# Contrary Inferences for Classical Histories within the Consistent Histories
Formulation of Quantum Theory

Adamantia Zampeli  Centro de Ciencias Matemáticas, Universidad Nacional
Autónoma de México, 58190, Morelia, Michoacán, Mexico<EMAIL_ADDRESS>Georgios E. Pavlou  Institute of Applied and Computational Mathematics (IACM),
Foundation for Research and Technology Hellas (FORTH), IACM/FORTH, GR-71110
Heraklion, Greece<EMAIL_ADDRESS>Petros Wallden  University of
Edinburgh, School of Informatics, Informatics Forum, 10 Crichton Street,
Edinburgh EH8 9AB, United Kingdom<EMAIL_ADDRESS>
###### Abstract

In the histories formulation of quantum theory, sets of coarse-grained
histories, that are called consistent, obey classical probability rules. It
has been argued that these sets can describe the semi-classical behaviour of
_closed_ quantum systems. Most physical scenarios admit multiple different
consistent sets and one can view each consistent set as a separate context.
Using propositions from different consistent sets to make inferences leads to
paradoxes such as the contrary inferences first noted by Kent [1]. Proponents
of the consistent histories formulation argue that one should not mix
propositions coming from different consistent sets in making logical
arguments, and that paradoxes such as the aforementioned contrary inferences
are nothing else than the usual microscopic paradoxes of quantum contextuality
as first demonstrated by Kochen and Specker theorem.

In this contribution we use the consistent histories to describe a macroscopic
(semi-classical) system to show that paradoxes involving contextuality (mixing
different consistent sets) persist even in the semi-classical limit. This is
distinctively different from the contextuality of standard quantum theory,
where the contextuality paradoxes do not persist in the semi-classical limit.
Specifically, we consider different consistent sets for the arrival time of a
semi-classical wave packet in an infinite square well. Surprisingly, we get
consistent sets that disagree on whether the motion of the semi-classical
system, that started within a subregion, ever left that subregion or not. Our
results point to the need for constraints, additional to the consistency
condition, to recover the correct semi-classical limit in this formalism and
lead to the motto ‘all consistent sets are equal’, but ‘some consistent sets
are more equal than others’.

##  1 Introduction

The consistent histories formulation of quantum theory [2, 3, 4, 5, 6, 7]
brings forward a proposal for dealing with the quantum mechanics of a closed
system by replacing the observer-dependent process of quantum measurement with
a _consistency condition_ , that determines when a set of (coarse-grained)
histories admits a classical description. The formalism allows for multiple
consistent sets to exist and the dominant view is that all these sets are to
be treated equally111An alternative view is that one consistent set (or a
family of consistent sets) are more _natural_ (in some suitably defined sense)
and should be given a preferred status in interpreting the theory. The
physical principle that selects such consistent set(s) is still elusive even
though certain proposals have been given [8, 9, 10]. Adrian Kent in ref. [1]
noted that, if we were to combine propositions belonging to different
consistent sets, we would reach “paradoxes” in classical logic (contrary
inferences). This reinforced the view, shared by proponents of the approach,
that each consistent set is a context and that one should reason within a
single consistent set each time. This would avoid the aforementioned paradoxes
[11]. Contextuality, that is, reaching to different conclusions depending on
the context in which the question is asked as well as the subsequent
“paradoxes” that appear when one mixes contexts is a well-known and
fundamental property of the (microscopic) quantum systems [12, 13, 14, 15]. It
is also true though that such paradoxes should _not_ persist in larger scales
when dealing, for example, with semi-classical systems.

In this contribution, we study a simple physical system at the semi-classical
limit within the consistent histories approach to show that the apparent
“contextuality” arising from the analysis of different consistent sets is more
paradoxical than the one of the “standard” quantum theory. We illustrate this
through a concrete example based on the time-of-arrival problem for an
initially suitably localised wave packet that moves within an infinite square
well, with positive average momentum. This can represent, for instance, a
tennis ball moving between two impenetrable walls. For this semi-classical
problem, we construct two different consistent histories sets and we find what
probability each consistent set assigns to the arguably simple question of
whether the ball ever crossed the middle of the interval. We show that one
consistent set concludes with certainty that the ball crossed the middle of
the interval, while the other consistent set concludes with certainty that the
ball never crossed it. Since there is no physical obstacle in the middle of
the interval, our intuition suggests that it is the former consistent set that
is “correct” while the latter gives the “wrong” answer. Note, however, that
within the consistent histories approach there is no a priori way to
distinguish between these two consistent sets.

#### Overview of the results.

Before going to the details, we give here an intuitive explanation of our
work. Perhaps the easiest way to view consistent histories is to consider the
sum-over-histories (Feynman) formulation of quantum theory. Then one simply
partitions the space of all histories (paths) to (disjoint) subsets of
histories. If all these coarse-grained histories pairwise have no
interference, then this partition is a consistent set. In the simplest case we
have partitions with only two cells, where the one cell is the question we ask
(e.g. did the wave packet remain in the negative axis) and the other is its
negation (e.g. the wave packet at some point was on the positive axis). Now,
in order for this partition to be a consistent set, the two cells need to have
zero interference. This leads to the condition that
$\bra{\Psi(\tau)}\Psi_{r}(\tau)\rangle=1$, where $\ket{\Psi(\tau)}$ is the
system unitarily evolved until the final time $\tau$ and
$\ket{\Psi_{r}(\tau)}$ is the evolution of the system with the restriction
that it did not leave the cell we ask for (which mathematically is equivalent
with normal evolution with extra infinite walls on the boundary of the region
in question) [16]. It is very important to note that in the consistent
histories formulation there is no actual physical measurement taking place –
we are dealing with closed quantum systems without any observer – and the
“infinite walls” is just a simpler way to describe the corresponding path
integral that will tell us if the consistency condition is satisfied (and thus
we can assign classical probabilities) or not (and thus this question involves
quantumness and we cannot assign classical probabilities to those histories).
Coming back to our example, we note that mathematically the consistency of the
set depends on the final time $\tau$ and the restricted evolution imposed by
the region we consider. In particular, the set is still consistent even if, in
some earlier time, this condition was not satisfied. Therefore, to find a
consistent set that contradicts our classical intuition, we simply need to
find a semi-classical system that after some time evolves to the same state as
if it was restricted to a smaller region, perhaps by “bouncing” and returning
to the same place.

We achieve this by looking at a free wave packet in an infinite square well
localised in the negative axis moving with positive average momentum. The
question asked (restriction) is whether it crosses to the positive axis. The
significance/effect of the infinite square well is that the “unrestricted”
evolution of the wave packet bounces back and coincides after some time with
the restricted evolution. We have now just constructed a consistent set that
concludes that the particle never crosses the axis, even though it is a semi-
classical object moving towards there. One might be led to believe that for
this question the consistent histories approach cannot recover the “right”
answer. However, we note that it is easy to find another consistent set that
actually gives the intuitive answer: we can consider a partition such that a
small region follows the semi-classical path of the wave packet throughout the
time interval. The surprising conclusion is that we have constructed two
consistent sets, disagreeing about a classical property (passing into the
positive axis/position) of a classical object (localised wave packet). We
recall at this point that in both sets, the “restrictions” come from the
question we ask, i.e. by simply partitioning the history space in two
different ways. There is no physical difference in the two scenarios. The
history space (possible paths) and the dynamics (that are encoded in the
unrestricted Hamiltonian that defines the decoherence functional) are
identical.

A final comment is that, in our analysis, we are using _approximate_
consistency (i.e. we require that there is _almost_ no interference between
paths). We do not expect that this limits the generality of our conclusions
since (i) for histories constructed with continuous variables, it is
approximate consistency/decoherence that is typically achieved and (ii) there
is theoretical work supporting this idea that relates approximate consistent
sets with exactly consistent sets [17].

#### Related works.

The interpretation of consistent histories has been a heated topic
(indicatively, examples of this dialogue can be found on the critical accounts
of [18, 19, 20] that were followed by responses of proponents [21, 22]). Our
work intends to add one more dimension in this scientific exchange
strengthening the argument for supplementing the consistency condition with a
“set-selection” criterion. This issue was first raised by Dowker and Kent [17]
and later by Kent [1] who noted that the problem of contrary inferences is the
result of a contextuality particular only to the histories formalism. He later
gave a set-selection criterion to restrict “ordered consistent sets” [23, 1,
24]. In [9] it was argued that contrary inferences originate from the
existence of zero covers on the histories space and brought forward another
set-selection criterion that restricts attention to preclusive consistent
sets. In [8], Anastopoulos connected the predictability of a consistent set
with the persistence in time; while recently, Halliwell proposed to classify
multiple consistent sets on whether or not there exists any unifying
probability for combinations of incompatible sets which replicates the
consistent histories result when restricted to a single consistent set [10].
Gell-Mann and Hartle [25] and Riedel, Zurek and Zwolak [26] on the other hand
gave set-selection criteria that were motivated by some physical principle
(e.g. thermodynamic considerations) or induced by the environment
respectively.

In the preclusive consistent sets criterion of [9], histories that are
precluded (have measure zero) have a special role, since such histories and
their subsets are not allowed to happen in any “legitimate” (i.e. preclusive)
consistent set. Precluded histories play crucial role in another histories
formulation, called co-events formulation or anhomomorphic logic [27, 28],
that interprets the decoherence functional (or quantum measure [29])
differently. Our results may have consequences in that approach too, but
analysing this is left for a future work.

