## 1. History



From the birth of the theory of quantum mechanics in 1925/6 to the
outbreak of war in Europe, a clear orthodoxy emerged in the conceptual
and ontological framework for understanding quantum theory. Now known
as the Copenhagen interpretation, this framework embodied the
positivistic tendencies of Heisenberg and Bohr, and was set opposed to
the more realist tendencies of de Broglie, Einstein, and
Schrodinger. It was not until Bell's theorem in the 1960s,
and its experimental tests in the 1970s and 1980s, that new energy was
breathed into this interpretational debate. However, beginning in the
mid-1940s, the first suggestions of retrocausality as part of the
conceptual and ontological framework in quantum theory had already
materialized.



There are two key ideas that punctuate the historical development of
the notion of retrocausality in quantum mechanics. The first proposal
of retroactive influence in quantum mechanics comes from a suggestion
made by Wheeler and Feynman (1945, 1949). They were led to this idea
while considering the potentially classical origins of some of the
difficulties of quantum theory. Consider the following problem from
classical electrodynamics: an accelerating electron emits
electromagnetic radiation and, through this process, the acceleration
of the electron is damped. Various attempts to account for this
phenomenon in terms of the classical theory of electrodynamics lacked
either empirical adequacy or a coherent physical interpretation.
Wheeler and Feynman attempted to remedy this situation by
reinterpreting Dirac's (1938) theory of radiating electrons.



The core of Wheeler and Feynman's proposed "absorber
theory of radiation" is a suggestion that the process of
electromagnetic radiation emission and absorption should be thought of
as an interaction between a source and an absorber rather than as an
independent elementary process. (This idea has its roots as far back
as Tetrode 1922 and G. Lewis 1926.) Wheeler and Feynman imagine an
accelerated point charge located within an absorbing system and
consider the nature of the electromagnetic field associated with the
acceleration. An electromagnetic disturbance can be imagined
"initially" to travel outwards from the source to perturb
each particle of the absorber. The particles of the absorber then
generate together a subsequent field. According to the Wheeler-Feynman
view, this new field is comprised of half the sum of the retarded
(forward-in-time) and advanced (backward-in-time) solutions to
Maxwell's equations. The sum of the advanced effects of all the
particles of the absorber then yields an advanced incoming field that
is present at the source simultaneous with the moment of emission
(although see
 SS5
 for more on how one should understand this "process").
The claim is that this advanced field exerts a finite force on the
source which has exactly the required magnitude and direction to
account for the observed energy transferred from source to absorber;
this is Dirac's radiative damping field. In addition, when this
advanced field is combined with the equivalent half-retarded,
half-advanced field of the source, the total observed disturbance is
the full retarded field known empirically to be emitted by accelerated
point charges.



The crucial point to note about the Wheeler-Feynman schema is that due
to the *advanced* field of the absorber, the radiative damping
field is present at the source at exactly the time of the initial
acceleration. This schema of advanced and retarded waves now forms the
basis for the most fully-formed retrocausal model of quantum
mechanics, the transactional interpretation (see
 SS5).



The second key idea in the historical development of retrocausality in
quantum mechanics occurs around the same time as Wheeler and
Feynman's absorber theory. French physicist Costa de Beauregard,
a student of de Broglie, noticed a potential objection to the
reasoning found in Einstein, Podolsky, and Rosen's famous paper
(1935) on the completeness of quantum mechanics (see the entry on
 the Einstein-Podolsky-Rosen argument in quantum theory),
 now widely known as the EPR argument. Einstein et al. argue that
quantum mechanics must be incomplete on the basis of the following
assumption: no reasonable definition of reality could be expected to
permit the reality of some system being dependent upon the process of
measurement carried out on some other distant system which does not in
any way disturb the first system. The key condition that Einstein et
al. suggest any reasonable definition of reality should satisfy is
that "there is no longer any interaction between the two
parts". (Later, Einstein 1948 accounted for this feature with an
explicit assumption of "locality", see
 SS2.2).
 Costa de Beauregard, however, was alert to a particular kind of
unorthodox alternative to this condition which upended its role in the
EPR argument. His proposal was that two distant systems could
"remain correlated by means of a successively advanced and
retarded wave" (Costa de Beauregard 1953: 1634); that is, one
system could influence, via an advanced wave, the state of the
combined systems in their common past, which then, via a retarded
wave, could influence the state of the distant system in a kind of
"zigzag" through spacetime. This way, there could be a
dependence between the two distant systems without any "spooky
action-at-a-distance". Thus, as Costa de Beauregard (1987b: 252)
puts it, 



