## 1. Introduction


The consistent histories interpretation of quantum mechanics was
introduced by Griffiths (1984), and
discussed by Omnes in a series of papers beginning
with (Omnes 1988). The decoherent
histories approach that first appeared
in Gell-Mann & Hartle (1990)
contains similar ideas. The single term "histories" will
be employed below, with the understanding that at points where the
"consistent" and "decoherent" approaches might
be slightly different, the former is intended.


The histories approach extends the calculational methods found in
standard textbooks and gives them a microscopic physical
interpretation which the textbooks lack. It is not intended as an
alternative, but as a fully consistent and clear statement of basic
quantum mechanics, "Copenhagen done right". In particular,
measurements are treated in the same way as all other physical
processes and play no special role in the interpretation. Thus there
is no measurement problem, and under appropriate conditions one can
discuss the microscopic properties revealed by measurements.


All the standard quantum paradoxes (double slit, Schrodinger cat,
Hardy, three box, etc.) are resolved using the histories approach, or
one might better say "tamed": they are no longer unsolved
difficulties casting doubt upon the reliability and completeness of
the quantum formalism, but instead they illustrate striking
differences between quantum and classical physics. In particular,
there is no conflict between quantum theory and relativity:
superluminal influences cannot carry information or anything else for
the simple reason that they do not exist. In appropriate regimes
(large systems, strong decoherence) the laws of classical physics
provide a very good approximation, valid for all practical purposes,
to the more fundamental and more exact laws of quantum mechanics.


But at what price? The histories approach introduces a variety of
concepts that go beyond textbook quantum theory, and these can be
summarized under two headings. First, quantum dynamics is treated as
stochastic or probabilistic. Not just when measurements are carried
out, but *always*. Born's rule, together with its extensions,
is a fundamental axiom of quantum theory. The task of Schrodinger's
time-dependent equation is to assist in assigning
probabilities. Second, following Sec. III.5
of J. von Neumann (1932), quantum
properties, which are what the probabilities refer to, are associated
with subspaces of the quantum Hilbert space. But then new logical
principles are required to consistently deal with the difference
between the quantum Hilbert space and a classical phase space. These
principles are related to, but not identical with, those of the
quantum logic initiated by Birkhoff &
Neumann (1936).


Since textbook quantum mechanics already contains certain rules for
calculating probabilities, the first innovation of the histories
approach, stochastic dynamics, is not very startling and by itself
causes few conceptual difficulties. It is the second innovation, in
the domain of logic and ontology, that represents the most radical
departure from classical thinking. However, the new quantum logic
reduces to the old familiar classical propositional logic in the same
domain where classical mechanics is a good approximation to quantum
mechanics. That is to say, the old logic is perfectly good for all
practical purposes in the same domain where classical mechanics is
perfectly good for all practical purposes, and a consistent, fully
quantum analysis explains why this is so.


## 2. Quantum Properties


### 2.1 Indicators, projectors, negation


A classical mechanical system can be described using a *phase
space* \(\Gamma\) with points denoted by \(\gamma\). A
classical *property* \(P\) is a collection of points \(\PC\) from
the phase space, and can be conveniently described by an *indicator
function* \(P(\gamma)\) which is equal to 1 if \(\gamma \in \PC\) and
0 otherwise. It will cause no confusion to use \(P\) for both the
property and its indicator function. The *negation* \(\lnot P\)
of a property \(P\) corresponds to the complement \(\PC^c\) of the set
\(\PC\), consisting of points in \(\Gamma\) that are not in \(\PC\). Its
indicator is \(I-P\), where the *identity* indicator \(I(\gamma)
=1\) for all \(\gamma\). For example, the phase space of a
one-dimensional harmonic oscillator is the real plane with a point
\(\gamma = (x,p)\) indicating that the particle is at \(x\) and has
momentum \(p\). The property that its energy



\[
E=p^2/2m + (1/2) m\omega^2 x^2
\label{eqn1}
\]

is less than some fixed value \(E\_r\) corresponds to the set of
points contained in the ellipse \(E=E\_r\) centered at the origin of the
\(x,p\) plane, and the indicator is the function that is 1 on the points
inside (and on the boundary) of the ellipse and 0 outside. Its
negation, the property that the energy is greater than \(E\_r\),
corresponds to all the points lying outside this ellipse.


The quantum counterpart of a phase space is a Hilbert space \(\HC\):
a complex vector space with an inner product. If infinite dimensional
it must be complete--Cauchy sequences have limits. But for our
purposes finite-dimensional spaces will suffice for discussing the
major conceptual difficulties of quantum theory and how the histories
approach resolves them. (Some examples use a harmonic oscillator with
an infinite dimensional Hilbert space simply because it is relatively
simple and familiar.) We use Dirac notation in which an element of
\(\HC\), a "ket", is denoted by \(\ket{\psi}\), where \(\psi\)
is a label, and \(\inpd{\phi}{\psi}\) denotes the inner product of
\(\ket{\phi}\) with \(\ket{\psi}\). The simplest quantum physical
property, the counterpart of a point \(\gamma\) in the classical phase
space, is a one-dimensional *subspace* of the Hilbert space,
or *ray*, consisting of all multiples \(c\ket{\psi}\) of some
nonzero \(\ket{\psi}\), with \(c\) an arbitrary complex number. The ray
uniquely determines and is uniquely determined by the
corresponding *projector*


\[P = [\psi] = \dya{\psi}, \label{eqn2}\]


assuming \(\ket{\psi}\) is normalized, \(\inp{\psi}=1\). In Dirac
notation \(\dyad{a}{b}\) is an operator which when applied to an
arbitrary ket \(\ket{\phi}\) yields the ket
\(\ket{a}\inpd{b}{\phi}=(\inpd{b}{\phi}) \ket{a}\). The square bracket
in \eqref{eqn2} is not standard Dirac notation, but is very convenient
and will be used later.


In addition to rays, a Hilbert space of dimension \(d\) also contains
subspaces of dimension 2, 3, etc., up to \(d\) (the entire space). These
larger subspaces also represent quantum properties and are the
analogues of sets of more than one point in a classical phase
space. Each subspace \(\PC\) corresponds to a unique projector \(P\), a
Hermitian operator \(P=P^\dagger =P^2\) such that
\(P\ket{\phi}=\ket{\phi}\) if and only if \(\ket{\phi}\) belongs to
\(\PC\). A quantum projector behaves in many ways like a classical
indicator function, e.g., its eigenvalues can only be 0 or 1. Using
the same symbol \(P\) for the property and the projector should cause no
confusion. Following Sec. III.5 of J. von
Neumann (1932), we identify the *negation* \(\lnot P\) of
a quantum property \(P\), subspace \(\PC\), with the *orthogonal
complement* \(\PC^\perp\) consisting of the kets in \(\HC\) which are
orthogonal to all the kets in \(\PC\). The projector on \(\PC^\perp\) is
\(I-P\), again analogous to the classical case.


Consider the example of a one-dimensional quantum harmonic
oscillator. As is well known, its energy \(E\) can take on only discrete
values \((n+1/2)\hbar\omega\), where \(n=0, 1,\ldots\) is any nonnegative
integer and \(\omega\) is its angular frequency, as in \eqref{eqn1}. We
denote the corresponding normalized ket by \(\ket{n}\). The projector
\([n]=\dya{n}\) then represents the property that the energy is equal to
\((n+1/2)\hbar\omega\). The property that the energy is less than or
equal to \((N+1/2)\hbar\omega\) for some integer \(N\) is given by the
projector


\[P = \sum\_{n=0}^N [n] = \sum\_{n=0}^N\dya{n}, \label{eqn3}\]


while the corresponding sum from \(n=N+1\) to \(\infty\) represents its
negation, \(I-P\), the property that \(E\) is greater than
\((N+1/2)\hbar\omega\).


Despite the close analogy between classical and quantum properties
there is actually a profound difference. For a particular classical
property \(P\), every point \(\gamma\) in the phase space lies either
inside the set \(\PC\), so that the property is true for this \(\gamma\),
or else it lies in the complementary set \(\PC^c\), and the property \(P\)
is false. However, given a nontrivial (neither 0 nor the entire space)
subspace \(\PC\) of the quantum Hilbert space \(\HC\), there are always
kets \(\ket{\phi}\) with corresponding rays \(\{c\ket{\phi}\}\) which lie
neither in \(\PC\) nor in \(\PC^\perp\). For example, \(\ket{\phi} =
\ket{N} + \ket{N+1}\) for \(P\) given by \eqref{eqn3}. How is one to
think about these? That is a key question in quantum foundations.


### 2.2 Conjunction and Disjunction


The conjunction \(P\land Q\), or \(P\) AND \(Q\), of two classical
properties corresponds to the intersection \(\PC\cap\QC\) of the sets
that represent them on the phase space, and the indicator for this set
is the product \(P(\gamma)Q(\gamma)\) of the two indicator
functions. What about the quantum case? In quantum logic \(P\land Q\) is
identified with the intersection \(\PC\cap\QC\) of the two Hilbert
subspaces, which is itself a subspace and thus a quantum property. The
histories interpretation identifies \(P\land Q\) with the product \(PQ\)
of the projectors \(P\) and \(Q\) *provided* the two
projectors *commute*, that is, provided \(PQ=QP\). When and only
when this condition is fulfilled is the product \(PQ\) itself a
projector, and it projects onto the subspace \(\PC\cap\QC\), in
agreement with quantum logic. However, if \(PQ\neq QP\) the conjunction
\(P\land Q\) is *undefined* or *meaningless* in the sense
that the histories interpretation assigns it no meaning. In the same
way the disjunction \(P\lor Q\), the nonexclusive \(P\) OR \(Q\), is
represented by the projector \(P+Q-PQ\) when \(PQ=QP\), but is otherwise
undefined.


The refusal to define the conjunction or disjunction when \(PQ\neq
QP\) should be thought of as a syntactical rule, analogous to that in
ordinary logic that says an expression like \(P\land\lor\, Q\) is
meaningless because it has not been formed according to the rules used
to construct meaningful sentences. In this connection it is important
to distinguish "meaningless" from "false". A
proposition which is false is meaningful and its negation is true,
whereas the negation of a meaningless statement is equally
meaningless. Once the commutativity restriction for meaningful
combinations of quantum propositions or properties is in place the
usual logical rules apply together with the intuition that goes along
with them. This will be further codified using the single framework
rule defined in subsection 3.4
below.


