*4.1. State on Category*

While an algebra embodies the intrinsic structure of a system, a state embodies the interface between that system and its environment. This view, which has been advocated by Ojima [30], is consistent with the mathematical framework of algebraic quantum field theory and quantum probability theory: states provide concrete representations of an algebra.

In general, a representation refers to an expression of intrinsic structures in a certain way, which corresponds to the concrete realization of the intrinsic properties of a system in the way it interacts with its environment. To be more specific, a state is a mapping which sends elements of an algebra to scalar values as "expectation values". In short, states define the statistical laws, which generalize the notion of probability measures to the noncommutative context. Conversely, if the algebra is a unital commutative *C*∗-algebra, we have the Radon measure on a compact Hausdorff space by the Riesz–Markov–Kakutani theorem [31]. In other words, a pair of an algebra and state on it is a generalized probability space: a noncommutative probability space.

As for the category algebras that reflect the structure of the possible dynamics, defining a state on it means evaluating arrows corresponding to the individual processes with expectation values. Conversely, for a category with a finite number of objects, the weighting of the arrows gives a state. Based on this fact, we call a state on a category algebra, a state on category by abuse of terminology.

The rest of this subsection is based on [10].

**Definition 32** (linear functional)**.** *Let A be an algebra over a rig R. An R-valued linear function on A, i.e., a function preserving addition and scalar multiplication, is called a linear functional on A. A linear functional on A is said to be unital if ϕ*() = 1*, where and* 1 *denote the multiplicative units in A and R, respectively.*

**Definition 33** (positivity)**.** *A pair of rigs with involution* (*<sup>R</sup>*, *R*+) *is called a positivity structure on R if R*+ *is a subrig with involution such that r*,*s* ∈ *R*+ *and r* + *s* = 0 *imply r* = *s* = 0*, and that a*<sup>∗</sup>*a* ∈ *R*+ *for any a* ∈ *R.*

**Definition 34** (state)**.** *Let R be a rig with involution and* (*<sup>R</sup>*, *R*+) *be a positivity structure on R. A state ϕ on an algebra A with involution over R with respect to* (*<sup>R</sup>*, *R*+) *is a unital linear functional ϕ* : *A* −→ *R, which satisfies ϕ*(*a*<sup>∗</sup>*a*) ∈ *R*+ *and ϕ*(*a*<sup>∗</sup>) = *ϕ*(*a*) *for any a* ∈ *R, where* (·)<sup>∗</sup> *and* (·) *denote the involutions on A and R, respectively (the last condition ϕ*(*a*<sup>∗</sup>) = *ϕ*(*a*) *follows from other conditions if R* = C*).*

**Definition 35** (noncommutative probability space)**.** *A pair* (*<sup>A</sup>*, *ϕ*) *consisting of an algebra A with involution over a rig R with involution and a state ϕ is called a noncommutative probability space.*

**Definition 36** (state on category)**.** *Let R be a rig with involution and* (*<sup>R</sup>*, *R*+) *be a positivity structure on R. A state on the category algebra R*[C] *over R with respect to* (*<sup>R</sup>*, *R*+) *is said to be a state on a category* C *with respect to* (*<sup>R</sup>*, *<sup>R</sup>*+)*.*

Given a state *ϕ* on a category C with involution, we have a function *ϕ*ˆ : C −→ *R* defined as *ϕ*<sup>ˆ</sup>(*c*) = *ϕ*(*ι<sup>c</sup>*). For the category with a finite number of objects, we can obtain the following theorem [10], which is a generalization of the result in [14] for groupoids.

**Theorem 7** (state and normalized positive semidefinite function)**.** *Let* C *be a category such that* |C| *is finite. Then, there is a one-to-one correspondence between states ϕ with respect to* (*<sup>R</sup>*, *R*+) *and normalized positive semidefinite Z*(*R*)*-valued functions φ with respect to* (*<sup>R</sup>*, *<sup>R</sup>*+)*, i.e., normalized functions such that*

$$\sum\_{\{ (\mathbf{c}, \mathbf{c'}) \mid \mathbf{d} \text{com}((\mathbf{c'})^\dagger) = \overline{\text{cod}(\mathbf{c})}} \overline{\xi^\mathbf{c'}} \overline{\phi}((\mathbf{c'})^\dagger \circ \mathbf{c}) \xi(\mathbf{c})$$

