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Consistent-history quantum theory was developed as an attempt to deal with closed systems in quantum mechanics. Some such innovation is needed since the standard Copenhagen interpretation is incapable of describing the universe as a whole, since the existence of an external observer is required.

Griffiths,[16], [21] Omn`es [18], [19], [20], [17] and Gell-Mann and Hartle [13], [15], [14] approached this problem by proposing a new way of looking at quantum mechanics and quantum field theory, in which the fundamental objects are `consistent' sets of histories. Using this approach it is then possible to make sense of the Copenhagen concept of probabilities even though no external observer is present. A key facet of this approach is that it is possible to assign probabilities to history propositions rather than just to propositions at a single time.

The possibility of making such an assignment rests on the so-called decoherence functional (see Section 2) which is a complex-valued functional, $ d:\mathcal{UP}\times\mathcal{UP}\rightarrow\Cl$ , where $ \mathcal{UP}$ is the space of history propositions. Roughly speaking, the decoherence functional selects those sets of histories whose elements do not `interfere' with each other pairwise (i.e., pairs of histories $ \alpha$ , $ \beta$ , such that $ d(\alpha,\beta)=0$ if $ \alpha\neq\beta$ ). A set $ C=\{\alpha,
 \beta,\cdots, \gamma\}$ of history propositions is said to be consistent if $ C$ is complete2 and $ d(\alpha,\beta)=0$ for all pairs of nonequal histories in $ C$ . Given a consistent set $ C$ , the value $ d(\alpha,\alpha)$ for any $ \alpha\in C$ is interpreted as the probability of the history $ \alpha$ being realized. This set can be viewed `classically' in so far as the logic of such a set is necessarily Boolean.

Although this approach overcomes many conceptual problems related to applying the Copenhagen interpretation of quantum mechanics to a closed system, there is still the problem of how to deal with the plethora of different consistent sets of histories. In fact a typical decoherence functional will give rise to many consistent sets, some of which are incompatible each other in the sense that they cannot be joined to form a larger set.

In the literature, two main ways have been suggested for dealing with this problem, the first of which is to try and select a particular set which is realized in the physical world because of some sort of physical criteria. An attempt along these lines was put forward by Gell-Mann and Hartle in [13] where they postulated the existence of a measure of the quasi-classicality of a consistent set, and which, they argued, is sharply peaked.

A different approach is to accept the plethora of consistent sets and interpret them in some sort of many-world view. This was done by Isham in [23]. The novelty of his approach is that, by using a different mathematical structure, namely `topos theory', he was able to give a rigorous mathematical definition of the concept of many worlds. In particular, he exploited the mathematical structure of the collection of all complete sets of history propositions to construct a logic that can be used to interpret the probabilistic predictions of the theory when all consistent sets are taken into account simultaneously, i.e., a many-worlds viewpoint.

The logic so defined has the particular feature that

  1. It is manifestly `contextual' in regard to complete sets of propositions (not necessarily consistent).

  2. It is multi-valued (i.e., the set of truth values is larger than just $ \{$true, false$ \}$ ).

  3. In sharp distinction from standard quantum logic, it is distributive.

Using this new, topos-based, logic Isham assigned generalised truth values to the probability of realizing a given history proposition. These type of propositions are called `second level' and are of the form ``the probability of a history $ \alpha$ being true is $ p$ ''. In defining these truth values Isham makes use of the notion of a `$ d$ -consistent Boolean algebras $ W^d$ ', which are the algebras associated with consistent sets. The philosophy of his approach, therefore, was to translate into the language of topos theory the existing formalism of consistent histories, but in such a way that all consistent sets are considered at once.

In this formalism the notion of probability is still involved because of the use of second-level propositions that refer to the probability of realizing a history. Therefore, the notion of a decoherence functional is still central in Isham's approach since, it is only in terms of this quantity that the probabilities of histories are determined

In the present paper the approach is different. We start with the topos formulation of physical theories as discussed in detail by Döring and Isham in [1], [2], [3], [4], [5] and [10]. In particular we start with these authors' topos formulation of standard quantum mechanics and extend it to become a new history version of quantum theory. As we shall see, this new formalism departs from consistent-history theory in that it does not make use of the notion of consistent sets, and thus of a decoherence functional. This result is striking since the notion of a decoherence functional is an essential feature in all of the history formalisms that have been suggested so far.

In deriving this new topos version of history theory, we had in mind that in the consistent-history approach to quantum mechanics there is no explicit state-vector reduction process induced by measurements. This suggests postulating that, given a state $ \ket\psi_{t_{1}}$ at time $ t_1$ , the truth value of a proposition $ A_1\in\Delta_1$ at time $ t_{1}$ should not influence the truth value of a proposition $ A_2\in\Delta_2$ with respect to the state, $ \ket\psi_{t_{2}}=\hat{U}(t_2,t_1)\ket\psi_{t_1}$ , at some later time $ t_2$ .