### 2.3 Example of spin half


Let us apply these ideas to a specific example, that of the spin
degree of freedom of a spin-half particle, such as an electron or a
proton, or a silver atom in its electronic ground state. The operator
\(S\_z\) for the \(z\) component of spin angular momentum acts on a
2-dimensional Hilbert space and has eigenvalues \(+1/2\) and \(-1/2\) in
units of \(\hbar\). If \(\ket{z^+}\) and \(\ket{z^-}\) are the corresponding
eigenkets, the projectors corresponding to \(S\_z=\pm 1/2\) are \([z^+]
=\dya{z^+}\) and \([z^-] =\dya{z^-}\). The product of these projectors in
either order is the zero operator \(0\), which is the property that is
always false, the quantum counterpart of the empty subset of a
classical phase space. Their sum \([z^+] + [z^-] =I\) is the identity
operator, the property that is always true. In physical terms, \(S\_z\)
is either positive or negative: it cannot be both (\([z^+]\cdot [z^-]
=0\)), and it must be one or the other (\([z^+] + [z^-] =I\)). This
explains the outcome of the Stern-Gerlach experiment.


In the same way the \(x\) component \(S\_x\) of angular momentum can
take only two values \(+1/2\) or \(-1/2\), corresponding to the projectors
\([x^+] = \dya{x^+}\) and \([x^-] = \dya{x^-}\) formed from its eigenkets
\(\ket{x^+}\) and \(\ket{x^-}\). Consequently, what was stated in the
previous paragraph about \(S\_z\) can also be said about
\(S\_x\). However--here we arrive at a central feature of the
histories interpretation--it is *not meaningful*
to *combine* discussions of \(S\_z\) with discussions of \(S\_x\),
because neither \([x^+]\) nor \([x^-]\) commutes with \([z^+]\) or
\([z^-]\). There is no way of associating the property "\(S\_z=+1/2\)
AND \(S\_x=+1/2\)" with a subspace of the two-dimensional quantum
Hilbert space. One way to see this is that every one-dimensional
subspace of this Hilbert space has a physical interpretation:
\(S\_w=+1/2\) for some direction \(w\) in space, so there are no
possibilities left over which could correspond to "\(S\_z=+1/2\)
AND \(S\_x=+1/2\)". But might one assign the zero-dimensional
subspace, the proposition that is always false, to "\(S\_z=+1/2\)
AND \(S\_x=+1/2\)"? This is the approach of quantum logic. However,
taking the negation of this proposition that is always false has the
consequence that the proposition "\(S\_z=-1/2\) OR
\(S\_x=-1/2\)" is *always true*, which to a physicist seems
extremely odd. See the
 Further Reading supplement
for more details.


For a more detailed discussion of the logic associated with the
histories approach, see (Griffiths
2014).


## 3. Quantum Probabilities


Probabilities in the histories interpretation of quantum mechanics
are standard Kolmogorov probabilities, not some new
invention that, for example, allows negative probabilities. But these
standard probabilities must be associated with a well-defined sample
space. While this is also true in classical applications of
probability theory, in quantum mechanics the penalty for carelessness
can be severe: confusion and unresolved paradoxes. The following
discussion indicates what steps need to be taken to construct
consistent probabilities and stay out of trouble. For more details see
Ch. 5 of Griffiths (2002a).


### 3.1 Probabilistic models


A *probabilistic model* as used in classical physics and
other sciences can be thought of as a triple \((\SC,\EC,\MC)\),
a *sample space* \(\SC\), an *event algebra* \(\EC\) and
a *probability measure* \(\MC\). When a die is rolled the sample
space \(\SC\) consists of six mutually exclusive possibilities, one and
only one of which actually occurs. The event algebra can be set equal
to the collection of all \(2^6\) subsets of \(\SC\), including the empty
set and \(\SC\) itself; it forms a Boolean algebra under complements and
intersections. Finally, \(\MC\) assigns probabilities, real numbers
between 0 and 1, to the different sets in \(\EC\) according to certain
rules. Again we need only consider the simplest situation in which
\(\SC\) is finite or countable, and a nonnegative number \(p\_j\) is
assigned to each \(j\in\SC\) in such a manner that \(\sum\_j p\_j = 1\). The
probability for some \(E\in \EC\) is then given by the formula


\[\Pr(E) = \sum\_{j\in E} p\_j. \label{eqn4}\]


Note that the mathematical rules of probability theory do not
constrain the choice of the \(p\_j\), aside from the requirement that
they be nonnegative and sum to 1. It is up to the scientist
constructing the model to come up with appropriate values, which may
be done using a variety of considerations; among them pure guesswork,
as well as carefully fitting parameters using results of previous
experiments.


### 3.2 Probabilities in quantum mechanics


The histories interpretation of quantum mechanics employs
probability theory in exactly the same way as in the other
sciences *once a sample space \(\SC\) has been specified*. A
quantum sample space \(\SC\) is always a *projective decomposition of
the identity operator \(I\)*, hereafter referred to as a
"projective decomposition" (or
"decomposition"), on the quantum Hilbert space \(\HC\) used
to model the situation of interest. That is to say, a collection
\(\{P\_j\}\) of projectors (\(P^{}\_j = P\_j^\dagger =P\_j^2)\) which are
mutually orthogonal (\(P\_jP\_k = 0\) for \(j\neq k\)) and sum to the
identity


\[I = \sum\_j P\_j. \label{eqn5}\]


These projectors represent *mutually exclusive* properties
(implied by orthogonality), one of which is true (implied by the sum
equal to \(I\)) in any given experiment or situation to which the model
is being applied.


The event algebra \(\EC\) consists of all projectors formed by taking
sums of some of the \(P\_j\) in the collection; i.e., \(\sum\_j\pi\_j P\_j\),
where each \(\pi\_j\) is either 0 or 1. This includes the identity
operator \(I\) and the zero operator 0 along with such things as \(P\_2\)
or \(P\_1+P\_3\). The event algebra is a Boolean algebra in the sense that
the complement \(I-P\) of any projector and the product \(PQ\) of any two
projectors belonging to \(\EC\) is an element of \(\EC\). The
term *framework* can refer to either the sample space or the
event algebra constructed from it in this way, since there is a
one-to-one correspondence between the two. Quantum theory does not in
general specify the probability \(p\_j\) to be assigned to the projector
\(P\_j\) in \(\SC\), apart from the requirement that it be nonnegative, and
that the \(p\_j\) sum to 1 (but see subsection 4.2
concerning time dependence). Thus, for example, to the element
\(P\_1+P\_3\) of \(\EC\) one assigns the probability
\(p\_1+p\_3\). The *intuitive interpretation* of probabilities
introduced in this way is the same as in the other sciences which
employ probability theory as long as a single framework is in
view. What is distinctly different in the quantum case is the
existence of multiple incompatible frameworks; given these consistency
requires that one adopt the single framework
rule, subsection 3.4.


### 3.3 Quantum observables


Quantum observables, represented by Hermitian operators, are the
counterpart of real-valued functions on a classical phase space. The
spectral decomposition


\[A = \sum a\_j P\_j \label{eqn6}\]


of an observable \(A\), where each eigenvalue \(a\_j\) of \(A\) occurs but
once in the sum, \(j\neq k\) implies \(a\_j\neq a\_k\), associates with \(A\)
a unique decomposition of the identity of the form \eqref{eqn5}. Thus
\(A\) takes on or has the value \(a\_j\) if and only if \(P\_j\) is true, and
the probability that \(A\) has the value \(a\_j\) is given by the
probability \(p\_j\) assigned to \(P\_j\). Hence *measuring* \(A\) is
equivalent to measuring the corresponding decomposition in the sense
of determining which of the mutually-exclusive properties \(P\_j\) is
true. (The subject of quantum measurements is taken up below
in section 7.) The property that the value of
\(A\) lies between \(a\) and \(a'\) corresponds to the sum of the projectors
\(P\_j\) for which the corresponding \(a\_j\) falls in this interval, which
is a projector belong to the corresponding event algebra, and its
probability is the sum of the corresponding \(p\_j\) values.


### 3.4 Refinement, coarsening, and the single framework rule


A *refinement* of a sample space is obtained by replacing
one or more of its projectors with other projectors which add up to
it; conversely a *coarsening* or coarse graining is obtain by
adding together two or more projectors in the projective decomposition
and using the sum as one of the elements in a new decomposition. Given
the connection between sample spaces and event algebras as defined
above, one sees that the event algebra of a refined sample space is
larger than the original event algebra: i.e., it contains all the
original projectors and others in addition; conversely, a coarsening
leads to a new event algebra which is a subset of the old event
algebra.


Two quantum sample spaces (projective decompositions) \(\{P\_j\}\) and
\(\{Q\_k\}\) associated with the same Hilbert space are said to
be *compatible* provided all the projectors in one
decomposition commute with all the projectors in the other
decomposition: \(P\_jQ\_k = Q\_k P\_j\) for all \(j\) and \(k\). Otherwise they
are *incompatible*. Note that all projectors in the two event
algebras commute if and only if this is true for the sample spaces, so
it makes sense to talk about compatible and incompatible
frameworks. When the frameworks are compatible one can combine
them. In particular, the *coarsest common refinement* is given
by the sample space or decomposition formed by all nonzero products of
the form \(P\_jQ\_k\), and the corresponding event algebra contains all
the projectors in the original algebras along with the additional ones
constructed from their products.


The *single framework rule* says that a specific quantum
probabilistic model employs just one sample space and its associated
event algebra, and in particular two incompatible
frameworks *cannot be combined*. This rule, together with its
extension to quantum dynamics, see subsection 4.2,
is a central principle of histories quantum mechanics. It provides a
guide for quantum reasoning that prevents one from falling into
paradoxes; conversely, most quantum paradoxes are constructed by
violating the single framework rule in some way.


As an illustration, consider the one-dimensional harmonic
oscillator of subsection 2.1 and a sample
space


\[\SC\_a = \{P,I-P\};\quad P = [0] + [1], \label{eqn7}\]


containing two projectors, where \(P\) corresponds to the case \(N=1\)
in \eqref{eqn3}. The physical interpretation of \(P\) is that the energy
of the oscillator is no greater than \((3/2)\hbar\omega\), whereas \(I-P\)
means the energy is at least \((5/2)\hbar\omega\). If \(P\) is true it
might seem reasonable to conclude that either \(n=0\) and the energy is
\((1/2)\hbar\omega\), or else \(n=1\) and the energy is
\((3/2)\hbar\omega\). After all, the energy can only take values
\((n+1/2)\hbar\omega\), so does it not follow from the truth of \(P\) that
\(n=0\) or \(n=1\)? No it does not, and understanding *why* it does
not will expose an important principle of quantum reasoning. The point
is that probabilistic reasoning, in quantum mechanics or in other
disciplines, does not allow one to make inferences about matters which
are outside the scope of discussion, i.e., which are not contained in
the event algebra. The event algebra \(\EC\_a\) corresponding to the
sample space \(\SC\_a\) in \eqref{eqn7} contains \(P\) and \(I-P\), but does
not contain either \([0]\) or \([1]\), the projectors that correspond to
the two lowest energy eigenstates. Consequently, this framework
does *not* allow for the statement of a result which might seem
intuitively obvious.


