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Contrary Inferences in Consistent Histories and a Set Selection Criterion


The best developed formulation of closed system quantum theory that handles multiple-time statements, is the consistent (or decoherent) histories approach. The most important weaknesses of the approach is that it gives rise to many different consistent sets, and it has been argued that a complete interpretation should be accompanied with a natural mechanism leading to a (possibly) unique preferred consistent set. The existence of multiple consistent sets becomes more problematic because it allows the existence of contrary inferences [1]. We analyse the conceptual difficulties that arise from the existence of multiple consistent sets and provide a suggestion for a natural set selection criterion. This criterion does not lead to a unique physical consistent set, however it evades the existence of consistent sets with contrary inferences. The criterion is based on the concept of preclusion and the requirement that probability one propositions and their inferences should be non-contextual. The allowed consistent sets turn-out to be compatible with coevents which are the ontology of an alternative, histories based, formulation [24].

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  1. Here we identify logical propositions with a collection of subsets of a sample space (Kolmogorov). In this notation set inclusion \(\subseteq \) means that there is a logical implication.

  2. A more general view of what a proposition is for a histories formulation will be given later.

  3. Note, that following Hartle and Sorkin, we are adopting the path integral view of histories that takes the stance that there exist a unique (preferred) fine grained description, i.e. paths in the generalised configuration space. Other points of view, such as Isham’s, are compatible with the one we take, at least in most ordinary cases.

  4. The initial state can be viewed as the initial condition, which in principle is the initial state of the universe. The decoherence functional and the quantum measure that we will later define, depend crucially on this initial state. The decoherence functional encodes both the dynamics and the initial condition of the system.

  5. In the original references, a weaker condition was given, namely that \(D(A,A)\ge 0\) which is called (weak) positivity, while the strong positivity condition was first defined in [26]. See Appendix 1 for details.

  6. The sum in the expression should be replaced with an integral if we consider continuous histories.

  7. Strictly speaking, from the real part of the off-diagonal terms.

  8. An example of the latter can be found in [40] where the quantum measure of a particle never leaving a region of the configuration space is unity because of the quantum Zeno effect, without belonging to any consistent set. Note however, that if a quantum measure one history does belong to a consistent set, then its negation has quantum measure zero.

  9. Essentially this example was first given by Aharonov and Vaidman [41].

  10. In this paper we have considered finite cardinality histories spaces. The case for infinite histories space is considerably more complicated, both conceptually and technically, since one needs to restrict attention to “measurable” sets. It is worth to note however, that in examples of discrete spacetimes, such as (finite) causal sets, the histories space is indeed finite.

  11. This compatibility means that if the coevent \(B\) is realised, then the statement “history \(A\) occurred” gets truth value “True”, while the negation \(\lnot A\) gets truth value “False”.

  12. Note that already deciding if a CS is PCS is a very demanding task even for history space \(\Omega \) of small cardinality.

  13. We should note here, that similarly with the OCSs it would be much more difficult to decide if a given CS satisfies this new criterion.


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This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) through Grant EP/K022717/1. Partial support from COST Action MP1006 is also gratefully acknowledged. The author is grateful to Robert Griffiths for his remarks concerning the conventional single-framework rule for consistent histories and to the anonymous reviewers for their constructive comments.

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Correspondence to Petros Wallden.


Appendix 1: Weak Versus Strong Positivity of Decoherence Functional

In Sect. 3 we gave the definition of the decoherence functional as either path integral or string of projection operators. Moreover we gave four conditions that the defined decoherence functional obeys. The third condition was the positivity condition and we noted that there exist two versions of this condition. The stronger one was used in this paper while the weaker one was given in the first papers [710]. One can show that for decoherence functionals defined by Eqs. (3) and (4), i.e. for standard quantum mechanics, the strong positivity condition is indeed obeyed.

It is important to show that the strong positivity condition is satisfied by all decoherence functionals allowed, because this is crucial in determining the relation between PCSs and OCSs.

When the initial state is pure, the decoherence functional for two histories \(\alpha \) and \(\beta \) can be expressed as the inner product between the branch state vectors \(\psi _\alpha \) and \(\psi _\beta \). But this means the decoherence functional over any set of histories \(\{\alpha \}\) is a Gram matrix, which is automatically a positive matrix. When the initial state is impure, it is just a mixture of pure state so that the decoherence functional is a mixture of (positive) Gram matrices, which must also be positive.

