Particles are irreps of the Poincaré group

2.3 Particles are irreps of the Poincaré group

The Poincaré group comprises all the physical symmetries of “flat” spacetime (i.e, without gravity), i.e. all the transformations which leave the laws of physics invariant. These include Lorentz transformations (boosts and rotations) and spacetime translations.

Particles can be defined as a “set of states which mix only among themselves under Poincaré transformations” (Schwartz [86] Ch. 8.1), leaving attributes like their mass and spin invariant. Elementary particles are those for which there is no smaller subset of states that also have this property. Thus, they correspond exactly to irreducible representations of the Poincaré group! That the physical and seemingly nebulous concept of a particle can be so precisely defined and characterized by a mathematical analysis of the symmetries of spacetime is one of the most beautiful results of fundamental physics.

In this section, we describe the irreps of the Poincaré group, starting first with the Lorentz group alone.

The (proper, orthochronous) Lorentz group

We know from special relativity that “flat” spacetime (i.e., without gravity) is described by 4D Minkowski space $ℝ^{1, 3}$ . This is a real vector space equipped with the metric $η_{μν} = diag (1, - 1, - 1, - 1)$ , which defines distances, or inner products $⟨⋅,⋅⟩$ , between 4-vectors $x_{μ} = (x_{0}, x_{1}, x_{2}, x_{3})$ as:

⟨ x, y ⟩ \equiv x_{μ} y^{μ} \equiv η_{μν} x^{μ} y^{ν} = x_{0} y_{0} - x_{1} y_{1} - x_{2} y_{2} - x_{3} y_{3} .

(2.3.1)

Definition 2.3.1. The Lorentz group is the group of all matrices $M$ orthogonal under the Minkowski metric $M^{T} ηM = η$ , and is called $O (1, 3)$ . This is the analog in flat spacetime to distance-preserving transformations in Euclidean space (e.g., $O (3)$ ).

Definition 2.3.2. The proper, orthochronous Lorentz group ${SO}^{+} (1, 3)$ is the subgroup of $O (1, 3)$ matrices continuously connected to the identity. Physically, these are the transformations that preserve the orientation of space and direction of time, and are typically what we refer to as Lorentz transformations. The two transformations of $O (1, 3)$ not included in ${SO}^{+} (1, 3)$ are parity $P = diag (1, - 1, - 1, - 1)$ and time reversal $T = diag (- 1, 1, 1, 1)$ (shown in the 4-vector representation), which flip the sign of spatial and temporal components of 4-vectors, respectively. Surprisingly, these are not symmetries of nature — they are violated by the weak interaction! Generally, in this chapter, when we talk about the Lorentz group or Lorentz invariance, we are referring only to the proper, orthochronous Lorentz group.

Generators of the Lorentz group

Lorentz transformations $Λ$ are generated by six antisymmetric matrices, three for boosts ( $K_{i}$ ) and three for rotations ( $J_{i}$ ). In the $4$ -vector representation, these are:

\begin{matrix} \begin{aligned} K_{x} & = - i (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), K_{y} = - i (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), K_{z} = - i (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}), \\ J_{x} & = i (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 \end{matrix}), J_{y} = i (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \end{matrix}), J_{z} = i (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) . \end{aligned} \end{matrix}

(2.3.2)

Lorentz transformations can thus be represented as

Λ (𝜃, β) = e^{i (𝜃_{i} J_{i} + β_{i} K_{i})},

(2.3.3)

where $𝜃$ and $β$ are the rotation and boost parameters, respectively, An important property of the Lorentz group is that it is not compact. This is related to the fact that the generators for boosts $K_{i}$ in the representation above are not Hermitian, which means the corresponding group elements $e^{i β_{i} K_{i}}$ are not unitary. In fact, there are no finite-dimensional unitary representations of the Lorentz group [87]. Unitarity of operators is an important condition for the invariance of physical properties under transformations in QM, and the consequences of this for the SM will be discussed in Chapter B.4.

