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 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)$ Def: A factor is a von Neumann algebra whose center consists of scalar multiples of I. Types:
- I (In, 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 (II1, II): No minimal projection, but it has a non-zero finite projection (I1 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.