Causal Categories: Relativistically Interacting Processes

Abstract

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a causal category. We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics.

Appendix

In this Appendix we expand on the assumptions made at the end of Sect. 2.1, that symbolic formulae are stated up to equivalence in the diagrammatic calculus. Our aim is to show how connectedness in the graphical language can be formalised symbolically.

The motivation for this is as follows. Firstly we recall from Sect. 2.1 that graphical connectedness distinguishes the two modes of composition symbolically, except when the tensor unit is involved. However, the two modes of composition have different, complementary, physical interpretations: parallel composition means independence. But, as we showed in Sect. 3.1, graphical disconnectedness also means independence. Since graphical disconnectedness subsumes parallel composition (as we just described), it also is the primary concept of independence. But the graphical language arose in a formal way from considering SMCs (see Theorem 4), which is usually defined symbolically. Hence it is desirable to understand what kind of symbolic language corresponds to this graphical language of connectedness: this will provide a ‘syntax for information flow’.

In other words, we want to ‘fix’ the language so that formulae faithfully reflect the connectedness of a corresponding diagram.

Standing Assumptions About Formulae

Since it is the non-atomic morphisms that would allow one to ‘hide’ disconnectedness, our two key standing assumptions in Sect. 2.1 were:

• 1. all morphism formulae contain only atomic morphisms;

• 2. a morphism is expressed as a formula using ⊗ instead of ∘ whenever possible.

This corresponds to:

• 1. ensuring that the ‘atoms’ of our language are connected morphisms, so that sequential composition corresponds to genuine information-flow

• 2. revealing genuine disconnectedness of a morphism in its corresponding formula (i.e. a formula reveals non-atomicity of the morphism).

In short, we force the syntax of the morphism language to faithfully represent the topology of the graphical language.

Remark 21

(Morphism language formulae containing only atomic morphisms)

Note that the assumption that all formulae contain only atomic morphisms can be made without any loss of generality, because we only consider protocols that involve a finite number of systems. An arbitrary formula can be turned into one made up of atomic morphisms simply by substituting all non-atomic morphisms by a non-trivial parallel composition, and repeat this procedure until all morphisms are atomic. Without finiteness this procedure may not terminate. This can be seen as a formally distinguishing between the following two views on the relation between a system and its subsystems. This first view is ‘compositional’: we describe a system by considering jointly its subsystems, and the latter are conceptually prior. Conversely, the second view is ‘decompositional’: Nature appears to us as a single entity, and we can tractably do physics by decomposing it into subsystems; now the global system is conceptually prior.

Definition 49

(‘Disconnectedness’ for symbolic language)

We say that a symbolic language expression $\mathcal{F}:\mathcal {A}\rightarrow\mathcal{B}$ factors through ⊗ if, by pre- and post-composition of generalized symmetry morphisms and by using the axioms of SMCs, $\mathcal{F}$ cannot be rewritten as a symbolic language expression $\mathcal{F}_{1}\otimes\mathcal{F}_{2}$.

For example, given ⊤:B→I and ψ:I→B, the expression (g∘f)⊗(ψ⊗⊤) does not factor through ⊗, whereas the expression (1 _I ⊗ψ)∘(⊤⊗1 _I ) does, because the axioms of SMCs ensure that

$$(1_I\otimes\psi)\circ(\top\otimes1_I) =(1_I\circ\top)\otimes(\psi\circ1_I).$$

Note that Definition 49 captures a notion that is distinct from non-atomicity, since the latter refers to whether a morphism is equal to the tensor of two other morphisms (and is therefore about the contingent equations of the category), whereas the former refers to whether a morphism formula can be manipulated syntactically to a particular form, using the axioms of symmetric monoidal categories. The requirement of pre- and post-composition in Definition 49 is included because of morphisms such as the symmetry isomorphism σ _A,B :A⊗B→B⊗A. This will be symbolically disconnected because post-composition with σ _B,A yields 1 _A ⊗1 _B .

Theorem 50

(Relationship between symbolic and graphical connectedness)

Given the standing assumptions above, the topological disconnectedness of a morphism in the graphical language is equivalent to its corresponding symbolic language expression factoring through ⊗.

Proof

This follows by Theorem 4 [33], which states that the graphical calculus corresponds fully and faithfully to the syntactic manipulations of the corresponding symbolic language expressions. □

The significance of Theorem 50 is that it provides a formal expression of ‘reading the topology from the syntax’, i.e. with the assumptions above, the syntax of ⊗ in an expression in the symbolic language corresponds to topological disconnectedness when translated to the graphical language.