A solution is readily at hand. In place of \(\SC\_a\) use a
refinement


\[\SC\_b = \{[0],[1],I-P\};\quad P = [0] + [1]. \label{eqn8}\]


Now because \([0]\) and \([1]\) belong to the event algebra \(\EC\_b\)
they can be discussed, and the truth of \(P\) indeed implies that either
\([0]\) or \([1]\) is true: if the energy is not greater than
\((3/2)\hbar\omega\) then it is either \(1/2\) or \(3/2\) \(\hbar\omega\). But
is not this insistence that \(\SC\_b\) be used in place of \(\SC\_a\) a
piece of sophistry irrelevant to the needs of the working scientist?
No, not if the scientist's work involves quantum mechanics. To see
why, consider an alternative refinement


\[\SC\_c = \{[+],[-],I-P\};\quad P = [+] + [-] = [0]+[1]
\label{eqn9}\]


of \(\SC\_a\), where


\[\ket{+} = (\ket{0} + \ket{1})/\sqrt{2},\quad \ket{-} = (\ket{0} -
\ket{1})/\sqrt{2}. \label{eqn10}\]


Since the event algebra \(\EC\_c\) contains \(P\) along with \([+]\) and
\([-]\), in this framework the truth of \(P\) implies that the oscillator
is either in the state \([+]\) or the state \([-]\). But neither of these
are energy eigenstates, and these projectors do not commute with \([0]\)
and \([1]\); the relationship is formally the same as that between
eigenstates of \(S\_x\) and those of \(S\_z\) for a spin-half
particle, subsection 2.3. Consequently, if
results using frameworks \(\EC\_b\) and \(\EC\_c\) are carelessly combined,
it is possible to infer for the oscillator something quite analogous
to the meaningless assertions combining properties of \(S\_x\) and
\(S\_z\). Hence the importance of the single framework rule which
prohibits combining \(\EC\_b\) and \(\EC\_c\) (equivalently, \(\SC\_b\) and
\(\SC\_c\)).


If \(\EC\_b\) and \(\EC\_c\) cannot be combined, which is
the *correct* framework to use in describing a harmonic
oscillator? The answer, again a key point in the histories
interpretation, which will be discussed in more detail
in section 5, and also section
11, is that in the quantum world there is never a single correct
framework, there are always many possibilities, something rather
strange from the perspective of classical physics. The single
framework rule prohibits *combining* incompatible frameworks;
it does *not* assert that there is just one correct framework
to use in modeling a particular physical situation.


One may add that in the early days of quantum theory there was a
tendency among the practitioners to think that quantum systems must
always be in states of well-defined energy, and this led to various
conundrums. Nowadays states which are superpositions of the ground
state and an excited state of an atom are produced routinely in
quantum optics experiments, and thus analogs of \(\EC\_c\) can be quite
relevant for a quantum description of what goes on in the
laboratory.


### 3.5 Quantum states as pre-probabilities


Once a quantum sample space has been chosen the question arises of
how probabilities are to be assigned to the different projectors of
which it is composed. As in other applications of probability theory
such assignments may represent pure guesswork, or contain parameters
which are to be adjusted according to experimental data, or be based
upon some other information about the system, such as prior
preparation or a later measurement. Information about the preparation
of a quantum system is often expressed using a *density
operator* \(\rho\), a positive semidefinite operator (Hermitian with
nonnegative eigenvalues) with trace (sum of the eigenvalues) equal to
1, used as a *pre-probability* in the terminology of Sec. 9.4
of Griffiths (2002a), a mathematical
object that can be used to generate a probability distribution. In
particular if \(\{P\_j\}\) is a projective decomposition of the identity,
a quantum sample space, one assigns a probability


\[p\_j = \text{Tr}(\rho P\_j) \label{eqn11}\]


to the property \(P\_j\). Very often \(\rho\) is referred to as the
"quantum state", since it is somewhat analogous to a
probability distribution, e.g., a "Gibbs state", in
classical statistical mechanics. The name
"pre-probability" refers to the fact that \(\rho\) can be
used to generate many different probability distributions for
different choices of the sample space.


Whereas the distinction between quantum properties and quantum
states in the sense of pre-probabilities is often evident from the
context and the use of different symbols, the case of a "pure
state" in which the density operator can be written as a rank
one projector or dyad, \(\rho =\dya{\psi}\), can lead to confusion in
that the same mathematical object can represent either a
pre-probability or a property. Indeed, there is a close connection in
that if for such a \(\rho\) one introduces a sample space
\(\{[\psi],I-]\psi]\}\), the probability of the property \([\psi]\) is 1,
which is to say that *if one uses this particular sample space*
the quantum system certainly has the property \([\psi]\). But if this
same \(\rho\) is used as a pre-probability for some other sample space,
and if one ignores the single framework rule, it is easy to fall into
the trap of thinking that \([\psi]\) and some other incompatible
property corresponding to a projector that does not commute with
\([\psi]\) can both be true, which makes no sense. The best known
example of the confusion that results is Schrodinger's infamous
dead-and-alive cat, subsection 10.1.


## 4. Quantum Time Development


### 4.1 Kinematics


Suppose a coin is tossed three times in succession, with results
heads (\(H\)) or tails (\(T\)). In the theory of stochastic processes one
constructs a sample space \(\SC\) containing 8 events, let us call
them *histories*: \(HHH\), \(HHT\), \(HTH\), .... Notice that
formally the same sample space is obtained if instead of a coin tossed
three times in a row, three different coins are tossed
simultaneously. Since in quantum mechanics the Hilbert space of
several separate systems is constructed as a tensor product of the
individual Hilbert spaces, this suggests that the Hilbert space for a
system at a sequence of times, the *histories* Hilbert space,
can be constructed as the tensor product of copies of the Hilbert
space for a single time. We write it as


\[\breve \HC = \HC\_1\odot\HC\_2\cdots\odot\HC\_f \label{eqn12}\]


for times \(t\_1 <t\_2 <\cdots < t\_f\), where \(\odot\) is
employed in place of the usual \(\otimes\) simply to emphasize that a
sequence of times is involved.


A single quantum history will typically correspond to a
projector


\[Y^\alpha = P\_1^{\alpha\_1}\odot P\_2^{\alpha\_2}\odot\cdots
P\_f^{\alpha\_f}, \label{eqn13}\]


where the superscript \(\alpha\_m\) labels different possible
projectors at time \(t\_m\), and \(\alpha =
(\alpha\_1,\alpha\_2,\ldots\alpha\_f)\) is a composite label for the
entire history. (Since the square of a projector is the same as the
projector, the superscript position is not needed for exponents, and
it is convenient to reserve the subscript to indicate the time.) If
for each time \(t\_m\) it is the case that \(\sum\_{\alpha\_m}
P\_m^{\alpha\_m} = I\_m\) it follows that \(\sum\_\alpha Y^\alpha = \breve
I\), the identity operator on \(\breve\HC\); thus the \(\{Y^\alpha\}\) form
a projective decomposition of the identity \(\breve I\), a quantum
sample space or *family* of histories, one and only one of
which will actually occur in any given experimental run. The physical
interpretation of \eqref{eqn13} is that the property \(P\_1^{\alpha\_1}\)
occurs or is true at time \(t\_1\), \(P\_2^{\alpha\_2}\) at time \(t\_2\), and
so forth. Indeed, since we live in a quantum world, the
"classical" coin-tossing history \(THT\) can be written in
the form \eqref{eqn13} using appropriate projectors to represent \(T\)
and \(H\). As will be discussed later in section
6, the histories approach employs the same mathematical procedures
for both microscopic and macroscopic physics, consistent with the
belief of most physicists that quantum theory applies to phenomena of
any size.


Several comments are in order. First, the projective decompositions
of the identity at different times need not be the same. Thus for a
spin half particle one could have the \(S\_z=\pm 1/2\) decomposition at
one time and the \(S\_x=\pm 1/2\) decomposition at some later
time. And more complicated history-dependent decompositions of
\(\breve I\) are possible; see the
 Further Reading supplement.

Second, there is
no reference to measurements. Events occur or do not occur; whether a
measuring apparatus is involved is entirely irrelevant, although
histories can be used to discuss what goes on in a measuring process;
see section 7. Third, the projectors in a
history can be coarse-grained and specify only a small amount of
information, e.g., the spin degree of freedom of an atom without
reference to its position; they do not and generally will not project
onto pure states. Fourth, there is no direct connection between the
choice of projectors entering a history and Schrodinger's
time-dependent equation, although in appropriate circumstances the
latter can be used to assign probabilities to the different histories
in a family, as discussed next.


### 4.2 Dynamics


Classical mechanics yields a deterministic dynamics only in the
case of a *closed* or *isolated* system, since an exact
prediction of the future or past for a system interacting with an
unknown environment is not possible. But even if one is interested in
the approximate dynamics of an open classical system it helps to start
off with the idealization that the environment and the system of
interest are parts of a total system which is isolated, and therefore
has a dynamics given by Hamilton's equations. Similarly, when
constructing the intrinsically stochastic dynamics appropriate to
quantum theory it is helpful to begin with a closed or isolated system
with well-defined boundary conditions in which Schrodinger's equation
has well defined solutions and induces a unitary time development
given by a collection of time-evolution operators \(T(t',t)\): if
\(\ket{\psi(t)}\) is any solution to Schrodinger's (time-dependent)
equation, then \(\ket{\psi(t')} = T(t',t)\ket{\psi(t)}\). In the case of
a time-independent Hamiltonian \(H\) one has \(T(t',t) =
\exp[-i(t'-t)H/\hbar]\).


