AQFT in Lean

4 Axiom 2 (Isotony)

Now let us consider Axiom 2 (Isotony) which states

Axiom 2 (Isotony) If \(\mathbf{B_1} \subset \mathbf{B_2}\) then \(\mathfrak {U}(\mathbf{B_1}) \subset \mathfrak {U}(\mathbf{B_2})\). We assume in addition that one of the two following situations prevails. Either \(\mathfrak {U}(\mathbf{B_1})\) and \(\mathfrak {U}(\mathbf{B_2})\) have a common unit element, or neither of them has a unit. The first situation can be obtained from the second by formal adjunction of a unit.

4.1 GNS Construction

Before examining Axiom 2 (Isotony), we need to take a moment to review how AQFT is operationalized.

Operationally, the (abstract) C*-algebras \(\mathfrak {U}(\mathbf{B})\) aren’t used directly. Instead they are operationalized through the “GNS construction”, which we now describe.

An AQFT state, sometimes simply called a state, is an element \(\omega \) of the dual space \(\mathfrak {U}(\mathbf{B})^*\) that is

  • Positive - for any \(a \in \mathfrak {U}(\mathbf{B})\) we have \(0 \le \omega (a^*a)\) and

  • Normalized - the operator norm satisfies \(\| \omega \| =1\).

Furthermore, an AQFT state \(\omega \) is said to be faithful if for any non-zero \(a\) in \(\mathfrak {U}(\mathbf{B})\) it follows that \(0 {\lt} \omega (a^*a)\).

The GNS construction is a procedure that given an AQFT state \(\omega \) on an (abstract) C*-algebra \(\mathfrak {U}(\mathbf{B})\) with a unit, produces a triple \((\mathcal{H}_\omega , \pi _\omega , \Omega _\omega )\) consisting of a Hilbert space \(\mathcal{H}_\omega \), a *-homomorphism (and thus representation) \(\pi _\omega \) of \(\mathfrak {U}(\mathbf{B})\) on the Hilbert space, and \(\Omega _\omega \) a distinguished vector in the Hilbert space. (Note 1 , without the unit in \(\mathfrak {U}(\mathbf{B})\), the GNS construction is unable to produce the distinguished vector \(\Omega _\omega \).)

Furthermore, the GNS construction’s representation \(\pi _\omega \) is faithful if the AQFT state \(\omega \) one starts with is faithful. (This justifies the name “faithful” being applied to an AQFT state.)

It is in the Hilbert space \(\mathcal{H}_\omega \) that “normal physics” occurs. In other words elements of \(\mathcal{H}_\omega \) are what one thinks of as normal QFT states, and self-adjoint elements of \(\pi _\omega (\mathfrak {U}(\mathbf{B}))''\) (the von Neumann algebra generated by \(\pi _\omega (\mathfrak {U}(\mathbf{B}))\)) are interpreted as the normal observables of QFT 2 . Finally, the distinguished vector \(\Omega _\omega \) is interpreted as being a vacuum state.

Also note that this GNS construction can in addition be applied to \(\mathfrak {U}\) the algebra of quasilocal observables that appears in Axiom 4 (Quasilocal Algebra). In other words one can have an AQFT state \(\omega \) that is an element of \(\mathfrak {U}^*\) and use this AQFT state and the GNS construction to create the triple \((\mathcal{H}_\omega , \pi _\omega , \Omega _\omega )\). (Again, without a unit in \(\mathfrak {U}\), the GNS construction is unable to produce the distinguished vector \(\Omega _\omega \).)

4.2 Isotony

With this understanding of how AQFT is operationalized, we can now examine isotony in a bit more detail. We will find that isotony, along with the requirement that \(\mathfrak {U}(\mathbf{B})\) be a C*-algebra, is a “natural” requirement.

As mentioned when we introduced the GNS construction, given an AQFT state \(\omega \) on an (abstract) C*-algebra \(\mathfrak {U}(\mathbf{B})\) with unit the GNS construction produces a triple \((\mathcal{H}_\omega , \pi _\omega , \Omega _\omega )\) consisting of a Hilbert space \(\mathcal{H}_\omega \), a *-homomorphism (and thus representation) \(\pi _\omega \) of \(\mathfrak {U}(\mathbf{B})\) on the Hilbert space, and \(\Omega _\omega \) a distinguished vector in the Hilbert space.

