6 Axiom 4 (Quasilocal Algebra)
Next let’s consider Axiom 4 (Quasilocal Algebra)
Axiom 4 (Quasilocal Algebra) The set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\) is a normed *-algebra. Taking its completion we get a C*-algebra which we denote by \(\mathfrak {U}\) and call the algebra of quasilocal observables. We maintain that \(\mathfrak {U}\) contains all observables of interest.
Most of this actually turns out to be straightforward.
6.1 Set-Theoretic Union
The first claim this axiom makes is that the set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\) is a normed *-algebra. Is this the case?
If the set-theoretic union of all \(\mathfrak {U}(\mathbf{B})\) is a normed *-algebra, then all of the standard properties of a normed *-algebra (e.g. finite vector sums, finite algebraic products, involution...) must be defined for elements of this set-theoretic union. Are such properties defined?
Consider two arbitrary elements \(a_1\) and \(a_2\) in this set-theoretic union. As \(a_1\) and \(a_2\) are elements of this set-theoretic union, there exists a \(\mathbf{B}_1\) such that \(a_1 \in \mathfrak {U}(\mathbf{B}_1)\) and a \(\mathbf{B}_2\) such that \(a_2 \in \mathfrak {U}(\mathbf{B}_2)\).
By definition \(\mathbf{B}_1\) is of the form \(I^+(p_1) \cap I^-(q_1)\) and \(\mathbf{B}_2\) is of the form \(I^+(p_2) \cap I^-(q_2)\). As we are in Minkowski spacetime, there always exists a \(p\) and \(q\) such that both \(I^+(p_1) \cap I^-(q_1)\) and \(I^+(p_2) \cap I^-(q_2)\) are contained in \(I^+(p) \cap I^-(q)\). Let us denote the set \(I^+(p) \cap I^-(q)\) as \(\mathbf{B}\), i.e. \(\mathbf{B} \equiv I^+(p) \cap I^-(q)\).
As both \(I^+(p_1) \cap I^-(q_1)\) and \(I^+(p_2) \cap I^-(q_2)\) are contained in \(I^+(p) \cap I^-(q)\), the isotony axiom implies that \(\mathfrak {U}(\mathbf{B}_1) \subset \mathfrak {U}(\mathbf{B})\) and \(\mathfrak {U}(\mathbf{B}_2) \subset \mathfrak {U}(\mathbf{B})\). This in turn implies that \(a_1, a_2 \in \mathfrak {U}(\mathbf{B})\), which in turn implies that all of the standard properties of a normed *-algebra (e.g. vector sum, algebraic product, involution...) hold for \(a_1\) and \(a_2\) viewed as elements of the algebra \(\mathfrak {U}(\mathbf{B})\).
As \(a_1\) and \(a_2\) were arbitrary elements of the set-theoretic union, the standard properties of a normed *-algebra (e.g. vector sum, algebraic product, involution...) hold for an arbitrary finite number of elements in the set-theoretic union.
The subtle point to note here is that limits of elements in this set-theoretic union may not be in the set-theoretic union. Completion solves this problem.
6.2 Completion of the Set-Theoretic Union
Completion of the set-theoretic union is relatively standard.
One considers any Cauchy sequence \(\{ a_i\} \) in the set-theoretic union. The limit of such a Cauchy sequence might not be in the set-theoretic union. So, we complete the set-theoretic union to the quasilocal algebra \(\mathfrak {U}\) by asserting that \(\mathfrak {U}\) contains, in addition to all the elements of the set-theoretic union, the limit of all Cauchy sequences in the set-theoretic union.
It turns out that all the standard properties of a C*-algebra are retained by the completion. For example, consider a Cauchy sequence \(\{ a_i\} \) that converges to \(a\). One has
So the C*-norm property holds for the quasilocal algebra \(\mathfrak {U}\). Furthermore, as a result of the following theorem (Corollary 2.2.6 of Bratteli and Robinson)
If \(\mathfrak {U}\) is a *-algebra and there exists a norm on \(\mathfrak {U}\) with the C*-norm property and with respect to which \(\mathfrak {U}\) is closed, then this norm is unique.
this C*-norm on the quasilocal algebra \(\mathfrak {U}\) is unique.
6.3 All Observables of Interest
The final assertion of this axiom states
We maintain that \(\mathfrak {U}\) contains all observables of interest.
This is the only element of the axiom that isn’t straightforward.
