbb{C} on the complex vector space n\mathbb{C}^{n}. This gives a splitting

n= φIV φ \mathbb{C}^{n}=\bigoplus_{\varphi\in I}V_{\varphi}

where II is a finite subset of Hom (,)\mathrm{Hom}_{\mathbb{R}}(\mathbb{C},\mathbb{R}) and V φV_{\varphi} is the subspace of n\mathbb{C}^{n} where \mathbb{C} acts via ρ\rho as

ρ(1,ζ)(v)=e iφ(ζ)v. \rho(1,\zeta)(\vec{v})=e^{\mathbf{i}\varphi(\zeta)}\vec{v}.

We now look at the U(1)U(1)-action. Since (1,z 2ζ)(z,0)=(z,0)(1,ζ)(1,z^{2}\zeta)\cdot(z,0)=(z,0)\cdot(1,\zeta) we have

ρ(z,0):V φV z 1 *φ, \rho(z,0):V_{\varphi}\to V_{{z^{-1}}^{*}\varphi},

for any φI\varphi\in I, where (z 1 *φ)(ζ)=φ(z 2ζ)({z^{-1}}^{*}\varphi)(\zeta)=\varphi(z^{-2}\zeta). Indeed, if v\vec{v} is an element of V φV_{\varphi}, then

ρ(1,ζ)(ρ(z,0)v)=ρ(z,0)(ρ(1,z 2ζ)v)=e iφ(z 2ζ)ρ(z,0)v. \rho(1,\zeta)\left(\rho(z,0)\vec{v}\right)=\rho(z,0)\left(\rho(1,z^{-2}\zeta)\vec{v}\right)= e^{\mathbf{i}\varphi(z^{-2}\zeta)}\rho(z,0)\vec{v}.

Therefore U(1)U(1) acts as a permutation group on the elemnets of the finite set II. Since U(1)U(1) is connected, this permutation action is trivial, so z 1 *φ=φ{z^{-1}}^{*}\varphi=\varphi for every φI\varphi\in I. But the only U(1)U(1)-invariant element in Hom (,)\mathrm{Hom}_{\mathbb{R}}(\mathbb{C},\mathbb{R}) is the zero morphism, hence the normal subgroup \mathbb{C} of U(1)\mathbb{C}\rtimes U(1) acts trivially on n\mathbb{C}^{n} and the representation ρ\rho factors through U(1)U(1)\mathbb{C}\rtimes U(1)\to U(1). We are therefore reduced to classify irreducible unitary representations of U(1)U(1). Since U(1)U(1) is an abelian Lie group, these are all 1-dimensional, and so everything boils down to the problem of describing Lie group homomorphisms from U(1)U(1) to itself. And it is well known (and easy to show) that these are all of the form zz kz\mapsto z^{k} for some integer kk. Writing k=2εk=2\varepsilon for an half-integer ε\varepsilon we finally find that irreducible unitary representations of the double cover SE˜(2)\widetilde{SE}(2) of the group SE(2)SE(2) of affine isometries of 2\mathbb{R}^{2} are classified by an half-integer ε\varepsilon, and that ε\varepsilon is an integer precisely when the given representation factors through SE˜(2)SE(2)\widetilde{SE}(2)\to SE(2). In the language of particle physics, the half-integer ε\varepsilon is called the helicity of the massless particle corresponding to the given irreducible representation of SE˜(2)\widetilde{SE}(2).

Irreducible representations of SU(2)SU(2)

Since the Lie group SU(2)SU(2) is compact, all its irreducible representations are finite dimensional. Moreover, since SU(2)SU(2) is simply connected, its representation theory is equivalent to the representation theory of its Lie algebra. A set of generators for 𝔰𝔲(2)\mathfrak{su}(2) is given by Pauli matrices

σ 1=(0 1 1 0),σ 2=(0 i i 0),σ 3=(1 0 0 1). \sigma_{1} = \left( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right),\qquad\sigma_{2} = \left( \begin{matrix} 0 & -i \\ i & 0 \end{matrix} \right),\qquad\sigma_{3}= \left( \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right).

By complexifying the Lie algebra 𝔰𝔲 2\mathfrak{su}_{2} we obtain

𝔰𝔲 2;=𝔰𝔲 2𝔰𝔩 2(). \mathfrak{su}_{2;\mathbb{C}}=\mathfrak{su}_{2} \otimes\mathbb{C} \cong\mathfrak{sl}_{2}(\mathbb{C}).

The latter is the Lie algebra of the complex Lie group SL(2;)SL(2;\mathbb{C}). Therefore, we are reduced to studying complex finite dimensional irreducible representations of 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C}). We ar going to show that there is exactly one such representation (up to isomorphism), for any dimension, which can be explicitly described. This is conveniently done by fixing the following set of generators for 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C}):

h=(1 0 0 1),e=(0 1 0 0),f=(0 0 1 0) h= \left( \begin{matrix}1 & 0 \\ 0 & -1 \end{matrix} \right), \qquad e= \left( \begin{matrix}0 & 1 \\ 0& 0 \end{matrix} \right), \qquad f= \left( \begin{matrix}0 & 0 \\ 1& 0 \end{matrix} \right)

with commutation rules

[h,e]=2e,[h,f]=2f,[e,f]=h. [h,e]=2e, \quad[h,f]=-2f, \quad[e,f]=h.

