Abstract
In the previous chapter we learned how to deal with projective representations: given a symmetry group, we are to find its universal cover and its most general central extension. Exact representations of this central extension then account for all projective representations of the original group. The remaining problem then is to write down explicit representations, so our goal in this chapter is to build Hilbert spaces of wavefunctions acted upon by a group of unitary transformations. Guided by group actions on homogeneous spaces, we will be led to the method of induced representations.
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Notes
- 1.
The concrete definition of integrals relies on a limiting procedure where the integrand is approximated by a sequence of locally constant functions, but we will not review these details here.
- 2.
We denote points of \({\mathcal M}\) as p, q, etc. to suggest thinking of them as possible momenta of a particle.
- 3.
This class has nothing to do with the ray (2.3) despite the identical notation.
- 4.
There exist infinitely many functions that all represent equally well the Radon–Nikodym derivative; the theorem ensures that these functions agree, except possibly on a set of zero measure. Accordingly, we call “the” Radon–Nikodym derivative any function that satisfies (3.10).
- 5.
As before elements of G are written as f, g, etc. and the identity is denoted e.
- 6.
This actually follows from the fact that typical momentum orbits for such groups are homotopic to a point, which then implies that the corresponding bundles \(G\rightarrow G/H\) are trivial.
- 7.
Recall that any continuous linear functional on a Hilbert space \({\mathcal E}\) is a scalar product \((v|\cdot )\) for some fixed vector \(v\in {\mathcal E}\).
- 8.
The symbol \(\sim \) denotes unitary equivalence of representations.
- 9.
We are assuming that \({\mathcal E}\) is a separable Hilbert space; N may be infinite.
- 10.
The notation “Z” stands for the German word Zustandssumme, meaning “sum over states.”
- 11.
Here the index a runs from one to r, the latter being essentially the rank of the symmetry group.
- 12.
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Oblak, B. (2017). Induced Representations . In: BMS Particles in Three Dimensions. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-61878-4_3
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