20 Representations of Semi-direct Products

In this chapter we will examine some aspects of representations of semi-direct products, in particular for the case of the Jacobi group and its Lie algebra, as well as the case of N ⋊ K, for N commutative. The latter case includes the Euclidean groups $E(d)$, as well as the Poincar´e group which will come into play once we introduce special relativity.

The Schr¨odinger representation provides a unitary representation of the Heisenberg group, one that carries extra structure arising from the fact that the symplectic group acts on the Heisenberg group by automorphisms. Each such automorphism takes a given construction of the Schr¨odinger representation to a unitarily equivalent one, providing an operator on the state space called an “intertwining operator”. These intertwining operators will give (up to a phase factor), a representation of the symplectic group. Up to the problem of the phase factor, the Schr¨odinger representation in this way extends to a representation of the full Jacobi group. To explicitly find the phase factor, one can start with the Lie algebra representation, where the sp(2d, $R$) action is given by quantizing quadratic functions on phase space. It turns out that, for a finite dimensional phase space, exponentiating the Lie algebra representation gives a group representation up to sign, which can be turned into a true representation by taking a double cover (called M p(2d, $R$)) of Sp(2d, $R$).

In later chapters, we will find that many groups acting on quantum systems can be understood as subgroups of this M p(2d, $R$), with the corresponding observables arising as the quadratic combinations of momentum and position operators determined by the moment map.

The Euclidean group $E(d)$ is a subgroup of the Jacobi group, and we saw in chapter 19 how some of its representations can be understood by restricting the Schr¨odinger representation to this subgroup. More generally, this is an example of a semi-direct product N ⋊ K with $N$ commutative. In such cases irreducible representations can be characterized in terms of the action of $K$ on irreducible representations of $N_{\ast }$ together with the irreducible representations of certain subgroups of $K$ .

The reader should be warned that much of the material included in this chapter is motivated not by its applications to non-relativistic quantum mechanics, a context in which such an abstract point of view is not particularly helpful. The motivation for this material is provided by more complicated cases in relativistic quantum field theory, but it seems worthwhile to first see how these ideas work in a simpler context. In particular, the discussion of representations of $N\times K$ for N commutative is motivated by the case of the Poincar´e group (see chapter 42). The treatment of intertwining operators is motivated by the way symmetry groups act on quantum fields (a topic which will first appear in chapter 38).

中文目录

Chapter contents

• 20.1 Intertwining operators and the metaplectic representation
• 20.2 Constructing intertwining operators
• 20.3 Explicit calculations
  • 20.3.1 The $SO(2)$ action by rotations of the plane for $d=2$
  • 20.3.2 An SO(2) action on the $d=1$ phase space
  • 20.3.3 The Fourier transform as an intertwining operator
  • 20.3.4 An R action on the $d=1$ phase space
• 20.4 Representations of N × K , N commutative
• 20.5 For further reading

Previous chapter: 19 The Quantum Free Particle as a Representation of the Euclidean 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 221、222、223、224、225、226、227、228、229、230、231、232、233

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：d6200b7008ca4a11108f700be118b9c166777e04875cc11a1d08323d8b0f03b6

OCR 产物 SHA-256：d6200b7008ca4a11108f700be118b9c166777e04875cc11a1d08323d8b0f03b6