The example we introduce, that within the consistent histories illustrates our
point (existence of contrary inferences for classical systems), is based on
the well-known time-of-arrival problem for a free particle [30]. The time-of-
arrival type of questions constitute prototypes for dealing with questions
that involve a non-trivial temporal aspect (c.f. time extended propositions).
Standard quantum theory cannot answer such questions easily, possibly due to
the fact that time is not an observable (no self-adjoint time operator), while
it only appears as an external parameter marking the evolution in Schrödinger
equation. Within the (consistent) histories framework temporal questions
without external observers (closed systems) can be addressed naturally. For
example, time-of-arrival histories are special cases of spacetime coarse-
grainings of the histories space [31].

There have been many works addressing temporally extended questions within the
consistent histories formulation. For instance Halliwell and Zafiris in [32]
showed that coupling of the particle with a thermal bath allows the definition
of coupling-dependent probabilities for the time-of-arrival problem;
Anastopoulos and Savvidou in [33] studied the issues of the definition of a
temporal probability density in the context of Copenhagen interpretation and
of histories formalism and attempted to construct POVMs by considering
measurements smeared in time, while in [34, 35, 16] it was investigated the
time-of-arrival problem for closed quantum systems in non-relativistic quantum
theory and in time-reparametrisation invariant theories.

The time-of-arrival histories are also related to the quantum Zeno effect [36,
37] which states that the evolution of a system “freezes” under continuous
observation (see for example [35]). In infinite-dimensional projection
operators, as the one we are dealing with in this work, it has been shown that
the evolution is restricted to a subspace of the Hilbert space [38]. These
cases usually refer to spatial projections but they also accommodate more
general questions regarding the assignment of some property of the quantum
system. In this case, the projector confines the system in some range of the
spectrum of the corresponding observable.

These ideas have also been used in the context of quantum gravity for the
definition of observables. Quantum gravity is a genuinely time-
reparametrisation invariant theory and treats time as a dynamical degree of
freedom, in contrast to quantum theory in which time is just a background
parameter. The time-of-arrival type of questions in quantum gravity can be
used to define the observables of the theory, by asking whether the system is
in some region of the configuration space (or equivalently in a subspace of
the Hilbert space) [39, 40]. Consistent histories formalism offers again a
suitable framework to work with this type of questions in quantum gravity
since not only can naturally be formulated, but also they can naturally
describe closed systems [41, 16, 42].

#### Organisation of the paper.

This paper is organised as follows: In section 2 we review the formalism of
the consistent histories, the zero covers of the histories space and the
formulation of the time-of-arrival problem in the histories language. In
section 3, we define the suitable consistent sets; we give a detailed account
of the assumptions that this construction is based on and demonstrate that the
consistency condition is indeed satisfied for both coarse-grainings. We
discuss and conclude in section 4.

##  2 Preliminaries

###  2.1 The consistent histories formalism

Figure 1: Paths of a non-relativistic particle on the two-dimensional
spacetime diagram. The particle starts from a point $x_{0}$ at $t_{0}$ and
ending at some $t_{f}$ after some time $t_{f}$. In every intermediate time
$t_{1},t_{2},...,t_{n}$ there is a spatial partition (exhaustive and
exclusive) depicted by the intervals $\Delta_{i}(t_{j})$.

In this section, we briefly review the histories formalism for completeness
and to fix the notation. We are dealing with a closed quantum system, with no
external environment222In cases that we have a quantum system interacting with
some, apparently, external environment, we simply include the degrees of
freedom of the “environment” in the definition of our quantum system. or
observer. The physics is fully captured by the kinematics and dynamics of the
quantum system. The kinematics defines the space of possible histories,
similarly to the space of possible “paths” in the path integral view of
quantum theory. The dynamics is determined by the Hamiltonian (or the action)
of the system, and mathematically enter the picture with the decoherence
functional (see later) that determines the “amplitudes” of different histories
and their “interference”. It is well known that due to interference, one
cannot in general assign classical probabilities to those histories (c.f.
double slit). The central aim of the consistent histories is to find under
what conditions can someone assign classical probabilities to (coarse-grained)
sets of histories. The main virtue of the approach is that those conditions
are derived from the quantum system (dynamics and kinematics) without any
further assumption about observers and/or measurements.

A history, that we will denote as $h_{i}$, is a generalised object that gives
a description of the system at different times. A finest-grained history gives
a complete description of the system at all moments of time. The set of all
finest-grained histories comprises the histories space $\Omega$. For instance,
in the case of the non-relativistic particle, fine-grained histories are all
the single-valued functions $x^{i}(t)$ on the configuration space of the
coordinate variables $(t,x^{i})$ that the particle can follow under certain
boundary conditions. We will use this example to clarify concepts in our brief
introduction since it is the most relevant for our work. In general a (coarse-
grained) history $h_{i}$ corresponds to a set of fine-grained histories (in
our example a set of trajectories $x^{i}(t)$). The space of histories $\Omega$
can be partitioned into a collection of coarse-grained histories $\\{h_{i}\\}$
that are exhaustive $\cup_{i}h_{i}=\Omega$ and disjoint $h_{i}\cap
h_{j}=\varnothing\ \forall\ i,j$. We call such set of histories a _partition_.
There is no unique way to partition the space of histories, and as we will see
not all partitions are suitable for predictions. Histories, as quantum
objects, interfere and in general it is not possible to assign consistently a
classical probability to a particular (fine-grained or coarse-grained)
history. Instead, one can define a bi-linear function on the space of
histories, called the decoherence functional

$D(h_{i},h_{j})=\int_{x^{i}(t)\in h_{i}}\mathcal{D}x^{i}\int_{x^{j}(t)\in
h_{j}}\mathcal{D}x^{j}\exp\\{i\left(S[x^{i}]-S[x^{j}\right)\\}\delta(x_{f}^{i}-x_{f}^{j})\rho(x_{0}^{i},x_{0}^{j})$
(1)

Here again we use the example of a non-relativistic particle for simplicity.
The decoherence functional is a complex-valued, bi-linear function defined on
the space of all (fine-grained or coarse-grained) histories $h_{i},h_{j}$. The
integral is defined over the paths $x^{i}(t)$ in the coarse-grained histories
$h_{i}$ and $h_{j}$ respectively. The paths start from $x_{0}$ at $t_{0}$ and
end at $x_{f}$, and are defined within the time interval $[t_{0},t_{f}]$,
while $\rho(x_{0}^{i},x_{0}^{j})$ is the initial density matrix; the
$\delta$-function ensures that histories with different end-point decohere (we
can mathematically think of having a final-time position measurement).

The decoherence functional essentially measures the interference between the
histories. When applied to a partition of the histories space, it measures
interference of different answers within the context defined by the
partition/coarse-graining. The decoherence functional satisfies the following
properties:

  1. (i)

hermiticity: $D(h_{i},h_{j})={D^{\dagger}}(h_{j},h_{i})$

  2. (ii)

positivity: $D(h_{i},h_{j})\geq 0$

  3. (iii)

normalisation: $\sum_{h_{i},h_{j}}D(h_{i},h_{j})=1$

  4. (iv)

superposition principle $D(h,h^{\prime})=\sum_{\begin{subarray}{c}h_{i}\in
h,\\\ h_{j}\in h^{\prime}\end{subarray}}D(h_{i},h_{j})$

An important special case is when one considers a partition (recall
$\cup_{i}h_{i}=\Omega\ ;\ h_{i}\cap h_{j}=\emptyset$), such that the non-
diagonal elements of the decoherence matrix vanish, that is

$D(h_{i},h_{j})=0,\ \forall\ i\neq j$ (2)

Then the (coarse-grained) histories that make this partition, form a
_consistent set_ of histories, and the diagonal elements of the decoherence
functional give the probabilities of the corresponding histories to happen,
that is

$p(h_{i}):=D(h_{i},h_{i}).$ (3)

The relation (2) is known as the consistency condition and it is the key
mathematical structure in consistent histories, giving us a tool to identify
when a coarse-grained set of histories (partition) of a closed quantum system
admits consistent classical probabilities.

The expression of (3), when applied to a general history (not one belonging to
a consistent set) is usually referred to as _candidate probability_. If a
history belongs to a consistent set (i.e. at a partition obeying (2) ensuring
that alternatives within that set have no interference), the candidate
probability is the actual probability of that history, while in general, it
cannot be treated as probability because of interference. The decoherence
functional and the related candidate probabilities inspired a generalisation
of standard measure theory to quantum measure theory [29], a mathematical
construct that replaces Kolmogorov sum rule with a weaker higher-order rule
$\mu(A\cup B\cup C)=\mu(A\cup B)+\mu(A\cup C)+\mu(B\cup
C)-\mu(A)-\mu(B)-\mu(C)$. It can be shown that the diagonal elements of a
decoherence functional always define a quantum measure (or more precisely a
“level-2 measure”, since higher level measures were also defined):

$\mu(h_{i}):=D(h_{i},h_{i}).$ (4)

In the following we will be using quantum measure notation to denote the
diagonal elements of the decoherence functional.

