Abstract
In this chapter we consider that the unit sphere \(\mathbb {S}^{2}\) of the Euclidean space ℝ3 with its canonical symplectic structure is a phase space. Then coherent states are labeled by points on \(\mathbb {S}^{2}\) and allow us to build a quantization of the two sphere \(\mathbb {S}^{2}\). They are defined in each finite-dimensional space of an irreducible unitary representation of the symmetry group SO(3) (or its covering SU(2)) of \(\mathbb {S}^{2}\) and give a semi-classical interpretation for the spin.
As an application we state the Berezin–Lieb inequalities and compute the thermodynamic limit for large spin systems.
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Notes
- 1.
An isometry in an Euclidean space is automatically linear so that O(3) is a subgroup of GL(ℝ3).
- 2.
Multiplication by i gives self-adjoint generators instead of anti-self-adjoint operators.
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Combescure, M., Robert, D. (2012). Spin-Coherent States. In: Coherent States and Applications in Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0196-0_7
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