Abstract
In the previous chapters we have seen how representation theory leads to geometric objects such as orbits. The purpose of this chapter is to describe the opposite phenomenon: starting from a coadjoint orbit of a group G, we will obtain a representation by quantizing the orbit. This construction will further explain why orbits of momenta classify representations of semi-direct products. In addition it will turn out to be a tool for understanding gravity in parts II and III.
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Notes
- 1.
Recall that the differential of a smooth map \({\mathcal F}:{\mathcal M}\rightarrow {\mathcal N}\) at \(p\in {\mathcal M}\) is the map \({\mathcal F}_{*p}:T_p{\mathcal M}\rightarrow T_{{\mathcal F}(p)}{\mathcal N}:\dot{\gamma }(0)\mapsto \frac{d}{dt}\big [{\mathcal F}(\gamma (t))\big ]\big |_{t=0}\), where \(\gamma (t)\) is a path in \({\mathcal M}\) such that \(\gamma (0)=p\).
- 2.
An integral curve of a vector field \(\xi \) on a manifold \({\mathcal M}\) is a path \(\gamma (t)\) on \({\mathcal M}\) such that \(\dot{\gamma }(t)=\xi _{\gamma (t)}\).
- 3.
From now on, real functions on \({\mathcal M}\) are denoted as \({\mathcal F}\), \({\mathcal G}\), \({\mathcal H}\), etc.
- 4.
A derivation of an algebra \({\mathcal A}\) is a linear map \(D:{\mathcal A}\rightarrow {\mathcal A}:a\mapsto D(a)\) that satisfies the Leibniz rule \(D(a\cdot b)=D(a)\cdot b+a\cdot D(b)\).
- 5.
Closedness means \(d\omega =0\), where d is the exterior derivative. Non-degeneracy means that for all \(p\in {\mathcal M}\), any vector \(v\in T_p{\mathcal M}\) such that \(\omega _p(v,w)=0\) for all \(w\in T_p{\mathcal M}\) necessarily vanishes.
- 6.
Recall that the pullback of a tensor field T of rank k on a manifold \({\mathcal N}\) by a map \(\phi :{\mathcal M}\rightarrow {\mathcal N}\) is defined by \((\phi ^*T)_p(v_1,\ldots ,v_k)\equiv T_{\phi (p)}(\phi _{*p}v_1,\ldots ,\phi _{*p}v_k)\) for any \(p\in {\mathcal M}\) and all \(v_1,\ldots ,v_k\in T_p{\mathcal M}\).
- 7.
\({\mathcal F}_{*p}\) is a linear map from \(T_p\mathfrak {g}^*\cong \mathfrak {g}^*\) to \(T_{{\mathcal F}(p)}\mathbb {R}\cong \mathbb {R}\) and therefore belongs to \((\mathfrak {g}^*)^*\cong \mathfrak {g}\).
- 8.
Here self-adjointness means that \(\langle {\mathcal I}(X),Y\rangle =\langle {\mathcal I}(Y),X\rangle \) for any two adjoint vectors X, Y.
- 9.
Property (5.31) does not contradict the fact that the adjoint and coadjoint representations of \(\mathfrak {g}\) are actual representations, i.e. for example that \(\text {ad}_X\text {ad}_Y-\text {ad}_Y\text {ad}_X=\text {ad}_{[X,Y]}\). Indeed, the vector fields in (5.31) are derivations acting on functions on \({\mathcal M}\), while \(\text {ad}_X\) and \(\text {ad}^*_X\) are generally understood as linear operators acting on \(\mathfrak {g}\) and \(\mathfrak {g}^*\), respectively.
- 10.
The map being “smooth” means that, given any two smooth sections \(\Phi ,\Psi \), the assignment \(q\mapsto (\Phi (q)|\Psi (q))\) is smooth.
- 11.
Here “closed” means “compact without boundary”.
- 12.
We denote by c the speed of light in the vacuum.
- 13.
Here the words “semi-direct product” refer to a group (4.1) with an Abelian vector group A.
- 14.
The sequence “group \(\leadsto \) adjoint \(\leadsto \) coadjoint” will be ubiquitous in this thesis.
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Oblak, B. (2017). Coadjoint Orbits and Geometric Quantization. In: BMS Particles in Three Dimensions. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-61878-4_5
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