> 
> 
> Einstein of course is right in seeing an incompatibility between his
> special relativity theory and the distant quantal correlations, but
> *only under the assumption that advanced actions are
> excluded*.
> 
> 
> 



When Costa de Beauregard in 1947 suggested this response to the EPR
argument to his then supervisor de Broglie, de Broglie was "far
from willing to accept" the proposal (1987b: 252) and forbade
Costa de Beauregard to publish his unorthodox idea (Price &
Wharton 2015). However, in 1948 Feynman had developed his eponymous
diagrams in which antiparticles were to be interpreted as particles
moving backward-in-time along the particle trajectories, and so by
1953 de Broglie had endorsed the publication of Costa de
Beauregard's response. On the seeming craziness of the proposal,
Costa de Beauregard claims, "[t]oday, as the phenomenon of the
EPR correlations is very well validated experimentally, and is in
itself a 'crazy phenomenon', any explanation of it must be
'crazy'" (1987b: 252; see also Costa de Beauregard
1976, 1977b, 1987a).


## 2. Metaphysical Preliminaries



Before addressing two of the main motivations for adopting the
hypothesis of retrocausality in the foundations of quantum theory, it
will be worthwhile to provide a few comments on two key notions that
play a significant role in the following discussion: causality and
locality. This will help to pin down what exactly is meant, and not
meant, by retrocausality.


### 2.1 Causality



There is a tradition that stretches back at least as far as Russell
(1913) that denies that there is any place for causal notions in the
fundamental sciences, including physics: the notion serves no purpose,
and simply does not appear, in the fundamental sciences. The argument
goes that, since at least the nineteenth century, the laws that govern
physical behavior in fundamental sciences such as physics are almost
always differential equations. Such equations are notable for
specifying, given some initial conditions, exact properties of systems
for all time. And thus if everything is specified for all time, there
is no place left for causality. Thus Russell advocates that
"causality" should be eliminated from the philosophers
lexicon, because it is certainly not a part of the scientific
lexicon.



In contrast to Russell's position, Cartwright (1979: 420) claims
that we do have a need and use for a causal vocabulary in science:
"causal laws cannot be done away with, for they are needed to
ground the distinction between effective strategies and ineffective
ones". One of the main contemporary accounts of causation, the
interventionist account of causation (Woodward 2003; see also the
entry on
 causation and manipulability),
 is an embodiment of Cartwright's dictum. In a nutshell, the
interventionist account claims that *A* is a cause of *B* if
and only if manipulating *A* is an effective means of
(indirectly) manipulating *B*. Causality in the present entry,
unless specified otherwise, should be understood along broadly
interventionist lines. According to accounts of quantum theory that
hypothesize retrocausality, manipulating the setting of a measurement
apparatus can be an effective means of manipulating aspects of the
past. A broadly interventionist view of causality indeed underlies
most contemporary attempts to harness the tool kit of causal modeling
(see the entry on
 causal models;
 Spirtes, Glymour, & Scheines 2000; Pearl 2009) in the foundations
of quantum theory (Leifer & Spekkens 2013; Cavalcanti & Lal
2014; Costa & Shrapnel 2016; Allen et al. 2017).



Using the notion of causality along broadly interventionist lines in
the foundations of quantum theory does not commit one to realism (or
anti-realism) about the causal relations at issue. Woodward combines
interventionism with realism about causality while acknowledging 



> 
> 
> important differences between, on the one hand, the way in which
> causal notions figure in common sense and the special sciences and the
> empirical assumptions that underlie their application and, on the
> other hand, the ways in which these notions figure in physics.
> (Woodward 2007: 67; although see Frisch 2014: chs. 4 and 5 for a
> response) 
> 
> 
> 



Another suggested strategy to take into account Russell's worry
while continuing to apply causal notions in physics in a consistent
manner is to understand interventionism in "perspectival"
terms (Price 2007; Price & Corry 2007; Price & Weslake 2010;
Ismael 2016). Perspectivalism is usually staged, as seems natural in
the setting of modern physics (although more will be said on this
below), in the framework of a block universe view where the past,
present, and future are equally real. In this framework, causality
cannot have anything to do with *changing* the future or the
past because both are--from an "external"
perspective--completely "fixed". But one can
understand causation in the block universe from an
"internal" perspective, according to which causal
correlations are precisely those that are stable under interventions
on those variables that we refer to as the "causes".



