Abstract
Consistent-history quantum theory was developed as an attempt to deal with closed systems in quantum mechanics. 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.
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Notes
- 1.
A set C={α,β,…,γ} is said to be complete if all history are pairwise disjoint and their logical ‘or’ forms the unit history.
- 2.
The acronym ‘HPO’ stands for ‘history projection operator’ and was the name given by Isham to his own (non-topos based) approach to consistent-history quantum theory. This approach is distinguished by the fact that any history proposition is represented by a projection operator in a new Hilbert space, that is the tensor product of the Hilbert spaces at the constituent times. In the older approaches, a history proposition is represented by a sum of products of projection operators, and this is almost always not itself a projection operator. Thus the HPO formalism is a natural framework with which to realise ‘temporal quantum logic’.
- 3.
The unit history is the history which is always true.
- 4.
We will denote the set of open subsets of a topological space, X, by Sub op (X).
- 5.
Arguably, it is more appropriate to represent propositions in classical physics with Borel subsets, not just open ones. However, we will not go into this subtlety here.
- 6.
Of course, in the case of temporal logic, the Hilbert spaces \(\mathcal {H}_{1}\) and \(\mathcal {H}_{2}\) are isomorphic and, hence, so are the associated topoi. However, their structural roles in the temporal logic are clearly different. In fact, in the closely related situation of composite systems it will generally be the case that \(\mathcal {H}_{1}\) and \(\mathcal {H}_{2}\) are not isomorphic. Therefore, in the following, we will not exploit this particular isomorphism.
- 7.
We are here exploiting the trivial fact that, for any pair of categories \(\mathcal{C}_{1},\mathcal{C}_{2}\), we have \((\mathcal{C}_{1}\times\mathcal{C}_{2})^{\mathit {op}} \simeq{\mathcal{C}_{1}}^{\mathit {op}}\times{\mathcal{C}_{2}}^{\mathit {op}}\).
- 8.
It should be noted that, in this context, by entangled algebra W we mean any algebra which can not be written in the form of a pure tensor product, i.e. W≠V 1⊗V 2.
- 9.
Since there is no state-vector reduction, the existence of an operation ⊓ between truth values that satisfies (17.67) is plausible. In fact, unlike the normal logical connective ‘∧’, the meaning of the temporal connective ‘⊓’ implies that the propositions it connects do not ‘interfere’ with each other, since they are asserted at different times: it is thus a sensible first guess to assume that their truth values are independent.
The distinction between the temporal connective ‘⊓’ and the logical connective ‘∧’ has been discussed in details in various papers by Stachow and Mittelstaedt [66–69]. In these papers they analyse quantum logic using the ideas of game theory. In particular, they define logical connectives in terms of sequences of subsequent moves of possible attacks and defences. They also introduce the concept of ‘commensurability property’, which essentially defines the possibility of quantities being measured at the same time or not. The definition of logical connectives involves both possible attacks and defences, as well as the satisfaction of the commensurability property since logical connectives relate propositions which refer to the same time. On the other hand, the definition of sequential connectives does not need the introduction of the commensurability properties, since sequential connectives refer to propositions defined at different times and, thus, can always be evaluated together. The commensurability property introduced by Stachow and Mittelstaedt can be seen as the game theory analogue of the commutation relation between operators in quantum theory. We note that the same type of analysis can be applied as a justification of Isham’s choice of the tensor product as temporal connective in the HPO theory.
- 10.
This is correct since the projectors which appear on the right hand side of the equation are pairwise orthogonal, thus the ‘or’, ∨, can be replaced by the summation operation + of projector operators.
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Flori, C. (2013). Topos History Quantum Theory. In: A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol 868. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35713-8_17
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