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What is the motivation for the use of coset spaces within the context of integral quantization?

My main confusion is with the fact that coset spaces are inherently linear algebraic and make sense to me within the context of creating some transformation with structural preservation between two vector spaces but then how does this influence integral kernels? Or the choice of integral kernel?

What is the cross-over between linear algebra and integral transformation in this context?

For context here are some references:

Brif and Mann on Phase Space Quantization

Bergeron and Gazeau's examples on Integral Quantization

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Let there be given a Hilbert space ${\cal H}$ that is an unitary irreducible representation of a symmetry group $G$. A coset space appears e.g. from the construction of Gilmore–Perelomov coherent states. Given a fixed normalized state vector $|\psi\rangle \in {\cal H}$, we may consider the corresponding projective little/isotropy group $H\subseteq G$, and thus we may construct its coset space $X=G/H$.

The moral justification/motivation is that we want to keep track of physically equivalent / physically inequivalent states under the $G$-action in order not to overcount.

A related situation takes place in BRST quantization where the physical states are the quotient of the BRST-closed states modulo the BRST-exact states.

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  • $\begingroup$ Really awesome answer thank you! You managed to unpack a lot of stuff in quite a brief answer. $\endgroup$ – Jake Xuereb Jun 5 '18 at 16:46

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