In the non-relativistic quantum mechanics, the decoherence functional can also
be formulated as a sequence of time-ordered projection operators that
represent physical events at different moments of time

$D(h,h^{\prime})=Tr(C_{h}^{\dagger}\rho C_{h^{\prime}})$ (5)

where

$C_{h}=P_{h_{n}}U(t_{n}-t_{n-1})...U(t_{2}-t_{1})P_{h_{1}}(t_{1})$ (6)

$U(t)=\exp(-iHt)$ is the unitary evolution operator, $H$ the Hamiltonian of
the system and $P_{h_{i}}$ is the projection operator on the subspace that the
history $h_{i}$ lies at time $t_{i}$.

Note that the operator and path-integral descriptions are not always
equivalent since they start from different points. For non-relativistic
quantum mechanics the Hamiltonian formulation is more general (e.g. the
projections at a given time can be either configuration space or momentum
space) but both approaches give same predictions. Specifically for our example
that involves paths on configuration space they are known to be equivalent
[43]. In this work, projections at any given time are on different ranges
$\Delta(t)$ of the position that in general depend on the time instance

$P=\int_{\Delta(t)}dx\outerproduct{x}{x}$ (7)

An important issue in the consistent histories approach is that there exist
many different partitions (sets of histories) that decohere (satisfy the
consistency condition) and do not arise as different coarse-grainings of a
single finest-grained consistent set. This could become problematic when
trying to infer what happened comparing predictions from incompatible
consistent sets. Here it is important to note that all consistent sets
correspond to the very same quantum system – the physics (e.g. Hamiltonian) is
identical while there is no measurement or observer involved to differentiate
the scenarios. It is simply a different view of the same system, each with
equal importance, unless the standard formalism of the consistent histories is
supplemented with some extra structure (e.g. a criterion selecting a preferred
consistent set).

A final point to note is that in real systems coarse-grained histories rarely
decohere exactly. Instead one deals with _approximate_ decoherence i.e.
partitions that obey the consistency condition up to some (negligible)
$\epsilon$. This is not considered a major issue for two reasons. First it is
difficult to test observables in arbitrary precision and it seems unlikely
that some negligible interference between histories would alter any observable
quantity. Second it has been shown that for any approximately consistent set,
it is possible to find an _exact_ set that is close in a suitably defined
sense [44, 17].

(a) Contrary propositions $P,Q$: they are orthogonal $PQ=QP=0$ and one
proposition implies the negation of the other $Q<1-P$ i.e $Q<\neg P$.

(b) Contradictory propositions $P,Q$ are those that are orthogonal $PQ=QP=0$
and one proposition is the complement of the other $Q=\neg P$.

Figure 2: Contrary and contradictory propositions. The letters represent fine-
grained histories, the sets $P,Q$ coarse-grained.

###  2.2 Classical logic and contrary inferences

Figure 3: Contrary inferences and zero cover measures: Assume that
$P=\\{a,e\\},\ Q=\\{a,b,c,d\\}$ are two sets (corresponding to propositions)
and that the transition amplitudes of the elements are
$\alpha(\textrm{a})=1,\alpha(b)=-1,\alpha(c)=1,\alpha(d)=-1,\alpha(e)=-1$. The
measures/probabilities of the propositions are $\mu(P)=\mu(Q)=0$ and that
$P\cup Q$ is a zero cover of $\Omega$. In addition $\\{P,\bar{P}\\},\
\\{Q,\bar{Q}\\}$ constitute two consistent sets, where $\bar{P},\bar{Q}$ are
contrary propositions and since $\mu(\bar{P})=\mu(\bar{Q})=1$ we have
“contrary inferences”.

The observation that there are more than one consistent sets of histories to
describe the same physical situation indicates that the formalism, unless
supplemented by additional criteria, may have multiple ways of making
predictions about the properties of the system. To ensure that this does not
contradict classical intuition and more importantly everyday experience, one
may need to consider what can be inferred from different partitions
(consistent sets) in the quantum and (semi)classical regimes. The latter is
the topic of this paper. Here we introduce some background on classical logic
and contrary inferences.

First, we note that sets of coarse-grained histories can be identified with
propositions. Then, contrary inferences appear due to contrary propositions
[1]. The definition in terms of projection operators is given as follows: if
we consider two propositions represented by the projection operators $P$, $Q$
at some moment of time, then, they are called:333We can write
$P_{A}P_{B}=P_{A}\wedge P_{B}:=P_{A\wedge B}$ only when $[P_{A},P_{B}]=0$. If
in addition $P_{A}P_{B}=P_{B}P_{A}=0$ we can write for the conjunction
$P_{A\vee B}:=P_{A}\vee P_{B}=P_{A}+P_{B}$, see e.g. [45] contradictory, if
they do not intersect but cover the full space jointly at that moment, i.e.
$PQ=QP=0$ and $P=1-Q$ and contrary if $PQ=QP=0$ and $P<1-Q$, if they do not
intersect and jointly do not span the full space at that moment. See Figure 2
for an example with coarse-grained histories described by sets, as it is in
the more general case, where we treat the histories space $\Omega$ as the set
that includes all the fine-grained histories as elements.

A proposition that happens with unit probability is assigned truth value True.
It is not hard to see that if one has two contrary propositions (say $P,Q$)
that both occur with probability unity, it means that they both have truth
value $v(P)=v(Q)=\textrm{True}$, something that leads to logical paradoxes. By
the law of inference (modus ponens), if a proposition $P$ is assigned truth
value True, so does any proposition that contains $P$. However, we note that
$1-Q>P$ so that the proposition $1-Q$ should also take value True which leads
to the situation that both $v(Q)$ and the negation $v(1-Q)$ are True. To reach
this paradox we had to make an inference (thus the term “contrary
inferences”). Interestingly, in consistent histories, one can get two contrary
propositions having probability unity (each defined in a different consistent
set).

In [9] it was highlighted that one can view the origin of contrary inferences
in the existence of a “two-histories zero measure” cover of histories
$\Omega$. A two-histories zero measure cover of $\Omega$ is a pair of (coarse-
grained) histories $h_{1},h_{2}$ that each of them has zero measure
$\mu(h_{1})=0,\mu(h_{2})=0$ and they cover $\Omega$ in the sense that
$h_{1}\cup h_{2}=\Omega$.

Let us denote $\bar{h}$ the negation of the history $h$. The existence of two-
histories zero measure cover means that we can find two consistent sets of
histories $C_{1}=\\{h_{1},\bar{h}_{1}\\}$ and $C_{2}=\\{h_{2},\bar{h}_{2}\\}$
such that the histories $h_{i},\bar{h}_{i}$ satisfy the relations

$\displaystyle\mu(h_{1})=0,\ \mu(\bar{h}_{1})=1,$ (8)
$\displaystyle\mu(h_{2})=0,\ \mu(\bar{h}_{2})=1$ (9)

and $h_{1},h_{2}$ cover $\Omega=h_{1}\cup h_{2}$. We can see that
$\bar{h}_{1}\cap\bar{h}_{2}=\varnothing$ which means that the two histories
$\bar{h}_{i}$ have no common fine-grained history and are contrary
propositions. Since both $\bar{h}_{i}$ occur with unit probability, they
should be assigned truth value True leading to contrary inferences. See Figure
3 for a pictorial representation on the language of propositions. A simple
physical example that arises is the three-slit experiment [9, 27]. Note,
however, that in the three-slit experiment, the histories (and the consistent
sets) “retrodict” the slit that a photon (or quantum particle) passed from.
Obtaining a contextual description for the history of a system in the quantum
realm is something widely accepted. What we show in this paper is that in the
consistent histories, such contextuality persists for semi-classical systems.

###  2.3 The time-of-arrival problem in the consistent histories approach

The time-of-arrival problem is the question “what is the probability that the
particle will be (“arrive”) in some region (range) $\Delta$ during some
interval $[0,\tau]$?”, given some initial conditions and with $\Delta$
corresponding to a region of the configuration space or more general a region
in the spectrum of some observable. A simple way to deal with this question is
to ask its complementary: “what is the probability that the particle always
remains in the region $\bar{\Delta}$”.

