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 & Zyczkowski 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 L^{2}(\(\mathbb R\), d*x*) ×
L^{2}(\(\mathbb R\), d*x*)
[@ 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];
> s.a. canonical approach; axioms for quantum theory;
quantum states.

**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].

@ __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.

@ __Projective__: in Boya &
Sudarshan FPL(91);
Stulpe a0708 [topologies].

@ __Hilbert superspace__: Valle IJTP(79) [using Grassmann numbers];
Rudolph CMP(00).

@ __Over finite fields__: Gurevich & Hadani JSG(09)-a0705 [and geometric quantization].

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