42.1 The Poincar´e group and its Lie algebra

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

Definition (Poincar´e group)

The Poincar´e group is the semi-direct product
$P = R^{4} ⋊ S O (3, 1)$
with double cover
$\tilde{P} = R^{4} ⋊ S L (2, C)$

The action of $S O (3, 1)$ or $S L (2, C)$ on $R^{4}$ is the action of the Lorentz group on Minkowski space.

We will refer to both of these groups as the “Poincar´e group”, meaning by this the double cover only when we need it because spinor representations of the Lorentz group are involved. The two groups have the same Lie algebra, so the distinction is not needed in discussions that only involve the Lie algebra. Elements of the group $P$ will be written as pairs $(a, Λ)$ , with $a \in δ R^{4}$ and $Λ \in S O (3, 1)$ . The group law is

(a_{1}, Λ_{1}) (a_{2}, Λ_{2}) = (a_{1} + Λ_{1} a_{2}, Λ_{1} Λ_{2})

The Lie algebra Lie $P = L i e \tilde{P}$ has dimension 10, with basis

t_{0}, t_{1}, t_{2}, t_{3}, l_{1}, l_{2}, l_{3}, k_{1}, k_{2}, k_{3}

where the first four elements are a basis of the Lie algebra of the translation group, and the next six are a basis of so(3, 1), with the $l_{j}$ giving the subgroup of spatial rotations, the $k_{j}$ the boosts. We already know the commutation relations for the translation subgroup, which is commutative so

[t_{j}, t_{k}] = 0

We have seen in chapter 40 that the commutation relations for $so (3, 1)$ are

[l_{1}, l_{2}] = l_{3}, [l_{2}, l_{3}] = l_{1}, [l_{3}, l_{1}] = l_{2}

[k_{1}, k_{2}] = - l_{3}, [k_{3}, k_{1}] = - l_{2}, [k_{2}, k_{3}] = - l_{1}

[l_{j}, l_{k}] = ϵ_{j k l} l_{l}, [k_{j}, k_{k}] = - ϵ_{j k l} l_{l}

and that the commutation relations between the $l_{j}$ and $k_{j}$ are

[l_{j}, k_{k}] = ϵ_{j k l} k_{l}

corresponding to the fact that the $k_{j}$ transform as a vector under spatial rotations.

The Poincar´e group is a semi-direct product group of the sort discussed in chapter 18 and it can be represented as a group of 5 by 5 matrices in much the same way as elements of the Euclidean group $E (3)$ could be represented by 4 by 4 matrices (see chapter 19). Writing out this isomorphism explicitly for a basis of the Lie algebra, we have

l_{1} \leftrightarrow 000000000000010 00 - 1 00 00000 l_{2} \leftrightarrow 00000 000 - 1 0 000000100000000 l_{3} \leftrightarrow 0000000100 0 - 1 000 0000000000

k_{1} \leftrightarrow 0100010000000000000000000 k_{2} \leftrightarrow 0010000000100000000000000 k_{3} \leftrightarrow 0001000000000001000000000

t_{0} \leftrightarrow 0000000000000000000010000 t_{1} \leftrightarrow 0000000000000000000001000

t_{2} \leftrightarrow 0000000000000000000000100 t_{3} \leftrightarrow 0000000000000000000000010 (42.1)

We can use this explicit matrix representation to compute the commutators of the infinitesimal translations $t_{j}$ with the infinitesimal rotations and boosts $(l_{j}, k_{j})$ . t commutes with the $l_{j}$ and $t_{1}, t_{2}, t_{3}$ transform as a vector under rotations, For rotations one finds

[l_{j}, t_{k}] = ϵ_{j k l} t_{l}

For boosts one has

[k_{j}, t_{0}] = t_{j}, [k_{j}, t_{j}] = t_{0}, [k_{j}, t_{k}] = 0 if j \neq = k, k \neq = 0 (42.2)

Note that infinitesimal boosts do not commute with infinitesimal time translation, so after quantization boosts will not commute with the Hamiltonian. Boosts will act on spaces of single-particle wavefunctions in a relativistic theory, and on states of a relativistic quantum field theory, but are not symmetries in the sense of preserving spaces of energy eigenstates.

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 444、445、446、447、448、449、450、451、452、453、454、455

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：77b4da6c2272b606e953ece8669bd1810946873c05ecfe49ade61de273a7678a

OCR 产物 SHA-256：77b4da6c2272b606e953ece8669bd1810946873c05ecfe49ade61de273a7678a

数学物理阅读

书架

42.1 The Poincar´e group and its Lie algebra

42.1 The Poincar´e group and its Lie algebra

关系图谱