41.1 Representations of the Lorentz group

41.1 Representations of the Lorentz group

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In the $SU(2)$ case we found irreducible unitary representations $({\pi }_n,V^n)$ of dimension $n+1$ for $n=0,1,2,\dots$ These could also be labeled by $\begin{array}{r}s=\frac{n}{2}\end{array}$ , called the “spin” of the representation, and we will do that from now on. These representations can be realized explicitly as homogeneous polynomials of degree $n=2s$ in two complex variables $z_1,z_2$ . For the case of $Spin(4)=SU(2)\times SU(2)$ , the irreducible representations will be tensor products

$$V^{s_1}\otimes V^{s_2}$$

of $SU(2)$ irreducibles, with the first $SU(2)$ acting on the first factor, the second on the second factor. The case $s_1=s_2=0$ is the trivial representation, $s_1=$ $\begin{array}{r}\frac{1}{2},s_2=0\end{array}$ is one of the half-spinor representations of $Spin(4)$ on ${\mathbf{C}}^2,s_1=0,s_2=$ $\frac{1}{2}$ 4 is the other, and $s_1=s_2=\frac{1}{2}$ is the representation on four dimensional (complexified) vectors.

Turning now to $Spin(3,1)=SL(2,\mathbf{C})$ , one can use the same construction using homogeneous polynomials as in the $SU(2)$ case to get irreducible representations of dimension $2s+1$ for $s=0,\frac{1}{2},1,\dots$ Instead of acting by $SU(2)$ on $z_1,z_2$ , one acts by $SL(2,\mathbf{C})$ , and then as before uses the induced action on polynomials of $z_1$ and $z_2$ . This gives representations $({\pi }_s,V^s)$ of $SL(2,\mathbf{C})$ . Among the things that are diferent though about these representations:

• They are not unitary (except in the case of the trivial representation). For example, for the defining representation $V^{\frac{1}{2}}$ on ${\mathbf{C}}^2$ , the Hermitian inner product

$$⟨\left(\genfrac{}{}{0ex}{}{\psi }{\varphi }\right),\left(\genfrac{}{}{0ex}{}{{\psi }^{\prime }}{{\varphi }^{\prime }}\right)⟩=(\overline{\psi }\overline{\varphi })\cdot \left(\genfrac{}{}{0ex}{}{{\psi }^{\prime }}{{\varphi }^{\prime }}\right)=\overline{\psi }{\psi }^{\prime }+\overline{\varphi }{\varphi }^{\prime }$$

is invariant under $SU(2)$ transformations $\Omega$ since

$$⟨\Omega \left(\genfrac{}{}{0ex}{}{\psi }{\varphi }\right),\Omega \left(\genfrac{}{}{0ex}{}{{\psi }^{\prime }}{{\varphi }^{\prime }}\right)⟩=\left(\begin{array}{cc}\overline{\psi }& \overline{\varphi }\end{array}\right){\Omega }^{†}\cdot \Omega \left(\genfrac{}{}{0ex}{}{{\psi }^{\prime }}{{\varphi }^{\prime }}\right)$$

and ${\Omega }^{†}\Omega =1$ by unitarity. This is no longer true for $\Omega \in SL(2,\mathbf{C})$

The representation $V^{\frac{1}{2}}$ of $SL(2,\mathbf{C})$ does have a non-degenerate bilinear form, which we’ll denote by $ϵ,$

$$ϵ\left(\left(\genfrac{}{}{0ex}{}{\psi }{\varphi }\right),\left(\genfrac{}{}{0ex}{}{{\psi }^{\prime }}{{\varphi }^{\prime }}\right)\right)=\left(\begin{array}{cc}\psi & \varphi \end{array}\right)\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\genfrac{}{}{0ex}{}{{\psi }^{\prime }}{{\varphi }^{\prime }}\right)=\psi {\varphi }^{\prime }-\varphi {\psi }^{\prime }$$

that is invariant under the $SL(2,\mathbf{C})$ action on $V^{\frac{1}{2}}$ and can be used to identify the representation and its dual. This is the complexification of the symplectic form on ${\mathbf{R}}^2$ studied in section 16.1.1, and the same calculation there which showed that it was $SL(2,\mathbf{R})$ invariant here shows that the complex version is $SL(2,\mathbf{C})$ invariant.

• In the case of $SU(2)$ representations, the complex conjugate representation one gets by taking as representation matrices $\pi (g)$ instead of $\pi (g)$ is equivalent to the original representation (the same representation, with a diferent basis choice, so matrices changed by a conjugation). To see this for the spin $\frac{1}{2}$ representation, note that $SU(2)$ matrices are of the form

$$\Omega =\left(\begin{array}{cc}\alpha & \beta \\ -\overline{\beta }& \overline{\alpha }\end{array}\right)$$

and one has

$$\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\left(\begin{array}{cc}\alpha & \beta \\ -\overline{\beta }& \overline{\alpha }\end{array}\right){\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)}^{-1}=\left(\begin{array}{cc}\overline{\alpha }& \overline{\beta }\\ -\beta & \alpha \end{array}\right)$$

so the matrix

$$\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

is the change of basis matrix relating the representation and its complex conjugate.