*is in R*+ *for any R-valued function ξ on* C *with finite support and that φ*(*c*†) = *φ*(*c*)*, where* (·)<sup>∗</sup> *and* (·) *denote the involutions on* C *and R, respectively.*

Conceptually, the theorem above means that states on a category with involution (with finite objects) are nothing but the weights on arrows, which are generalizations of probability distributions on a (finite) set as the discrete category (with finite objects). More generally, we can say that to define a state on a category whose support is contained in a subcategory with finite numbers of objects is equivalent to defining the corresponding function which assigns the weight to each arrow.

For a state on a category whose support is not contained in a subcategory with finite number of objects, we will need some topological structures (or coarse geometric structures [32]). Nonstandard-analytical methods (see [33], for example) will provide useful tools.

### *4.2. States of Quantum Fields as States on Categories*

As we see quantum fields as category algebras, it is quite natural to model physical states of quantum fields as states on category algebras.

**Definition 37** (state of quantum field)**.** *Let* C *be a causal category with partial involution structure* C∼*and* (*<sup>R</sup>*, *R*+) *be a positivity structure on a rig R with involution. A unital linear functional on* C*, which is also a state on* C∼ *with respect to* (*<sup>R</sup>*, *<sup>R</sup>*+)*, whose image is contained in a subrig R with involution of R, is said to be an R-valued state of the quantum field on* C *over R with respect to* (*<sup>R</sup>*, *<sup>R</sup>*+)*.*

In conventional cases, *R* is supposed to be a ∗-algebra over C and *R* = C. The phrase "with respect to (*<sup>R</sup>*, *R*+)" will be omitted if it is clear in the context. In general, given a state *ϕ* on ∗-algebra *A* over a ∗-rig *R* with involution, we can construct the representation of the algebra into the algebra consisting of endomorphism on a certain module (generalized GNS representation [10]). In particular, when *R* is a ∗-rig over C and *ϕ* is C-valued, we have a representation called the GNS (Gelfand–Naimar–Segal) representation [34,35] into a pre-Hilbert space consisting of the equivalence class of the elements in *A*, equipped with the inner product structure induced by the sesquilinear form *a*, *a* = *ϕ*((*a*)<sup>∗</sup>*a*) (*<sup>a</sup>*, *a* ∈ *A*). The unit of the algebra plays a role of a "vacuum" vector (see [10] and reference therein, for example).

In sum, a noncommutative probability space, i.e., a pair (*<sup>A</sup>*, *ϕ*) of a ∗-algebra over C and a C-valued state on it, is sufficient to reconstruct the ingredients in conventional quantum physics based on the Hilbert spaces. In fact, the approach based on the noncommutative probability space is more general than the conventional approach: if we focus onto the local structures of quantum fields, it is known that we cannot use one a priori Hilbert space as a starting point of the theory (actually, this fact itself was one of the historical motivations of AQFT, the pioneer of noncommutative probabilistic approach; see [4], for example).

Our category algebraic approach is a new unification of the noncommutative probabilistic approach and the category theoretic viewpoint. As we observed in the previous section, for †-category (or in general, categories with involution), its category algebra is a ∗-algebra (or in general, an algebra with involution). Note that our algebra is unital, even if C has infinitely many objects. Then, the construction above holds and we can see the unit as a "vacuum" in our theory.

The concept of states on categories also shed light on the foundation of quantum mechanics as a part of the quantum field theory. From our viewpoint, a quantum mechanical system of finite degrees of freedom can be defined as a noncommutative probability space whose algebra is a subalgebra of a category algebra on a causal category with partial involution and whose state satisfies the condition that the support is contained in a subcategory with finite numbers of objects.

In general, quantum fields as category algebras together with states "contain" vast numbers of quantum mechanical systems, or, more precisely, considering a state whose support is contained in some subcategory with finite numbers of objects, is focusing on a quantum mechanical system as an aspect of the quantum field. Note that quantum mechanical systems in the above meaning are not necessarily contained in a single point but can have spatial degrees of freedom, e.g., a system in the double slit experiment, in which the support of the state can be considered to be contained in a subcategory with finite objects. Understanding the situation in which multiple observers are involved in a single quantum field—such as the EPR (Einstein–Podolsky–Rosen) situation [36]—through the concepts of local algebra and local states also seems to be an important research topic.