Having fixed this notation, we have


For each nonnegative half-integer ll there exists a unique (up to isomorphism) irreducible complex linear representation of 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C}) on a complex vector space of dimension 2+12\ell+1. Moreover, there is a basis

{v 2,v 2+2,,v 22,v 2} \{ v_{-2\ell},v_{-2\ell+2},\dots,v_{2\ell-2},v_{2\ell}\}

of VV such that:

  1. hv 2(j)=2(j)v ihv_{2(\ell-j)} = 2(\ell-j) v_{i},

  2. ev 2(j)=j(2j+1)v 2(j+1)e v_{2(\ell-j)} = j(2\ell-j+1)v_{2(\ell-j+1)}, with ev 2=0e v_{2\ell}=0,

  3. fv 2(j)=v 2(j1)f v_{2(\ell-j)}=v_{2(\ell-j-1)}, with fv 2=0f v_{-2\ell}=0.

Let VV be a complex linear irreducible representation of 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C}) of dimension 2+12\ell+1, and denote by V λV_{\lambda} the λ\lambda-eigenspace of VV with respect to the operator hh. We begin by showing that

e:V λV λ+2;f:V λV λ2. e:V_{\lambda}\to V_{\lambda+2}; \qquad f:V_{\lambda}\to V_{\lambda-2}.


h(ev)=ehv+[h,e]v=e(λv)+2ev=(λ+2)ev, h(e v) = ehv + [h,e] v = e (\lambda v) + 2 e v = (\lambda+ 2) e v,

and similarly for ff. Assume V λ{0}V_{\lambda}\neq\{0\}. By finite dimensionality of VV, only finitely many eigenspaces V λ+2kV_{\lambda+2k} can be nonzero, so there exists k 0k_{0}\in\mathbb{N} with V λ+2k 00V_{\lambda+2k_{0}}\neq{0} and V λ+2k 0+2=0V_{\lambda+2k_{0}+2}=0. Pick a nonzero vector v 2v_{2\ell} in V λ+2k 0V_{\lambda+2k_{0}} and for any j0j\geq0, set v 2(j)=f jv 2v_{2(\ell-j)}=f^{j}v_{2\ell}. Since v 2(j)v_{2(\ell-j)} is an element of V λ+2(k 0j)V_{\lambda+2(k_{0}-j)} there exist a minimum kk such that v 2((k+1))=0v_{2(\ell-(k+1))}=0. Then the vectors v 2,v 2(1),,v 2(k)v_{2\ell},v_{2(\ell-1)},\dots,v_{2(\ell-k)} are linearly independent (since they are nonzero and belong to hh-eigenspaces for distinct eigenvalues). Now we show that V=span{v 2(k),,v 2}V = \mathrm{span}\{ v_{2(\ell-k)}, \dots, v_{2\ell}\} . Since by hypothesis VV is irreducible, it is sufficient to show that span{v 2(k),,v 2}\mathrm{span}\{ v_{2(\ell-k)}, \dots, v_{2\ell}\} is stable under e,f,he,f,h. The ff- and hh-stability is obvious, so we needonly to check the ee-stability. This results from

ev 2(j)=j(λ+2k 0j+1)v 2(j+1), ev_{2(\ell-j)} = j(\lambda+2k_{0}-j+1) v_{2(\ell-j+1)},

with v 2(j)=0v_{2(\ell-j)} = 0 for j<0j\lt 0 and j>kj\gt k, which we prove inductively. Assume we have proved the statement for jj. Then for j+1j+1 we have:

ev 2((j+1)) =efv 2(j) =hv 2(j)+fev 2(j) =(λ+2k 02j)v 2(j)+j(λ+2k 0j+1)fv 2(j+1) =(j+1)(λ+2k 0j)v 2(j) \begin{aligned} e v_{2(\ell-(j+1))}&= ef \cdot v_{2(\ell-j)}\\ & = h v_{2(\ell-j)} + fev_{2(\ell-j)} \\ &= (\lambda+2k_{0}- 2j) v_{2(\ell-j)} + j(\lambda+2k_{0}-j+1)f v_{2(\ell-j+1)}\\ &= (j+1)(\lambda+2k_{0}-j)v_{2(\ell-j)} \end{aligned}

So V=span{v 2(k),,v 2}V=\mathrm{span}\{ v_{2(\ell-k)}, \dots, v_{2\ell}\}. In particular dimV=k+1\dim V=k+1 and so k=2k=2\ell. This means that our basis for VV is actually

{v 2,v 2+2,,v 22,v 2}. \{ v_{-2\ell},v_{-2\ell+2}, \dots, v_{2\ell-2}, v_{2\ell}\}.