This is no longer true for $SL(2,\mathbf{C})$ . Conjugation by a fixed matrix will not change the set of eigenvalues of the matrix, and the two eigenvalues are not necessarily complex conjugates (unlike for the case of $SU(2))$ . So such a conjugation cannot change all $SL(2,\mathbf{C})$ matrices to their complex conjugates, since in general (complex) conjugation will change the set of eigenvalues.

The classification of irreducible finite dimensional SU(2) representation was done in chapter 8 by considering its Lie algebra $\mathfrak{su}(2)$ , complexified to give us raising and lowering operators, and this complexification is $\mathfrak{sl}(2,\mathbf{C})$ . If one examines that argument, one finds that it mostly also applies to irreducible finite dimensional $\mathfrak{sl}(2,\mathbf{C})$ representations. There is a diference though: now flipping positive to negative weights (which corresponds to change of sign of the Lie algebra representation matrices, or conjugation of the Lie group representation matrices) no longer takes one to an equivalent representation. It turns out that to get all irreducibles, one must take both the representations we already know about and their complex conjugates. One can show (we won’t prove this here) that the tensor product of one of each type of irreducible is still an irreducible, and that the complete list of finite dimensional irreducible representations of $\mathfrak{sl}(2,\mathbf{C})$ is given by:

Theorem (Classification of finite dimensional ${ \mathfrak { s l } } ( 2 , \mathbf { C } )

$representations).Theirreduciblerepresentationsof${ \mathfrak { s l } } ( 2 , \mathbf { C } )$arelabeledby$( s _ { 1 } , s _ { 2 } )$for$s _ { j } = 0 , { \textstyle \frac { 1 } { 2 } } , 1 ,$ These representations are given by the tensor product representations

$$\left({\pi }_{s_1}\otimes {\overline{\pi }}_{s_2},V^{s_1}\otimes V^{s_2}\right)$$

where $({\pi }_s,V^s)$ is the irreducible representation of dimension $2s+1$ and $({\overline{\pi }}_s,V^s)$ its complex conjugate. Such representations have dimension $(2s_1+1)(2s_2+1)$

All these representations are also representations of the group $SL(2,\mathbf{C})$ and one has the same classification theorem for the group, although we will not try and prove this. We will also not try and study these representations in general, but will restrict attention to the cases of most physical interest, which are

$(0,0)$ : The trivial representation on $C$, also called the “spin $0^{†}$ or scalar representation.

$\left(\frac{1}{2},0\right)$ : These are called left-handed (for reasons we will see later on) “Weyl spinors”. We will often denote the representation space ${\mathbf{C}}^2$ in this case as $S_L$ , and write an element of it as ${\psi }_L$

$(0,\frac{1}{2})$ : These are called right-handed Weyl spinors. We will often denote the representation space ${\mathbf{C}}^2$ in this case as $S_R$ , and write an element of it as ${\psi }_R$

$\left(\frac{1}{2},\frac{1}{2}\right)$ : This is called the “vector” representation since it is the complexification of the action of $SL(2,\mathbf{C})$ as $SO(3,1)$ transformations of space-time vectors that we saw earlier. It is a representations of $SO(3,1)$ as well as $SL(2,\mathbf{C})$

$(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ : This reducible 4 complex dimensional representation is known as the representation on “Dirac spinors”.

One can manipulate these Weyl spinor representations $\left(\frac{1}{2},0\right)$ and $(0,\frac{1}{2})$ in a similar way to the treatment of tangent vectors and their duals in tensor analysis. Just like in that formalism, one can distinguish between a representation space and its dual by upper and lower indices, in this case using not the metric but the $SL(2,\mathbf{C})$ invariant bilinear form $ϵ$ to raise and lower indices. With complex conjugates and duals, there are four kinds of irreducible $SL(2,\mathbf{C})$ representations on ${\bar{\mathbf{C}}}^2$ to keep track of:

$S_L:$ : This is the standard defining representation of $SL(2,\mathbf{C})$ on ${\mathbf{C}}^2$ , with $\Omega \in SL(2,\mathbf{C})$ acting on ${\psi }_L\in S_L$ by

$${\psi }_L\to \Omega {\psi }_L$$

A standard index notation for such things is called the “van der Waerden notation”. It uses a lower index A taking values 1, 2 to label the components with respect to a basis of $S_L$ as

$${\psi }_L=\left(\genfrac{}{}{0ex}{}{{\psi }_1}{{\psi }_2}\right)={\psi }_A$$

and in this notation Ω acts by

$${\psi }_A\to {\Omega }_A^B{\psi }_B$$

For instance, the element

$$\Omega =e^{-i\frac{\theta }{2}{\sigma }_3}$$

corresponding to an $SO(3)$ rotation by an angle $\theta$ around the z-axis acts on $S_L$ by

$$\left(\genfrac{}{}{0ex}{}{{\psi }_1}{{\psi }_2}\right)\to e^{-i\frac{\theta }{2}{\sigma }_3}\left(\genfrac{}{}{0ex}{}{{\psi }_1}{{\psi }_2}\right)$$