As we’ve seen, given a state \(\omega \) we can use the GNS construction to create the von Neumann algebra \(\pi _\omega (\mathfrak {U})''\). “Observables” then correspond to self-adjoint elements of \(\pi _\omega (\mathfrak {U})''\).
Tracing the definitions of \(\pi _\omega (\mathfrak {U})'\) and \(\pi _\omega (\mathfrak {U})'' \equiv (\pi _\omega (\mathfrak {U})')'\) we see that for a generic state \(\omega \)
i.e. \(\pi _\omega (\mathfrak {U})\) is generically a proper subset of \(\pi _\omega (\mathfrak {U})''\) and the complement of \(\pi _\omega (\mathfrak {U})\) in \(\pi _\omega (\mathfrak {U})''\) may not be the empty set.
For the generic state \(\omega \) self-adjoint elements \(a\) of \(\mathfrak {U}\) then correspond to “observables” as follows: \(\pi _\omega \) is a *-homomorphism; hence, \(\pi _\omega \) maps a self-adjoint element \(a\) to a self-adjoint element \(\pi _\omega (a)\). By construction \(\pi _\omega (a)\) is within \(\pi _\omega (\mathfrak {U})\) and thus within \(\pi _\omega (\mathfrak {U})''\). So \(\pi _\omega (a)\) is a self-adjoint element of \(\pi _\omega (\mathfrak {U})''\) and thus corresponds to an “observable”. It is in this sense that \(\mathfrak {U}\) “contains observables”.
However, generically \(\pi _\omega (\mathfrak {U})\) is only a proper subset of \(\pi _\omega (\mathfrak {U})''\). Hence, generically there exist “observables” corresponding to self-adjoint elements of \(\pi _\omega (\mathfrak {U})''\) that do not correspond to an element of \(\mathfrak {U}\), i.e. generically self-adjoint elements of \(\pi _\omega (\mathfrak {U})'' \backslash \pi _\omega (\mathfrak {U})\) exist.
What are we to make of this in light of the axiom’s statement “We maintain that \(\mathfrak {U}\) contains all observables of interest”?
What this actually means becomes clear if we examine a related theorem. But before doing so we must dispense with a few preliminary results.
The GNS construction provides a *-homomorphism \(\pi _\omega \). Furthermore, our slight modification of Axiom 2 (Isotony) implies that \(\mathfrak {U}\) is unital, i.e. there exists a unit \(\mathbf{1}\) in \(\mathfrak {U}\). Thus, \(\pi _\omega (\mathfrak {U})\) is a *-subalgebra of \(\mathcal{B}(\mathcal{H}_\omega )\) ( i.e. the set of bounded operators on the Hilbert space \(\mathcal{H}_\omega \) ) and \(\pi _\omega (\mathfrak {U})\) contains the identity operator on the Hilbert space \(\pi _\omega (\mathbf{1})\).
These facts together with the following theorem (Lemma 4.1.4 of Murphy)
Let \(\mathcal{H}\) be a Hilbert space and \(\mathcal{S}\) a *-subalgebra of \(\mathcal{B}(\mathcal{H})\) that contains the identity. Then \(\mathcal{S}\) is strongly dense in \(\mathcal{S}''\).
imply that \(\pi _\omega (\mathfrak {U})\) is strongly dense in \(\pi _\omega (\mathfrak {U})''\).
As \(\pi _\omega (\mathfrak {U})\) is strongly dense in \(\pi _\omega (\mathfrak {U})''\), the definition of strongly dense implies that if one is presented with any \(A''\) in \(\pi _\omega (\mathfrak {U})''\) then for any \(\varepsilon \) such that \(0 {\lt} \varepsilon \) there exists a \(\pi _\omega (a)\) in \(\pi _\omega (\mathfrak {U})\) such that
for all \(\psi \) in \(\mathcal{H}_\omega \). In other words the distance between the Hilbert space state \(A''\psi \) and the Hilbert space state \(\pi _\omega (a)\psi \) can be made as small as one likes.
This implies that experimentally one is unable to measure if one is working with the operators \(\pi _\omega (\mathfrak {U})\) or with the operators \(\pi _\omega (\mathfrak {U})''\). So there may indeed exist self-adjoint elements of \(\pi _\omega (\mathfrak {U})'' \backslash \pi _\omega (\mathfrak {U})\), but physically they are indistinguishable from elements in \(\pi _\omega (\mathfrak {U})\).
So, in this sense \(\mathfrak {U}\) “contains all observables of interest”.