The important difference between the two viewpoints--internal and
external to the block--is that there is a discrepancy between the
parts of the spacetime block that are epistemically accessible from
each perspective. The spatiotemporally constrained perspective by
which we are bound permits us only limited epistemic accessibility to
other spatiotemporal regions. This is the perspective in which,
according to causal perspectivalism, causal notions are perfectly
serviceable. Once, on the other hand, we imagine ourselves to be
omniscient beings that have epistemic access to the whole
spatiotemporal block, it should not come as a surprise that our causal
intuitions get confused when we attempt to consider how a
spatiotemporally bound agent can deliberate about whether or not to
affect a particular event that is already determined from our imagined
omniscient perspective. It is because we do not *know* which
events are determined to occur and are ignorant about many others that
we can be deliberative agents at all. Again, these considerations are
relevant just as much to ordinary forward-in-time causation as they
are to backward-in-time causation.



Many of the retrocausal approaches to quantum theory considered in
 SS6
 are best understood with some type of perspectival interventionist
account of causality in mind. A notable exception is the transactional
interpretation
 (SS5),
 in which causality might be best understood in terms of processes
underscored by conserved quantities. The possibilist extension of the
transactional interpretation, defended by Kastner (2006, 2013),
moreover eschews the block universe picture.


### 2.2 Locality



According to Bell's theorem (Bell 1964; Clauser et al. 1969; see
also the entry on
 Bell's theorem)
 and its descendants (e.g., Greenberger, Horne, & Zeilinger 1989;
see also the entry on
 Bell's theorem, SS6;
 Goldstein et al. 2011; Brunner et al. 2014 for overviews), any theory
that reproduces all the correlations of measurement outcomes predicted
by quantum theory must violate a principle that Bell calls *local
causality* (Bell 1976, 1990; see also Norsen 2011; Wiseman &
Cavalcanti 2017). In a locally causal theory, probabilities of
spatiotemporally localized events occurring in some region 1 are
independent of what occurs in a region 2 that is spacelike separated
from region 1, given a complete specification of what occurs in a
spacetime region 3 in region 1's backward light cone that
completely shields off region 1 from the backward light cone of region
2. (See, for instance, Figs. 4 and 6 in Bell 1990 or Fig. 2 in the
entry on
 Bell's theorem.)



In a relativistic setting, then, the notion of locality involves
prohibiting conditional dependences between spacelike separated
events, provided that the region upon which these spacelike separated
events are conditioned constitutes their common causal (Minkowski)
past. This characterization of locality explicitly assumes causal
asymmetry. Thus locality is the idea that there are no causal
relations between spacelike separated events.



There is another sense of "local" that is worth explicitly
noting for the purposes of avoiding ambiguity. This is the idea that
causal influences are mediated continuously along timelike
trajectories. Thus, given Costa de Beauregard's suggestion of
"zigzag" causal influences, it is perfectly possible for a
retrocausal model of quantum phenomena to be nonlocal in the sense
that causal relations exist between spacelike separated events, but
"local" in the sense that these causal influences are
constrained to timelike trajectories. For clarity, this latter notion
is best understood as set opposed to action-at-a-distance, and is
variously delineated as "action-by-contact" (Evans, Price,
& Wharton 2013) or "continuous action" (Wharton &
Argaman 2020; Adlam 2022).


## 3. Motivation I: Exploiting Loopholes in the "No-Go Theorems"



The first of two main motivating considerations for invoking
retrocausality in the foundations of quantum mechanics derives from
the exploitation of what is essentially the same loophole in a range
of theorems collectively known as "no-go theorems".
According to these theorems, any theory or model that is able to
account for the empirically confirmed consequences of quantum theory
must be unavoidably *nonlocal*, *contextual*, and
*\(\psi\)-ontic* (i.e., ascribe reality to the quantum states
\(\psi\)).



One way to understand the role that retrocausality plays in
circumventing the results of the no-go theorems is to consider each
theorem to be underpinned by what is known as the ontological models
framework (Harrigan & Spekkens 2010; Leifer 2014; Ringbauer 2017).
This is a general framework that can be applied to a wide variety of
"realist" models, including theories of classical physics
as well as local hidden variable approaches to quantum mechanics. The
framework consists of an operational description of a
"process" which describes observed statistics for outcomes
of measurements given both preparations and transformations, along
with an ontological model (or "ontic extension")
accounting for the observed statistics. In the quantum context, when a
preparation procedures corresponds to a quantum state \(\psi\), a
quantum system subjected to this procedure actually ends up in an
"ontic" state \(\lambda\), chosen from a set of states
\(\Lambda\), which completely specifies the system's properties.
The framework leaves open (and is ultimately used to define) whether
the quantum state \(\psi\) is itself an ontic or epistemic state (if
\(\psi\) is ontic, \(\lambda\) either includes additional ontic
degrees of freedom, or is in one-to-one correspondence with \(\psi\);
see
 SS3.3).
 Each preparation is assumed to result in some \(\lambda\) via a
classical probability density over \(\Lambda\), and a set of
measurement procedures that determine conditional probabilities for
outcomes dependent upon \(\lambda\) (which thus screens off the
preparation procedure \(\psi\)); explicitly, \(\lambda\) does not
causally depend on any future measurement setting \(\alpha\). Finally,
the operational statistics must reproduce the quantum statistics.