It is now easy to show that λ+2k 0=2\lambda+2k_{0}=2\ell. Indeed, since h=[e,f]h=[e,f] is a commutator, hh is traceless in any representation. Computing the trace of hh on the vector space span{v 2(j)} j=0,,2\mathrm{span}\{ v_{2(\ell-j)}\}_{j=0,\dots,2\ell} we therefore find

0= j=0 2(λ+2k 02j)=(λ+2k 02)(2+1). 0=\sum_{j=0}^{2\ell}(\lambda+2k_{0}-2j)=(\lambda+2k_{0}-2\ell)(2\ell+1).

Therefore, our basis vectors satisfy:

  • hv 2(j)=2(j)v ihv_{2(\ell-j)} = 2(\ell-j) v_{i},

  • ev 2(j)=j(2j+1)v 2(j+1)e v_{2(\ell-j)} = j(2\ell-j+1)v_{2(\ell-j+1)}, with ev 2=0e v_{2\ell}=0,

  • fv 2(j)=v 2(j1)f v_{2(\ell-j)}=v_{2(\ell-j-1)}, with fv 2=0f v_{-2\ell}=0. This shows that up to isomorphisms, a 2+12\ell+1-dimensional irreducible representation od 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C}) is completely determined by its dimension. To prove the existence of an irreducible 2+12\ell+1-dimensional representation of 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C}), consider the free vector space over the set {v 2,v 2+2,,v 22,v 2}\{ v_{-2\ell},v_{-2\ell+2}, \dots, v_{2\ell-2}, v_{2\ell}\} and define an 𝔰𝔩 2()\mathfrak{sl}_{2}(\mathbb{C})-action by defining the action of the generators e,f,he,f,h on the basis elements v 2(j)v_{2(\ell-j)} by the above formulas. To see that the representation defined this way is irreducible, let UU be a nonzero invariant subspace. Since UU is invariant under hh \cdot, UU is spanned by a nonempty subset of the basis {v 2(j)}\{ v_{2(\ell-j)}\}. By repeatedly applying ee and ff to any vector in the basis {v 2(j)}\{ v_{2(\ell-j)}\} one obtains all the others (up to nonzero scalar multiples), and so U=VU = V.


In Section \ref{casimir} we anticipated that the momentum operator J 2\| J\|^{2} acts as the scalar (+1)\ell(\ell+1) on the 2+12\ell+1 irreducible representation VV of SU(2)SU(2). We can now prove this statement. Let {v 2(j)} j=0,,2\{ v_{2(\ell-j)}\}_{j=0,\dots,2\ell} be a distinguished basis of VV as described above. Since J 2\| J\|^{2} acts as a scalar on VV, we can compute this scalar simply by looking at the action of J 2\| J\|^{2} on the vector v 2v_{2\ell}. We have

J 1=12h;J 2=i2(ef);J 3=12(e+f) J_{1}=-\frac{1}{2}h;\qquad J_{2}=\frac{\mathbf{i}}{2}(e-f);\qquad J_{3}=-\frac{1}{2}(e+f)

and so

J 2v 2=14(h 2+2ef+2fe)v 2=14(4 2+4)v 2=(+1)v 2. \| J\|^{2}v_{2\ell}=\frac{1}{4}(h^{2}+2ef+2fe)v_{2\ell}=\frac{1}{4}(4\ell^{2}+4\ell)v_{2\ell}=\ell(\ell+1)v_{2\ell}.

The Klein-Gordon equation

As an illustrative example of the above construction, we investigate the solutions of the Klein-Gordon equation on 1,3\mathbb{R}^{1,3}. We begin by considering the defining action of the Poincaré group on 1,3\mathbb{R}^{1,3}, lifted to an action of 4SL(2;)\mathbb{R}^{4}\rtimes SL(2;\mathbb{C}). Since the standard Lesbegue measure on 1,3\mathbb{R}^{1,3} is translation- and Lorentz-invariant, we have an induced unitary representation on the Hilbert space =L 2( 1,3)\mathcal{H}=L^{2}(\mathbb{R}^{1,3}) of square-integrabel functions on 1,3\mathbb{R}^{1,3}. Passing to Lie algebras, the (universal cover of the) Poincaré group acts by vector fields, i.e., by derivations. A major technical point to be stressed is that the first-order differential operator one gets this way, i.e., corresponding to elements in 𝔭 \mathfrak{p}_{\mathbb{C}} are only densely defined on \mathcal{H}. For instance, the element P μP^{\mu} corresponds to the derivation i μ\mathbf{i}\partial^{\mu}. This means that 𝔭 \mathfrak{p}_{\mathbb{C}} can not be handled as a Lie algebra of operators on \mathcal{H} within the framework of classical Hilbert spaces. A rigorous treatment can be given within the framework of rigged Hilbert spaces/Gelfan’d triples, see gelfand-vilenkin. Yet, such a rigorous treatment is not the aim of this note, so we will kindly ask the reader to pretend that xe i(p|x)\vec{x}\mapsto e^{-\mathbf{i}(\vec{p}|\vec{x})} is a square-integrable function on 1,3\mathbb{R}^{1,3}.

