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!

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