The set-up of the problem, shown in Figure 4(a), is that of a free particle
with support in the interval $\Delta$ and its complement $\bar{\Delta}$
defined respectively by $\Delta=[0,+\infty)$ and $\bar{\Delta}=(-\infty,0]$,
which evolves under the action of its Hamiltonian

$H=-\frac{\hbar^{2}}{2M}\frac{d^{2}}{dx^{2}}$ (10)

The particle is initially located in the negative axis with positive average
momentum and it is thus expected to cross the surface $x=0$ at some moment
$t\in[0,\tau]$. Let us consider histories that frequently check if the
particle is still in $\bar{\Delta}$, i.e. apply frequently the corresponding
projection that mathematically is equivalent with performing frequent
measurements with the continuous projection operator

$\bar{P}=\int_{\bar{\Delta}}dx\outerproduct{x}{x}$ (11)

to check whether the particle leaves the subspace $\bar{\Delta}$. At the limit
of infinitely many projections $N\rightarrow\infty$ and $\delta
t=T/N\rightarrow 0$, it has been shown that the evolution of the system is
restricted in the subregion $\bar{\Delta}$ with evolution propagator [16]

$g_{r}(t)=\exp(-iH_{r}t)\bar{P}$ (12)

where the restricted Hamiltonian $H_{r}$ is given by

$H_{r}\equiv\bar{P}H\bar{P}$ (13)

and is defined on the proper subspace $L^{2}(\bar{\Delta})$. Its domain is

$D(H_{r})=\\{\Psi\in\mathcal{L}^{2}(\mathbb{R})|\partial_{x}\Psi\in\mathcal{L}^{2}(\mathbb{R}),\Psi(\partial\bar{\Delta}=0)\\}$
(14)

This can alternatively be written as [38]

$H_{r}=-\frac{\hbar^{2}}{2M}\frac{d^{2}}{dx^{2}}+\bar{V}(x),\quad\bar{V}(x)=\begin{cases}0,&x\in\bar{\Delta},\\\
\infty,&\textrm{otherwise}\end{cases}$ (15)

which means that the particle behaves as if it were confined in $\bar{\Delta}$
by rigid barriers when one makes infinitely many measurements.444We will call
the walls induced by infinitely many projections barriers while we keep the
term “wall” for the boundaries of the infinite square well in which all the
dynamics happens. This construction has also been called “Zeno box” [46],
because it is the analogue of the quantum Zeno effect for continuous variables
systems. We note, again, that this is only a mathematical equivalence, since
in our considerations there are no measurements, the projections arise in our
effort to compute the amplitude of a particular (restricted) set of histories.

(a) Time-of-arrival problem for a free particle

(b) Histories in the time-of-arrival problem

Figure 4: Left Panel: Instant of the coarse-graining for the time-of-arrival
problem for a free particle. The question is whether the particle will have
for at least some moment $t$ positive position coordinate. A natural coarse-
graining is to select the intervals $\Delta(t)=[0,+\infty)$ and
$\bar{\Delta}(t)=(-\infty,0]$ as shown here.
Right Panel: Various histories in the time-of-arrival problem in a specific
partitioning – red paths remain always in the region $\bar{\Delta}$, blue
paths have being in $\Delta$ at some moment of time.

To express this problem in the consistent histories approach, we construct the
coarse-grained alternative histories

$\displaystyle h=\\{x(t)\ |\ x(t)\in\bar{\Delta}\ \forall t\in[0,\tau]\\},$
(16a) $\displaystyle\bar{h}=\\{x(t)\ |\ x(t)\notin\bar{\Delta}\text{ for some
}t\in[0,\tau]\\}.$ (16b)

These histories define the set $\mathcal{C}=\\{h,\bar{h}\\}$ and form a two-
histories partition since they satisfy the relations

$h\cup\bar{h}=\Omega,\quad h\cap\bar{h}=\varnothing.$ (17)

Note that while this pair of histories form a partition, in general these
histories would _not_ decohere and this partition would _not_ constitute a
consistent set.

The evolution restricted in $\bar{\Delta}$ is given by
$g_{r}(t,t_{0})\ket{\Psi_{0}}$ that corresponds to history $h$, while the
evolution that is not always in the region $\bar{\Delta}$ that corresponds to
the history $\bar{h}$ is
$\left(g(t,t_{0})-g_{r}(t,t_{0})\right)\ket{\Psi_{0}}$ as it can easily be
seen from the path integral.

In order to assign probabilities to the histories, the set of the alternative
histories $\mathcal{C}$ must be consistent, i.e. the non-diagonal terms of the
decoherence functional must vanish,

$\displaystyle D(h,\bar{h})=0$ $\displaystyle=$
$\displaystyle\bra{\Psi_{0}}g^{\dagger}_{r}(t,t_{0})\left(g(t,t_{0})-g_{r}(t,t_{0})\right)\ket{\Psi_{0}}$
$\displaystyle=$
$\displaystyle\bra{\Psi_{0}}g^{\dagger}_{r}(t,t_{0})g(t,t_{0})\ket{\Psi_{0}}-\bra{\Psi_{0}}g^{\dagger}_{r}(t,t_{0})g_{r}(t,t_{0})\ket{\Psi_{0}}$
$\displaystyle=$
$\displaystyle\bra{\Psi_{0}}g^{\dagger}_{r}(t,t_{0})g(t,t_{0})\ket{\Psi_{0}}-\bra{\Psi_{0}}\bar{P}\ket{\Psi_{0}}$
$\displaystyle\bra{\Psi_{0}}\bar{P}\ket{\Psi_{0}}$ $\displaystyle=$
$\displaystyle\bra{\Psi_{0}}g^{\dagger}_{r}(t,t_{0})g(t,t_{0})\ket{\Psi_{0}}.$
(18)

We note that if $\ket{\Psi_{0}}$ is initialised in the region $\bar{\Delta}$
(as it will be done later) then we have
$\bra{\Psi_{0}}\bar{P}\ket{\Psi_{0}}=1$ simplifying the expression.

This relation, with the above assumption therefore gives:

$\bra{\Psi_{0}}g_{r}^{\dagger}(t,t_{0})g(t,t_{0})\ket{\Psi_{0}}=\bra{\Psi_{r}(t)}\ket{\Psi(t)}=1$
(19)

The condition (19) states that, for the set of histories to be consistent, the
overlap between the full and the restricted propagator at some time $t$ must
be equal to unity (or more generally, if the state was not localised in
$\bar{\Delta}$, equal to the overlap at the initial time $t_{0}$). In that
case we have a consistent set and classical probabilities can be assigned.

We note here that the calculation of the overlap between the full and
restricted wave functions involves the inner product of two functions defined
in different Hilbert spaces;
$\mathcal{L}^{2}(\Delta)\neq\mathcal{L}^{2}(\Delta_{2L})$. Thus, one has to
extend the definition of the restricted wave function on the full Hilbert
space of the problem by embedding the restricted wave function in this larger
Hilbert space. Thus we have

$\mathcal{L}^{2}(\Delta_{2L})=\mathcal{L}^{2}(\Delta)\oplus\mathcal{L}^{2}(\bar{\Delta})$
(20)

where $\bar{\Delta}=\Delta_{2L}\backslash\Delta$ is the complement of
$\Delta$.

In the case that the initial wave function is not fully localised in
$\bar{\Delta}$, we also require that it vanishes at the boundary of the
region, i.e.

$\innerproduct{x}{\Psi_{0}}_{\partial\bar{\Delta}}=0$ (21)

This presentation makes also clear the connection between the quantum Zeno
effect and the time-of-arrival problem as the latter being the infinite-
dimensional case of the first for spatial projections and quadratic
Hamiltonian, e.g. [35, 47].

##  3 Contrary inferences in a semi-classical system

In this section we will show that there exist (at least) two different
_consistent_ sets, where each of the sets gives a different answer to the
time-of-arrival question. We explicitly _construct_ them and show that they
satisfy the consistency conditions (19) and (21). In our example we use a
quantum particle in one dimension, in an infinite square well $[-L,L]$ with
Hamiltonian

$H=-\frac{\hbar^{2}}{2M}\frac{d^{2}}{dx^{2}}+V(x),\quad
V(x)=\begin{cases}0,&\absolutevalue{x}<L,\\\ \infty,&\absolutevalue{x}\geq
L\end{cases}$ (22)

where $M$ denotes the mass of the particle and $\hbar$ the reduced Planck
constant, with boundary conditions such that the wave function vanishes at
$x=-L$ and $x=L$. This means that at these points there are impenetrable and
perfectly reflective walls (Dirichlet boundary conditions). This Hamiltonian
is a self-adjoint operator $\mathcal{L}^{2}(\Delta_{2L})$ and describes the
free evolution in $\Delta_{2L}=[-L,L]$.

We ask whether a semi-classical particle (suitably localised wave packet)
initially located (centred) at $x_{0}=-\frac{L}{2}$ with positive average
momentum $p_{0}$ will cross the surface $x=0$ at some moment of time during
the interval $[0,\tau]$. As already discussed, one way to deal with this
question is to ask the complementary question “will the particle always remain
in the subregion $[-L,0]$?” and construct “Zeno” histories [35].

Specifically, we use an initial state that (i) is a Gaussian wave packet
centred at $-L/2$ and normalised in the interval $\Delta_{2L}$, (ii) is also a
Gaussian in the momentum space, with average momentum $p_{0}$, (iii) can be
chosen to have minimum uncertainty and (iv) the width of the well $2L$ is much
bigger than $\Delta x$:

$\displaystyle\Psi(x,0)$ $\displaystyle=$ $\displaystyle
A\exp\left(-(x+L/2)^{2}/4\sigma^{2}\right)\times\exp\left(ip_{0}\left(x+L/2\right)/\hbar\right)$
(23) $\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\sigma\sqrt{\frac{\pi}{2}}\left({{\erf}\left({\frac{L}{{2\sqrt{2}\sigma}}}\right)+{\erf}\left({\frac{{3L}}{{2\sqrt{2}\sigma}}}\right)}\right)\right)^{-1/2}$
(24)

We can easily see that for a suitable normalisation constant $A^{\prime}$ it
has the same form in the momentum space:

$\Phi(k,0)=A^{\prime}\exp\left[{-\frac{{{{(k-{p_{0}})}^{2}}{\sigma^{2}}}}{{{\hbar^{2}}}}+i\frac{{kL}}{{2\hbar}}{\rm{}}}\right]$
(25)

The classical trajectory of a free particle in 1-dimension, with momentum
$p_{0}$, starting position $x_{0}=-L/2$ moving between two walls at $\pm L$,
is given by

$\displaystyle
x_{cl}(t)=\begin{cases}&x_{\rightarrow}^{(n)}\equiv-2(2n)L+(x_{0}+\frac{p_{0}}{M}t),\quad
t_{2n}\leq t\leq t_{2n+1},\ n=0,1,2...\\\ &x_{\leftarrow}^{(n)}\equiv
2(2n-1)L-(x_{0}+\frac{p_{0}}{M}t),\quad t_{2n-1}\leq t\leq t_{2n}\
n=1,2,...\end{cases}$ (26)

where $t_{n}=nT_{cl}$, with

$T_{cl}=\frac{2M(2L)}{p_{0}}$ (27)

the classical period of the system and $x_{\leftrightarrows}$ denotes the
direction of the motion of the particle. It is not hard to see that the
evolution of the centre of our wave packet indeed follows the classical
trajectory. It is known that the free evolution maintains the shape of the
initial state, provided that the centre is far from the walls that the
“bounce”, that momentarily distorts the shape, happens. We have also confirmed
this numerically.

