JBW-algebraic quantum mechanics

It can be nice to describe the kinematics of a quantum system using a JBW-algebra. (Then some extra structure is needed for dynamics.)

$JBW$-algebras are arrived at by the confluence of several lines of motivation.

To begin with, the algebraic approach to quantum mechanics (which is also the basis of AQFT) is a common generalization of classical mechanics using phase spaces (or even Poisson manifolds or even Poisson algebras) and quantum mechanics using Hilbert spaces. In this approach, we have an algebra of observables, of which special cases are the commutative algebra underlying a Poisson algebra (or its complexification) and the algebra of operators on a Hilbert space. This also allows one to apply superselection rules at a fundamental level by restricting to a subalgebra?.

More particularly, one may restrict to *bounded* observables, that is those for which there is a largest possible absolute value that could be observed. There is a philosophical reason for this, that no measuring device in reality is unbounded; but it also makes the algebra more tractable, and an arbitrary observable $O$ can still be described by a sequence of bounded observables with increasing cut-offs.

At this point, it's natural to expect the algebra of observables to be a Banach space (and so, being an algebra, some kind of Banach algebra), with the norm giving this largest possible absolute value. In fact, complex $C^*$-algebras are usually used; both the algebra of bounded continuous functions on a Poisson manifold (or indeed any local compactum) and the algebra of bounded operators on a Hilbert space are $C^*$-algebras.

Then we may further refine from $C^*$-algebras to $W^*$-algebras. This actually amounts to generalizing the notion of observable. Whereas a typical commutative $C^*$-algebra is an algebra of continuous functions, a typical commutative $W^*$-algebra is an algebra of measurable functions. This is necessary if we want to include characteristic functions, reflecting an observation of whether the system is in some measurable subset of phase space. (If it is on the boundary, then we won't be able to tell; this is reflected in that the $W^*$-algebra of functions identifies functions that are equal almost everywhere.) The functional calculus of $W^*$-algebras also allows for composition of an observable with a measurable function on its spectrum, allowing us to (for example) turn continuous observables into less-precise discrete ones. Of course, the algebra of bounded operators on a Hilbert space is also a $W^*$-algebra. Finally, the algebra of observables that respect a class of superselection rules will also typically form a $W^*$-algebra.

There is, however, an objection to all of these algebras of observables, which is that they have many elements that are not observable. Only the self-adjoint elements of these algebras are considered to be observable. (Even if we allow for complex-valued observables, only the normal element?s can be observables.) By themselves, the self-adjoint operators form a Jordan algebra. Instead of recovering the classical case when the algebra of operators is commutative, we recover it when the Jordan algebra of self-adjoint operators is associative (and specifically we get the classical algebra of real-valued observables immediately). Thus, the algebra of observables is a Jordan algebra, even if it is viewed as embedded within a larger algebra.

The analytic structure on a Jordan algebra that makes it the algebra of self-adjoint elements in a $W^*$-algebra is captured in the definition of a JW-algebra (Jordan $W$-algebra, which may be read as a Jordan $W^*$-algebra with trivial involution) or JBW-algebra (Jordan–Banach $W$-algebra, which allows exceptional cases such as the Albert algebra). This is a loss of structure from the full $W^*$-algebra; in fact, a complex $W^*$-algebra can be recovered from a $JBW$-algebra by giving it an additional compatible Lie algebra structure, making it into a JLBW-algebra, which (as explained there) is equivalent to a complex $W^*$-algebra. However, the $JBW$-algebra already identifies the observables and the states, so it is a sufficient place to start from.

This is hastily copied from elsewhere and minimally edited. More work should be done to spell this out.

We have:

- a Banach space $A$ whose elements are thought of as (bounded, real-valued) observables;
- a Banach space $A_*$, some of whose norm-$1$ elements (those to be identified later as ‘positive’) are thought of (normal, mixed) states;
- with $A$ expressed as the dual of $A_*$;
- with a commutative (but nonassociative) multiplication operation on $A$;
- with the resulting nonassociative Banach algebra $A$ satisfying a few technical conditions:
- power-associativity: all ways of parenthesizing $x^n$, for $x$ an element of $A$ and $n$ a natural number, are equal;
- positivity: ${\|x^2\|} \leq {\|x^2 + y^2\|}$; and
- the B-identity: ${\|x^2\|} = {\|x\|^2}$.

In particular, the positivity allows us to define a partial order on $A$ according to which an element of $A$ is (weakly) *positive* if it is a limit of sums of squares, and then an element $\psi$ of $A_*$ is *positive* (and hence a *normal state* if it has norm $1$) if every positive element $x$ of $A$ takes $\psi$ to a nonnegative real number.

If the multiplication on $A$ happens to be associative, then there exists a measure space $X$ such that $A_*$ is the Lebesgue space $L^1(X)$ and $A$ is $L^\infty(X)$. It is always possible to take $X$ to be localizable, in which case the Radon–Nikodym theorem applies and we can identify $L^1(X)$ with the space of absolutely continuous measures on $X$. At this point, the measure on $X$ is irrelevant except for its specification of full (or null) subsets, so we may treat $X$ as simply a localizable measurable space (which includes this data). Then the choice of $X$ is actually essentially unique.

In this way, the associative case reduces to probability theory. We have a measurable space $X$, an observable is an essentially bounded function, and a state is a probability measure. It is true that we only have localizable spaces, but these are the only ones that satisfy the nice theorems (such as Radon–Nikodym, aforementioned) anyway. (Similarly, the observables are defined only up to almost equality, and the normal states must be absolutely continuous measures.)

In the general (nonassociative) case, we still think of the elements of $A$ (the observables) as essentially bounded functions, but now on a sort of noncommutative (or rather nonassociative) space, and we may still think of the elements of $A_*$ (the states) as probability measures on that space. (This is particularly appropriate in the Bayesian interpretation of quantum mechanics.) Because $A$ is a Jordan algebra instead of an associative algebra, *all* of its elements may be thought of as observables, but this is more of a convenience than an essential feature.

We might also want to consider generalized (non-normal) states, that is norm-$1$ positive linear functionals on $A$. (Every state in $A_*$ may be so interpreted, using the natural map $A_* \to (A_*)^{**} = A^*$). One may go further and consider quasistate?s on $A$, if any exist (besides the states themselves).

In particular, if $A$ is the space of bounded self-adjoint operators on some Hilbert space $H$, then $A_*$ is the space of trace-class self-adjoint operators on $H$. Then a normal state is a density matrix, and the pure states (those that cannot be written as nontrivial convex combinations of other states) are those of the form ${|\psi\rangle \langle\psi|}$ for $\psi$ a unit vector in $H$.

We can add dynamics by giving the $JBW$-algebra the structure of a pointed JLBW-algebra, that is by defining a kind of commutator (the ‘L’) and identifying one of the operators in the algebra as the Hamiltonian (the ‘pointed’). This is mathematically equivalent to a complex $W^*$-algebra equipped with a self-adjoint element.

Last revised on September 10, 2018 at 19:50:07. See the history of this page for a list of all contributions to it.