Hilbert Space  

In General > s.a. functional analysis; operator.
$ Def: A complete inner product space, over a field which is usually \(\mathbb R\) or \(\mathbb C\).
* History: The theory was motivated by the development of quantum physics, but it is now an important tool in functional analysis.
* Remark: The inner product is a special case of the action of elements of the dual space, ψ(ψ') = \(\langle\)ψ|ψ'\(\rangle\).
* Separable: One in which all complete orthonormal sets are countable.
* Operations on Hilbert spaces: > see Direct Sum; tensor product.
@ Books: Riesz & Nagy 55; Von Neumann 55; Cirelli 72; Reed & Simon 72; Halmos 82; Debnath & Mikusiński 05; Hansen 06 [II].
@ Different realizations: Schenker & Aizenman LMP(00)mp [functions on a graph]; Kyukov IJMMS(05)a0704 [and linear algebra and differential geometry on a Hilbert space]; > s.a. Holomorphic Functions.
@ Related topics: Hu & Yu a0705-wd [infinite-dimensional, Schmidt decomposition theorem]; Bengtsson & Życzkowski a1701-ch [discrete structures in finite Hilbert spaces]; > s.a. complex structure.

In Physics > s.a. Koopman-von Neumann Formalism; modified quantum mechanics [including discrete and real Hilbert space]; space of connections.
* Non-relativistic quantum mechanics: The usual one is L\(^2({\mathbb R}^n,{\rm d}^n x)\) over \(\mathbb C\), but other choices are possible (> see Bohr Compactification), and necessary if the classical configuration space is different; Notice that, e.g., L\(^2({\mathbb R}^2,{\rm d}^2 x)\) is isomorphic to L2(\(\mathbb R\), dx) × L2(\(\mathbb R\), dx) [@ in Reed & Simon 72].
* Linear field theory: Given a phase space (Γ, Ω) with a complex structure J, compatible with Ω in the sense that μ( · , · ):= Ω( · , J · ) is a positive-definite inner product (Kähler structure), define

\[\langle \cdot , \cdot \rangle:= (2\hbar)^{-1} \mu(\cdot,\cdot) + {\rm i}\,(2\hbar)^{-1} \Omega(\cdot,\cdot)\;.\]

@ General references: Mostafazadeh CQG(03)mp/02 [space of solutions of Klein-Gordon field theory]; Saller ht/05 [for unstable particles]; Dutkay & Jorgensen JMP(06) [multi-scale problems]; Barbero et al CQG(17)-a1701 [for systems with boundaries, and trace operators]; Pollack & Singh a1801 [emergence of a space lattice].
@ In quantum mechanics: Brunner et al PRL(08) [testing the dimension]; Amrein 09; Fields Axi(14)-a1205 [physical systems do not have well-defined Hilbert spaces]; Brunet Axioms(13)-a1309 [motivation, using orthomatroids]; Gallone 15; Curcuraci a1708 [motivation]; Hoff da Silva & Caires da Rocha a2105 [representation theory, role of continuity]; > s.a. canonical approach; axioms for quantum theory; quantum states.
@ Field theories: Okołów a2102 [metrics of fixed signature].

Rigged Hilbert Space > s.a. formulations of quantum mechanics / dirac quantization; Perturbation Methods.
* Idea: A (Gel'fand) triplet, consisting of a Hilbert space \(\cal H\) together with a choice of dense subspace Ω and its dual Ω* (Ω ⊂ \(\cal H\) ⊂ Ω*), and a map η: Ω → Ω*, the rigging map.
* Applications: Time irreversibility in quantum mechanics; Refined algebraic quantization for systems with constraints.
@ General references: de la Madrid EJP(05)qp [pedestrian intro]; Celeghini a1502-conf [constructive presentation]; Celeghini et al a1907 [and special functions and Lie groups]; Káninský a2007 [symplectic transformations and observables].
@ Examples: Castagnino et al IJTP(97)qp/00 [inverted oscillator]; de la Madrid JPA(02)qp/01 [Schrödinger equation], et al FdP(02)qp/01 [continuous spectrum], IJTP(03)qp/02-proc [free particle]; de la Madrid JPA(04)qp [1D rectangular barrier]; Celeghini et al a1711 [and representations of SO(2)].
@ Irreversibility: Schulte et al qp/95; Bohm et al IJTP(99)qp/97; Bohm PRA(99)qp, & Harshman LNP(98)qp; Gadella & De La Madrid IJTP(99); Bohm & Scurek in(00)qp [in decays]; > s.a. arrow of time.
@ Related topics: Wickramasekara & Bohm JPA(02) [symmetries]; Rowe & Repka JMP(02)mp [coherent triplets]; Deotto et al JMP(03)qp/02, JMP(03)qp/02 [in classical mechanics]; Gadella & Gómez IJTP(03) [spectral decompositions]; > s.a. localization; representations; resonances [Gamow vectors].

Other Generalized Types of Hilbert Spaces > s.a. fock space [exponential Hilbert space]; generalized coherent states [on Hilbert modules over C*-algebras].
* Projective: The set of rays –one-dimensional linear subspaces– of a Hilbert space; It can be considered as an infinite-dimensional version of the complex projective space \({\mathbb C}{\rm P}^n\), with a metric (and a compatible symplectic structure); In quantum theory, it is the space of pure states of a quantum system described by an operator algebra on the Hilbert space.
@ General references: Gurevich & Hadani JSG(09)-a0705 [over finite fields, and geometric quantization]; Gudder a2005 [finite-dimensional]; Hsu a2007 [discrete].
@ Projective: in Boya & Sudarshan FPL(91); Stulpe a0708 [topologies].
@ Hilbert superspace: Valle IJTP(79) [using Grassmann numbers]; Rudolph CMP(00).

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