The idea at the heart of the above discussions is that we are free to think of "localized states" (not just "global" states such as vacuum states). The concept corresponding to these kinds of states is particularly important in the context of AQFT and is called "local states" [15,16]. We can define the local states in our framework, which is a conceptual counterpart of the local states in AQFR, as follows.

**Definition 38** (local state)**.** *Let* C *be a causal category equipped with partial involution structure* C∼ *and R be a rig with involution. A state on R*[C∼(O)] *for a subset* O *of* |C| *is called a local state of the quantum field R*[C] *on* O*.*

From a physical point of view, the notion of local state is quite natural. A macroscopic setting of the environment for the quantum field basically concerns only a bounded spacetime domain and the global state should be seen as an idealization of it. Considering a family of local states, instead of a single state, can be seen as a sheaf theoretic extension of the conventional quantum field theory. The extension will lead to the notion of consistent families of Hilbert spaces and operators on them through the GNS construction, which will be mathematically interesting.

By translating the previous study of [15] into our context, we will be able to construct the generalized sector theory, which is the generalization of DHR (Doplicher–Haag– Roberts)–DR (Doplicher–Roberts) Theory [17–23], as well as develop Ojima's micro–macro duality [24,25] and quadrality scheme [26] from the viewpoint of category algebras and states on categories.

### *4.3. Remarks on the Comparison to TQFT (Continued)*

In Section 3, we constructed for any †-category C a functor (·)*R* : C −→ *Mod*(*R*) by *cR* = *ι <sup>c</sup>*(·) for *c* ∈ C. For quantum physical studies, we need to induce a functor into *Hilb*, which is a category of Hilbert spaces over C. Let us explain the role of states in this induction.

Given any state *ϕ* on the †-category C, *CR* := *CR*[C] for each object *C* ∈ C can be equipped with the "almost inner product" (semi-Hilbert space structure) by defining the sesquilinear form ·|·*ϕ* by *α* |*α<sup>ϕ</sup>* := *ϕ*((*α* )<sup>∗</sup>*α*) (generalized GNS construction, see [10] and references therein). When *R* is a ∗-algebra over C and *ϕ* is a C-valued "good" state *ϕ* on the category given, this functor induces a functor into *Hilb*. More precisely, suppose that *R* is a ∗-algebra over C and the C-valued state *ϕ* satisfies the condition

$$
\varphi(\mathfrak{a}^\*\mathfrak{a}) \ge \varphi((\mathfrak{a}^\varepsilon\mathfrak{a})^\*(\mathfrak{a}^\varepsilon\mathfrak{a}))
$$

for any *α* ∈ *R*[C] and *c* ∈ C. Then, the functor (·)*R* : C −→ *Mod*(*R*) induces the functor (·)*R<sup>ϕ</sup>* : C −→ *preHilb*, where *preHilb* denotes the category of the pre-Hilbert spaces, taking the quotient of *CR* equipped with ·|·*ϕ* by *Nϕ* = {*α* ∈ *CR*|*ϕ*(*α*<sup>∗</sup>*α*) = <sup>0</sup>}, which can be shown as a submodule of *CR* (note that by the assumption *ϕ*(*α*<sup>∗</sup>*α*) = 0 =⇒ *ϕ*((*ι <sup>c</sup>α*)∗(*ι <sup>c</sup>α*)) = 0 holds; this kind of construction itself has a certain generalization to a more general *R* by using this condition directly). Then, by the assumption of *ϕ*, (*c*)*Rϕ* extends uniquely to the morphism in *Hilb* by completion and we have the corresponding functor from C to *Hilb*, which sends *c* to the unique bounded extension of (*c*)*Rϕ* (note that a bounded operator between pre-Hilbert spaces extends to the unique bounded operator between Hilbert spaces). By applying this construction for C = *nCob*, we have a version of TQFT. Note that our framework is naturally incorporated with the causal structure and it is quite interesting to the counterpart of the structure in TQFT.
