Hilbert
Space| Hilbert
Space |
In General > s.a. complex
structure; fock
space [exponential Hilbert space];
operator.
$ Def: A complete inner product space.
* Remark: The inner product
is a special case of the action of elements of the dual space, 
(')
= 
|'
.
* 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 & Mikusinski 90.
@ 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].
@ Related topics: Rudolph CMP(00)
[super Hilbert space]; Hu & Yu a0705 [infinite-dimensional,
Schmidt decomposition theorem].
Examples in Physics > s.a. modified
quantum mechanics [including discrete Hilbert space]; space
of connections.
* Non-relativistic quantum
mechanics:
The usual one is L2(Rn,
dnx),
but other choices are possible (> see Bohr
Compactification), and necessary
if the classical configuration space is different; Notice that, e.g., L2(R2,
d2x)
is isomorphic to L2(R, dx)
× L2(R, dx)
[@ in Reed & Simon
72].
* Linear field theory:
Given a phase space (
, 
)
with a complex structure J, compatible with in
the sense that 
(
· , · ):= (
· , J · ) is a
positive-definite inner product (Kähler structure), define
· , · := (2
)–1 ( · , · )
+ i (2)–1 ( · , · )
.
@ 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].
@ In quantum mechanics: Brunner et al PRL(08) [testing the dimension];
> s.a. states
in quantum mechanics.
Rigged Hilbert Space > s.a. [formulations
of quantum mechanics]; dirac quantization; Perturbation
Methods.
* Idea: A (Gel'fand) triplet, consisting of a Hilbert space 
together
with a choice of dense subspace and
its dual * ( 
*),
and a map 
: → *,
the
rigging map.
* Applications: Time irreversibility
in quantum mechanics; Refined algebraic quantization
for systems with constraints.
@ Intros: de la Madrid EJP(05)qp [pedestrian].
@ 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-in
[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 qp/98-in;
Gadella & De La
Madrid
IJTP(99);
Bohm & Scurek qp/00-in
[in decays]; > s.a.
arrow of time.
@ Related topics: Wickramasekara & Bohm JPA(02)
[symmetries]; Rowe & Repka
mp/02 [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
* Projective: The set
of rays of a Hilbert space; Can be considered as an
infinite-dimensional
version of the complex projective space CPn,
with
a
metric (and a compatible symplectic structure).
@ Projective: in Boya & Sudarshan FPL(91);
Stulpe a0708 [topologies].
@ Over finite fields: Gurevich & Hadani
a0705 [and geometric quantization].
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send feedback and suggestions to bombelli at olemiss.edu – modified 5 jul
2008