Like most people, I don’t really know anything about quantum field theory. But the other day I stumbled across this paper by Stefano Gogioso, Maria E. Stasinou, and Bob Coecke that provides a very nice framework for what sort of thing a “quantum field theory” (or really, any “field theory”) is. It certainly doesn’t mean I understand quantum field theory, but knowing what sort of thing it is helps me categorize it in my brain.
The definition is as follows: Suppose we’re given a partial order where we are meant to interpret elements as points in spacetime and where means could causally affect .
We say that are space-like separated if neither nor and we say that a subset is a space-like slice if it’s an antichain (that is, where any two distinct elements are space-like separated).
Given a subset we say that a subset is a path to if:
- is a linear ordering
- has a maximum element and that maximum element is in
- is a maximal subset with the above two properties
Now, we can form a partial symmetric monoidal category as follows:
- The objects are the space-like slices of
- The monoidal product is defined just when all elements of are space-like separated with every element of . In that case, it’s defined to be
- The category is a partial order with a morphism from to just in case every path to intersects
In other words, there is a morphism between two space like slices and just in case the state of the world at should determine the state of the world at .
Now, given that representation of spacetime, to form a field theory you simply pick some other symmetric monoidal category and a monoidal functor from to . The authors of the paper point out that you have different choices for depending on the type of field theory you want: for example, in a finite-dimensional context you could pick a category of finite dimensional Hilbert spaces and completely positive maps between them, or in an infinite-dimensional context you could pick the category of Hilbert spaces and bounded linear maps.
I find just that definition by itself illuminating. Of course, the paper doesn’t stop there: you can define other concepts from this definition (spacetime regions, foliations, etc.) as well as put restrictions on what slices you allow if you need them to, e.g., be nice topologically. They also relate their approach to other frameworks for quantum field theory. Neat!