Lie algebra of the Lorentz group

From Eq. 2.3.2, we can derive the commutation relations of the generators and, hence, the Lie algebra:

\begin{matrix} \begin{aligned} [K_{i}, K_{j}] & = - i 𝜖_{ijk} J_{k}, \\ [J_{i}, J_{j}] & = i 𝜖_{ijk} J_{k}, \\ [J_{i}, K_{j}] & = i 𝜖_{ijk} K_{k} . \end{aligned} \end{matrix}

(2.3.4)

Moreover, if we define the operators

J_{i}^{+} = \frac{1}{2} (J_{i} + i K_{i}), J_{i}^{-} = \frac{1}{2} (J_{i} - i K_{i}),

(2.3.5)

we find that $𝔰𝔬 (1, 3)$ contains two mutually commuting $𝔰𝔲 (2)$ subalgebras:

\begin{matrix} \begin{aligned} [J_{i}^{+}, J_{j}^{+}] & = i 𝜖_{ijk} J_{k}^{+}, \\ [J_{i}^{-}, J_{j}^{-}] & = i 𝜖_{ijk} J_{k}^{-}, \\ [J_{i}^{+}, J_{j}^{-}] & = 0 . \end{aligned} \end{matrix}

(2.3.6)

This implies the irreps of $𝔰𝔬 (1, 3)$ are simply two copies of the irreps of $𝔰𝔲 (2)$ from Section 2.2, indexed as $(j_{1}, j_{2})$ with $j_{1}, j_{2} \in ℤ^{\geq 0} ∕ 2$ and dimension $(2 j_{1} + 1) (2 j_{2} + 1)$ . With this, we can easily obtain the generators $J_{i}$ , $K_{i}$ for the smallest few irreps:

\begin{align} (0, 0) : & J_{i}^{+} = J_{i}^{-} = 0 & \Rightarrow & J_{i} = K_{i} = 0 . & (2.3.7) \\ (\frac{1}{2}, 0) : & J_{i}^{+} = \frac{1}{2} σ_{i}, J_{i}^{-} = 0 & \Rightarrow & J_{i} = \frac{1}{2} σ_{i}, K_{i} = - \frac{i}{2} σ_{i} . & (2.3.8) \\ (0, \frac{1}{2}) : & J_{i}^{+} = 0, J_{i}^{-} = \frac{1}{2} σ_{i} & \Rightarrow & J_{i} = \frac{1}{2} σ_{i}, K_{i} = \frac{i}{2} σ_{i} . & (2.3.9) \\ ⋮ \end{align}

The $(\frac{1}{2}, \frac{1}{2})$ irrep is actually our familiar $4$ -vector representation, but it is more involved to recover the generators in the same form as Eq. 2.3.2.⁶

Representations of the Lorentz group

It turns out the above four irreps of the Lorentz group are all we need for the SM. Their nomenclature and corresponding elementary particle fields are listed in Table 2.2. Notably, fermions are classified as those with half-integer total spin $j = j_{1} + j_{2}$ , and bosons with integer $j$ . Their radically different behavior is a consequence of the Spin-Statistics theorem [89] (a notoriously difficult theorem to prove [90]), which states that half-integer spin particles obey Fermi-Dirac statistics and integer spin particles Bose-Einstein statistics.

All known fermionic particle fields live in the $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ , or Dirac spinor, representation. The $(\frac{1}{2}, 0)$ and $(0, \frac{1}{2})$ representations are called the left- and right-handed Weyl spinors respectively, where the handedness refers to the direction of their spin angular momentum relative to their momentum. Physically, this means there is a left-handed and right-handed copy of each fermion, and they have to be packaged together in a Dirac spinor to have masses without violating parity, as we discuss in Chapter B.4. We will also see that left- and right-handed representations can be equivalently thought of as particles and antiparticles.

The $(\frac{1}{2}, 0) \oplus (0, \frac{1}{2})$ representation technically also includes real Majorana spinors as a subspace, which can represent neutral fermions. The only candidate for these in the SM are right-handed neutrinos and, in fact, the existence of such Majorana neutrinos could potentially explain the curiously small left-handed neutrino masses through a process called the seesaw mechanism [91, 92]. To date, however, no experimental evidence for these, such as neutrinoless double beta decay [93] or same-sign charged dilepton decays [94], has been observed.

