Observable
Algebras| Observable
Algebras |
In General > s.a. operator theory;
symmetries in quantum theory.
* Idea: Observables form
an algebra with addition and operator product as the algebra operations; In
the quantum theory, this is usually the Banach
algebra 
(
)
of bounded self-adjoint operators on some Hilbert
space , with norm
|A|:=
sup{|Ax|, |x| =
1}, or some sub-algebra thereof (C*-algebra, W*-algebra).
* Remark: They can be made
into a commutative algebra with a · b:= (ab+ba)/2.
* Quantum-classical correspondence:
A classical Poisson algebra of observables is promoted to an algebra of quantum
operators in a Hilbert space, with the condition that commutators correspond
to Poisson brackets in the sense that [A^, B^] = i
[A, B]^,
at least in the → 0
limit.
$ Von Neumann algebra:
A subalgebra R of ()
which is weak-operator closed, contains I, and is closed under adjoints; It
is a
C*-algebra, and if Abelian it is isomorphic to some algebra C(X)
of
real functions on a compact Hausdorff extremely disconnected manifold.
* Relationships: Von Neumann algebra theory can be considered as a
non-commutative generalization of analysis (i.e. of C(X)) and of measure
theory (projection-valued
measures).
@ Quantum-classical correspondence: Kryvohuz & Cao PRL(05)
[commutators, ito response theory].
Factors (Murray & von Neumann terminology)
$ Def: A factor is a
von Neumann algebra whose center consists of scalar multiples of I. Types:
- I (In,
Iinfty): The simplest
possibility, perfectly
adequate for quantum systems with a finite number of degrees of freedom;
It has a minimal projection (and is thus isomorphic to some (),
with of dimension n;
The Hilbert space can
be split into 1 
2
such that ...
- II (II1,
IIinfty):
No minimal projection, but it has a non-zero finite projection (I1 if
I is finite relative to the factor, Iinfty otherwise);
In some sense nicer than I, seems to be the right framework for
describing
a system in a thermal bath; Was claimed to be more generally useful
for quantum field theory,
but this idea does not seem to work.
- III: All non-zero
projections are infinite; A pathological case, essential for quantum field
theory and statistical mechanics of infinite systems.
* Decomposable algebras: An algebra 
is
said to be decomposable in two factors if there are two commuting subalgebras
which together generate
(this can happen only
if there are no superselection rules).
* Coupled factors: The factors are called coupled factors if ...
@ References: Yngvason RPMP(05)mp/04 [type
III].
Other Topics and Specific Theories > s.a. causality; GNS
construction; lattice
gauge theory; quantum
probability.
@ General references: Dimock CMP(80)
[on a Lorentzian manifold]; Balian & Vénéroni AP(88)
[expectation values]; Zafiris IJTP(07)gq/04 [sheaf
theoretical,
and abstract differential geometry]; Hamhalter IJTP(04)
= IJTP(04)
[states].
@ Representations: Gotay & Grundling dg/97-in [on
non-compact symplectic
manifold]
@ Subalgebras: Halvorson & Clifton IJTP(99) [maximal beable]; Olkiewicz
AP(00) [stable wrt environment interaction].
@ Causal nets: Baumgärtel & Wollenberg 92; Thomas & Wichmann
JMP(98); Ruzzi RVMP(05)
[net
cohomology of posets]; > s.a. causal
sets.
References > s.a. approaches
to quantum field theory; C*-Algebra; W*-Algebra;
formulations of quantum mechanics.
@ General: Jordan, von Neumann & Wigner AM(34);
Murray & von Neumann TAMS(37); in Hermann
66, ch16; Haag in(71); Sunder 87.
@ Operator algebras: Segal 52; Takesaki 79; Bratteli & Robinson
81,
87; Kadison & Ringrose 83; Kadison in(90); Murphy 90; Schroer mp/01-ln.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
19 jul 2008