Let now m 2>0m^{2}\gt 0 and consider the m 2m^{2}-eigenspace equation

P 2ϕ=m 2ϕ \| P\|^{2}\phi=m^{2}\phi

for the Casimir element P 2\| P\|^{2}. Since P μP^{\mu} acts on \mathcal{H} as i μ\mathbf{i}\partial^{\mu}, this equation is the Klein-Gordon equation on 1,3\mathbb{R}^{1,3}:

(+m 2)ϕ=0, (\square+m^{2})\phi=0,

where = μ μ\square=\partial_{\mu}\partial^{\mu} is the D’Alembert operator on 1,3\mathbb{R}^{1,3}. Therefore, we see that the space of solutions of the Klein-Gordon equation is a representation of the Poincaré group. Obviously, this could be directly seen by the manifest Poincaré-invariance of the D’Alembert operator. What we have gained by deriving this from the abstract nonsense of Casimir operators is that we now see the Klein-Gordon equation as a stand-alone equation, but as a piece of the larger picture of Wigner’s investigation of unitary representations of the Poincaré group. And in the larger picture we know that the representation of the Poincaré group given by solutions of the Klein-Gordon equation is induced by a representation of the little group at p\vec{p} on p\mathcal{H}_{\vec{p}}, where p\vec{p} is a point in 1,3\mathbb{R}^{1,3} with p 2=m 2\|\vec{p}\|^{2}=m^{2}. By definition p\mathcal{H}_{\vec{p}} is the p\vec{p}-eigenspace for the action of the infinitesimal translations P μP^{\mu}, hence it is defined as the space of the joint solutions of the first-order differential equations

i μϕ=p μϕ. \mathbf{i}\partial^{\mu}\phi=p^{\mu}\phi.

Therefore we find that p\mathcal{H}_{\vec{p}} consists of the functions

ϕ(x)=e i(p|x)ϕ 0,ϕ 0. \phi(\vec{x})=e^{-\mathbf{i}(\vec{p}|\vec{x})}\phi_{0}, \qquad\phi_{0}\in\mathbb{C}.

In particular it is a 1-dimensional space, and so for m 2>0m^{2}\gt 0 the Klein-Gordon equation describes a spin 0 mass mm elementary particle. We will come back to this when discussing the massive scalar field.

Since p\mathcal{H}_{\vec{p}} is 1-dimensional, the general discussion in Section \ref{wigner} tells us that the space of solutions of the Klein-Gordon equation is naturally identified with the space of sections of a line bundle over the mass m 2m^{2} hyperboloid. This can be nicely interpreted as the Fourier transform of the Klein-Gordon equation,

(p 2m 2)ϕ^(p)=0 (\| p\|^{2}-m^{2})\hat{\phi}(\vec{p})=0

telling us that ϕ^\hat{\phi} is a distribution on 1,3\mathbb{R}^{1,3} which is supported on the mass m 2m^{2} hyperboloid p 2m 2\| p\|^{2}-m^{2}. If p 0\vec{p}_{0} is a vector in 1,3\mathbb{R}^{1,3}, the p 0\vec{p}_{0}-eigenspace equations become, via the Fourier transform,

(p μp 0 μ)ϕ^(p)=0, (p^{\mu}-{p_{0}}^{\mu})\hat{\phi}(\vec{p})=0,

which show that p 0\vec{p}_{0}-eigenstates are distributions supported at the point p\vec{p}. Since a distribution supported on a point is a finite linear combination of derivatives of δ\delta-functions at that point, we have

ϕ^= |I|na I Iδ p 0 \hat{\phi}=\sum_{|I|\leq n}a_{I}\partial_{I}\delta_{\vec{p}_{0}}

for suitable coefficients a Ia_{I} in \mathbb{C}, see, e.g., vladimirov. The condition that ϕ^\hat{\phi} is annihilated by the ideal (p μp 0 μ) μ=0,,3(p^{\mu}-{p_{0}}^{\mu})_{\mu=0,\dots,3} then implies that all the coefficients a Ia_{I} with |I|>0|I|\gt 0 vanish, i.e. ϕ^\hat{\phi} is a scalar multiple of the Dirac’s δ\delta at p 0\vec{p}_{0}:

ϕ^(p)=σ(p)δ p 0(p), \hat{\phi}(\vec{p})=\sigma(\vec{p})\delta_{\vec{p}_{0}}(\vec{p}),

and σ\sigma is naturally interpreted as a section of the trivial line bundle over the mass m 2m^{2} hyperboloid.


Tim van Beek: …(obsolete comment deleted)…

The construction of a free scalar real field is now an example on the page “Wightman axioms” on the nLab.

Giuseppe: Thank you a lot for your comment, I just corrected. I wrote a statement too careless. The costruction of yours for a quantum field is the so-said “second quantization” isn’t it? I know it’s an inevitable approach for quantum field theory, so I planned to introduce it for the interaction part of the thesis. By the way I’m consulting about this with my advisor. Thank you a lot for your contribution.

