Cecilia Flori

Group Action in Topos Quantum Theory

When analysing the origin of the twisted presheaves, it is clear that the reason we do get a twist is because the group moves the abelian algebras around, i.e. the group action is defined on the base category itself. Thus a possible way of avoiding the occurrence of twists is by imposing that the group does not act on the base category. The category of abelian von-Neumann sub-algebras with no group acting on it will be denoted \(\mathcal{V}_{f}(\mathcal{H})\), where we have added the subscript f (for fixed) to distinguish this situation from the case in which the group does act. Obviously, if one just defined sheaves over \(\mathcal{V}_{f}(\mathcal{H})\), then there would be no group action at all, therefore something extra is needed. As we will see this ‘extra’ will be the introduction of an intermediate category which will be used as an intermediate base category. On such an intermediate category the group is allowed to act, thus the sheaves defined over it will admit a group action. Once this is done, everything is “pushed down” to the fixed category \(\mathcal{V}_{f}(\mathcal{H})\). As we will see, the sheaves defined in this way will admit a group action which now takes place at an intermediate stage, but will not produce any twists since the final base category stays fixed.

Probabilities in Topos Quantum Theory

The main idea in the topos formulation of quantum theory it to have logic as a fundamental concept and try and derive other concepts in terms of it. This is what is done in the case of probabilities.

Introducing Category Theory

Roughly speaking a category is a collection of objects and relations between these objects. These relations are required to satisfy certain properties which make the set of all such relations ‘coherent’. Given a category, it is not the case that every two objects have a relation between them, some do and others don’t. For the ones that do, the number of relations can vary depending on which category we are considering.

Quantity Value Object and Physical Quantities

In this chapter we will introduce the quantity value object. As the name suggests such an object will be used to assign values to quantities. In classical theory the quantity value object is simply the real numbers since each quantity has, as its value, an element of the Reals. Similarly, in standard quantum theory we have the reals as the quantity value object.

Topos Analogue of Propositions

In this chapter we will explain how propositions are represented in topos quantum theory. Such a representation requires the very important notion of daseinisation which, literally translated, means “to bring into existence”. Such an operation is used when trying to represent a proposition which is locally defined by a projection operator, in all contexts. In fact it could be the case that, given a context V, the proposition represented by the projection operator \(\hat{P}\) does not belong to V, i.e. \(\hat{P}\notin V\). It is in such cases that the process of daseinisation is used. What it does is essentially to approximate the proposition \(\hat{P}\) as best as possible so that it now belongs to V. This process of approximation is called coarse graining.

Topos Analogue of the State Space

In this chapter we will describe how topos quantum theory can be seen as a contextual quantum theory, in the sense that each element is defined as a collection of ‘context dependent’ descriptions. Such context dependent descriptions will turn out to be classical snapshots.

One-Parameter Group of Transformations and Stone’s Theorem

In this chapter we will define the topos notion of a one-parameter group taking values in the complex number object and in the real number objects. To this end we first of all need to upgrade the monoids \(\underline{\mathbb{C}}^{\leftrightarrow}\) and \(\underline{\mathbb {R}}^{\leftrightarrow}\) to groups. This can be done using a standard method called Grothendieck k-Construction already mentioned in Doring and Isham (2008). Having done that the construction of a one-parameter group can be defined. This in turn allows us to define the topos analogue of the Stone’s theorem which uniquely associates to each self adjoint operator \(\breve{\delta}(\hat{A}): {\underline{\varSigma}\rightarrow \underline{\mathbb{R}}}\) a one parameter group \(\underline{Q}^{\hat{A}}\). This is of particular importance in the view of defining a unique time evolution. In fact, given a Hamiltonian operator \(\underline{H}\), the topos analogue would be \(\breve{\delta}(\hat {H})\) with associated the unique one-parameter group of transformations \(\underline{Q}^{\hat{H}}\). This group would represent the group of time evolutions in topos quantum theory. The detailed analysis of such a group and how it acts had not yet been carried out but it would be of particular interest to do so.

Kochen-Specker Theorem

In this chapter we will analyse one of the main theorems (another one would be Bell’s inequality) which states the impossibility of quantum theory, as it is canonically expressed, to be a realist theory. This theorem is know as the Kochen-Specker theorem and will be explained in details in Sect. 3.3. However, in order to fully understand this theorem, we first of all have to introduce the concept of a valuation function. We will do so first for classical theory and then quantum theory.

Topos Analogues of States

In classical physics a pure state, s, is a point in the state space. It is the smallest subset of the state space which has measure one with respect to the Dirac measure δ s .

The Category of Functors

From previous paragraphs we have understood what a category is, how to define maps between categories, namely functors, and finally how to define maps between functors, which are natural transformations. It is now possible to group all these definitions in a coherent way and define the category of functors.