In this paper we consider general probabilistic theories pertaining to circuits that satisfy two very natural assumptions. We provide a formalism that is local in the following very specific sense: calculations pertaining to any region of space–time employ only mathematical objects associated with that region. We call this formalism locality. It incorporates the idea that space and time should be treated on an equal footing. Formulations that use a foliation of space--time to evolve a state do not have this property, nor do histories-based approaches. An operation has inputs and outputs (through which systems travel), for example,
A circuit is built by wiring together operations such that we have no open inputs or outputs left over. A fragment is a part of a circuit and may have open inputs and outputs, for example,
We show how each operation is associated with a certain mathematical object, which we call a duotensor
(this is like a tensor but with a bit more structure). The following diagram shows how a duotensor is represented graphically:
We can link duotensors together such that black and white dots match up to get the duotensor corresponding to any fragment. The following diagram shows the duotensor for the above fragment:
Links represent summation over the corresponding indices. We can use such duotensors to make probabilistic statements pertaining to fragments. Since fragments are the circuit equivalent of arbitrary space–time regions, we have formalism locality. The probability for a circuit is given by the corresponding duotensorial calculation (which is a scalar since there are no indices left over). We show how to put classical probability theory and quantum theory into this framework.