Tim van Beek: You ‘re welcome! Feel free to discuss any topic in the corresponding thread on the nForum. “Second Quantization”: Quantum mechanics (QM) is not relativistic, that is it does not take into account special relativity. One core feature of special relativity (SR) is the famous equation E=mc 2E = mc^2, which says that matter is a manifestation of energy. Quantum field theory (QFT) combines QM and SR, so one effect QFT has to incorporate is particle creation and annihilation (energy transforms into matter, matter transforms into energy). QM does not have that. QFT needs a state space that contains states with any prescribed number of particles, that is what second quantization builds out of the state space with a fixed number of particles of quantum mechanics.

First quantization is “classical one particle state space” -> “quantum one particle state space”.

Second quantization is “quantum one particle state space” -> “multiple particle state space” (multiple means any positive integer, not one in particular).

The “second quantization” step is the part where I wrote “construct the (bosonic) Fock space”. You start with a one particle state space, that is a Hilbert space H. Two non interacting particles would live in HHH \otimes H. Three non interacting particles in HHHH \otimes H \otimes H. So the general Fock space is

F(H)= i=1 H i F(H) = \oplus_{i=1}^{\infty} H^{\otimes i}

Now there are Bosons and Fermions in nature, Bosons live in the subspace of F(H)F(H) of all symmetric states (interchanging two particles does not change anything), Fermions in the subspace of all asymmetric subspaces (interchanging two particles produces a -1 factor).

Tim van Beek:

A book that I recommend to every mathematician who would like to learn QFT for the first time is this:

  • Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, Math. Surveys and Monographs 149 (ZMATH)

Yang-Mills theory

Representations of U(N)U(N) and SU(N)SU(N)

We exhibit irreps for SU(N)SU(N) and U(N)U(N) for every NN \in \mathbb{N}. First we compute irreps for U(1)U(1) and SU(2)SU(2) directly.

Irreducible representations for U(1)U(1)

The group U(1)U(1) is also called the circle group since it is the set of the complex number of norm 1:

U(1)={a||a|=1}=S 1. U(1) = \{ a \in \mathbb{C} | |a| = 1 \} = S^1.

The group U(1)U(1) is not simply connected, but it is compact and connected. Hence we can’t rely on infinitesimal method for calculating irreps. The fundamental tool in this case is the Schur’s Lemma.

Lemma. Let VV and WW be two complex irreducible representations for a group GG then

Hom G(V,W)={0ifVW 1ifVW Hom_G (V,W) = \left\{ \begin{matrix} 0 if V \ncong W \\ 1 if V \cong W \end{matrix} \right.

Corollary 1. When V=WV=W, Hom G(V,W)=IHom_G(V,W)=\mathbb{C}I

Corollary 2. All complex irreducible representations VV of an abelian group are one dimensional

Proof of the Corollary 2. Every representation of a gGg \in G, as an element of GL(V)GL(V) is in Hom G(V,V)Hom_G(V,V) and commute with the representation. Then by Corollary 1 it is a multiple of the identity. The invariant subspaces of the identity are all one dimensional. Finally the only allowed irreducible representation must be one dimensional.

This result can be applied to U(1)U(1), since it is a commutative group. Then any VV complex representation is one dimensional. Moreover the compactness of U(1)U(1) implies that all irreps are unitary. Then a representation for U(1)U(1) is a map:

π:U(1)U(1). \pi : U(1) \to U(1).

The problem is now to classify all possible continuous group automorphism of U(1)U(1).

Proposition. Every continous automorphism of U(1)U(1) can be written as:

f n(θ)=e inθ,θ[0,2π] f_n(\theta)= e^{i n \theta}, \theta \in [0, 2 \pi ]

for a choice of n.n \in \mathbb{Z}.

Proof.Obviously f nf_n is a continuous automorphism for S 1S^1. The viceversa can be proven by remarking that {f n} n\{ f_n \}_{n \in \mathbb{Z}} is a basis for L 2[S 1]L^2[S^1], then any continuous function S 1S^1-valued can be written as linear combination of f nf_n. Remark that the sum may be infinite and the S 1S^1-valued continuous function are contained in L 2[S 1]L^2[S^1]. Now the condition

(1)gf(θ)=f(gθ), g \bullet f(\theta) = f(g \bullet \theta),

compels the sum to be over a single element f nf_n. Otherwise let ff be the sum of at least two terms f mf_m and f nf_n:

f(θ)=af n(θ)+bf m(θ), f(\theta) = af_n(\theta) + bf_m(\theta),

Then we can prove that ff does not satisfy the (1)

f(g ηθ)=f(θ+η)=af n(θ+η)+bf m(θ+η)=af n(θ)f n(η)+bf m(θ)f m(η)f(η)f(θ)=g ηf(θ). f(g_{\eta} \bullet \theta ) = f(\theta + \eta) = af_n(\theta + \eta) + bf_m(\theta + \eta) = a f_n(\theta)f_n(\eta) + bf_m(\theta)f_m(\eta) \ne f(\eta)f(\theta) = g_{\eta} \bullet f(\theta).