Consider a family \(\FC=\{Y^\alpha\}\) of histories of a closed
quantum system, for a finite (possibly very large) collection of times
\(t\_1, t\_2, \ldots t\_f\). Under appropriate conditions one can use
\(T(t',t)\) to assign probabilities in a consistent way to the histories
in \(\FC\), using a multi-time generalization of the Born rule. To
simplify the exposition we restrict attention to situations in which
all histories begin with the same normalized initial pure state
\(\ket{\Psi\_0}\) at a time \(t\_0 <t\_1\), and modify the definition in
\eqref{eqn13} so that it reads


\[Y^\alpha = [\Psi\_0]\odot P\_1^{\alpha\_1}\odot
P\_2^{\alpha\_2}\odot\cdots P\_f^{\alpha\_f}. \label{eqn14}\]


In addition, add an extra history \(Y^0=(I-[\Psi\_0])\odot I\_1\odot
I\_2\cdots\) so that the sum of all the history projectors together is
the identity \(\breve I\) on the Hilbert space
\(\HC\_0\odot\HC\_1\odot\cdots\). (Here, consistent with the notation used
elsewhere in this article, \([\Psi\_0]\) denotes the projector
\(\dya{\Psi\_0}\).) Then define a collection of "chain kets"
\(\ket{\Phi^\alpha}\),


\[\ket{\Phi^\alpha} = P^{\alpha\_f}\_f T(t\_f,t\_{f-1})
 P^{\alpha\_{f-1}}\_{f-1} T(t\_{f-1},t\_{f-2})\cdots P^{\alpha\_1}\_1
 T(t\_1,t\_0) \ket{\Psi\_0}, \label{eqn15}\]


one for each history \(\alpha\). Provided the *consistency
conditions*


\[\inpd{\Phi^\alpha}{\Phi^{\alpha'}} = 0 \text{ for }
\alpha\neq\alpha', \label{eqn16}\]


are satisfied, where \(\alpha\) and \(\alpha'\) are considered unequal
if \(\alpha\_m\neq \alpha'\_m\) for at least one value of \(m\) between 1
and \(f\), the history \(Y^\alpha\) is assigned the probability


\[\Pr(Y^\alpha) = \inp{\Phi^\alpha}. \label{eqn17}\]


In the case \(f=1\) the consistency condition \eqref{eqn16} is
automatically satisfied because the projectors at \(t\_1\) are orthogonal
to each other, and the probabilities \(\inp{\Phi^\alpha}\) are those
given by the usual Born rule. For \(f=2\) or more the consistency
conditions are not trivial, and when they are satisfied \eqref{eqn17}
provides a *generalization* of the Born
rule.[1] Insight
into the meaning of the consistency conditions is best obtained by
working through examples; see the 
 Further Reading supplement.

Families
for which the consistency conditions are satisfied are
called *consistent families*, and it is only for this
restricted class that the laws of quantum dynamics for a closed system
can be used to assign a meaningful set of probabilities.


Some additional comments. First, two consistent families \(\FC\_a\)
and \(\FC\_b\) may be such that the (history) projectors in one commute
with those in the other, but still the common coarsest refinement does
not satisfy the dynamical consistency conditions, in which case they
are said to be *incommensurate* and cannot be combined, at
least if one wishes to use the dynamical laws of quantum mechanics to
assign them probabilities. Second, *measurements* play no role
in terms of the fundamental mathematical formulation or its physical
interpretation. In classical physics there is never any difficulty in
imagining events actually occurring inside a closed system, even with
no observers or measuring apparatus present. The same is true for
quantum physics once one removes "measurement" from its
spurious role as an interpretive principle, and instead treats it like
any other quantum process, see section 7. Third,
the histories approach is *time symmetrical* in the sense that
the rules for assigning probabilities do not single out a particular
"direction" or sense of time. (This symmetry is evident in
the more complete discussion in Ch. 10
of Griffiths (2002a), which allows more
general types of family. That generality has been sacrificed here in
the interests of a simpler and more intuitive exposition based on the
assumption of a single pure state at the initial time.)


## 5. Quantum Reasoning


### 5.1 General principles


The histories interpretation of quantum mechanics, as noted
earlier, is probabilistic, and hence the processes of reasoning are
those that scientists generally use for probabilistic models. This
requires a method of assigning probabilities along with procedures for
drawing inferences, usually in the form of conditional
probabilities. In both these respects quantum theory resembles other
applications of probabilistic reasoning. The main difference is the
necessity in the quantum case of paying close attention to the problem
of choosing a sample space (which then determines the event algebra)
or framework, and then confining the logical reasoning process, from
assumptions to conclusions, to a *single* framework. We have
already encountered some applications of this single framework rule,
and additional ones are discussed later, but at this point it is
useful to summarize some of the basic principles involved, which can
be conveniently placed under four headings:


* (R1)

Liberty. The physicist is free to employ as many frameworks as desired
when constructing descriptions of a particular quantum system,
provided the principle R3 below is strictly observed.
* (R2)

Equality. No framework is more fundamental than any other; in
particular, there is no "true" framework, no framework
that is "singled out by nature".
* (R3)

Incompatibility. The single framework rule: incompatible frameworks
are *never* to be combined into a single quantum
description. The (probabilistic) reasoning process starting from
assumptions (or data) and leading to conclusions must be carried out
using a *single* framework.
* (R4)

Utility. Some frameworks are more useful than others for answering
particular questions about a quantum system.


The reasoning process begins with some *initial data*, which
may reflect some knowledge of the system under discussion or may be
purely hypothetical. Here "initial" refers to the
beginning of a logical argument, not necessarily to the initial state
of a quantum system, though the latter is often included in the
initial data. These data must then be expressed in appropriate quantum
terms as some sort of property or properties, subspaces of an
appropriate Hilbert space, which could be a space of histories, all
belonging to a *single* framework or event algebra, principle
R3. Generally there will be more than one framework that can contain
the initial data, and the physicist is at liberty to choose among
these possibilities, R1. However, the types of conclusions which can
be drawn using a particular framework (and thus event algebra) will
depend upon the framework, and for this reason the physicist's
interests can, and in general will, influence the choice of
framework. It is utility, R4, which determines this choice; from a
fundamental point of view all frameworks are equally good, R2.


The link between initial data and the conclusions one can draw from
them will be probabilistic (sometimes with probabilities 0 or 1, hence
deterministic). One might be concerned that the flexibility in
framework choice provided by R1 and R2 could lead to different results
when using different frameworks. However, it is straightforward to
show, see Ch. 16 of Griffiths (2002a),
that as long as both frameworks contain all the events needed to
express the initial data and also those needed for drawing
conclusions, the probabilities linking data and conclusions will be
identical even if the two frameworks in question are mutually
incompatible. This ensures an overall consistency. The discussion of
quantum preparation and measurement found in
 section 7 below will show how principles
of quantum reasoning apply to
measurements, a topic that has given rise to a lot of confusion.


### 5.2 Counterfactual reasoning


A counterfactual argument begins by imagining another world which
resembles the actual world in certain respects and differs from it in
others, and asking the question: "What *would* be the
case *if* instead of ...it were true that
...". For example, "What would the weather be like in
Pittsburgh in January if the city were 1000 kilometers closer to the
equator?" In general, counterfactual reasoning is not easy to
analyze, and this gives rise to difficulties when trying to understand
quantum paradoxes stated in counterfactual form. The basic principle
governing counterfactual reasoning in the histories approach is the
single framework rule, with the requirement that a comparison between
the real and the counterfactual world be carried out using
the *same* framework for both. For details, see the
Further Reading supplement
and the applications in Sec. 10.4 below.




## 6. Classical Physics


The histories approach assumes that the same fundamental quantum
mechanical laws apply to systems of any size, from quarks to jaguars
to quasars. Classical mechanics is a very good approximation to the
more exact quantum laws in appropriate circumstances. The following
remarks 
provide only a broad outline of how the classical world
emerges from quantum theory; see the 
 Further Reading supplement for additional
material.




The first task is to identify suitable *quasiclassical*
frameworks, appropriate choices of projective decompositions of the
identity at a single time corresponding to macroscopic properties,
together with suitable quantum histories that involve such properties,
with results which correspond closely to the time development of
classical physics. The desired decompositions will be coarse grained,
with individual projectors corresponding to subspaces of enormous
dimension (e.g., 10 raised to the power \(10^{16}\)). In such a
situation it is plausible that there are families containing quantum
histories having a probability very close to 1, and which exhibit a
time development that closely approximates that of classical
mechanics.


To be sure, the task of showing that classical laws emerge to a
good approximation in this manner is a nontrivial one, and it cannot
be claimed that it has been completed. However, the effort expended
thus far has not revealed any fundamental difficulty which would
undermine this approach. One also expects to find cases in which the
relevant quantum probabilities are not close to 0 or 1; in particular
in a regime where classical analysis predicts chaos (positive Lyapunov
exponents) quantum "fluctuations" are likely to be
amplified. But this is also a situation in which the deterministic
aspect of classical time development is not to be taken too seriously,
due to the sensitive dependence upon initial conditions, so there is
no reason to suppose that an appropriate quantum description cannot be
constructed, at least in principle.


Understanding the emergence of classical physics from the quantum
world does not require a unique quasiclassical framework; there are
undoubtedly a large number of different possibilities, any one of
which could for the circumstances of interest provide
"classical" results within a suitable approximation
(further comments on nonuniqueness will be found
in subsection 11.3 below). In constructing a
quasiclassical description *decoherence* plays a useful role in
the sense that it removes the effects of quantum interference which
might otherwise render a quasiclassical family inconsistent,
preventing the application of the Born rule and its
generalizations. Once again, this statement can be supported by model
calculations of an appropriate sort which make it plausible, but work
still remains to be done in coming to a full understanding of
decoherence in quantum terms. Invoking decoherence by itself outside
the framework provided by the histories approach does not solve the
conceptual difficulties of quantum theory; see,
e.g., Adler (2003).


## 7. Measurement and Preparation


The principles discussed above provide a resolution of
the *measurement problem* of quantum foundations: that is, how
to discuss the measurement process itself in fully quantum mechanical
terms. It is convenient to think of this as made up of two
problems:


* (M1)

How can the macroscopic *outcome* of the measurement,
traditionally thought of as a pointer position, be described in
quantum terms?
* (M2)

How is this outcome related to the earlier microscopic property the
apparatus was designed to measure?


The reliable *preparation* of a microscopic system in a
particular quantum state raises similar conceptual issues; indeed,
what in textbooks are often called "measurements" would be
better thought of as "preparations". Both are discussed
below.


### 7.1 Schematic model


The issues are most simply addressed by using a simple schematic
quantum model of measurement and preparation that goes back
to J. von Neumann (1932: sec. VI.3). Let
\(\HC\_s\) be the Hilbert space of the system to be measured, henceforth
referred to as a *particle*, and \(\HC\_m\) that of the measuring
device or *apparatus*. For example, \(\HC\_s\) could be the
2-dimensional Hilbert space of the spin of a spin-half particle,
whereas the quantum description of its position might be included in
\(\HC\_m\), along with all the many degrees of freedom needed to describe
the apparatus itself. Let \(\{\ket{s^j}\}\) be an orthonormal basis for
\(\HC\_s\), with states labeled by a superscript so subscripts can label
time. Let


\[\ket{s\_0} = \sum\_j c\_j \ket{s^j},\quad \ket{m\_0},\quad
\ket{\Psi\_0}=\ket{s\_0}\otimes\ket{m\_0} \label{eqn18}\]


be the normalized initial states of the particle, the measuring
device, and the total system at time \(t\_0\). Here \(c\_j\) are complex
numbers satisfying \(\sum\_j|c\_j|^2=1\).


