A formalism-local framework for general probabilistic theories including quantum theory
In this paper we consider general probabilistic theories that pertain to circuits which satisfy two very natural assumptions. We provide a formalism that is local in the following very specific sense: calculations pertaining to any region of spacetime 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 spacetime to evolve a state do not have this property nor do histories-based approaches. An operation (see figure on left) has inputs and outputs (through which systems travel). A circuit is built by wiring together operations such that we have no open inputs or outputs left over. A fragment (see figure on right) is a part of a circuit and may have open inputs and outputs. 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). In the figure on the left we show 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 figure on the right is the duotensor for the above fragment. Links represent summing over the corresponding indices. We can use such duotensors to make probabilistic statements pertaining to fragments. Since fragments are the circuit equivalent of arbitrary spacetime 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. [Note: the abstract in the paper has pictures.]