Observable
Algebras |

**In General** > s.a. C*-Algebra; W*-Algebra; 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 \(\cal B\)(\(\cal H\))
of bounded self-adjoint operators on some Hilbert space \(\cal H\), 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 \([\hat A,\,\hat B] = {\rm i}\hbar\, \widehat{[A,\, B]}\),
at least in the \(\hbar\) → 0 limit.

$ __Von Neumann algebra__:
A subalgebra *R* of \(\cal B\)(\(\cal H\))
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;
> s.a. types of metric spaces.

* __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).

@ __General references__: Jordan, von Neumann & Wigner AM(34);
Murray & von Neumann TAMS(37); in Hermann 66, ch16;
Haag in(71); Sunder 87;
Li Bing-Ren 92 [intro];
Lledó CM-a0901 [modular theory, introduction]; Zalamea a1612-PhD [observables as Jordan-Lie algebras, etc].

@ __Operator algebras__: Segal 52;
Takesaki 79; Bratteli & Robinson 81,
87; Kadison & Ringrose 83;
Kadison in(90); Murphy 90;
Schroer mp/01-ln; Guido et al JFA(17)-a1512 [Gromov-Hausdorff distance between von Neumann algebras].

@ __Quantum-classical correspondence__: Kryvohuz & Cao PRL(05)
[commutators, in terms of response theory].

**Factors** (Murray & von Neumann terminology) > s.a. modified approaches to quantum gravity.

$ __Def__: A factor is a
von Neumann algebra whose center consists of scalar multiples of I. Types:

- __I__ (I_{n},
I_{∞}): 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 \(\cal B\)(\(\cal H\)),
with \(\cal H\) of dimension *n*; The Hilbert space \(\cal H\) can be split into
\(\cal H\)_{1} ⊗ \(\cal H\)_{2}
such that ...

- __II__ (II_{1},
II_{∞}):
No minimal projection, but it has a non-zero finite projection (I_{1} if
I is finite relative to the factor, I_{∞} 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 \(\cal A\) is said to be decomposable in two factors if there are
two commuting subalgebras which together generate \(\cal A\) (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];
Nobili a0809 [introduction, and extension];
Valente SHPMP(08)
[von Neumann's program and difficulties].

**Other Topics and Specific Theories** > s.a. causality;
GNS construction; lattice gauge theory;
quantum probability.

* __More general situations__: Non-associative algebras
of observables may appear in certain physical situations, but they cannot be represented as operators on a Hilbert space.

@ __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]; Bojowald et al JHEP(15)-a1411 [non-associative algebras in quantum theory]; Bodendorfer et al JHEP(16)-a1510 [in Gaußian normal spacetime coordinates]; Piparo a1707 [pseudo-observable algebra].

@ __Representations__: Gotay & Grundling in(99)dg/97 [on a non-compact symplectic manifold].

@ __Subalgebras__: Halvorson & Clifton IJTP(99)
[maximal beable]; Olkiewicz
AP(00)
[stable with respect to environment interaction].

@ __Causal nets__: Baumgärtel & Wollenberg 92; Thomas & Wichmann
JMP(98); Ruzzi RVMP(05)
[net cohomology of posets]; > s.a. causal sets.

@ __Types of theories__: Rudolph & Schmidt JMP(09)-a0807 [gauge theories].

> __Related topics__:
see approaches to quantum field theory; formulations of quantum mechanics.

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