# A Nice Definition of “Field Theory”

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 $\Omega$ where we are meant to interpret elements $x\in\Omega$ as points in spacetime and where $x\leq y$ means $x$ could causally affect $y$.

We say that $x,y\in\Omega$ are space-like separated if neither $x\leq y$ nor $y\leq x$ and we say that a subset $\Sigma\subseteq\Omega$ is a space-like slice if it’s an antichain (that is, where any two distinct elements are space-like separated).

Given a subset $\Sigma\subseteq\Omega$ we say that a subset $p\subset\Omega$ is a path to $\Sigma$ if:

• $p$ is a linear ordering
• $p$ has a maximum element and that maximum element is in $\Sigma$
• $p$ is a maximal subset with the above two properties

Now, we can form a partial symmetric monoidal category $C$ as follows:

• The objects are the space-like slices of $\Omega$
• The monoidal product $\Sigma\otimes\Gamma$ is defined just when all elements of $\Sigma$ are space-like separated with every element of $\Gamma$. In that case, it’s defined to be $\Sigma\cup\Gamma$
• The category is a partial order with a morphism from $\Sigma$ to $\Gamma$ just in case every path to $\Gamma$ intersects $\Sigma$

In other words, there is a morphism between two space like slices $\Sigma$ and $\Gamma$ just in case the state of the world at $\Sigma$ should determine the state of the world at $\Gamma$.

Now, given that representation of spacetime, to form a field theory you simply pick some other symmetric monoidal category $D$ and a monoidal functor from $C$ to $D$. The authors of the paper point out that you have different choices for $D$ 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 $\Sigma$ 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!