On another technical note, the Lorentz group, similar to $SO (3)$ , does not itself admit half-integer, fermionic representations. Thus, the true spacetime symmetry group is actually the double cover of $SO (1, 3)$ , $Spin (1, 3)$ ! Indeed, there are many subtleties to the Lorentz group, some of which will be revisited in the context of Lorentz-group equivariant neural networks in Chapter 16. To conclude, however, it is worth emphasizing again the remarkable physical insight these seemingly abstract group-theoretic concepts deliver. We are able to classify a fundamental dichotomy of particle physics — bosons versus fermions, and their completely different behavior — simply by their representation under the Lorentz (or, rather, the $Spin (1, 3)$ ) group!

---------------------------------------------------------------------------------------

Representation (j1,j2) Name Elementary Fields
---------------------------------------------------------------------------------------

(0,0) Scalar Higgs boson

(1,0 ) Left-handed Weyl spinor —
2

(0, 1) Right -handed Weyl spinor —
2

(1,0 ) ⊕ (0, 1 ) Dirac spinor All fermions
2 2

(1, 1) Vector g,γ, W ±, and Z gauge bosons
-----------2--2------------------------------------------------------------------------ — Table 2.2. Representations of the Lorentz group and their associated particle fields in the SM.

Lie algebra of the Poincaré group

The Poincaré group is Lorentz transformations plus spacetime translations. Just as angular momentum generates rotations, translations are generated by the momentum operator $P_{μ}$ . $P_{μ}$ and the Lorentz generators $J_{i}$ and $K_{i}$ together comprise the generators of the Poincaré group, and its algebra is thus the Lorentz algebra (Eq. 2.3.4) plus the commutation relations with the $P_{μ}$ s:⁷

\begin{matrix} \begin{aligned} [P_{μ}, P_{ν}] & = 0, \\ [J_{i}, P_{0}] & = 0, \\ [J_{i}, P_{j}] & = i 𝜖_{ijk} P_{k}, \\ [K_{i}, P_{0}] & = - i P_{i}, \\ [K_{i}, P_{j}] & = i η_{ij} P_{0} . \end{aligned} \end{matrix}

(2.3.10)

As is conventional, the Greek indices run over all four spacetime dimensions, while the Latin indices only the three spatial. The Poincaré algebra can be expressed more compactly by first combining the Lorentz generators into the antisymmetric tensor $M_{μν}$ :

M_{μν} = (\begin{matrix} 0 & K_{x} & K_{y} & K_{z} \\ - K_{x} & 0 & J_{z} & - J_{y} \\ - K_{y} & - J_{z} & 0 & J_{x} \\ - K_{z} & J_{y} & - J_{x} & 0 \end{matrix}) \Rightarrow Λ (ω) = e^{\frac{i}{2} ω^{μν} M_{μν}},

(2.3.11)

with $ω_{μν}$ another antisymmetric tensor containing the six rotation and boost parameters. The algebra can be then written as:

\begin{matrix} \begin{aligned} [M_{μν}, M_{ρσ}] & = i (η_{νρ} M_{μσ} - η_{μρ} M_{νσ} - η_{νσ} M_{μρ} + η_{μσ} M_{νρ}), \\ [M_{μν}, P_{ρ}] & = i (η_{νρ} P_{μ} - η_{μρ} P_{ν}), \\ [P_{μ}, P_{ν}] & = 0 . \end{aligned} \end{matrix}

(2.3.12)

Irreps of the Poincaré group

As we saw from Section 2.2, we can derive the irreps of an algebra using its Casimir invariants (Definition 2.2.4). Each set of their eigenvalues uniquely labels an irrep, while each basis state within the irreps is indexed by eigenvalues of a maximal set of commuting operators (e.g., ${J^{2}, J_{z}} \to | j, m ⟩$ for $𝔰𝔬 (3)$ ). Note that in the following, we simply provide a sketch of the derivations and point to, for example, Zee GT [79] Chapter VII.2 and Tong SM [76] Chapter 1.1.2 for more detailed proofs and discussion.