Let \(T(t',t)\), subsection 4.2, be the unitary
time development operator for the total system, and assume it is
trivial--equal to the identity operator \(I=I\_s\otimes
I\_m\)--for \(t\) and \(t'\) both less than some \(t\_1\) or both greater
than \(t\_2\), whereas for the interval from \(t\_1\) to \(t\_2\) during which
the particle and the apparatus interact,


\[T(t\_2,t\_1) \bigl( \ket{s^j}\otimes\ket{m\_0}\bigr) =
\ket{w^j}\otimes \ket{m^j}. \label{eqn19}\]


Here the \(\ket{m^j}\) are orthonormal states of the apparatus,
\(\inpd{m^j}{m^k}=\delta\_{jk}\), corresponding to different pointer
positions, and the \(\ket{w^j}\) are normalized, \(\inp{w^j}=1\),
but *not necessarily orthogonal*; it may be that
\(\inpd{w^j}{w^k}\neq 0\) when \(j\neq k\). (In von Neumann's original
model the \(\ket{w^j}\) were the same as the \(\ket{s^j}\), and this is
also possible, but is not being assumed, for the situation we are
considering.) The transformation \eqref{eqn19} can be extended to a
unitary transformation on \(\HC\_s\otimes\HC\_m\), because the
orthogonality of the \(\ket{m^j}\) ensures that the states on the right
side of \eqref{eqn19} are normalized and mutually orthogonal. Unitary
time development leads to states



\[

\begin{align}
\label{eqn20}
\ket{\Psi\_1} &= T(t\_1,t\_0) \ket{\Psi\_0} =\ket{\Psi\_0}, \\
\ket{\Psi\_2} &= T(t\_2,t\_1) \ket{\Psi\_1} = \sum\_j c\_j\ket{w^j}\otimes\ket{m^j} \nonumber
\end{align}

\]

for the total system at times \(t\_1\) and \(t\_2\).


### 7.2 Measurement


To study the physical implications of this model we need to
introduce a family of histories. From a mathematical perspective the
simplest such family is based solely on unitary time development and
takes the form


\[\FC\_u:\;\;[\Psi\_0]\;\odot\; \{[\Psi\_1],I-[\Psi\_1]\}\;\odot\;
\{[\Psi\_2], I-[\Psi\_2]\}, \label{eqn21}\]


where \([\Psi\_1]\) and \([\Psi\_2]\) are projectors onto the states
\(\ket{\Psi\_1}\) and \(\ket{\Psi\_2}\) defined in \eqref{eqn20}, and the
family \eqref{eqn21} should be interpreted as a set of four mutually
exclusive histories obtained by choosing at \(t\_1\) one of the
projectors inside the first pair of curly brackets, and at \(t\_2\) one
inside the second pair. The corresponding chain
kets, subsection 4.2, are zero except for the
single history \([\Psi\_0]\odot [\Psi\_1] \odot [\Psi\_2]\), which is
assigned probability 1 by \eqref{eqn17}.


The family \(\FC\_u\) is of no use for resolving the first measurement
problem M1, because the family of histories does not include the
projectors \(\{[m^j]\}\) for the pointer positions at time \(t\_2\) needed
to discuss the measurement outcome, nor can it be refined to include
them, because \([\Psi\_2]\) will not commute with some of the \([m^j]\),
assuming at least two of the \(c\_j\) in \eqref{eqn18} are nonzero. This
is the basic reason why the first measurement problem M1 is an
enormous difficulty for quantum interpretations in which the only
possible time dependence for a closed quantum system is that given by
Schrodinger's equation. By contrast, the histories approach gives the
physicist Liberty to use a different family


\[\FC\_1:\;\;[\Psi\_0]\;\odot\;[\Psi\_1] \;\odot\; \{[m^j]\}
\label{eqn22}\]


in place of \(\FC\_u\). Note that \([m^j]\) is understood as
\(I\_s\otimes[m^j]\), using a common physicist's convention. In addition
a projector \(R=I\_m-\sum\_j[m^j]\) should be included among the
possibilities at the final time \(t\_2\), but as it occurs with zero
probability it can be ignored. The consistency of the family \(\FC\_1\)
is easily demonstrated, and the probability \(|c\_j|^2\) for the final
position \([m^j]\) is an immediate consequence of \eqref{eqn17}.


However, the second measurement problem M2, relating the outcome of
the measurement to the property of the particle before the
measurement, cannot be solved using the family \(\FC\_1\), because the
properties of interest, the \([s^j]\), are not included among the
possibilities at time \(t\_1\). To discuss these requires another
family


\[\FC\_2:\;\;[\Psi\_0]\;\odot\; \{ [s^j]\}\;\odot\; \{[m^k]\}.
\label{eqn23}\]


Again, \([s^j]\) can be understood as \([s^j]\otimes I\_m\), a property
of the particle without reference to the apparatus. Showing that
\(\FC\_2\) is consistent is straightforward, as is calculating the joint
probability distribution


\[\Pr(\,[s^j]\_1,[m^k]\_2) = |c\_j|^2 \delta\_{jk} \label{eqn24}\]


using \eqref{eqn17}. Here the subscripts refer to the time: the
particle property at time \(t\_1\) before the measurement takes place,
and the outcome at \(t\_2\) when the measurement is over. The marginal
probabilities are


\[\Pr([s^j]\_1) = \Pr([m^j]\_2) = |c\_j|^2. \label{eqn25}\]


Thus for every case in which \(c\_k\) is not zero (so \([m^k]\) occurs
with finite probability) one has the conditional probability


\[\Pr([s^j]\_1 \boldsymbol{\mid} [m^k]\_2) = \delta\_{jk}.
\label{eqn26}\]


In other words, given a measurement outcome \([m^k]\) at \(t\_2\) one
can be certain that the prior state of the particle at \(t\_1\) was
\([s^k]\); i.e., the apparatus did what it was designed to do.


### 7.3 Preparation and wave function collapse


The reader may have noticed that the particle states \(\ket{w^j}\) in
\eqref{eqn19} played no role in our discussion of measurement
in subsection 7.2. Von Neumann's
original (1932: sec. VI.3) presentation
used \(\ket{w^j}=\ket{s^j}\), as he thought it important that a second
measurement could confirm the first. This practice was followed in
later textbooks, and has resulted in some confusion. The \(\ket{w^j}\)
are in fact irrelevant when considering measurements, but they are
useful when considering a *preparation procedure* in which
macroscopic apparatus is used to produce a quantum system--again
we shall speak of it as a particle--in some well-defined
state. To see how this works let us again employ the model
of subsection 7.1 with unitary development given
by \eqref{eqn19}, but now introduce a family


\[\FC\_p:\;\;[\Psi\_0]\;\odot\;[\Psi\_1] \;\odot\;
\{[v^l]\otimes[m^j]\} \label{eqn27}\]


that is a refinement of \(\FC\_1\) in \eqref{eqn22}, with \(I\_s\)
replaced by a projective decomposition onto some orthonormal basis
\(\{\ket{v^j}\}\) of the particle Hilbert space \(\HC\_s\). The joint
probability distribution at time \(t\_2\) of the particle and pointer
states is then



\[

\begin{align}
\label{eqn28}
\Pr([m^j],[v^l]) &= \mte{\Psi\_2}{[v^l]\otimes[m^j]} \\
 &= |c\_j|^2 \mte{w^j}{[v^l]} \nonumber \\
 &= |c\_j|^2 |\inpd{v^l}{w^j}|^2. \nonumber
\end{align}

\]

Since \(\Pr([m^j]) = |c\_j|^2\), \eqref{eqn25}, one obtains the
conditional probability


\[\Pr([v^l]\boldsymbol{\mid} [m^j]) = |\inpd{v^l}{w^j}|^2
\label{eqn29}\]


(assuming \(|c\_j|^2\) is not 0). That is to say, if the measurement
outcome is known to be \([m^j]\), then the probability of any property
of the particle is precisely that given by assuming the particle is in
the state \(\ket{w^j}\), with "state" having its usual
meaning: a pre-probability from which the probability distribution for
the properties represented by any projective decomposition of its
identity can be calculated; see subsection
3.5.


Note that this discussion remains valid when \(\{\ket{w^j}\}\) is not
a collection of orthogonal states, even though this means that the
corresponding projectors \(\{[w^j]\}\) cannot all belong to the same
projective decomposition of \(I\_s\). The point is that \(\ket{w^j}\) is
assigned to the particle only in association with the projector
\([m^j]\), and the collection \(\{\ket{w^j}\otimes\ket{m^j}\}\) is
orthogonal. In such circumstances the \([w^j]\) are what in Ch. 14
of Griffiths (2002a) are referred to
as *dependent* or *contextual*
("conditional" would be equally good) *properties*:
they must be understood in conjunction with other properties, which in
this case are the pointer positions \([m^j]\), when forming part of a
quantum description or history.


What has here been deduced from fundamental quantum principles,
including the Born rule, is in textbooks stated as a mysterious and
separate principle of "wave function collapse": if the
measurement outcome is \([m^j]\) then the particle state
"collapses" onto a particular state associated with this
outcome. To make closer contact with textbook discussions of this
topic, let


\[S = \sum\_j \sigma\_j [s^j] \label{eqn30}\]


be a Hermitian operator with nondegenerate eigenvalues \(\sigma\_j\),
and let \(\ket{w^j}=\ket{s^j}\). Then the measurement outcome \([m^j]\)
corresponds to measuring the eigenvalue \(\sigma\_j\)--see the
remarks in subsection 3.3--and
\eqref{eqn29} and the remarks following it justify assigning to the
particle at time \(t\_2\) the ket \(\ket{w^j}=\ket{s^j}\). One may add that
the more general form of wave function collapse embodied in the
Luder's rule (Busch & Lahti 2009) is
likewise a consequence of the general quantum principles governing
stochastic time development set forth in section
4, so need not be added as an additional hypothesis.



In any case there is no need to set \(\ket{w^j}\) equal to
\(\ket{s^j}\) in order to discuss quantum measurements, and a better
way of viewing the family \(\FC\_p\) in \eqref{eqn27} is as a model of
a quantum preparation process in which the macroscopic pointer
position tells one the future rather than the past state of the
particle. The conditional probability in \eqref{eqn29} refers
specifically to properties at the time \(t\_2\), but given the
assumption that \(T(t',t)\) is trivial for times after \(t\_2\), the
argument leading to \eqref{eqn29} is easily extended to \([v^l]\) and
\([m^j]\) at any times (not necessarily the same) later than \(t\_2\).
In general the outcome \([m^j]\) is random, and if the wrong \(j\)
occurs \(\ket{w^j}\) may not be the desired state. But then the
preparer can throw the particle away (run it into a baffle) and repeat
the experiment until the desired outcome occurs. There are more
sophisticated ways to do preparations, but our model illustrates the
general idea, and in fact discarding unwanted outcomes is a procedure
often used in quantum optics. The essential point is that an
apparatus can be constructed using quantum principles to prepare a
particle whose future state is known on the basis of macroscopic
(pointer) information.