$S_L^{\ast }$ : This is the dual of the defining representation, with $\Omega \in SL(2,\mathbf{C})$ acting on ${\psi }_L^{\ast }\in S_L^{\ast }$ by

$${\psi }_L^{\ast }\to ({\Omega }^{-1}{)}^T{\psi }_L^{\ast }$$

This is a general property of representations: given any finite dimensional representation $(\pi (g),V)$ , the pairing between V and its dual $V^{\ast }$ is preserved by acting on $V^{\ast }$ by matrices ${\overline{(\pi (g{)}^{-1})}}^T$ , and these provide a representation $((\pi (g{)}^{-1}{)}^T,V^{\ast })$ . In van der Waerden notation, one uses upper indices and writes

$${\psi }^A\to (({\Omega }^{-1}{)}^T{)}_B^A{\psi }^B$$

Writing elements of the dual as row vectors, our example above of a particular Ω acts by

$$\left(\begin{array}{cc}{\psi }^1& {\psi }^2\end{array}\right)\to \left(\begin{array}{cc}{\psi }^1& {\psi }^2\end{array}\right)e^{i\frac{\theta }{2}{\sigma }_3}$$

Note that the bilinear form $ϵ$ gives an isomorphism of representations between $S_L$ and $S_L^{\ast }$ , written in index notation as

$${\psi }^A={ϵ}^{AB}{\psi }_B$$

where

$${ϵ}^{AB}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

$S_R:$ This is the complex conjugate representation to $S_L$ , with $\Omega \in SL(2,\mathbf{C})$ acting on ${\psi }_R\in S_R$ by

$${\psi }_R\to \overline{\Omega }{\psi }_R$$

The van der Waerden notation uses a separate set of dotted indices for these, writing this as

$${\psi }_{\dot{A}}\to {\overline{\Omega }}_{\dot{A}}^{\dot{B}}{\psi }_{\dot{B}}$$

Another common notation among physicists puts a bar over the $\psi$ to denote that the vector is in this representation, but we’ll reserve that notation for complex conjugation. The Ω corresponding to a rotation about the z-axis acts as

$$\left(\genfrac{}{}{0ex}{}{{\psi }_{\dot{1}}}{{\psi }_{\dot{2}}}\right)\to e^{i\frac{\theta }{2}{\sigma }_3}\left(\genfrac{}{}{0ex}{}{{\psi }_{\dot{1}}}{{\psi }_{\dot{2}}}\right)$$

• $S_R^{\ast }i$ : This is the dual representation to $S_R$ , with $\Omega \in SL(2,\mathbf{C})$ acting on ${\psi }_R^{\ast }\in S_R^{\ast }$ by

$${\psi }_R^{\ast }\to ({\overline{\Omega }}^{-1}{)}^T{\psi }_R^{\ast }$$

and the index notation uses raised dotted indices

$${\psi }^{\dot{A}}\to (({\overline{\Omega }}^{-1}{)}^T{)}_{\dot{B}}^{\dot{A}}{\psi }^{\dot{B}}$$

Our standard example of a Ω acts by

$$\left(\begin{array}{cc}{\psi }^{\dot{1}}& {\psi }^{\dot{2}}\end{array}\right)\to \left(\begin{array}{cc}{\psi }^{\dot{1}}& {\psi }^{\dot{2}}\end{array}\right)e^{-i\frac{\theta }{2}{\sigma }_3}$$

Another copy of $ϵ$

$${ϵ}^{\dot{A}\dot{B}}=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$

gives the isomorphism of $S_R$ and $S_R^{\ast }$ as representations, by

$${\psi }^{\dot{A}}={ϵ}^{\dot{A}\dot{B}}{\psi }_{\dot{B}}$$

Restricting to the $SU(2)$ subgroup of $SL(2,\mathbf{C})$ , all these representations are unitary, and equivalent. As $SL(2,\mathbf{C})$ representations, they are not unitary, and while the representations are equivalent to their duals, $S_L$ and $S_R$ are inequivalent (since as we have seen, one cannot complex conjugate $SL(2,\mathbf{C})$ matrices by a matrix conjugation).

For the case of the $(\frac{1}{2},\frac{1}{2})$ representation, to see explicitly the isomorphism between $S_L\otimes S_R$ and vectors, recall that we can identify Minkowski space with 2 by 2 self-adjoint matrices. $\Omega \in SL(2,\mathbf{C})$ acts by

$$\left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right)\to \Omega \left(\begin{array}{cc}{x}_0+x_3& x_1-i{x}_2\\ x_1+i{x}_2& x_0-x_3\end{array}\right){\Omega }^{†}$$

We can identify such matrices as linear maps from $S_R^{\ast }$ to $S_L$ (and thus isomorphic to the tensor product $S_L\otimes (S_R^{\ast }{)}^{\ast }=S_L\otimes S_R.$ see chapter 9).

Chapter contents · Next section: 41.2 Dirac \gamma matrices and Clif(3, 1) · 中文版本

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