40.2 The Lorentz group and its Lie algebra

40.2 The Lorentz group and its Lie algebra

Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · complexification

Recall that in 3 dimensions the group of linear transformations of ${\mathbf{R}}^3$ preserving the standard inner product was the group $O(3)$ of 3 by 3 orthogona matrices. This group has two disconnected components: $SO(3)$ , the subgroup of orientation preserving (determinant +1) transformations, and a component of orientation reversing (determinant −1) transformations. In Minkowski space, one has:

Definition (Lorentz group)

The Lorentz group $O(3,1)$ is the group of linear transformations preserving the Minkowski space inner product on ${\mathbf{R}}^4$

In terms of matrices, the condition for a 4 by 4 matrix $\Lambda$ to be in $O(3,1)$ will be

$${\Lambda }^T\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\Lambda =\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

The Lorentz group has four components, with the component of the identity a subgroup called $SO(3,1)$ (which some call $S{O}^{+}(3,1),$ ). The other three components arise by multiplication of elements in $SO(3,1)$ by $P,T,PT$ where

$$P=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)$$

is called the ${}^6\mathrm{parit}{\mathrm{y}}^5$ transformation, reversing the orientation of the spatial variables, and

$$T=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

reverses the time orientation.

The Lorentz group has a subgroup $SO(3)$ of transformations that just act on the spatial components, given by matrices of the form

$$\Lambda =\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& & & \\ 0& & R& \\ 0& & & \end{array}\right)$$

where $R$ is in $SO(3)$ . For each pair j, $k$ of spatial directions one has the usual $SO(2)$ subgroup of rotations in the $jk$ plane, but now in addition for each pair $0,j$ of the time direction with a spatial direction, one has $SO(1,1)$ subgroups of matrices of transformations called “boosts” in the $j$ direction. For example, for $j=1$ , one has the subgroup of $SO(3,1)$ of matrices of the form

$$\Lambda =\left(\begin{array}{cccc}\cosh \varphi & \sinh \varphi & 0& 0\\ \sinh \varphi & \cosh \varphi & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$

for $\varphi \in \mathbf{R}$

The Lorentz group is six dimensional. For a basis of its Lie algebra one can take six matrices $M_{\mu \nu }$ for $\mu ,\nu \in L_{0,1,2,3}$ and $j<k$ . For the spatial indices, these are

$$M_{12}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& -1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 0\end{array}\right),M_{13}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& -1& 0& 0\end{array}\right),M_{23}=\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)$$

which correspond to the basis elements of the Lie algebra of $SO(3)$ that we first saw in chapter 6. These can be renamed using the same names as earlier

$$l_1=M_{23},l_2=M_{13},l_3=M_{12}$$

and recall that these satisfy the $\mathfrak{so}(3)$ commutation relations

$$[l_1,l_2]=l_3,[l_2,l_3]=l_1,[l_3,l_1]=l_2$$

and correspond to infinitesimal rotations about the three spatial axes.

Taking the first index 0, one gets three elements corresponding to infinitesimal boosts in the three spatial directions

$$M_{01}=\left(\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),M_{02}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),M_{03}=\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right)$$

These can be renamed as

$$k_1=M_{01},k_2=M_{02},k_3=M_{03}$$

One can easily calculate the commutation relations between the $k_j$ and $l_j.$ , which show that the $k_j$ transform as a vector under infinitesimal rotations. For instance, for infinitesimal rotations about the $x_1$ axis, one finds

$$[l_1,k_1]=0,[l_1,k_2]=k_3,[l_1,k_3]=-k_2\text{\hspace{1em}(40.1)}$$

Commuting infinitesimal boosts, one gets infinitesimal spatial rotations

$$[k_1,k_2]=-l_3,[k_3,k_1]=-l_2,[k_2,k_3]=-l_1\text{\hspace{1em}(40.2)}$$

Digression. A more conventional notation in physics is to use $J_j=i{l}_j$ for infinitesimal rotations, and $K_j=i{k}_j$ for infinitesimal boosts. The intention of the diferent notation used here is to start with basis elements of the real Lie algebra so(3, 1), $(the{l}_j$ and $k_j)$ which are purely real objects, before complexifying and considering representations of the Lie algebra.

Taking the following complex linear combinations of the $l_j$ and $k_j$

$$A_j=\frac{1}{2}(l_j+i{k}_j),B_j=\frac{1}{2}(l_j-i{k}_j)$$

one finds

$$[A_1,A_2]=A_3,[A_3,A_1]=A_2,[A_2,A_3]=A_1$$

and

$$[B_1,B_2]=B_3,[B_3,B_1]=B_2,[B_2,B_3]=B_1$$

This construction of the $A_j,B_j$ requires that we complexify (allow complex linear combinations of basis elements) the Lie algebra $\mathfrak{so}(3,1)$ of $SO(3,1)$ and work with the complex Lie algebra $s$ $\mathfrak{o}(3,1)\otimes \mathbf{C}$ . It shows that this Lie algebra splits into a sum of two sub-Lie algebras, which are each copies of the (complexified) Lie algebra of $SO(3)$ , so $\left(3\right)\otimes \mathbf{C}$ . Since

$$\mathfrak{so}(3)\otimes \mathbf{C}=\mathfrak{su}(2)\otimes \mathbf{C}=\mathfrak{sl}(2,\mathbf{C})$$

we have

$$\mathfrak{so}(3,1)\otimes \mathbf{C}=\mathfrak{sl}(2,\mathbf{C})\oplus \mathfrak{sl}(2,\mathbf{C})$$

In section 40.4 we’ll see the origin of this phenomenon at the group level.

Previous section: 40.1 Minkowski space · Next section: 40.3 The Fourier transform in Minkowski space · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 425、426、427、428、429、430、431、432、433、434

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：346edf4cbbddffa109e24945f1ff7006ef82167adc590089cedba91e2d050f4d

OCR 产物 SHA-256：346edf4cbbddffa109e24945f1ff7006ef82167adc590089cedba91e2d050f4d