### 7.4 Generalized and Weak Measurements


Von Neumann's measurement model in 
 section 7.1
is often referred to as a "projective measurement", since its outcome
\(k\), see , indicates an earlier quantum property corresponding to
the projector \([s^k]\). It is also called a "strong measurement" in
contrast to a weak measurement as defined below. In practice one often
uses a more general measurement model, referred to as
a *generalized measurement* or POVM (positive operator value
measure) where the measurement outcome corresponds not to a specific
property but instead to a positive (semi-definite) operator for the
measured system at an earlier time. 




Generalized measurements are often thought of in the following
way. Instead of interacting directly with the measurement apparatus,
the system to be measured first interacts with another (microscopic)
system, a *probe*, prepared in a known quantum state. Then the
probe is subjected to a projective measurement, and from its outcome
one learns something about the original system. If the interaction
between the probe and the system of interest is very weak this is
called a *weak measurement*. Sometimes a hybrid measurement is
employed, in which, following its interaction with a probe, the
original system is subjected to another (typically projective)
measurement. Both the weak measurement and the hybrid are examples of
generalized measurements, and can be discussed in the manner indicated
below. A weak measurement must be carefully distinguished from
a *weak value*; see the comment at the end of this section.



The generalized measurement model starts, as in Sec. [sbct7.1], with a
Hilbert space \({{\mathcal H}}={{\mathcal H}}\_s\otimes{{\mathcal
H}}\_m\), where \({{\mathcal H}}\_s\) represents the measured system
(hereafter referred to as a particle), and \({{\mathcal H}}\_m\) the
measuring apparatus, including the probe if that is part of the
arrangement. And as in and assume an initial state




\[
{|\Psi\_0\rangle }={|s\_0\rangle }\otimes{|m\_0\rangle }, \label{eqn31}
\]


 and a unitary time development from \(t\_0\) to later times \(t\_1 < t\_2\) given by 




\[
{|\Psi\_1\rangle } = T(t\_1,t\_0) {|\Psi\_0\rangle } ={|\Psi\_0\rangle },\quad 
{|\Psi\_2\rangle } = T(t\_2,t\_1) {|\Psi\_1\rangle }, \label{eqn32}
\]


where, as previously, we are assuming that the time interval from
\(t\_0\) to \(t\_1\) is so short that \(T(t\_1,t\_0)\) is the identity
operator.



But instead of introducing pure states \({|m^j\rangle }\) for the
different positions of the measurement pointer, we assume a projective
decomposition of the identity of \({{\mathcal H}}\),



\[I =\sum\_k M^k,
\label{eqn33}\]


where \(M^k\) projects onto the entire (macroscopic) subspace in which
the apparatus pointer is in position \(k\). Since any two situations
which are macroscopically distinct correspond to mutually orthogonal
subspaces of the quantum Hilbert space, the \(M^k\) for different
\(k\) will be orthogonal to each other. In addition one can include a
special projector, say \(\hat M\), in the sum to represent any
additional possibilities (e.g., the pointer has fallen off the
apparatus), to make the sum add up to \(I\). The \({|m^j\rangle }\) of
Sec. [sbct7.1] are related to the \(M^k\) through



\[M^k {|m^j\rangle } = {\delta }\_{jk} {|m^j\rangle }, 
\label{eqn34}\]


i.e., \({|m^j\rangle }\) belongs to the subspace associated with
\(M^j\).



Since we are interested in what the measurement tells us about the
particle, we can regard the initial apparatus state \({|m\_0\rangle }\)
in as fixed, and indicate the dependence of \({|\Psi\_2\rangle }\) on
\({|s\_0\rangle }\) using the formula



\[{|\Psi\_2\rangle } = J {|s\_0\rangle },
\label{eqn35}\]


where \(J\) is an isometry (it preserves inner products) that maps
\({{\mathcal H}}\_s\) onto a subspace of \({{\mathcal H}}\). This
provides a convenient way to define operators



\[Q^k = J{^\dagger }M^k J.
\label{eqn36}\]


acting on \({{\mathcal H}}\_s\). Because the \(M^k\) are positive
(semi-definite) operators, the same is true of the \(Q^k\), and
because \(J{^\dagger }J= I\_s\) (\(J{^\dagger }\) is the left inverse
of \(J\)), it follows from that



\[\sum\_k Q^k = I\_s. \label{eqn37}\]


(We assume that \(\hat Q=J{^\dagger }\hat M J=0\) for the special
projector \(\hat M\) introduced following .) Thus \(\{Q^k\}\) is a
collection of positive operators which sum to the identity, which is
the definition of a POVM.



If every \(Q^k\) is a projector, then implies that they are mutually
orthogonal, and form a projective decomposition of the identity of
\({{\mathcal H}}\_s\). This means we are dealing with a projective
measurement, though more general than the von Neumann model of
Sec. [sbct7.1], as the \(Q^k\) may project on subspaces of dimension
greater than 1. However, in general \(Q^k\) is not a projector or
proportional to a projector, but since it is Hermitian it can be
written in the form



\[Q^k = \sum\_j q\_{jk} \xi^{jk}, \quad \sum\_j \xi^{jk} = I\_s,
\label{eqn38}\]


where for a fixed \(k\) the \(\{\xi^{jk}\}\) form a projective
decomposition of \(I\_s\), and the \(\{q\_{jk}\}\) are the nonnegative
eigenvalues of \(Q^k\). It is then not hard to show that the family of
histories



\[Y^{jk} = [s\_0]\otimes[m\_0] \odot \xi^{jk}\otimes I\_m \odot M^k,
\label{eqn39}\]


labeled by both \(j\) and \(k\), is consistent. It gives rise to
conditional probabilities



\[\Pr(\xi^{jk}\_1 {\boldsymbol{\mid}}M^k\_2) = p\_{jk} = q\_{jk} \mte{s\_0}{\xi^{jk}}/\mte{s\_0}{Q^k},
\label{eqn40}\]


where the subscripts 0, 1, and 2 indicate the time. In words, what
tells us is that if at the end of a run the pointer is in position
\(k\), then at the earlier time before the measurement took place the
particle had a property represented by a projector \(\xi^{jk}\) for
some value of \(j\). If in addition the initial state \({|s\_0\rangle
}\) is known, that can be used to assign a probability \(p\_{jk}\),
note that \(\sum\_j p\_{jk}=1\), to the property \(\xi^{jk}\) at time
\(t\_1\). While not as "clean" as in the case of a projective
measurement, this can still be informative.



Let us close this section with two additional remarks about weak
measurements, including the hybrid case. First, there is an abundant
literature in which weak measurements are discussed in terms of
so-called *weak values* of certain quantum observables. The
physical significance (if any) of weak values has been the subject of
an ongoing controversy. Without entering that discussion it is worth
pointing out that apart from exceptional cases a weak value has
nothing to do with a quantum property in the sense used in the
histories interpretation, related to some subspace of the quantum
Hilbert space; e.g., one of the \(\xi^{jk}\) in .



Second, a single weak measurement corresponding to a weak interaction
between particle and probe will not yield much information about the
particle, and hence it is usually assumed that such a measurement will
be repeated a large number of times, with the particle prepared in the
same initial state for each run, so that useful information can be
extracted from the random outcomes. It is sometimes assumed that
because the interaction is weak, in none of the runs will the state of
the particle change significantly. However, that is not necessarily
the case. Weakness of the interaction is also consistent with the
particle's state being drastically altered in a number of runs,
provided these are but a small fraction of the total. A useful example
is provided by Feynman (Feynman, Leighton,
& Sands: Chapter 1) in his discussion of a weak light
source used to determine which of two holes an electron passes through
on its way to a distant screen where successive arrivals display a
two-hole (double slit) interference pattern. If the light source is
very weak, only for a small fraction of the runs does a scattered
photon reveals which hole the electron went through. But if one
collects data for these rare instances one finds a very large effect:
in these runs the interference pattern is completely washed out.


See the 
 Further Reading supplement for additional
details of the histories approach to generalized and weak
measurements.


### 7.5 Concluding remarks



In summary, the histories approach resolves both quantum measurement problems
M1 and M2 by *introducing appropriate stochastic families of histories*.
In order to describe the measurement outcome as a pointer position one needs an
appropriate projective decomposition of the identity--\(\{[m^j]\}\) in the
simplest model or the more general \(\{M^k\}\) of 
 section 7.4--associated
with the pointer positions. Similarly, in order to discuss how properties of
the microscopic particle before the measurement are related to measurement
outcomes, one needs a decomposition at \(t\_1\) that includes these properties,
the \([s^j]\) in \eqref{eqn23} for the simple model or the \(\xi^{jk}\) in
\eqref{eqn38} for a generalized measurement. There is no fundamental quantum
principle that forces one to use these decompositions. The choice is based on
utility, and illustrates the importance of principle R4 in 
 section 5. If
one wants to discuss measurements *as measurements*, one needs
to adopt a framework in which the associated concepts make sense. And,
since it is often misunderstood, it is worth emphasizing that neither
the validity nor the utility of a particular framework is in any way
reduced by the existence of other frameworks which are incompatible
(or incommensurate) with it.



The models employed here are a bit simplistic compared to the experimental
situation in a laboratory. However, extending them to something more realistic
poses difficulties no worse than those encountered when introducing
quasiclassical descriptions. Thus, rather than a pure initial
state \(\ket{m\_0}\) for the apparatus it is more realistic to use a macroscopic
projector of the type mentioned in 
 section 6, or else an initial density
operator. These changes present no difficulty in principle, and some
of them are discussed in Ch. 17 of Griffiths (2002a). The role of
decoherence is to allow a consistent quasiclassical description, as
noted in
 section 6. By itself decoherence does not
resolve either of the measurement problems; see, e.g., Adler
(2003).


Finally, while wave function collapse is a legitimate calculational
procedure when describing a preparation in quantum mechanical terms,
it amounts to nothing more than a convenient way of calculating
conditional probabilities useful in the quantum description of a
preparation. Trying to view it as some sort of mysterious physical
process leads to confusion and the incorrect notion that quantum
mechanics is somehow inconsistent with special relativity, which is
our next topic.


## 8. Locality and Special Relativity


According to some interpretations of quantum
mechanics, e.g., 
 d'Espagnat (2006),
Maudlin (2011),
the quantum world is infested with nonlocal influences:
systems which are far apart, even spacelike separated in the sense of
special relativity, can influence each other. This has given rise to
the notion that quantum theory is incompatible with special
relativity. No such nonlocal influences are present in the histories
interpretation, which is perfectly compatible with special
relativity. See the 
Further Reading supplement for the detailed discussions
underlying the following remarks.