The Casimirs of the Poincaré algebra are the operators

P^{2} = P_{μ} P^{μ} and W^{2} = W_{μ} W^{μ},

(2.3.13)

where

W_{μ} = \frac{1}{2} 𝜖_{μνρσ} M^{νρ} P^{σ}

(2.3.14)

is the Pauli-Lubanski vector, the relativistic analog of the angular momentum operator $J_{i}$ . Furthermore, $P_{μ}$ commutes with both, and we can label its eigenstates as $| p ⟩$ ,

P_{μ} | p ⟩ = p_{μ} | p ⟩,

(2.3.15)

which represent single-particle states with 4-momentum $p_{μ}$ . These are therefore eigenstates of $P^{2}$ as well, with eigenvalues $m^{2}$ , the squared mass of the particle:

P^{2} | p ⟩ = p_{μ} p^{μ} | p ⟩ = m^{2} | p ⟩ .

(2.3.16)

Thus, we see that the mass of a particle, $m$ , is one label of the irreps, with states therein indexed by $p_{μ}$ . The other label, the particle spin $j$ , is based on the eigenvalue of $W^{2}$ :

W^{2} | p, j ⟩ \propto j (j + 1) | p, j ⟩ .

(2.3.17)

The easiest way to see this is to, for a given $m > 0$ , pick a single eigenstate $| p^{*} ⟩$ . The simplest is the rest frame $p_{μ}^{*} = (m, 0, 0, 0)$ . The subgroup of Poincaré transformations which leave $| p^{*} ⟩$ invariant is called its little group. In this case, it comprises all 3D rotations — i.e., $SO (3)$ . Indeed, if we look at the Pauli-Lubanski vector acting on $| p^{*} ⟩$ ,

W_{μ} | p^{*} ⟩ = \frac{1}{2} 𝜖_{μνρσ} M^{νρ} p^{* σ} | p^{*} ⟩ \Rightarrow W_{0} = 0, W_{i} = - m J_{i},

(2.3.18)

we simply recover the generators $J_{i}$ of $𝔰𝔬 (3)$ .⁸ Therefore,

W^{2} | p^{*}, j ⟩ = m^{2} J^{2} | p^{*}, j ⟩ = m^{2} j (j + 1) | p^{*}, j ⟩,

(2.3.19)

using the eigenvalues of $J^{2}$ from Eq. 2.2.14. Although we chose here to look at a specific state $| p^{*} ⟩$ , this can be shown to hold for all states $| p ⟩$ in the irrep.⁹ Thus, we see that irreps of the Poincaré group, and, hence, particles, are characterized by their mass $m$ and spin $j$ .

Massive versus massless particles

Continuing with the massive, $m > 0$ , particle case, we know as well from Section 2.2 that the eigenstates within the $| p, j ⟩$ irreps are further labeled by their spin along a particular axis: $J_{z} | p, j, m_{j} ⟩ = m_{j} | p, j, m_{j} ⟩$ , with $m_{j} \in {- j, - j + 1, \dots, j - 1, j}$ . Thus, massive particles $| m, j ⟩$ exist in $2 j + 1$ spin states $\otimes$ an infinite number of momentum states $| p ⟩, p_{μ} p^{μ} = m^{2}$ .

However, this is not the case for massless particles, which have a different little group. Recall that we can never boost into the rest frame of a massless particle to define the simple $| p^{*} ⟩$ we did above. Instead, let us consider the next-best state $| p^{'} ⟩$ , $p_{μ}^{'} = (E, 0, 0, E)$ . Its little group turns out to be $E (2)$ , the Euclidean group in 2D, whose representations and implications for massless particles are considerably more involved. However, the upshot is that it as well has irreps characterized by spin $j$ (and mass $m = 0$ ), but with only two helicity eigenstates therein.

As it turns out, physically, the mass of particles is based on the strength of their interactions (or lack thereof) with the Higgs boson, the particle at the center of this dissertation. We will discuss the mechanisms for these and all fundamental interactions in the next chapter, and see that symmetries are crucial in understanding them.

⁶See e.g. Ref. [88].

⁷See Appendix A.1.1 for a derivation.

⁸This also motivates why $W_{μ}$ can be thought of as relativistic angular momentum.

⁹By choosing an eigenstate $| p^{*}, j ⟩$ of $P^{2}$ and looking for transformations which leave it invariant, we “induced” a subgroup, $SO (3)$ , and used its representation theory to derive the irreps of the Poincaré group. Such a representation is hence called an induced representation.