We have now justified that our system, with the choice of Eq (23) as our
initial state, describes a semi-classical system and we would not expect to
observe any of the quantum paradoxes that arise when considering microscopic
systems. We proceed by giving the two different coarse-grainings that, within
the consistent histories formulation, lead to a contrary inferences for this
macroscopic system.

###  3.1 Construction of the coarse-grainings: The first consistent set

(a) The time-of-arrival problem in the infinite square well

(b) Various histories

Figure 5: Left Panel: The time-of-arrival problem in the infinite square well;
this is an instant of the partition for the first set of histories. The two
subregions are $\Delta_{1}=[0,L]$ and $\bar{\Delta}_{1}=[-L,0]$. The question
is whether the particle, equipped with positive momentum will be even for only
one moment in $\Delta_{1}$. The barrier at $x=0$ is fictitious and corresponds
to infinitely many projections with the operator (7). The projection takes
place also at $x=-L$ but this also coincides with the real barrier of the
square well.
Right Panel: Various histories: the blue set belongs to $h_{1}$ while the red
in $\bar{h}_{1}$

#### Coarse-graining and histories.

The first coarse-graining we consider for the time-of-arrival problem takes
the simpler approach. Let us define the intervals $\Delta_{1}=[0,L]$ and
$\bar{\Delta}_{1}=[-L,0]$. To answer the question whether the particle ever
arrived in $\Delta_{1}$ within the time interval $[0,\tau]$, we can group the
possible paths into two sets (coarse-grained histories). The first history
$h_{1}$ that “never arrives” contains the paths that remained at all times
within $\bar{\Delta_{1}}$. The second history $\bar{h_{1}}$ contains all the
other histories, i.e. those that at some point entered the region $\Delta_{1}$
(see Figure 5(a)). In other words:

$\displaystyle h_{1}=\\{x(t)\ |\ x(t)\in\bar{\Delta}_{1}\forall\
t\in[0,\tau]\\}$ (28a) $\displaystyle\bar{h}_{1}=\\{x(t)\ |\
x(t)\notin\bar{\Delta}_{1}\text{ for some }t\in[0,\tau]\\}.$ (28b)

It is important to note here, that our conclusion is valid even if the
consistency condition is satisfied for a single specific choice of $\tau$. In
the following we will treat $\tau$ as variable and see for which choices of
$\tau$ the condition is satisfied. Interestingly, we will find that the
consistency condition is satisfied for large “intervals” so it is “robust” in
time. For example a good choice for our aims is $\tau=0.75T_{cl}$, where the
centre of the wave packet has returned to its initial position but with
opposite average momentum.

#### Consistency condition

Figure 6: The square of the overlap of the two wave packet with respect to
time for the first two classical periods. The parameters chosen are :
${p_{0}}L/\hbar=40\pi,\sigma/L=0.05,{x_{0}}/L=-1/2$. For the details of the
calculation see the Appendix.

To show that this set is consistent, we need to check the relations (19) and
(21). We first examine whether the boundary condition for the wave packet
$\Psi_{2\Delta}(x,t_{0})$ is satisfied. Indeed, it is trivially satisfied at
$x=-L$ since we have normalised the wave packet in such a way that it vanishes
exactly; this is actually the reason of the appearance of error functions in
the normalisation. At $x=0$, the initial wave packet should also be
approximately zero in order for the consistency condition to be satisfied.
This also holds true for the specific values of the parameters we choose for
the semi-classical approximation; these are shown in Figure 6. The initial
condition (21) is satisfied with good precision, $\Psi(0,0)\rightarrow 0$,
even though our histories are constructed by continuous projection operators
and we would expect to be satisfied approximately. Note, that in our example
we took somehow arbitrary choices of the parameters to illustrate our main
point. We only had to ensure that the width of the well is much larger than
the spread of the wave packet (in either position or momentum space). With
these choices we achieved (as we will see later) approximate consistency,
where this approximation $\epsilon$ depended on the aforementioned ratio. If
we required an even smaller $\epsilon$ we could have chosen that ratio to be
even smaller, and in general we could have achieved any arbitrarily small
$\epsilon$ value.

Then, we examine whether the overlap between the free state in $\Delta_{2L}$
given by (23) and the restricted one in $\bar{\Delta}_{1}$ coincide. The
evolution of the restricted state is given by
$\mathcal{H}_{r}=\bar{P_{1}}\mathcal{H}\bar{P_{1}}$, where $\bar{P}_{1}$ is
the projection operator in $\bar{\Delta}_{1}$ defined as in (7) and the
corresponding restricted Gaussian wave packet is given by

$\bar{\Psi}_{1}(x,0)=\frac{e^{-(x+L/2)^{2}/4\sigma^{2}}e^{ip_{0}(x+L/2)/\hbar}}{\sqrt{\sigma\sqrt{2\pi}\erf\left(\frac{L}{2\sqrt{2}\sigma}\right)}}$
(29)

It is the overlap

$A(t)=\innerproduct*{\bar{\Psi}_{1}(x,t)}{\Psi(x,t)}$ (30)

between the free and the restricted wave packet that shows when the two wave
packets coincide. At $t=t_{0}$, this becomes (see Appendix A):

$\displaystyle
A(t_{0})=\frac{\sqrt{2}\sqrt{\erf\left(\frac{L}{2\sqrt{2}\sigma}\right)}}{\sqrt{\erf\left(\frac{L}{2\sqrt{2}\sigma}\right)+\erf\left(\frac{3L}{2\sqrt{2}\sigma}\right)}}$
(31)

which gives the right-hand side of (19). This is expected to be close to unity
since both the free and the restricted evolution are contained in
$\bar{\Delta}_{1}$ at $t=t_{0}$. We are interested in finding values (or even
better ranges of values) of $\tau$ that this overlap is close to unity so that
the consistency condition is (approximately) satisfied. To understand better,
we just observe how the overlap changes in time, and subsequently conclude
which value of $\tau$ to use to demonstrate the existence of semi-classical
contrary inferences. To this end, we calculate the time-evolved overlap of
these wave packets by expanding them in their corresponding energy
eigenstates. The result is (see Appendix A)

$\displaystyle
A(t)=\sum_{n,m=1}^{\infty}b_{k}^{*}c_{n}e^{iEt(k^{2}-n^{2}/4)/\hbar}I_{nk},$
(32)

where

$I_{nk}=\int_{-L}^{0}dxu_{n}(x)v_{k}(x)$ (33)

and we have denoted with $u_{n}(x)$ the eigenfunctions (43) and

$v_{k}=\sqrt{\frac{2}{L}}\sin\frac{k\pi x}{L},\quad x\in\bar{\Delta}_{1}.$
(34)

The corresponding energy eigenvalues and their relation are
$E_{k}\equiv\frac{\hbar^{2}\pi^{2}k^{2}}{2ML^{2}}=k^{2}E$ and
$E_{n}\equiv\frac{\hbar^{2}\pi^{2}n^{2}}{8ML^{2}}=\frac{n^{2}}{4}E$. The
explicit form of the coefficients $b_{k}$ and $c_{n}$ is also given in the
Appendix A. It can be seen that it is only the odd parts of the overlap that
contribute to its time-dependence.

The time scale we are interested in is the first time that the centre of the
wave packet of the unrestricted evolution returns in the region
$\bar{\Delta}_{1}$. Indicatively we could consider the $\tau=0.75T_{cl}$ which
is when the centre of the wave packet is at its initial position, but the
momentum has flipped sign, or we could also consider the full classical period
of the system $\tau=T_{cl}$. If we let the system evolve for longer time we
can see that for a few classical periods it still returns to its initial
shape, but this effect is lost after very few periods because of the
diffusion. Such results are presented in Figure 6 where we plot the first
classical periods. This overlap shows the spatial coincidence of the full and
restricted evolution, which obviously happens when the overlap is at least
approximately one.

Note that the quantum problem has two time scales; one is the classical period
at which the wave packet has returned at its initial position and momentum;
the other is the revival time at which the wave packets recover their shape.
For this problem, the revival time is approximately $160$ times the classical
period [48], which is large enough as expected for a classical system and it
will not concern us further here.