Thus for a history proposition of the form ``the quantity $ A_1$ has a value in $ \Delta_1$ at time $ t_1$ , and then the quantity $ A_2$ has a value in $ \Delta_2$ at time $ t_1=2$ , and then $ \cdots$ '' it should be possible to determine its truth value in terms of the individual (generalised) truth values of the constituent single-time propositions as in the work of Döring and Isham. Thus our goal is use topos theory to define truth values of sequentially-connected propositions, i.e., a time-ordered sequence of proposition, each of which refers to a single time.

As we will see, the possibility of doing this depends on how entanglement is taken into consideration. In fact, it is possible to encode the concept of entanglement entirely in the elements (which, for reasons to become clear, we will call `contexts') of the base category with which we are working. In particular, when entanglement is not taken into account the context category is just a product category. In this situation it is straightforward to exhibit a direct dependence between the truth values of a history proposition, both homogeneous and inhomogeneous, and the truth values of its constituent single-time components.

Moreover, in this case it is possible to identify all history propositions with certain subobjects which are the categorical products of the appropriate pull-backs of the subobjects that represent the single-time propositions. It follows that, when entanglement is not considered, a precise mathematical relation between history propositions and their individual components subsists, even for inhomogeneous propositions. This is an interesting feature of the topos formalism of history theories which we develop since it implies that, in order to correctly represent history propositions as sequentially connected proposition, it suffice to use a topos in which the notion of entanglement is absent. However, if we were to use the full topos in which entanglement is present then a third type of history propositions would arise, namely entangled inhomogeneous propositions. It is precisely such propositions that can not be defined in terms of sequentially connected single-time propositions. This is a consequence of the fact that projection operators onto entangled states cannot be viewed, in the context of history theory, as inhomogeneous propositions.

Our goal is to construct to a topos formulation of quantum history theory as defined in the HPO formalism3. In particular, HPO history propositions will be considered as entities to which the Döring-Isham topos procedure can be applied. Since the set of HPO history-propositions forms a temporal logic, the possibility arises of representing such histories as subobjects in a certain topos which contains a temporal logic formed from Heyting algebras of certain subobjects in the single-time topoi. In this paper we will develop such a logic. Moreover, we will also develop a temporal logic of truth values and discuss the extent to which the evaluation map, which assigns truth values to propositions, does or does not preserve all the temporal connectives. An interesting feature of the topos analogue of the HPO formalism of quantum history theory is that, although it is possible to represent such a formalism within a topos in which the notion of entanglement is present (full topos), in order to correctly define history propositions and their truth values, we have to resort to the intermediate topos. Specifically we need to pull back history propositions as expressed in the full topos to history propositions as expressed in the intermediate topos in which the notion of entanglement is not present. This is necessary since history propositions per se are defined as sequentially connected single-time propositions and such a definition makes sense only in a topos in which the context category is a product category (intermediate topos). It is precisely to such an intermediate topos that the correct temporal logic of Heyting algebras belongs.

The absence of the concept of probability is consistent with the philosophical motivation that underlines the idea in the first place of using topos theory to describe quantum mechanics. Namely, the need to find an alternative to the instrumentalism that lies at the heart of the Copenhagen interpretation of quantum mechanics. In this respect, to maintain the use of a decoherence functional would conflict with the basic philosophical premises of the topos approach to quantum theory. In fact, as will be shown in the present paper, the topos formulation of quantum history theory does not employ a decoherence functional, and the associated concept of `consistency' is absent.

This is an advantage since it avoids the problem of the plethora of incompatible consistent history sets. In fact, the novelty of this approach rests precisely on the fact that, although all possible history propositions are taken into consideration, when defining the logical structure in terms of which truth values are assigned to history propositions, there is no need to introduce the notion of consistent sets.

The outline of the paper is as follows. In Section 2 we give a brief introduction to the theory of consistent histories. Section 3 is devoted to a description of the HPO formulation of history theory. Then, in Section 4, we outline the topos formulation of quantum theory put forward by Isham and Döring, describing in detail how truth values of single-time propositions emerge from the formalism. In section 5 we generalize the above-mentioned formalism to sequentially-connected propositions. In particular, we assign truth values to history propositions in terms of the truth values of single-time propositions for non-entangled settings. We also define the temporal logics of the Heyting algebra of subobjects and of truth values, and we discuss the extent to which the evaluation map preserve temporal connectives. Finally, in Section 6, the issue of entanglement leads us to introduce the topos version of the HPO formalism of quantum history theory.

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