This completes the proof. We have a one on one correspondance between \mathbb{Z} and the irreducible representations of U(1)U(1).

Irreducible representations for SU(2)SU(2).

The group SU(2)SU(2) is the closed sub-group of U(2)U(2) with determinant one. Since U(2)U(2) is compact, so it is SU(2)SU(2). Moreover, as a Lie group it is three dimensional and the generators are the followings:

(2)σ 1=(0 1 1 0),σ 2=(0 i i 0),σ 3=(1 0 0 1). \sigma_1 = \left( \begin{matrix} 0 && 1 \\ 1 && 0 \end{matrix} \right), \sigma_2 = \left( \begin{matrix} 0 && -i \\ i && 0 \end{matrix} \right), \sigma_3= \left( \begin{matrix} 1 && 0 \\ 0 && -1 \end{matrix} \right).

We don’t want to work with 𝔰𝔲(2)\mathfrak{su}(2): by the general theory it is equivalent to consider the complexification 𝔰𝔩(2)\mathfrak{sl}(2) generated by:

(3)H=(1 0 0 1),E=(0 1 0 0),F=(0 0 1 0) H = \left( \begin{matrix} 1 && 0 \\ 0 && -1 \end{matrix} \right), E = \left( \begin{matrix} 0 && 1 \\ 0 && 0 \end{matrix} \right), F= \left( \begin{matrix} 0 && 0 \\ 1 && 0 \end{matrix} \right)

with commutation rules:

(4)[H,E]=2E [H,E]= 2E
(5)[H,F]=2F [H,F]= -2F
(6)[E,F]=H. [E,F]= H.

Finally the group SU(2)SU(2) is topologically equivalent to a sphere, then it is simply connected. Utilizing the infinitesimal method for Lie group representation, we can study the representations of its algebra complexified 𝔰𝔩(2)\mathfrak{sl}(2) generated by H,E,FH,E,F and extend the results directly to SU(2)SU(2). Remark that we have already given a concrete realization for the irreducible representation of 𝔰𝔩(2)\mathfrak{sl}(2) in Lie groups and algebras representations, so that the majority of the following computations could result in a repetition. We want to proceed anyway for emphasizing the connection between group and algebra representation for compact and simply connected groups. First we introduce a typical SU(2)SU(2) representation. Define

V n( 2)={p[z 1,z 2]|pishomogeneous,deg(p)=n}.V_n(\mathbb{C}^2)= \{ p \in \mathbb{C}[z_1,z_2] | p is homogeneous, deg(p)=n\}.

The set V n( 2)V_n(\mathbb{C}^2) is actually a vector space with base {z 1 kz 2 nk|okn}\{ z_1^k z_2^{n-k} | o \le k \le n \} with Dim(V n( 2))=n+1Dim(V_n(\mathbb{C}^2)) = n+1.

Let zz be z=(z 1,z 2)z=(z_1,z_2). The SU(2)SU(2) action is the following:

(7)gp(z)=p(g 1z). g \bullet p (z) = p(g^{-1} \bullet z).


g=(a b¯ b a¯),g = \left( \begin{matrix} a && - \bar{b} \\ b && \bar{a} \end{matrix} \right),


gz 1 kz 2 nk=(a¯z 1+b¯z 2) k(bz 1+az 2) nk. g \bullet z_1^k z_2^{n-k}= (\bar{a} z_1 + \bar{b}z_2)^k(-bz_1 + az_2)^{n-k}.

The action (7) defines a representation since with an easy calculation the following relation can be proven:

(8)[g 1(g 2p=](z)=[g 1g 2P](z). [g_1 \bullet(g_2 \bullet p =](z) = [g_1 g_2 \bullet P](z).

Naturally the algebra too acts over V n( 2)V_n(\mathbb{C}^2). Given an element

X=(ix w¯ w ix)SU(2),X = \left( \begin{matrix} ix && - \bar{w} \\ w && -ix \end{matrix} \right) \in SU(2),

the action is explicitly written:

(9)X(z 1 kz 2 nk)=kw¯z 1 k1z 2 nk+1+i(n2k)xz 1 kz 2 nk+(kn)wz 1 k+1z 2 nk1. X \bullet (z_1^k z_2^{n-k}) = k\bar{w}z_1^{k-1} z_2^{n-k+1}+i(n-2k)xz_1^k z_2^{n-k} + (k-n)w z_1^{k+1} z_2^{n-k-1}.

This is easily calculated by doing the exponential of the matrix tXtX, t t \in \mathbb{C}, then deriving in tt and finally evaluating the expression in t=0t=0.

Proposition. The representation V n( 2)V_n(\mathbb{C}^2) for 𝔰𝔲(2)\mathfrak{su}(2), with action (9), is irreducible.