#### Probabilities

We have thus shown that for the selected coarse-graining, the constructed
histories form a consistent set since the consistency condition is satisfied.
The candidate probability, for remaining during all the interval $[0,\tau]$ in
the region $\bar{\Delta}_{1}$ is given by the expression
$D(h_{1},h_{1})=\mu(h_{1})$ and it is not difficult to see that is unity:

$\mu(h_{1})=\bra{\Psi_{0}}g_{r}^{\dagger}(\tau,0)g_{r}(\tau,0)\ket{\Psi_{0}}\approx
1$ (35)

This result, that mathematically is the quantum Zeno effect for continuous
variables, is well known. However, in general, this “Zeno” history does not
decohere with its negation, and therefore the consistent histories cannot
assign (this or another) probability to such history. After all the “real
quantum Zeno” effect arises due to the act of continuous measurements, while
in our case there is no observer making such measurements. In contrast, our
example with the given choices of walls and choosing $\tau\approx 0.75T_{cl}$,
does decohere as we saw in the previous section. Therefore unlike other “Zeno-
type” histories we get:

$\displaystyle\mu(\bar{h}_{1})$ $\displaystyle=$
$\displaystyle\bra{\Psi_{0}}(g^{\dagger}(\tau,0)-g_{r}^{\dagger}(\tau,0))\times(g(\tau,0)-g_{r}(\tau,0)\ket{\Psi_{0}}$
(36) $\displaystyle=$
$\displaystyle\bra{\Psi_{0}}g^{\dagger}(\tau,0)g(\tau,0)\ket{\Psi_{0}}-\bra{\Psi_{0}}g_{r}^{\dagger}(\tau,0)g(\tau,0)\ket{\Psi_{0}}-$
$\displaystyle-\bra{\Psi_{0}}g^{\dagger}(\tau,0)g_{r}(\tau,0)\ket{\Psi_{0}}+\bra{\Psi_{0}}g_{r}^{\dagger}(\tau,0)g_{r}(\tau,0)\ket{\Psi_{0}}$
$\displaystyle=$ $\displaystyle
2-2Re\bra{\Psi_{0}}g_{r}^{\dagger}(\tau,0)g(\tau,0)\ket{\Psi_{0}}\approx 0$

where for the last equality we used the previous result from the consistency
condition. We can see that indeed $\\{h_{1},\bar{h}_{1}\\}$ form a consistent
set with probabilities

$\displaystyle p(h_{1})=1,$ (37a) $\displaystyle p(\bar{h}_{1})=1-p(h_{1})=0.$
(37b)

These show that the answer to the initial question of the particle crossing
the origin within the time interval $[0,\tau]$, is negative.

###  3.2 Construction of the coarse-grainings: The second consistent set
(time-dependent)

(a) Instant of the coarse-graining for the second histories set.

(b) Various histories

Figure 7: Left Panel: Instant of the coarse-graining for the second histories
set. In this case, the partition is
$\Delta_{2}(t)=[x_{cl}(t)-\lambda,x_{cl}(t)+\lambda]$ and changes in time,
following the classical path of the particle. The two barriers move uniformly
and continuously in the infinite square well due to the semi-classical nature
of the problem, see section 4 and this one for detailed discussion on the
continuity.
Right Panel: Various histories: the blue set belongs to $h_{2}$ while the red
in $\bar{h}_{2}$

#### Coarse-graining and histories

Here we are interested in histories that remain within a (time-dependent)
interval that at all times contains the centre of the wave packet, and is
sufficiently wide to include essentially the full wave packet. Note that our
state does not spread as it evolves and remains localised. One can define such
a time-dependent interval in different ways and care needs to be taken when
the classical trajectory (and the centre of the wave packet) hits one of the
walls placed at $\pm L$. The first important point is that this interval
should be much larger than the width of the wave packet at all times, and this
can be achieved if the moving interval is $O(L)$ since $L\gg\sigma$.

The approach we will take is that at each moment of time we will apply the
position projection operator

$P_{2}(t)=\int_{\Delta_{2}(t)}dx\outerproduct{x}{x}$ (38)

in the (time-dependent) interval
$\Delta_{2}(t)=[x_{cl}(t)-\lambda,x_{cl}(t)+\lambda]$ that follows the
trajectory of the wave packet in the well. We will take $\lambda$ to be the
width of the interval and its centre is the instantaneous mean position of the
wave packet or, equivalently, the classical trajectory, $x_{cl}(t)$. An
instant of this situation is depicted in Figure 7(a).

Following the same procedure as described before, the interval $\Delta_{2}$ is
constructed by very frequent application of the position projection operator
$P_{2}(t)$, as a “Zeno” box. At each moment of time, the position of
$\Delta_{2}(t)$ changes since it follows the motion of the particle.

We take the interval $\Delta_{2}(t)$ to have width $\lambda=L/8$ and the
mathematical effect of these projections is to introduce “fictitious” infinite
potential walls at the boundary of this region. It is clear that when
$x_{cl}(t)+L/8>L$ or $x_{cl}-L/8<-L$, the fictitious wall is further than the
real walls that exist in $\pm L$. This means that the box (infinite square
well) that the restricted evolution takes place, in those cases, changes size
to reach $L/8$ when $x_{cl}(t)=L$ from the initial $L/4$ width. Since
$L\gg\sigma$ this does not change significantly the analysis. Alternatively,
one could use a different interval that “freezes” when $x_{cl}(t_{1})+L/8=L$
and then starts following the wave packet at $t_{2}$ when the centre has
returned at $7L/8$ but is moving the opposite direction. We focus on the first
case, but the consistency condition and probabilities would be the same in
that case too.

The interval $[-L,L]$ is divided in three subregions; we are interested in the
histories in the second one, those that never leave $\Delta_{2}$. These are
constructed by all the paths reflected from the boundaries of the interval. In
this way, the set of histories belonging in the interval $\Delta_{2}(t)$ are:

$\displaystyle h_{2}=\\{x(t)\ |\ x(t)\in\Delta_{2}(t)\ \forall
t\in[0,\tau]\\}$ (39a) The contributions of the paths outside $\Delta_{2}$
constitute the set of histories that do not belong to $\Delta_{2}$ at least
for some moment, that is $\displaystyle\bar{h}_{2}=\\{x(t)\ |\
x(t)\notin\Delta_{2}(t)\text{ for some }t\in[0,\tau]\\}.$ (39b)

#### Consistency condition

Now, we have to establish that the consistency condition is satisfied, i.e.
that the initial wave function vanishes at the boundaries of $\Delta_{2}$ and
that the overlap between the restricted and full solution of the Schrödinger
equation is one initially as well as at later times, since the interval
$\Delta_{2}$ is time-dependent.

It can be seen that at the initial moment $t_{0}=0$, the overlap between the
wave packet defined in $\Delta_{2L}$ and its restriction in
$[x_{cl}-\lambda,x_{cl}+\lambda]$ is equal to one:

$\int_{x_{cl}-\lambda}^{x_{cl}+\lambda}\Psi^{*}(x,t_{0})\Psi_{2}(x,t_{0})dx=1$
(40)

because we have selected the interval $\Delta_{2}(t_{0})$ such that it
completely contains the initial wave packet. The translational symmetry
guarantees that this value will be approximately preserved in time while it
moves in the infinite square well because of the semi-classical properties of
the Gaussian state. Viewing this differently, the unrestricted evolution of
our semi-classical system does not spread and keeps the same width in position
and momentum space, while the centre of the wave packet follows the classical
trajectory. By construction the same holds for the restricted evolution.
Moreover, the wave packet bounces at the real walls $\pm L$ in both cases,
while the fictitious walls of the restricted evolution are always at point
that the wave function is vanishingly small because $L/8\gg\sigma$.

#### Probabilities

Through the above procedure we have thus constructed a consistent set, since
it satisfies the two sub-conditions into which the consistency conditions
breaks into: i) the boundary condition and ii) the overlap between the full
and restricted evolution of the wave packet. We can now calculate the
probability the particle is always in $\Delta_{2}(t)$. By construction, the
particle never leaves the subspace $\Delta_{2}(t)$, thus the probability to
find it there is one:

$\displaystyle p(h_{2})=1,$ (41a) $\displaystyle p(\bar{h}_{2})=1-p(h_{2})=0$
(41b)

What is crucial to note here is that all the paths that remained through all
the interval $[0,\tau=3/4T_{cl}]$ in $\Delta_{2}(t)$ passed through $x=0$.
This is true since, for example, at $t=T_{cl}/4$, the interval
$\Delta_{2}(t)=[3L/8,5L/8]$ and is fully in the positive axis meaning that
paths that at that moment are within $\Delta_{2}(t)$ have been in the $x>0$
region. In other words $h_{2}$ is a strict subset of the histories that
crossed $x=0$. We can therefore infer from this consistent set, that the
particle passed $x=0$ with certainty (probability unity). Interestingly, while
this consistent set gives an indirect way to answer the arrival question, it
actually recovers the answer that one would intuitively expect.

##  4 Discussion

In this work, we considered the construction of two different consistent sets
and showed that _it is possible to have contrary inferences even at the semi-
classical limit within the consistent histories formalism_. In standard
quantum theory, contextuality related paradoxes exist only in the
microscopic/quantum regime and these are resolved when considering macroscopic
(classical) objects. Therefore in consistent histories, the context/consistent
set dependence is more significant and points to the need of supplementing the
formalism with certain selection criteria.

