42.2 Irreducible representations of the Poincar´e group

42.2 Irreducible representations of the Poincar´e group

Concept links · terms present in this machine draft; source roles are unverified: eigenvalue · Lie algebra · Lie algebra representation · group representation · irreducible representation · Hilbert space · group action · Schur’s lemma

We would like to construct unitary irreducible representations of the Poincar´e group. These will be given by unitary operators $u(a,\Lambda )$ on a Hilbert space ${\mathcal{H}}_1$ which will have an interpretation as a single-particle relativistic quantum state space. In the analogous non-relativistic case, we constructed unitary irreducible representations of $E(3)$ (or its double cover $\stackrel{\sim }{E}(3))$ as

• The space of wavefunctions of a free particle of mass $m,$ with a fixed energy $E$ (chapter 19). These are solutions to

$$D\psi =E\psi$$

where

$$D=-\frac{1}{2m}{\nabla }^2$$

and the $\psi$ are single-component wavefunctions. $E(3)$ acts on wavefunctions by

$$\psi \to u(\mathbf{a},R)\psi (\mathbf{x})=\psi (R^{-1}(\mathbf{x}-\mathbf{a}))$$

• The space of solutions of the “square root” of the Pauli-Schr¨odinger equation (see section 34.2). These are solutions to

$$D\psi =\pm \sqrt{2mE}\psi$$

where

$$D=-i\sigma \cdot \nabla$$

and the $\psi$ are two-component wavefunctions. $\widetilde{E(3)}$ acts on wavefunctions by

$$\psi \to u(\mathbf{a},\Omega )\psi (\mathbf{x})=\Omega \psi (R^{-1}(\mathbf{x}-\mathbf{a}))$$

(where $R$ is the $SO(3)$ element corresponding to $\Omega \in SU(2)$ in the double cover).

In both cases, the group action commutes with the diferential operator, or equivalently one has $uD{u}^{-1}=D,$ and this is what ensures that the operators $u(\mathbf{a},\Omega )$ take solutions to solutions.

To construct representations of $\mathcal{P},$ we would like to generalize this construction from ${\mathbf{R}}^3$ to Minkowski space $M^4$ . To do this, one begins by defining an action of $\mathcal{P}$ on $n$-component wavefunctions by

$$\psi \to u(a,\Lambda )\psi (x)=S(\Lambda )\psi ({\Lambda }^{-1}(x-a))$$

This is the action one gets by identifying n-component wavefunctions with

$$(\text{functions on}{M}^4)\otimes {\mathbf{C}}^n$$

and using the induced action on functions for the first factor in the tensor product, on the second factor taking $S(\Lambda )$ to be an n dimensional representation of the Lorentz group.

One then chooses a diferential operator $D$ on $n$-component wavefunctions, one that commutes with the group action, so

$$u(a,\Lambda )Du(a,\Lambda {)}^{-1}=D$$

The $u(a,\Lambda )$ then give a representation of $\mathcal{P}$ on the space of solutions to the wave equation

$$D\psi =c\psi$$

for c a constant, and in some cases this will give an irreducible representation. If the space of solutions is not irreducible an additional set of “subsidiary conditions” can be used to pick out a subspace of solutions on which the representation is irreducible. In later chapters we will consider several examples of this construction, but now will turn to the general classification of representations of $\mathcal{P}$ .

Recall that in the $E(3)$ case we had two Casimir operators:

$$P^2=P_1^2+P_2^2+P_3^2$$

and

$$\mathbf{J}\cdot \mathbf{P}$$

Here $P_j$ is the representation operator for Lie algebra representation, corresponding to an infinitesimal translation in the j-direction. $J_j$ is the operator for an infinitesimal rotation about the j-axis. The Lie algebra commutation relations of $E(3)$ ensure that these two operators commute with the action of $E(3)$ and thus, by Schur’s lemma, act as a scalar on an irreducible representation. Note that the fact that the first Casimir operator is a diferential operator in position space and commutes with the $E(3)$ action means that the eigenvalue equation

$$P^2\psi =c\psi$$

has a space of solutions that is a $E(3)$ representation, and potentially irreducible. In the Poincar´e group case, we can easily identify:

Definition (Casimir operator)

The Casimir (or first Casimir) operator for the Poincar´e group is the operator

$$P^2=-P_0^2+P_1^2+P_2^2+P_3^2$$

A straightforward calculation using the Poincar´e Lie algebra commutation relations shows that $P^2$ is a Casimir operator since one has

$$[P_0,P^2]=[P_j,P^2]=[J_j,P^2]=[K_j,P^2]=0$$

for $j=1,2,3$ . Here $J_j$ is a Lie algebra representation operator corresponding to the $l_j$ , and $K_j$ the operator corresponding to $k_j$

The second Casimir operator is more dificult to identify in the Poincar´e case than in the $E(3)$ case. To find it, first define:

Definition (Pauli-Lubanski operator)

The Pauli-Lubanski operator is the fourcomponent operator

$$W_0=-\mathbf{P}\cdot \mathbf{J},\mathbf{W}=-P_0\mathbf{J}+\mathbf{P}\times \mathbf{K}$$

By use of the commutation relations, one can show that the components of $W_{\mu }$ behave like a four-vector, $i$.e.,

$$[W_{\mu },P_{\nu }]=0$$

and the commutation relations with the $J_j$ and $K_j$ are the same for $W_{\mu }$ as for $P_{\mu }$ . One can then define:

Definition (Second Casimir operator)

The second Casimir operator for the Poincar´e Lie algebra is

$$W^2=-W_0^2+W_1^2+W_2^2+W_3^2$$

Use of the commutation relations shows that

$$[P_0,W^2]=[P_j,W^2]=[J_j,W^2]=[K_j,W^2]=0$$

so $W^2$ is a Casimir operator.

To classify Poincar´e group representations, we have two tools available. We can use the two Casimir operators $P^2$ and $W^2$ and characterize irreducible representations by their eigenvalues. In addition, recall from chapter 20 that irreducible representations of semi-direct products $N\times K$ are associated with pairs of a K-orbit ${\mathcal{O}}_o$ for $\alpha \in \hat{N}$ , and an irreducible representation of the corresponding little group $K_{\alpha }$

For the Poincar´e group, $\hat{N}={\mathbf{R}}^4$ is the space of characters (one dimensional representations) of the translation group of Minkowski space. Elements $\alpha$ are labeled by

$$p=(p_0,p_1,p_2,p_3)$$

where the $p_{\mu }$ are the eigenvalues of the energy-momentum operators $P_{\mu }$ . For representations on wavefunctions, these eigenvalues will correspond to elements in the representation space with space-time dependence.

$$e^{i(-p_0{x}_0+p_1{x}_1+p_2{x}_2+p_3{x}_3)}$$

Given an irreducible representation, the operator $P^2$ will act by the scalar

$$-p_0^2+p_1^2+p_2^2+p_3^2$$

which can be positive, negative, or zero, so given by $m^2,-m^2,0$ for various m. The value of the scalar will be the same everywhere on the orbit, so in energy-momentum space, orbits will satisfy one of the three equations

$$-p_0^2+p_1^2+p_2^2+p_3^2=\{\begin{array}{l}-m^2\\ m^2\\ 0\end{array}$$

The representation can be further characterized in one of two ways:

• By the value of the second Casimir operator $W^2$

• By the representation of the stabilizer group $K_p$ on the eigenspace of the momentum operators with eigenvalue $p.$

At the point $p$ on an orbit, the Pauli-Lubanski operator has components

$$W_0=-\mathbf{p}\cdot \mathbf{J},\mathbf{W}=-p_0\mathbf{J}+\mathbf{p}\times \mathbf{K}$$

In the next chapter we will find the possible orbits, then pick a point $p$ on each orbit, and see what the stabilizer group $K_p$ and Pauli-Lubanski operator are at that point.

Previous section: 42.1 The Poincar´e group and its Lie algebra · Next section: 42.3 Classification of representations by orbits · 中文版本

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