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},
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},
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

\(\langle\) · , · \(\rangle\):= (2\(\hbar\))^{–1} *μ*( · , · )
+ i (2\(\hbar\))^{–1} Ω( · , · )
.

@ __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 a1205 [physical systems do not have well-defined Hilbert spaces]; Brunet a1309 [motivation, using orthomatroids]; Gallone 15; Curcuraci a1708 [motivation]; > s.a. canonical quantum mechanics; axioms; 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].

@ __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 \(\mathbb C\)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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