“Observables” in the region \(\mathbf{B}\) then correspond to self-adjoint elements of the von Neumann algebra \(\pi _\omega (\mathfrak {U}(\mathbf{B}))''\). Furthermore, the possible values one can obtain when “measuring” the “observable” corresponding to a self-adjoint \(A\) in \(\pi _\omega (\mathfrak {U}(\mathbf{B}))''\) are the values in the spectrum \(\sigma _{\pi _\omega (\mathfrak {U}(\mathbf{B}))''}(A)\) of \(A\) when it is viewed as an element of the von Neumann algebra \(\pi _\omega (\mathfrak {U}(\mathbf{B}))''\).

Consider again our main theme, isotony. It implies that if \(\mathbf{B_1} \subset \mathbf{B_2}\) then \(\mathfrak {U}(\mathbf{B_1}) \subset \mathfrak {U}(\mathbf{B_2})\).

So, if we have a state \(\omega \) in \(\mathfrak {U}(\mathbf{B_2})^*\) we can consider the “observables” corresponding to the self-adjoint elements of \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))''\).

As \(\mathfrak {U}(\mathbf{B_1}) \subset \mathfrak {U}(\mathbf{B_2})\) we can restrict the action of \(\omega \) to \(\mathfrak {U}(\mathbf{B_1})\) and use this restriction, which we also denote by \(\omega \), to define a state in \(\mathfrak {U}(\mathbf{B_1})^*\).

So, with this state \(\omega \) in \(\mathfrak {U}(\mathbf{B_1})^*\) we can consider the “observables” corresponding to the self-adjoint elements of \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_1}))''\).

If one traces through definitions of taking the commutator, one finds

\begin{align} \pi _{\omega }(\mathfrak {U}(\mathbf{B_2})) \subset \pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))”. \end{align}

However, as \(\mathfrak {U}(\mathbf{B_1}) \subset \mathfrak {U}(\mathbf{B_2})\), one also has \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_1})) \subset \pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))\). This in turn implies that

\begin{align} \pi _{\omega }(\mathfrak {U}(\mathbf{B_1})) \subset \pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))”. \end{align}

Tracing definitions of the commutators again one finds this implies

\begin{align} \pi _{\omega }(\mathfrak {U}(\mathbf{B_1}))” \subset \pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))”. \end{align}

So we can view an “observable’s” self-adjoint element \(A\) in \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_1}))''\) equivalently as a self-adjoint element \(A\) in \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))''\). This allows us to view the “observable” as being “measured” in \(\mathbf{B}_1\) or being “measured” in \(\mathbf{B}_2\).

However, as one will recall, the possible values one can obtain when “measuring” the “observable” corresponding to a self-adjoint \(A\) in \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_1}))''\) are the values in the spectrum \(\sigma _{\pi _\omega (\mathfrak {U}(\mathbf{B}_1))''}(A)\). Similarly, the possible values one can obtain when “measuring” the self-adjoint \(A\) viewed as an element of \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))''\) are the values in the spectrum \(\sigma _{\pi _\omega (\mathfrak {U}(\mathbf{B}_2))''}(A)\).

Observation indicates that experimental results do not depend on whether we consider an experiment as occurring in \(\mathbf{B}_1\), the city of Cambridge, Massachusetts, or whether we consider it as occurring in \(\mathbf{B}_2\), the state of Massachusetts. So this means that the spectrum \(\sigma _{\pi _\omega (\mathfrak {U}(\mathbf{B}_1))''}(A)\) and the spectrum \(\sigma _{\pi _\omega (\mathfrak {U}(\mathbf{B}_2))''}(A)\) should be identical.

