Abstract
It is well known from the work of Berezin (Commun. Math. Phys. 40:153–174, 1975) in 1975 that the quantization problem of a classical mechanical system is closely related with coherent states. In particular coherent states help to understand the limiting behavior of a quantum system when the Planck constant ħ becomes negligible in macroscopic units. This problem is called the semi-classical limit problem.
In this chapter we discuss properties of quantum systems when the configuration space is the Euclidean space ℝn, so that in the Hamiltonian formalism, the phase space is ℝn×ℝn with its canonical symplectic form σ. The quantization problem has many solutions, so we choose a convenient one, introduced by Weyl (The Classical Groups, 1997) and Wigner (Group Theory and Its Applications to Quantum Mechanics of Atomic Spectra, 1959).
We study the symmetries of Weyl quantization, the operational calculus and applications to propagation of observables.
We show that Wick quantization is a natural bridge between Weyl quantization and coherent states. Applications are given of the semi-classical limit after introducing an efficient modern tool: semi-classical measures.
We illustrate the general results proved in this chapter by explicit computations for the harmonic oscillator. More applications will be given in the following chapters, in particular concerning propagators and trace formulas for a large class of quantum systems.
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- 1.
Recall that \(f \in\mathcal{S}(\mathbb{R}^{n})\) means that f is a smooth function in ℝn and for every multiindices α, β, \(x^{\alpha}\partial_{x}^{\beta}u\) is bounded in ℝn. It has a natural topology. \(\mathcal{S}^{\prime}(\mathbb{R}^{n})\) is the linear space of continuous linear form on \(\mathcal{S}(\mathbb{R}^{n})\).
- 2.
This means that for every γ, sup ħ∈]0,1]∥∂ γ A∥∞<+∞.
- 3.
That λ is a non-critical value for H means that ∇H(z)≠0 if H(z)=λ.
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Combescure, M., Robert, D. (2012). Weyl Quantization and 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_2
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DOI: https://doi.org/10.1007/978-94-007-0196-0_2
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