### 8.1 Singlet state of spin-half particles


The Bohm version (Bohm 1951: Chapter
22) of the Einstein-Podolsky-Rosen
paradox (Einstein, Podolsky, & Rosen
1935) is based on the spin singlet state


\[\ket{\psi\_0} = \bigl( \ket{z^+\_a,z^+\_b} - \ket{z^-\_a,z^-\_b}
\bigr) \label{eqn41}\]


of two spin-half particles \(a\) and \(b\), using the same notation as
in subsection 2.3. This is said to be
an *entangled* state because it is not a product state of the
form \(\ket{a,b} = \ket{a}\otimes\ket{b}\). It is easy to show that the
corresponding projector \([\psi\_0]\) does not commute with any projector
on a nontrivial property of particle \(a\), such as \([z^+\_a]\), or on a
nontrivial property of \(b\). (The trivial properties are the identity
and the zero projector.) Thus a quantum description that contains
\([\psi\_0]\) as a property cannot at the same time ascribe any
nontrivial properties to either of the particles.


This is a mathematical fact which makes no reference to where the
particles are located in space. It is true for the ground state of a
hydrogen atom, in which the electron and proton are on top of each
other, as well as for the situation we consider next in which particle
\(a\) is in Alice's laboratory and \(b\) in Bob's laboratory some distance
away. Suppose that Alice measures \(S\_{az}\) for her particle and Bob
measures \(S\_{bz}\) for his. Then using \([\psi\_0]\) *as a
pre-probability*, subsection 3.5, one
can show that the outcomes, although they are random, will always be
opposite: if Alice obtains \(+1/2\) (in units of \(\hbar\)), Bob will find
\(-1/2\); if Alice obtains \(-1/2\), Bob will find \(+1/2\).


Correlated outcomes are by themselves not surprising once one
accepts the idea that quantum mechanics is stochastic, not
deterministic. Here is a classical analogy: Charlie in Chicago places
a red slip of paper in an opaque envelope and a green slip in a second
identical envelope, and after shuffling them chooses one at random and
mails it to Alice in Atlanta and the other to Bob in Boston. Upon
opening her envelope and finding a red slip, Alice can at once
conclude from her knowledge of the protocol that Bob's envelope
contains a green slip of paper. There is no need to invoke some
superluminal influence to explain this.


Her reasoning is exactly the same in the case where Charlie
prepares a spin singlet state at the center of the laboratory and
sends the two particles in opposite directions towards apparatuses
constructed by Alice and Bob, both competent experimentalists, and
Alice measures \(S\_{az}\). From her measurement outcome and because she
is a competent experimentalist she is entitled to conclude that before
the measurement took place particle \(a\) possessed the value indicated
later by the pointer position; see the discussion
in subsection 7.2. Assume for the sake of
discussion that this was \(S\_{az} = +1/2\). Then from her knowledge of
the protocol (initial singlet state) she can assign a state
\(S\_{bz}=-1/2\) to Bob's particle and use this (as a pre-probability) to
assign a probability to the outcome of a measurement of \(S\_{bz}\). She
employs a scheme of probabilistic inference appropriate to the quantum
world; there is no need to invoke wave function collapse except,
perhaps, as a handy calculational tool.


But suppose Bob rather than measuring \(S\_{bz}\) measures some other
component \(S\_{bw}\) of the spin of particle \(b\), where \(w\) could be \(x\)
or \(y\) or some arbitrary direction in space? Again, Alice on the basis
of her measurement outcome can still assign an \(S\_{bz}\) value to Bob's
particle before measurement, and use it as a pre-probability to
calculate a probability for Bob's measurement outcome. Conversely,
given the outcome of his \(S\_{bw}\) measurement (and knowledge of the
protocol) Bob can assign a state \(S\_{aw}\) to Alice's particle before
it is measured, and use that to predict the outcome of a measurement
of \(S\_{az}\). Both reasoning processes follow the rules indicated above
in subsection 5.1, but when \(w\) is neither \(z\)
nor \(-z\) the single framework rule means they cannot be combined. It
is at this point that quantum principles must be used, unlike the
analysis applicable to colored slips of paper, where one can employ
the single quasiclassical framework (section 6)
appropriate for macroscopic physics.


If Alice measured \(S\_{az}\), then surely she *could* have
instead measured \(S\_{ax}\) perhaps making the choice between this and
\(S\_{az}\) at the very last instant, even after Bob has completed his
measurement; what *would* have occurred in that situation? This
is a counterfactual question of the form discussed
in subsection 5.2, 
and we refer the reader to the references given there.


### 8.2 Bell inequalities and Einstein locality


Quantum theory predicts correlations in the spin-singlet state that
do not satisfy Bell inequalities (Shimony
2009), and by now there is ample evidence from experiments on
the analogous property of pairs of correlated photons that quantum
theory is correct, and therefore one or more of the assumptions that
go into the derivation of a Bell inequality must be faulty. While the
claim has been made that the key assumption is locality, and therefore
a violation of Bell inequalities implies that the real (quantum) world
is nonlocal, a histories analysis identifies the problem as a
different assumption made by Bell: the existence of classical hidden
variables that are inconsistent with Hilbert space quantum
mechanics. For more details see the 
 Further Reading supplement.




In addition, the histories approach makes it possible to establish
on the basis of quantum mechanics itself a principle of *Einstein
locality* (Griffiths 2011b):



> 
> Objectively real internal properties of an isolated individual
> system do not change when something is done to another non-interacting
> system.
> 
> 
> 


This statement is to be understood in the following way. Let
\(\HC\_S\) be the Hilbert spaces of an isolated individual system \(S\),
\(\HC\_R\) that of the rest of the world, so \(\HC=\HC\_S\otimes\HC\_R\) is
the Hilbert space of the total system. The properties of \(S\) must be
represented by a projective decomposition of \(\HC\_S\), or by a
consistent family of such properties. That \(S\) is isolated means there
is no interaction between \(S\) and \(R\), which is to say that for any
two times \(t\) and \(t'\) during the period of interest the time
development operator of the total system factors:


\[T(t',t)=T\_S(t',t)\otimes T\_R(t',t). \label{eqn42}\]


Something done to another system might be, for example, the choice
between measuring \(S\_{bx}\) and \(S\_{bz}\) for a particle \(b\) which is
part of \(R\), with the measuring apparatus and whatever makes the
measurement choice, e.g., a quantum
coin (Sec. 19.2 of Griffiths 2002a) that
is also part of \(R\).


## 9. Quantum Information


Quantum information is at present a very active field of research,
one in which quantum foundational issues continually arise. One reason
for current interest is the hope of building quantum computers that
are capable, assuming they can be made to work properly, of carrying
out certain information-processing tasks faster and more efficiently
than is possible using ordinary classical computers. To be sure, if
the entire world is quantum mechanical then our current
"classical" computers must in some real sense be quantum
computers. So what is the difference? At least to a first
approximation, classical computation is that kind where a single
quasiclassical framework, section 6, suffices
for describing the carriers of information (e.g., on-off states of
transistors) and the way information is processed, whereas quantum
computation is what cannot be so described. Further clarifying this
distinction is one of the tasks of quantum information theory.


Classical information theory as developed by Shannon and his
successors employs standard (Kolmogorov) probability theory. The
histories approach to quantum mechanics also uses standard probability
theory, and thus it is plausible that as long as one sticks to a
single quantum framework, both the mathematical structure and the
intuition associated with classical information theory can be directly
translated into the quantum domain. This is indeed the case, and such
a quantum framework is not limited to quasiclassical states; it can
also include microscopic quantum states.


This makes it possible for the histories approach to answer two
questions raised by J. S. Bell (1990) as
objections to using information theory to address foundational issues
in quantum mechanics: "*Whose* information? Information
about *what*?" The answers in reverse order: information
is always *about* quantum properties or, more generally,
properties at different times, thus histories. And it is
possessed *by* someone who, like Alice
in subsection 8.1, can build a reliable
piece of apparatus and interpret the (macroscopic) outcome by applying
consistent quantum principles.


Not all problems in quantum information can be simply and
immediately mapped in a useful way onto a single framework, and those
that cannot are what make quantum information an interesting, as well
as a rather difficult, field of research. Note that the single
framework rule forbids *combining* incompatible frameworks, but
allows one to compare them or work out relationships that hold between
results in different frameworks. Therefore one can say that, at least
in some general sense, problems at the frontiers of quantum
information theory have to do with comparisons of incompatible
frameworks. See the 
 Further Reading supplement for additional discussions of these
issues.




In summary, the histories approach provides a consistent conceptual
framework for the full scope of quantum information theory, including
classical (Shannon) information as a special case.


## 10. Paradoxes Resolved


The histories approach resolves, or one might say
"tames", all the usual quantum paradoxes in the sense of
providing a clear explanation of why certain results which seem
peculiar from the perspective of classical physics are perfectly
consistent with quantum theory, and can thus serve as useful
illustrations of its basic principles. As the explanations have for
the most part already been published, what is found below is mostly
just a few comments along with references.


### 10.1 Schrodinger's cat


The paradox of Schrodinger (1935)'s
cat is a particularly striking way of stating the unresolved
difficulties associated with the first measurement problem in standard
(i.e., textbook) quantum mechanics. It corresponds to using the
framework \(\FC\_u\), \eqref{eqn21} in subsection 7.2,
with \(\ket{\Psi\_2}\) the superposition of dead and live cat states. The
histories approach addresses this paradox by first noting that there
is nothing wrong from the point of view of fundamental quantum theory
(principle R2 of subsection 5.1) with using
\(\FC\_u\), but it is a serious misunderstanding to suppose that
\(\ket{\Psi\_2}\) or the corresponding property \([\Psi\_2]\) represents a
cat which is both dead and alive. First, the projectors, let us call
them \(P\_d\) and \(P\_a\), that correspond to the cat being dead and alive
do not commute with \([\Psi\_2]\), so they cannot, by the single
framework rule, be brought into the discussion (and, indeed, it is
doubtful that any projector associated with what one might call
"catness" will commute with \([\Psi\_2]\)). Second, since
\(P\_d P\_a=0\) (distinct macroscopic states) 
both properties cannot be true simultaneously, so it is false to say that the cat is
both dead and alive. In addition the historian may
remark that the existence of other frameworks in no way invalidates a
discussion based on the quasiclassical one that lies behind the
ordinary language discussion of cats, which is distinguished by its
utility and not by some framework selection rule,
see subsection 11.3. In this case a useful
framework is the counterpart of \(\FC\_1\), \eqref{eqn22}
in subsection 7.2, with quasiclassical projectors
\(P\_d\) and \(P\_a\) at time \(t\_2\), and \(\ket{\Psi\_2}\) understood not as a
property but as a pre-probability useful for assigning probabilities
that the cat is dead or alive. At this point the paradox has
disappeared.