In consistent histories framework as considered by Omnes, Griffiths, and Gell-Mann and Hartle, branch state vectors are obtained by applying the class operators—i.e. strings of time-evolved projectors—to the initial state. Importantly, this does not rely on having a set \(\Omega \) of fine-grained histories from which all other histories are a coarse-graining.

However one could, instead of starting with definitions of Eqs. (3) and (4), take a more radical view and use conditions (1–4) of Sect. 3 as defining the decoherence functional. This view can be adapted in order to generalise quantum mechanics in such a way that is desirable for cases that time (and thus time ordered projections) may not be well defined, as for example in certain approaches to quantum gravity. Another reason that this type of generalisations are interesting is because they do not presuppose the Hilbert space structure.

In attempting to generalise quantum mechanics one is free to choose to use either the strong or the weak positivity condition. In [44] it was shown that if one adopts the strong positivity condition, then some Hilbert space structure can be recovered starting solely from conditions (1–4). Interestingly, in [45] it was shown that even if one is restricted to strongly positive decoherence functionals, there are some correlations allowed that are not predicted by standard quantum theory. Therefore, even the stronger positivity condition constitutes a generalisation of standard quantum mechanics. Moreover in [45] it was also pointed out that the weak positivity condition is not closed under composition. In other words it is possible to have two uncorrelated, non-interacting systems described by weak positive decoherence functionals and if one attempted to construct a joint decoherence functional for those systems, it no longer obeys the positivity condition. On the other hand, strongly positive decoherence functionals are closed under composition (two systems described by strongly positive decoherence functionals lead to a joint decoherence functional that is also strongly positive).

For the above reasons, if one was to generalise quantum mechanics using conditions (1–4) as starting point, then there are good physical reasons to believe that it is more appropriate to adopt the strong positivity condition.

Appendix 2: Example of PCS that is Not OCS

Here we will give an explicit example that a CS is Preclusive but not Ordered. Let us consider an example with three possible histories \(\Omega =\{h_1,h_2,h_3\}\). Consider the following decoherence functional

$$\begin{aligned} D=\left( \begin{array}{c@{\quad }c@{\quad }c@{}} 1/3 &{} -7/24 &{} 7/24\\ -7/24 &{} 1/2 &{} -7/24\\ 7/24 &{} -7/24 &{} 3/4\end{array} \right) \!. \end{aligned}$$

One can easily check that it obeys the requirements of a strongly positive decoherence functional (is a positive matrix, symmetric, normalised to unity with all diagonal terms non negative). There are two CS, namely \(C_1=\{\{h_1\},\{h_2,h_3\}\}\) which leads to the coarse grained decoherence functional

$$\begin{aligned} D=\left( \begin{array}{c@{\quad }c} 1/3 &{} 0\\ 0 &{} 2/3\end{array} \right) \!. \end{aligned}$$

and \(C_2=\{\{h_1,h_2\},\{h_3\}\}\) which leads to the coarse grained decoherence functional

$$\begin{aligned} D=\left( \begin{array}{c@{\quad }c} 1/4 &{} 0\\ 0 &{} 3/4\end{array} \right) \!. \end{aligned}$$

Since there is no precluded history, both CSs are PCSs. However, \(C_1\) is not an OCS since history \(\{h_1\}\) has quantum measure \(\mu (\{h_1\})=1/3\) which is greater than the quantum measure of \(\{h_1,h_2\}\) which is \(\mu (\{h_1,h_2\})=1/4\). Both \(\{h_1\}\) and \(\{h_1,h_2\}\) are consistent histories and (evidently) \(\{h_1\}\subset \{h_1,h_2\}\). Similarly the consistent set \(C_2\) is also not an OCS. This means that the only OCS is the trivial CS, while both \(C_1\) and \(C_2\) are PCSs.

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Wallden, P. Contrary Inferences in Consistent Histories and a Set Selection Criterion. Found Phys 44, 1195–1215 (2014).

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  • Consistent histories
  • Decoherent histories
  • Contrary inferences