Proof. Through a substitution we compute the action of H,EH,E and FF:

(10)E(z 1 kz 2 nk)=kz 1 k1z 2 nk+1. E \bullet (z_1^k z_2^{n-k}) = -k z_{1}^{k-1} z_2^{n-k+1}.
H(z 1 kz 2 nk)=(n2k)(z 1 kz 2 nk) H \bullet (z_1^k z_2^{n-k}) = (n-2k) (z_1^k z_2^{n-k})
F(z 1 kz 2 nk)=(kn)(z 1 k+1z 2 nk+1). F \bullet (z_1^k z_2^{n-k}) = (k-n)(z_1^{k+1} z_2^{n-k+1}) .

By these three relations, the representation V n( 2)V_n(\mathbb{C}^2) results irreducible by construction: with the action of EE and FF we obtain all n+1n+1 possible elements of the base, so that the only allowed invariant sub-space are {0} \{0\} and V n( 2) V_n(\mathbb{C}^2).

Now given any irreducible representation VV for 𝔰𝔲(2)\mathfrak{su}(2), we want identify VV with V n( 2)V_n(\mathbb{C}^2), existn\exist n \in \mathbb{N}. The vector space V n( 2)V_n(\mathbb{C}^2) is a good candidate since by varying nn it can cover any finite dimension. We remark again that we can work with a representation for 𝔰𝔲(2)\mathfrak{su}(2) without losing any generality, since SU(2)SU(2) is connected and simply connected (see Lie groups and algebras representations).

Let VV be a representation for 𝔰𝔲(2)\mathfrak{su}(2) then existv 0V \exist v_0 \in V such that Hv 0=λv 0Hv_0=\lambda v_0, λ\lambda \in \mathbb{C} and Ev 0=0Ev_0 = 0. Remark that we are not writing explicitly the action of 𝔰𝔲(2)\mathfrak{su}(2) over VV. Since EE is nihilpotent, so it is the action over VV. Then existk\exist k such that E kv=0E^k v =0 and E k1v0E^{k-1}v \ne 0, for a given vVv \in V. Define E k1v=v 0E^{k-1} v = v_0. Now using commutation rules we show that v 0v_0 is an eigenvector for the action of HH:

(11)0=Ev 0=[H,E]v 0=HEv 0EHv 0=EHv 00 = Ev_0 = [H,E]v_0 = HE v_0 - EH v_0 = - EH v_0

Hence Hv 0=λv 0Hv_0 = \lambda v_0, λ\lambda \in \mathbb{C}.

Define v i=F iv 0v_i = F^i v_0. We want to compute Hv iHv_i. First remark that

Hv 1=HFv 0=[H,F]v 0+FHv 0=2Fv 0+λFv 0=(λ2)v 1. Hv_1 = HFv_0 = [H,F] v_0 + FHv_0 = -2Fv_0 + \lambda Fv_0 = (\lambda - 2)v_1.

Then by iterating

Hv i=(λ2i)v i.Hv_i = (\lambda - 2i)v_i.

An analogous computation can be made for EE. It results:

Ev i=i(λi+1)v i1.Ev_i = i(\lambda -i+1)v_{i-1}.

Call nn the smallest natural number satisfying v n+1=0v_{n+1} = 0. The set {v i} i=0..n\{v_i\}_{i=0..n} is a basis for VV. If W=span({v i} i=0..n)W= span(\{v_i\}_{i=0..n}) is such that WVW \subsetneq V then WW is a sub-representation for VV, but VV is irreducible, then a contraddiction occurs. Finally, we know that the trace of the action of HH is zero, also by making the sum over the eigen-value of v iv_i we obtain the following expression for the trace:

Tr(H)=(λn)n. Tr(H) = (\lambda-n)n.

Then λ=n\lambda=n. We can conclude that VV n( 2)V \cong V_n(\mathbb{C}^2) as representations by mapping each v iv_i with z i=z 1 iz 2 miz_i = z_1^{i}z_2^{m-i} and remarking that such a isomorphism between vector space commute with the action of 𝔰𝔲(2)\mathfrak{su}(2) due to relations (10). It results that every irrep VV is of the type V n( 2)V_n(\mathbb{C}^2), for a given nn. Then we classified any irrep for 𝔰𝔲(2)\mathfrak{su}(2) and, consequently, for SU(2)SU(2).

Irreducible representations for SU(N)SU(N)

The group SU(N)SU(N) is connected, simply connected and compact for every NN. Hence we can study the representation of the algebra 𝔰𝔲(n)\mathfrak{su}(n). As for 𝔰𝔲(2)\mathfrak{su}(2), the algebra 𝔰𝔲(n)\mathfrak{su}(n) is:

𝔰𝔲(n)={AGL n()|Tr(A)=0,A *=A}. \mathfrak{su}(n) = \{ A \in GL_n(\mathbb{C}) | Tr(A)=0, A^* = A \} .