A similar but somewhat more complex analysis can be applied
to Wigner (1961)'s friend; the solution
is to use an appropriate framework.


### 10.2 Interference paradoxes


The best known of the interference paradoxes is the double
slit. How can successive quantum particles passing through the slit
system and later detected at specific points in the interference zone
build up a pattern showing interference of a sort that depends on the
separation of the slits, even though measurements carried out directly
behind the slits will show that the particle passed through one or the
other, but not both? Feynman's
discussion (Feynman, Leighton, & Sands
1965: Chapter 1, where two holes replace the two slits) is
superb. What is essentially the same paradox, but in a form somewhat
easier to analyze, is the one in which a particle passes through the
two arms of Mach-Zehnder interferometer. Chapter 13
of Griffiths (2002a) provides an
extensive discussion of how the histories approach deals with
different possibilities in a consistent way that avoids paradoxes
while producing analogs of all the interesting effects discussed by
Feynman.


### 10.3 Incompatibility paradoxes


The Bell-Kochen-Specker family of
paradoxes (Bell 1966; Kochen & Specker
1967) are based on a relatively simple idea, although its
execution is sometimes complicated: if one chains together a
sufficient number of incompatible frameworks it is possible to arrive
at a logical contradiction. The chaining consists in finding a
projector which belongs to two (or more) incompatible decompositions
of the identity, and then assuming it has the same truth value in both
frameworks, whereas in any single decomposition one and only one
projector is true. The histories approach blocks paradoxes of this
sort because they (obviously) violate the single framework rule. See
the Further Reading supplement for details.




The three-box paradox of Aharonov &
Vaidman (1991) is similar to Bell-Kochen-Specker, except that
in this case quantum dynamics is involved, and the issue has to do
with incommensurate families of histories (combining them violates the
consistency conditions) rather than incompatible sample spaces. Again,
the histories resolution of the paradox, Sec. 22.5
of Griffiths (2002a), consists in
applying the single framework rule: a contradiction can only be
reached by violating quantum rules of reasoning.


### 10.4 Counterfactual paradoxes


In the histories approach the rule governing counterfactual
reasoning, as indicated in subsection 5.2, is
that a single framework must be used for comparing the actual and
counterfactual world. Counterfactual paradoxes are constructed by
ignoring this rule. One of the best known is
Wheeler's (1978) delayed choice paradox
in which in a Mach-Zehnder interferometer the second beam splitter is
or is not removed at a time when the quantum particle (or wave) has
already passed the first beam splitter, so the particle is already
physically present inside the interferometer. With the second beam
splitter absent one can determine the arm through which the particle
passed, whereas the interference exhibited when the beam splitter is
present shows that the particle cannot have been in a particular
arm. 

There are several other paradoxes which hinge upon the
(incorrect) use of counterfactuals, or at least include a
counterfactual element in the way they are stated. Among these
are Hardy (1992)'s paradox

and the somewhat similar Greenberger-Horne-Zeilinger
paradox (Greenberger, Horne, Shimony, &
Zeilinger 1990; Greenberger, Horne, & Zeilinger 1989). 
For comments on how the histories approach deals with these and 
similar paradoxes, see the 
 Further Reading supplement.




### 10.5 Nonlocality paradoxes


As discussed in section 8, the Bohm (spin
singlet) version of the Einstein-Podolsky-Rosen paradox has been
interpreted, by Bell and others, as indicating that quantum theory is
nonlocal. Similar conclusions have been drawn from the Hardy and the
Greenberger-Horne-Zeilinger paradoxes stated in a form which involves
measurements at distinct physical locations. One can add to this list
the Elitzur-Vaidman indirect measurement
paradox (Elitzur & Vaidman 1993),
which again suggests some form of nonlocal influence. When critically
examined using the tools provided by the histories interpretation all
evidence for nonlocality vanishes, for reasons discussed in
subsection 8.2.




## 11. Difficulties and Objections


Any physical theory can give rise to various kinds of conceptual
difficulty. Problems may arise because of lack of familiarity with, or
misunderstanding of, new ideas. Most students upon their first
encounter with calculus or with special relativity have difficulties
of this sort, which can be cleared up by additional study, working
through examples, and the like. In addition theories may be internally
inconsistent. This is often the case when they are relatively new and
under development, but can also be true of more mature
theories. Indeed certain theories turn out to be of great usefulness
despite the fact that they contain serious flaws or
inconsistencies. (Standard, which is to say textbook, quantum
mechanics is a prime example!) And even theories which are properly
understood and are internally consistent can be rejected, either by
individual scientists or the community as a whole, because they are
not "good physics", a term whose importance should not be
underestimated even if it cannot be given a precise definition. The
histories interpretation of quantum theory has given rise to
conceptual difficulties that fall in all of these areas, and while it
is impossible in a few words to discuss each of them in detail, the
following remarks are intended to indicate the general nature of some
of the more serious issues raised by critics, along with responses
made by advocates of the histories approach. 




### 11.1 Internal consistency


The issue of internal consistency has been raised
by Kent (1997) and
by Bassi & Ghirardi (1999, 2000) on
the grounds that the rules for reasoning using histories, summarized
in section 5, lead to contradictions: certain
assumptions allow one to derive both a proposition and its
contradiction. Equivalently, a quantum property can be assigned, on
the basis of the same initial data, probabilities of both 0 and
1. These criticisms have been answered
in Griffiths & Hartle (1998) and in 
Griffiths (2000a, 2000b).
In addition there is a general argument for the
consistency of the histories approach in Ch. 16
of Griffiths (2002a). The key point,
easily overlooked as it has no close analogy in classical physics, is
the single framework rule R3 of section 5. In
situations that involve quantum properties at a single time the use of
a single framework means the use of a particular projective
decomposition of the identity and the associated Boolean event
algebra. When reasoning is restricted to this algebra the usual rules
of propositional logic apply, so an inconsistency at this level would
imply the inconsistency of standard logic. If the framework in
question consists of a family of histories of a closed system, it
cannot be assigned probabilities using the extended Born
rule, subsection 4.2, unless the consistency
conditions are satisfied. But in those cases where probabilities can
be assigned they satisfy the usual rules of probabilistic reasoning,
thus ruling out contradictions.


### 11.2 Single framework rule


The single framework rule itself lies at the center of a large
number of objections to the histories approach, which is hardly
surprising since in an important sense it marks the boundary between
classical and quantum physics. As it is absent from classical physics
and from most quantum textbooks, the concept is unfamiliar, and one
has to work through various examples in order to gain an intuitive as
well as a formal understanding of how it deals with the various
paradoxes and other conceptual difficulties of quantum theory. The
four principles R1 to R4 found in subsection 5.1
should help in understanding how frameworks should be thought of and
how they should be used. It is of particular importance that the
single framework rule does *not* imply that there is only one
way available to the physicist to think about or discuss a particular
situation. What it forbids is not the construction of many, perhaps
mutually incompatible, frameworks, but *combining* them in the
fashion characteristic of the incompatibility paradoxes discussed
above in subsection 10.3. Indeed, the single
framework rule rather than being a restriction actually extends the
Liberty of the theoretical physicist to describe the quantum world in
a consistent way without the danger of running into insoluble
paradoxes. But Liberty is coupled to a denial of the unicity of the
world, subsection 11.4, which is perhaps the most
important philosophical implication of quantum mechanics from the
histories perspective, as well as the principal stumbling block in the
way of a wider acceptance of that approach as a satisfactory
interpretation of quantum theory.


### 11.3 Nonuniqueness of quasiclassical frameworks


An objection first raised by Dowker &
Kent (1996), and subsequently repeated by 
Kent (1998),
Schlosshauer (2004),
and Wallace (2008), among others, is that
quasiclassical frameworks are by no means the only possible consistent families
of histories in a closed quantum system. In particular, one can have
consistent families in which there is quasiclassical behavior up to some time
followed by something which is not at all quasiclassical at a later time. This
would be a serious objection if historians were claiming that there must be a
unique family of histories which can be used to describe the world, but such is
not the case. The physicist is at Liberty to choose a family whose histories
closely resemble those of classical mechanics, and the existence of alternative
frameworks in no way diminishes the validity of such a quasiclassical
description, for reasons indicated in section 4.2 of Griffiths (2013).
One senses that underlying such criticisms is the expectation that any good
physical theory must satisfy the principle of unicity, to which
we now turn.


### 11.4 Unicity


In a probabilistic theory the limiting cases of a probability equal
to 1 or 0 are equivalent to asserting that the corresponding
proposition (e.g., "the system has property \(P\)") is,
respectively, true or false. In the histories approach probabilities
are linked to frameworks, and for this reason the notions of
"true" and "false" are also framework
dependent. This cannot lead to inconsistencies, a proposition being
both true and false, because of the single framework rule. But it is
contrary to a deeply rooted faith or intuition, shared by
philosophers, physicists, and the proverbial man in the street, that
at any point in time there is one and only one state of the universe
which is "true", and with which every true statement about
the world must be consistent. In Sec. 27.3
of Griffiths (2002a) this belief is
referred to as *unicity*, and it is what must be abandoned if
the histories interpretation of quantum theory is on the right
track.


As an example, consider the measurement model
in subsection 7.2, where in the family \(\FC\_u\) at
time \(t\_2\) the measurement apparatus is in a superposition state
corresponding to a projector which is incompatible with the projectors
representing different pointer positions, which are employed in the
family \(\FC\_1\). The histories approach says the physicist is at
Liberty to employ either family, both of which are valid ways of
describing the quantum world, but Incompatibility forbids combining
them into a single description. The prohibition on combining families
is based on the fact that certain quantum projectors do not commute,
something quite specific to quantum mechanics with no classical
counterpart. In classical mechanics the true state of affairs for a
physical system is always represented by a single point in the phase
space. It is this feature that has to be abandoned in quantum
mechanics interpreted using histories. See the 
Further Reading supplement for additional
details.




Abandoning unicity is certainly a radical proposal, comparable in
the history of science to the radical step our intellectual ancestors
took when they replaced the centuries old notion of an immovable earth
with the modern concept of a spinning planet in motion around the
sun. In making that transition it was important to develop an
understanding of why the earlier ideas worked so well; e.g., why one
did not feel the earth was moving. By analogy it is helpful to note,
as discussed in section 6, that from a quantum
perspective the macroscopic world of everyday experience can be
understood for all, or almost all, practical purposes using a single
quasiclassical framework. Within that framework the principle of
unicity holds, and one can thus understand why it was quite
satisfactory before the quantum revolution, and remains so in sciences
where specific quantum effects do not need to be taken into
account. See the 
 Further Reading supplement for a more
extended discussion.