Coherent states in mathematical physics

Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see also). However, they have generated a huge variety of generalizations, which have led to a tremendous literature in mathematical physics.In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys. Let H {displaystyle {mathfrak {H}},} be a complex, separable Hilbert space, X {displaystyle X} a locally compact space and d ν {displaystyle d u } a measure on X {displaystyle X} . For each x {displaystyle x} in X {displaystyle X} , denote | x ⟩ {displaystyle |x angle } a vector in H {displaystyle {mathfrak {H}}} . Assume that this set of vectors possesses the following properties: holds in the weak sense on the Hilbert space H {displaystyle {mathfrak {H}}} , i.e., for any two vectors | ϕ ⟩ , | ψ ⟩ {displaystyle |phi angle ,|psi angle } in H {displaystyle {mathfrak {H}}} , the following equality holds: A set of vectors | x ⟩ {displaystyle |x angle } satisfying the two properties above is called a family of generalized coherent states. In order to recover the previous definition (given in the article Coherent state) of canonical or standard coherent states (CCS), it suffices to take X ≡ C {displaystyle Xequiv mathbb {C} } , the complex plane and d ν ( x ) ≡ 1 π d 2 x . {displaystyle d u (x)equiv {frac {1}{pi }}d^{2}x.} Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors | x ⟩ {displaystyle |x angle } simply forming a total set in H {displaystyle {mathfrak {H}},} and the functions Ψ ( x ) = ⟨ x | ψ ⟩ {displaystyle Psi (x)=langle x|psi angle } , as | ψ ⟩ {displaystyle |psi angle } runs through H {displaystyle {mathfrak {H}}} , forming a reproducing kernel Hilbert space.The objective in both cases is to ensure that an arbitrary vector | ψ ⟩ {displaystyle |psi angle } be expressible as a linear (integral) combination of these vectors. Indeed, the resolution of the identity immediately implies that where Ψ ( x ) = ⟨ x | ψ ⟩ {displaystyle Psi (x)=langle x|psi angle } . These vectors Ψ {displaystyle Psi } are square integrable, continuous functions on X {displaystyle X} and satisfy the reproducing property where K ( x , y ) = ⟨ x | y ⟩ {displaystyle K(x,y)=langle x|y angle } is the reproducing kernel, which satisfies the following properties