It turns out that for C*-algebras we have the following theorem (Proposition 2.2.7 of Bratteli and Robinson)

Theorem 1 C*-Spectrum is Invariant Under Inclusion
#

Let \(\mathfrak {U}_1\) be a C*-subalgebra of the C*-algebra \(\mathfrak {U}_2\). If \(a\) is an element of \(\mathfrak {U}_1\), then the spectrum \(\sigma _{\mathfrak {U}_1}(a)\) of \(a\) when viewed as an element of \(\mathfrak {U}_1\) is the same as the spectrum \(\sigma _{\mathfrak {U}_2}(a)\) of \(a\) when viewed as an element of \(\mathfrak {U}_2\)*

\begin{align} \sigma _{\mathfrak {U}_1}(a) = \sigma _{\mathfrak {U}_2}(a). \end{align}

This allows us to drop the subscript from \(\sigma _{\mathfrak {U}_1}(a)\) and \(\sigma _{\mathfrak {U}_2}(a)\) and to write \(\sigma (a)\) unambiguously.

As the von Neumann algebras \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_1}))''\) and \(\pi _{\omega }(\mathfrak {U}(\mathbf{B_2}))''\) are both also C*-algebras, we can apply this theorem to conclude

\begin{align} \sigma _{\pi _\omega (\mathfrak {U}(\mathbf{B}_1))''}(A) = \sigma _{\pi _\omega (\mathfrak {U}(\mathbf{B}_2))''}(A) \end{align}

which is a check on our use of C*-algebras (The same doesn’t hold for Banach *-algebras for example.) and a check on the isotony axiom.

4.3 Common Unit

The isotony axiom also contains this rather “odd” addendum:

We assume in addition that one of the two following situations prevails. Either \(\mathfrak {U}(\mathbf{B_1})\) and \(\mathfrak {U}(\mathbf{B_2})\) have a common unit element, or neither of them has a unit. The first situation can be obtained from the second by formal adjunction of a unit.

If, as suggested, it is always possible, through formal adjunction, to add a unit, then why even consider the case in which there is no unit?

For example, using the state \(\omega \) to apply the GNS construction to an (abstract) C*-algebra \(\mathfrak {U}(\mathbf{B})\) that doesn’t have a unit is impossible 3 . The presence of a unit in \(\mathfrak {U}(\mathbf{B})\) leads directly to the distinguished vector \(\Omega _\omega \) in the Hilbert space \(\mathcal{H}_\omega \). This distinguished vector \(\Omega _\omega \) is interpreted as being a vacuum. So having a unit in \(\mathfrak {U}(\mathbf{B})\) seems indispensable.

That being said, we will simply assume that all such \(\mathfrak {U}(\mathbf{B})\) contain a unit.

So if we adopt the position that a unit is always present, one convenient way to include it, and the method we adopt, is to assume that the empty set \(\emptyset \) is assigned the C*-algebra \(\mathbb {C} \mathbf{1}\) generated by the unit \(\mathbf{1}\)

\begin{align} \emptyset \rightarrow \mathfrak {U}(\emptyset ) \equiv \mathbb {C} \mathbf{1}. \end{align}

As the empty set \(\emptyset \) is a member of any \(\mathbf{B}\), isotony implies \(\mathfrak {U}(\emptyset ) \subset \mathfrak {U}(\mathbf{B})\) which in turn implies that all \(\mathfrak {U}(\mathbf{B})\) contain a unit.

  1. Technically, one only requires the existence of a “bounded, approximate identity” and not a unit to produce a distinguished vector \(\Omega _\omega \). However, physical motivation for the existence of a “bounded, approximate identity” as opposed to a unit is lacking.
  2. Given \(\mathcal{N}\) a self-adjoint subset of \(\mathcal{B}(\mathcal{H}_\omega )\) the set of bounded operators on \(\mathcal{H}_\omega \) ( i.e. \(\mathcal{N} \subseteq \mathcal{B}(\mathcal{H}_\omega )\) such that \(\mathcal{N}^*=\mathcal{N}\) ) we define \(\mathcal{N}' \equiv \{ N' \in \mathcal{B}(\mathcal{H}_\omega ) : NN' - N'N = 0 \; \forall N \in \mathcal{N} \} \) and similarly define the shorthand \(\mathcal{N}'' \equiv (\mathcal{N}')'\).
  3. Again, technically, one only requires the existence of a “bounded, approximate identity” and not a unit to produce a distinguished vector \(\Omega _\omega \). However, physical motivation for the existence of a “bounded, approximate identity” as opposed to a unit is lacking.