The two scenarios we considered were:

  1. (i)

The system does not leave the subregion $\bar{\Delta}$.

  2. (ii)

The system remains always within a small box that follows the classical
evolution of the centre of the classical object/wave packet.

We considered a particle in the infinite square well, initially positioned at
the negative axis and moving to the right and asked whether the particle will
pass from the origin $x=0$ at some moment $t\in[0,\tau]$. The particle was
semi-classical (a gaussian state) with average position in the middle of the
negative part, and average momentum positive.

We selected two different coarse-grainings corresponding to the situations
discussed above. More specifically, we constructed the following histories:

$\displaystyle h_{1}=\\{x(t)\ |\ x(t)\in\bar{\Delta}_{1}\ \forall
t\in[0,\tau]\\}$ $\displaystyle\bar{h}_{1}=\\{x(t)\ |\
x(t)\notin\bar{\Delta}_{1}\ \text{ for some }t\in[0,\tau]\\}$ $\displaystyle
h_{2}=\\{x(t)\ |\ x(t)\in\Delta_{2}(t)\ \forall t\in[0,\tau]\\}$
$\displaystyle\bar{h}_{2}=\\{x(t)\ |\ x(t)\notin\Delta_{2}(t)\ \text{ for some
}t\in[0,\tau]\\}$

that belong in $\Omega$ and pairwise ($\\{h_{1},\bar{h_{1}}\\}\ ;\
\\{h_{2},\bar{h_{2}}\\}$) form partitions. In sections 3.1 and 3.2, we
demonstrated that the consistency condition (19) is satisfied. We were thus
able to find the following probabilities for the histories:

$\displaystyle p(h_{1})=D(h_{1},h_{1})=\mu(h_{1})=1,$ $\displaystyle
p(\bar{h}_{1})=D(\bar{h}_{1},\bar{h}_{1})=\mu(\bar{h}_{1})=0,$ $\displaystyle
p(h_{2})=D(h_{2},h_{2})=\mu(h_{2})=1,$ $\displaystyle
p(\bar{h}_{2})=D(\bar{h}_{2},\bar{h}_{2})=\mu(\bar{h}_{2})=0$

We emphasise that these were the candidate probabilities, which by virtue of
the consistency of the sets ($D(h_{i},\bar{h}_{i})=0,\ i=1,2$), satisfy the
rules of the classical probability theory; that is
$\mu(h_{i})+\mu(\bar{h}_{i})=1,\ i=1,2$.

We now show how contrary inferences appear in this setting. First, we recall
that we have contrary inferences when two (coarse-grained) histories have both
zero quantum measure and cover the histories space $\Omega$. In our example
zero cover of $\Omega$ is the set $\\{\bar{h}_{1},\bar{h}_{2}\\}$. Their
quantum measure is zero, while it is clear that any trajectory belongs to
either $\bar{h}_{1}$ or $\bar{h}_{2}$ or to both sets, and thus it forms a
cover. Second, we also have that $\mu(h_{1})=1$ and $\mu(\bar{h}_{2})=0$. If
we assign a truth value $\mathsf{True}$ to events/histories with probability
one and $\mathsf{False}$ to events/histories with probability zero, we are led
to contrary inferences because of the modus ponens (implication inference)
which states that if $A=\mathsf{True}$ and $A\implies B$ (i.e. subset) jointly
implies $B=\mathsf{True}$ something that does not hold in our example.

It is important at this point to emphasise that the initial motivation of the
consistent histories formalism is the description of closed quantum systems
without measurements or external observers. This is achieved by replacing the
process of measurement with the consistency condition that must be satisfied
in order for a question to be (classically) answered. As it was demonstrated,
the same question can be answered by considering different partitions of the
histories space. For each of those partitions, to calculate the amplitudes and
their interference one needs to use projections and/or restrictions in the
path integral. Mathematically the different partitions appear as if different
measurements took place, but in consistent histories there is no “real”
measurement taking place, or external observer. The physical system (paths and
true Hamiltonian) are identical in all partitions, therefore reaching to a
different conclusion about properties of the same classical system seems
problematic and gives further evidence that consistent histories without a
set-selection mechanism may not be a viable alternative.

## Acknowledgements

P.W. thanks Fay Dowker and Rafael Sorkin for early discussions and
acknowledges support by the UK Hub in Quantum Computing and Simulation with
funding from the EPSRC grant EP/T001062/1.

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##  Appendix A Calculation of the time-dependent overlap

To find the time evolution of the wave packet, driven by the evolution
operator $U(t)=\exp(-iHt/\hbar)$ we will expand it in energy eigenstates basis
which are solutions of the Schrödinger equation in $[-L,L]$. The wave function
expansion is

$\Psi(x,t)=\sum_{n=1}^{\infty}c_{n}u_{n}(x)e^{-iE_{n}t/\hbar}$ (42)

and the complete set of the eigenfunctions $u_{n}(x)$ in $\Delta_{2L}$ is

$u_{n}=\sqrt{\frac{1}{L}}\sin\frac{n\pi(x+L)}{2L}$ (43)

with energy eigenvalues

$E_{n}=\frac{\hbar^{2}\pi^{2}n^{2}}{2M(2L)^{2}}\equiv\frac{n^{2}}{4}E.$ (44)

The coefficients $c_{n}$ are given by the relation555Their explicit
calculation is presented in Appendix A.

$c_{n}=\int_{-L}^{L}dxu_{n}^{*}(x)\Psi(x,0)$ (45)

For the interval $\Delta_{2L}$ the coefficients appearing in eq. (42) can be
calculated using eq. (45):

$\begin{split}{c_{n}}&=i\frac{{{\pi^{1/4}}}}{{{2^{3/4}}}}{e^{\frac{{in\pi}}{4}-\frac{{{\sigma^{2}}}}{{{L^{2}}}}\left({\frac{{n\pi{p_{0}}L}}{\hbar}+{{\frac{{{n^{2}}\pi}}{4}}^{2}}+\frac{{p_{0}^{2}{L^{2}}}}{{{\hbar^{2}}}}}\right)}}\sqrt{\frac{\sigma}{{L\left({{\erf}\left({\frac{L}{{2\sqrt{2}\sigma}}}\right)+{\erf}\left({\frac{{3L}}{{2\sqrt{2}\sigma}}}\right)}\right)}}}\times\\\
&\left({{e^{-\frac{{in\pi}}{2}+\frac{{2n\pi{p_{0}}{\sigma^{2}}}}{{\hbar
L}}}}\left({{\erf}\left[{\frac{L}{{4\sigma}}-i\frac{\sigma}{L}\left({\frac{{n\pi}}{2}-\frac{{{p_{0}}L}}{\hbar}}\right)}\right]+{\erf}\left[{\frac{L}{{4\sigma}}+i\frac{\sigma}{L}\left({\frac{{n\pi}}{2}-\frac{{{p_{0}}L}}{\hbar}}\right)}\right]}\right)}\right.-\\\
&\left.{\left({{\erf}\left[{\frac{L}{{4\sigma}}-i\frac{\sigma}{L}\left({\frac{{n\pi}}{2}+\frac{{{p_{0}}L}}{\hbar}}\right)}\right]+{\erf}\left[{\frac{L}{{4\sigma}}+i\frac{\sigma}{L}\left({\frac{{n\pi}}{2}+\frac{{{p_{0}}L}}{\hbar}}\right)}\right]}\right)}\right)\end{split}$
(46)

For the interval $[-L,0]$ we define the relative expansion and define the
coefficients $b_{k}$. They are given by the expression:

$\begin{split}{b_{k}}&=i\frac{{{\pi^{1/4}}}}{{{2^{3/4}}}}{e^{\frac{{ik\pi}}{2}+\frac{{{\sigma^{2}}}}{{{L^{2}}}}\left({\frac{{2k\pi{p_{0}}L}}{h}-{k^{2}}{\pi^{2}}-\frac{{p_{0}^{2}{L^{2}}}}{{{h^{2}}}}}\right)}}\sqrt{\frac{\sigma}{{L{\erf}\left({\frac{L}{{2\sqrt{2}\sigma}}}\right)}}}\times\\\
&\left({{\erf}\left[{\frac{L}{{4\sigma}}+i\frac{\sigma}{L}\left({k\pi-\frac{{{p_{0}}L}}{h}}\right)}\right]+{\erf}\left[{\frac{L}{{4\sigma}}-i\frac{\sigma}{L}\left({k\pi-\frac{{{p_{0}}L}}{h}}\right)}\right]-}\right.\\\
&{{e^{-ik\pi-\frac{{4k\pi{p_{0}}{\sigma^{2}}}}{{\hbar
L}}}}\left({{\erf}\left[{\frac{L}{{4\sigma}}+i\frac{\sigma}{L}\left({k\pi+\frac{{{p_{0}}L}}{h}}\right)}\right]+{\erf}\left[{\frac{L}{{4\sigma}}-i\frac{\sigma}{L}\left({k\pi+\frac{{{p_{0}}L}}{h}}\right)}\right]}\right)}\end{split}$
(47)

For the initial time $t_{0}$ we can easily find that the overlap is:

$\displaystyle A(t_{0})$
$\displaystyle=\innerproduct{\Psi_{r}(x,t_{0})}{\Psi(x,t_{0})}$
$\displaystyle=\int_{-L}^{0}dx\frac{e^{-(x+L/2)^{2}/4\sigma^{2}}e^{-ip_{0}(x+L/2)/\hbar}}{\sqrt{\sigma\sqrt{2\pi}\erf\left(\frac{L}{2\sqrt{2}\sigma}\right)}}\frac{e^{-(x+L/2)^{2}/4\sigma^{2}}e^{ip_{0}\left(x+L/2\right)/\hbar}}{\sqrt{\sigma\sqrt{2\pi}\frac{1}{2}\left(\erf\left(\frac{L}{2\sqrt{2}\sigma}\right)+\erf\left(\frac{3L}{2\sqrt{2}\sigma}\right)\right)}}$
(48)

or, after direct integration:

$\displaystyle
A(t_{0})=\frac{\sqrt{2}\sqrt{\erf\left(\frac{L}{2\sqrt{2}\sigma}\right)}}{\sqrt{\erf\left(\frac{L}{2\sqrt{2}\sigma}\right)+\erf\left(\frac{3L}{2\sqrt{2}\sigma}\right)}}$
(49)

Another way to calculate the overlap is via the expansion in the eigenstates
(see equation 42) for both wave functions considered. Let us begin by
repeating the calculation for $t=t_{0}$

$\begin{split}A({t_{0}})&=\int_{-L}^{0}dx\Psi_{r}^{*}(x,{t_{0}})\Psi(x,{t_{0}})\\\
&=\frac{{\sqrt{2}}}{L}\sum\limits_{k,n=1}^{\infty}{b_{k}^{*}{c_{n}}{e^{i\left({{k^{2}}-{n^{2}}/4}\right)E{t_{0}}/\hbar}}\int_{-L}^{0}dx\sin\frac{{k\pi
x}}{L}\sin\frac{{n\pi(x+L)}}{{2L}}}\\\
&\equiv\frac{{\sqrt{2}}}{L}\sum\limits_{k,n=1}^{\infty}{b_{k}^{*}{c_{n}}{e^{i\left({{k^{2}}-{n^{2}}/4}\right)E{t_{0}}/\hbar}}{I_{kn}}}\end{split}$
(50)

To calculate the integral we split it in two parts:

$\begin{split}I&=\int_{-L}^{0}dx\sin\frac{{k\pi
x}}{L}\sin\frac{{n\pi(x+L)}}{{2L}}\\\
&=\cos\frac{{n\pi}}{2}\int_{-L}^{0}dx\sin\frac{{k\pi x}}{L}\sin\frac{{n\pi
x}}{{2L}}+\sin\frac{{n\pi}}{2}\int_{-L}^{0}dx\sin\frac{{k\pi
x}}{L}\cos\frac{{n\pi x}}{{2L}}\\\ &=I_{1}+I_{2}\end{split}$ (51)

For $I_{1}$, we have:

  * •

The case of even $n$ ($n=2l$):

$\begin{split}I_{1}^{{\rm{even}}}&=\cos\frac{{2l\pi}}{2}\int_{-L}^{0}dx\sin\frac{{k\pi
x}}{L}\sin\frac{{2l\pi x}}{{2L}}{\rm{}}\\\ &=\frac{{\cos
l\pi}}{2}\int_{-L}^{L}dx\sin\frac{{k\pi x}}{L}\sin\frac{{l\pi
x}}{L}=\frac{{\cos l\pi}}{2}{\delta_{k,l}}\end{split}$ (52)

  * •

The case of odd $n$ ($n=2l+1$):

$\begin{split}I_{1}^{{\rm{odd}}}&=\cos\frac{{(2l+1)\pi}}{2}\int_{-L}^{0}dx\sin\frac{{k\pi
x}}{L}\sin\frac{{(2l+1)\pi x}}{{2L}}\\\
&=\sin(l\pi)\int_{-L}^{0}dx\sin\frac{{k\pi x}}{L}\sin\frac{{(2l+1)\pi
x}}{{2L}}=0\end{split}$ (53)

due to the $\sin(l\pi)$ term because the integral has no poles.

Similarly, for $I_{2}$, we have

  * •

The case of even $n$ ($n=2l$):

$\displaystyle I_{2}^{even}=\sin l\pi\int_{-L}^{0}dx\sin\frac{k\pi
x}{L}\cos\frac{2l\pi x}{2L}=0$ (54)

for the same reasons as the previous case.

  * •

The case of odd $n$ ($n=2l+1$):

$\begin{split}I_{2}^{odd}&=\sin\frac{{(2l+1)\pi}}{2}\int_{-L}^{0}dxsin\frac{{k\pi
x}}{L}\cos\frac{{(2l+1)\pi x}}{{2L}}\\\ &=-\frac{{4kL\cos
l\pi}}{{\pi\left[{2\left({k-l}\right)-1}\right]\left[{(2\left({k+l}\right)+1)}\right]}}\end{split}$
(55)

since the integral has no poles

Therefore, if we combine the above results, we have

$\begin{split}A({t_{0}})&=\sum\limits_{l=1,k=1}^{\infty}{b_{k}^{*}}{c_{2l}}\frac{1}{{\sqrt{2}}}\cos(l\pi){\delta_{k,l}}-\sum\limits_{l=0,k=1}^{\infty}{b_{k}^{*}}{c_{2l+1}}\frac{{4\sqrt{2}k\cos(l\pi)}}{{\pi\left[{2\left({k-l}\right)-1}\right]\left[{2\left({k+l}\right)+1}\right]}}\\\
&=\frac{{\sqrt{2}\sqrt{{\erf}\left({\frac{L}{{2\sqrt{2}\sigma}}}\right)}}}{{\sqrt{{\erf}\left({\frac{L}{{2\sqrt{2}\sigma}}}\right)+{\erf}\left({\frac{{3L}}{{2\sqrt{2}\sigma}}}\right)}}}.\end{split}$
(56)

We then repeat the calculation for $t>{t_{0}}$:

$\begin{split}A(t)&=\sum\limits_{l=1,k=1}^{\infty}{b_{k}^{*}}{c_{2l}}\frac{1}{{\sqrt{2}}}\cos(l\pi){\delta_{k,l}}\\\
&-\sum\limits_{l=0,k=1}^{\infty}{b_{k}^{*}}{c_{2l+1}}{e^{i4Et(4{k^{2}}-{{(2l+1)}^{2}})/\hbar}}\frac{{4\sqrt{2}k\cos(l\pi)}}{{\pi\left[{2\left({k-l}\right)-1}\right]\left[{2\left({k+l}\right)+1}\right]}}\end{split}$
(57)

where $E=\frac{\pi^{2}\hbar^{2}}{2ML^{2}}$. With the help of equation 56 we
can eliminate the first sum from the above equation:

$\begin{split}A(t)&=\frac{{\sqrt{2}\sqrt{{\erf}\left({\frac{L}{{2\sqrt{2}\sigma}}}\right)}}}{{\sqrt{{\erf}\left({\frac{L}{{2\sqrt{2}\sigma}}}\right)+{\erf}\left({\frac{{3L}}{{2\sqrt{2}\sigma}}}\right)}}}\\\
&-\sum\limits_{l=0,k=1}^{\infty}{b_{k}^{*}}{c_{2l+1}}\left({{e^{i\frac{{{\pi^{2}}\hbar(4{k^{2}}-{{(2l+1)}^{2}})}}{{8M{L^{2}}}}t}}-1}\right)\frac{{4\sqrt{2}k\cos(l\pi)}}{{\pi\left[{2\left({k-l}\right)-1}\right]\left[{2\left({k+l}\right)+1}\right]}}\end{split}$
(58)

##  Appendix B Making the equations dimensionless and numerical calculation

From all the above we see that we have four parameters in this problem:
$\sigma$, $L$, $p_{0}$ and the classical period $T_{cl}=\frac{2M(2L)}{p_{0}}$.
Therefore it makes sense to rescale our variables making them dimensionless:

$\displaystyle a\equiv\frac{\sigma}{L},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
q\equiv{p_{0}}\frac{L}{\hbar},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \tau\equiv\frac{t}{{T_{cl}}}.$ (59)

With this choice we can handle our equations easier numerically. For example
equation 58 takes the following form:

$\begin{split}A(t)&=\frac{{\sqrt{2}\sqrt{{\erf}\left({\frac{1}{{2\sqrt{2}a}}}\right)}}}{{\sqrt{{\erf}\left({\frac{1}{{2\sqrt{2}a}}}\right)+{\erf}\left({\frac{1}{{2\sqrt{2}a}}}\right)}}}\\\
&-\sum\limits_{l=0,k=1}^{\infty}{b_{k}^{*}}{c_{2l+1}}\left({{e^{i\frac{{{\pi^{2}}(4{k^{2}}-{{(2l+1)}^{2}})}}{{2q}}\tau}}-1}\right)\frac{{4\sqrt{2}k\cos(l\pi)}}{{\pi\left[{2\left({k-l}\right)-1}\right]\left[{2\left({k+l}\right)+1}\right]}}.\end{split}$
(60)

One could use this equation to perform the numerical calculations. However,
after defining the dimensionless parameters, we opted to solve the Schrödinger
equation numerically for both wells using the Crank–Nicholson scheme [49],
thus calculating the wave functions and, from them, the overlap $A(t)$ using
the definition from eq. (30).