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 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].

@ __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].

@ __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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2017