The complexified algebra 𝔰𝔲(n) =𝔰𝔲(n)\mathfrak{su}(n)_\mathbb{C} = \mathfrak{su}(n) \otimes \mathbb{C} is 𝔰𝔩(n)={AGL n()|Tr(A)=0},\mathfrak{sl}(n) = \{ A \in GL_n(\mathbb{C}) | Tr(A) = 0 \}, , as already seen for SU(2)SU(2). The problem is now classifying the irreducible representations for 𝔰𝔩(n)\mathfrak{sl}(n). Since it is a semi-simple Lie algebra we can follow the general theory (see Lie groups and algebras representations).

First we find a maximal Cartan’s sub algebra 𝔥𝔰𝔩(n)\mathfrak{h} \in \mathfrak{sl}(n), i.e. the biggest commutative subalgebra. This is easily achieved by considering the algebra 𝔥\mathfrak{h} generated by the n1n-1 traceless diagonal matrices: they commute each other, form a vector space and it is maximal (any try of adding another element of 𝔰𝔲(n)\mathfrak{su}(n) to 𝔥\mathfrak{h} results in losing the commutative property).

Given a representation VV for 𝔰𝔩(n)\mathfrak{sl}(n), as usual we decompose VV in eigenspaces V αV_\alpha such that hv=αvh \bullet v = \alpha v for vV αv \in V_\alpha and h𝔥h \in \mathfrak{h} . When an highest weight is found between the eigenvalues, say β\beta, we can generate an irreducible representation WVW \subset V by the action of g𝔰𝔩(n)g \in \mathfrak{sl}(n) such that gv0g \bullet v \ne 0 for vV βv \in V_\beta. This representation is irreducible by construction.

We present now a more specific characterization of such an irreducible representation. We introduce a more fitting notation: let L i:𝔰𝔩(n)L_i : \mathfrak{sl}(n) \to \mathbb{C} be the linear functional such that

L i(a 1 a 1 a 1 a 1 a 1 a 1 a 1 a n)=a i. L_i \left( \begin{matrix} a_1 & \phantom{a_1} & \phantom{a_1} \\ \phantom{a_1} & \ddots & \phantom{a_1} \\ \phantom{a_1} & \phantom{a_1} & a_n \end{matrix} \right) = a_i .


𝔰𝔩(n)={AGL n()|L 1(A)+L n(A)=0)}. \mathfrak{sl}(n) = \{ A \in GL_n(\mathbb{C}) | L_1(A)+ \dots L_n(A) = 0) \}.

Let VV be the standard representation. Given the standard basis of VV, e 1,,e ne_1, \dots , e_n this is a basis of eigenvectors with weights L iL_i. Choose an L iL_i, say L 1L_1, then the representation VV has highest weight L 1L_1. Following the general theory, the set {L i} i=1,..,n\{L_i\}_{i=1,..,n} generates a lattice in n\mathbb{R}^n, so that every possible weight for any representation lays in this lattice. According to the algebra representation theory, a typical highest weight could be written as: α=(a 1+..+a n1)(L 1)+..+a n1L n1 \alpha = (a_1+ .. +a_{n-1})( L_1 ) + .. + a_{n-1} L_{n-1}. Such an α\alpha could be realized as eigenvalue of an eigenvector in the vector space X=Sym a 1(V)..Sym a n1(Λ n1V) X = Sym^{a_1}(V)\otimes .. \otimes Sym^{a_{n-1}}(\Lambda^{n-1}V) (since the eigenvalue of a tensor product of eigenvectors is the sum of each eigenvalue). Remark that the representation XX is not irreducible in general, but it contains an irrep WW, generated through the action of 𝔰𝔩(n)\mathfrak{sl}(n) over vX α v \in X_{\alpha} , as previously stated.

Referring to Lie groups and algebras representations, we know that the Schur functor applied to the standard representation VV of GL n()GL_n(\mathbb{C}) furnishes all finite dimensional irreducible representations 𝕊 λ(V)\mathbb{S}_{\lambda}(V) for each λ=(λ 1,...,λ n)\lambda= (\lambda_1,..., \lambda_n), such that λ 1...λ n \lambda_1 \ge ... \ge \lambda_n . It follows that 𝕊 λ(V) \mathbb{S}_{\lambda}(V) is irreducible for SL n()SL_n(\mathbb{C}) too, since it is a subgroup of GL n()GL_n(\mathbb{C}). In particular it is an irrep for the Lie algebra 𝔰𝔩(n)\mathfrak{sl}(n). Now we have to check that every possible highest weight value for a representation of 𝔰𝔩(n)\mathfrak{sl}(n) is assumed in 𝕊 λ(V)\mathbb{S}_{\lambda}(V). The answer is the following proposition:

Proposition. The representation 𝕊 λ(V)\mathbb{S}_{\lambda}(V), with VV the standard representation, is the irreducible representation of 𝔰𝔩(n)\mathfrak{sl}(n) with highest weight λ 1L 1+...+λ nL n\lambda_1 L_1 + ... + \lambda_n L_n.

Representations of SU(3)SU(3) and quarks (the eightfold way)

The Goldstone theorem

The Higgs mechanism

The Higgs boson

Revised on March 8, 2011 at 11:45:56 by Domenico Fiorenza