By Jean-Pierre Gazeau

This self-contained advent discusses the evolution of the suggestion of coherent states, from the early works of Schr?dinger to the latest advances, together with sign research. An built-in and glossy method of the application of coherent states in lots of assorted branches of physics, it moves a stability among mathematical and actual descriptions. cut up into elements, the 1st introduces readers to the main well-known coherent states, their beginning, their development, and their software and relevance to varied chosen domain names of physics. half II, quite often in accordance with contemporary unique effects, is dedicated to the query of quantization of varied units via coherent states, and indicates the hyperlink to techniques in sign research.

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72) s s This distribution appears in the Fock–Bargmann representation of ρ: R(α, β) e – 2 (|α| 1 ρ= C2 = C2 2 +β|2 ) |α β| d2α d2β π π R(α, β) |α β| μs (dα) μs (dβ) . 73) s In the case of a pure number state, ρ = |n n|, this distribution reads as R(α, β) = (αβ)n . n! 74) In the case of a coherent state ρ = |z z|, 2 R(α, β) = e –|z| e zα+¯z β . 2 The Density Matrix and the Representation “Q” The distribution “Q”, or “lower symbol” of ρ, or Husimi function of the state ρ, is the set of expected values of ρ in coherent state representation: def Q(α, α) = α|ρ|α .

52) of the ﬁrst-order correlation function factorize into independent terms: (1) (–) (+) G i j (r, t; r , t ) = Ei (r, t)E j (r , t ) . 58). Nevertheless, one should notice that such a deﬁnition does not imply monochromaticity. 1 Introduction In this third chapter we develop, on an elementary level, the mathematical formalism of the standard coherent states: Hilbertian properties, resolution of the unity, Weyl–Heisenberg group. We also describe some probabilistic aspects of the coherent states and their essential role in the existence of four important quantum distributions, namely, the “R”, “Q”, and “P” distributions and the Wigner distribution.

Moreover, the map θ → P θ furnishes a noncommutative version of the unit circle since P θ P θ – P θ P θ = sin (θ – θ ) 0 1 –1 0 . 12) More generally, we ﬁnd in this example the notion of a positive-operator-valued measure (POVM) on the unit circle, which means that to any measurable set Δ ⊂ [0, 2π) there corresponds the positive 2 ~ 2 matrix: def Δ → P (Δ) = = 1 π 1 π dθ |θ θ| Δ 1 π Δ dθ cos2 θ dθ cos θ sin θ Δ 1 π dθ cos θ sin θ dθ sin2 θ Δ 1 π Δ . 13) This matrix is obviously positive since, for any nonzero vector |v in the plane, we have v |P (Δ)|v = π1 Δ dθ | v |θ |2 > 0.