42 The Poincaré Group and its Representations

In chapter 19 we saw that the Euclidean group $E(3)$ has infinite dimensional irreducible unitary representations on the state space of a quantum free particle. The free particle Hamiltonian plays the role of a Casimir operator: to get irreducible representations one fixes the eigenvalue of the Hamiltonian (the energy), and then the representation is on the space of solutions to the Schr¨odinger equation with this energy. There is also a second Casimir operator, with integral eigenvalue the helicity, which further characterizes irreducible representations. The case of helicity $\pm \frac{1}{2}$ (which uses the double cover $\widetilde{E(3)})$ ) occurs for solutions of the Pauli equation, see section 34.2.

For a relativistic analog, treating space and time on the same footing, we will use instead the semi-direct product of space-time translations and Lorentz transformations, called the Poincar´e group. Irreducible representations of this group will again be labeled by eigenvalues of two Casimir operators, giving in the cases relevant to physics one continuous parameter (the mass) and a discrete parameter (the spin or helicity). These representations can be realized as spaces of solutions for relativistic wave equations, with such representations corresponding to possible relativistic elementary particles.

For an element $(a,\Lambda )$ of the Poincar´e group, with a a space-time translation and $\Lambda$ an element of the Lorentz group, there are three diferent sorts of actions of the group and Lie algebra to distinguish:

• The action

$$x\to \Lambda x+a$$

on a Minkowski space vector $x$. This is an action on a real vector space, it is not a unitary representation.

• The action

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

on $n$-component wavefunctions, solutions to a wave equation (here $S$ is an n dimensional representation of the Lorentz group). These will be the unitary representations classified in this chapter.

• The space of single-particle wavefunctions can be used to construct a quantum field theory, describing arbitrary numbers of particles. This will come with an action on the state space by unitary operators $U(a,\Lambda )$ . This will be a unitary representation, but very much not irreducible.

For the corresponding Lie algebra actions, we will use lower case letters $(e.g.$ 2 $t_j,l_j)$ to denote the Lie algebra elements and their action on Minkowski space, upper case letters $(e.g.,P_j,L_j)$ to denote the Lie algebra representation on wavefunctions, and upper case hatted letters $(e.g.,{\hat{P}}_j,{\hat{L}}_j)$ to denote the Lie algebra representation on states of the quantum field theory.

中文目录

Chapter contents

• 42.1 The Poincar´e group and its Lie algebra
• 42.2 Irreducible representations of the Poincar´e group
• 42.3 Classification of representations by orbits
  • 42.3.1 Positive energy time-like orbits
  • 42.3.2 Negative energy time-like orbits
  • 42.3.3 Space-like orbits
  • 42.3.4 The zero orbit
  • 42.3.5 Positive energy null orbits
  • 42.3.6 Negative energy null orbits
• 42.4 For further reading

Previous chapter: 41 Representations of the Lorentz Group

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