2 Axiom 0 (Minkowski Space)

“Minkowski space” is referred to, directly or indirectly, in most of these axioms. However, a crisp, mathematical definition of “Minkowski space” is never given in Haag and Kastler’s original work. So, to clarify the axioms, we should provide one.

Normally this level of pedanticism isn’t warranted. But, for example, Axiom 1 (Local Algebras) refers to the \(\mathbf{B}\) as “open sets with compact closure in Minkowski space”. So, the topology of “Minkowski space” is relevant, and not all topologies may yield “physical” results.

All that being said, to ensure that the axioms are built on a solid foundation, we need a formal definition of “Minkowski space” to build them on.

2.1 Standard Topology on Minkowski Space

Physicists usually think of Minkowski space as the set \(\mathbb {R}^4\) equipped with a Euclidean topology, a Minkowski metric, and the “standard” time-orientation. The “standard” time-orientation in this context is simply the non-vanishing vector \((1,0,0,0)\) in the standard coordinate system on \(\mathbb {R}^4\).

This may be what Haag and Kastler mean by “Minkowski space”, but then again it may not. As they don’t explicitly define the term “Minkowski space”, it’s uncertain what they mean. Let’s consider some alternatives.

2.2 Indiscrete Topology on Minkowski Space

Other topologies are possible on \(\mathbb {R}^4\), and as Haag and Kastler don’t explicitly specify a topology, we can explore other possibilities.

For example, one can equip \(\mathbb {R}^4\) with the indiscrete topology. The indiscrete topology is the topology in which the only open sets are the empty set and \(\mathbb {R}^4\) itself.

In this case, as we require our \(\mathbf{B}\) to be an open, proper subset of \(\mathbb {R}^4\), the only possible value that \(\mathbf{B}\) can take on is the empty set. (Note, we require \(\mathbf{B}\) to be a proper subset instead of just a subset to avoid the inclusion of global observables. See Section I Paragraph 5 of Haag Kastler.) As \(\mathbf{B}\) is intended to be a region in which one performs a “measurement”, requiring \(\mathbf{B}\) be the empty set doesn’t capture that intention at all. The empty set is too “small”. So, we can assume that the indiscrete topology isn’t implicitly being used.

But this leaves open the question: Exactly which topology is being used? The Euclidean topology?

2.3 Euclidean Topology on Minkowski Space

If the Euclidean topology is being used, then this raises another question: Why use the \(\mathbf{B}\), open sets with compact closure, based on a topology derived from a Euclidean metric on \(\mathbb {R}^4\)? Euclidean metrics have nothing to do with physics which is determined by a Minkowski metric. A more “natural” thing to do would be to base \(\mathbf{B}\) on a “topology derived from a Minkowski metric”, whatever that might mean.

2.4 Alexandrov Topology on Minkowski Space

To consider what that might mean let \(p\) and \(q\) be any two points in what physicists think of as Minkowski space. With these two points, one can define two natural sets: \(I^+(p)\), the chronological future of \(p\) (i.e. the set of points reachable from \(p\) via a future-directed, time-like curve), and \(I^-(q)\), the chronological past of \(q\) (i.e. the set of all points that reach \(q\) via a future-directed, time-like curve).

It turns out that \(I^+(p)\) is an open set in the Euclidean topology for any \(p\) in what physicists think of as Minkowski space (Proposition 2.8 of Penrose). Similarly, the time-reversed version of Proposition 2.8 of Penrose implies that \(I^-(q)\) is an open set in that same topology. Hence, \(I^+(p) \cap I^-(q)\) is also open in that topology.

This suggests that sets of the form \(I^+(p) \cap I^-(q)\) might be able to be used as the basis of a new topology on \(\mathbb {R}^4\). As it so happens, sets of this form can be used as the basis for a topology on \(\mathbb {R}^4\).

Sets \(I^+(p) \cap I^-(q)\) with \(p,q \in \mathbb {R}^4\) form the basis for what’s known as the Alexandrov topology of \(\mathbb {R}^4\) (Definition 4.22 of Penrose). (This is also sometimes known as the interval topology of \(\mathbb {R}^4\) (Remark 4.23 of Penrose).) Furthermore, it turns out that the Alexandrov topology of \(\mathbb {R}^4\) agrees with the Euclidean topology of \(\mathbb {R}^4\) (Paragraph 4.23 of Penrose), a welcome happenstance that “explains” why physicists had not confronted this earlier.

All that being said, we will equip \(\mathbb {R}^4\) with the Alexandrov topology as it is the “natural” topology of relativity. So for us Minkowski space is \(\mathbb {R}^4\) with the Alexandrov topology, the Minkowski metric, and standard time-orientation.

In this case the Euclidean topology and the Alexandrov topology agree, but on other manifolds this need not be the case.