20.5 For further reading
Concept links · terms present in this machine draft; source roles are unverified: Lie algebra · Lie algebra representation · group action
For more on representations of semi-direct products, see section 3.8 of [85], chapter 5 of [95], [9], and [39]. The general theory was developed by Mackey during the late 1940s and 1950s, and his lecture notes on representation theory [58] are a good source for the details of this. The point of view taken here, that emphasizes constructing representations as solution spaces of diferential equations, where the diferential operators are Casimir operators, is explained in more detail in [47].
The conventional derivation found in most physics textbooks of the operators coming from an infinitesimal group action uses Lagrangian methods and Noether’s theorem. The purely Hamiltonian method used here treats configuration and momentum variables on the same footing, and is useful especially in the case of group actions that mix them (such as the example of section 20.3.2)
For another treatment of these operators along the lines of this chapter, see section 14 of [37].
For a concise but highly insightful discussion of the metaplectic representation, see chapters 16 and 17 in Graeme Segal’s section of [14]. For a discussion of this topic emphasizing the role of the Fourier transform as an intertwining operator, see [49]. The issue of the phase factor in the intertwining operators and the metaplectic double cover will be discussed later in the context of the harmonic oscillator, using a diferent realization of the Heisenberg Lie algebra representation. For a discussion of this in terms of the Schr¨odinger representation, see part I of [56].
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