Important for our purposes here is the qualification that, in the
ontological models framework, \(\lambda\) is assumed to be
"conditionally independent" of \(\alpha\)
("measurement independence"). This assumption effectively
rules out, among other things, that the future measurement setting
\(\alpha\) causally influences the earlier state \(\lambda\). Thus, in
so far as the no-go theorems are underpinned by the ontological models
framework, the no-go theorems are not applicable to models that do
allow retrocausality from \(\alpha\) to \(\lambda\). And in so far as
there is motivation to avoid the consequences of the no-go theorems
for quantum theory, retrocausality is well placed to provide such a
model (or so the argument goes). Notably, it has been argued that
admitting retrocausality (i) makes it possible to account for the
correlations entailed by quantum theory using action-by-contact causal
influences (and so ensures Lorentz invariance); (ii) undermines as
implausible the assumption of (certain types of) noncontextuality from
the outset; and (iii) may enable an independently attractive
\(\psi\)-epistemic interpretation of the wavefunction underpinned by
local hidden variables. These arguments are addressed in turn.


### 3.1 Nonlocality Theorems



The principle of local causality, according to Bell, is meant to spell
out the idea that 



> 
> 
> [t]he direct causes (and effects) of events are near by, and even the
> indirect causes (and effects) are no further away than permitted by
> the velocity of light. (1990: 105) 
> 
> 
> 



Violation of this principle, according to some researchers in the
foundations of quantum theory, indicates a fundamental incompatibility
between quantum theory and the spirit, perhaps even the letter, of
relativity theory (Maudlin 2011). That the correlations entailed by
quantum theory which violate local causality actually occur in nature
has been experimentally documented many times (for example, by
Freedman & Clauser 1972; Aspect, Dalibard, & Roger 1982; and
Aspect, Grangier, & Roger 1982; see the entry on
 Bell's theorem,
 SS4 for an overview on early experiments).



Bell's result crucially depends not only on the assumption of
local causality, but also on the assumption that whatever variables
\(\lambda\) describes in some spacetime region, which Bell calls
"local beables", do not depend probabilistically on which
measurement setting \(\alpha\) some experimenters choose in the future
of that region: 

 
\[\tag{1}
P(\lambda \mid \alpha) = P(\lambda)\,. \label{eq:independence}
\]


This is the aforementioned assumption of measurement independence. It
is also sometimes referred to as "no superdeterminism"
because it is incompatible with a particularly strong form of
determinism ("superdeterminism") according to which the
joint past of the measurement setting \(\alpha\) and the measured
system state \(\lambda\) determines them both completely and induces a
correlation between them. But, as pointed out in the first instance by
Costa de Beauregard (1977a) and then by Price (1994, 1996),
superdeterminism is not the only way in which Eq.
(\ref{eq:independence}) can be violated: if there is retrocausality
(understood along interventionist lines as outlined in
 SS2.1),
 the choice of measurement setting \(\alpha\) may causally influence
the physical state \(\lambda\) at an earlier time and thereby also
render Eq. (\ref{eq:independence}) incorrect. In this picture, as
pointed out by Price and Wharton (2021), it might be possible to
interpret the local causality-violating correlations between distant
events as reflecting a type of retrocausality-induced "collider
bias". 



Without Eq. (\ref{eq:independence}), in turn, Bell's theorem can
no longer be derived. Thus, admitting the possibility of
retrocausality in principle reopens the possibility of giving a causal
account of the nonlocal correlations entailed by quantum theory as
mediated by purely action-by-contact, spatiotemporally contiguous,
Lorentz invariant causal influences (of the type envisaged by Costa de
Beauregard) acting between systems described by local beables.
Concrete steps towards a model that might fulfill this promise will be
reviewed in
 SS6.



Contrary to this narrative, Adlam (2022) argues that the attempt to
rescue *continuous-action*-type locality (of the sort discussed
in
 SS2.2)
 is not a good motivation for retrocausality. According to Adlam, to
the extent that the independently most attractive picture of causality
is an interventionist one in combination with a block universe view,
there is no principled reason for assuming that causal influences are
confined to continuous action. Adlam provides her own motivation for
retrocausality, based on an argument that "accepting the
existence of genuine, unmediated nonlocality in and of itself leads us
to accept retrocausality" (2022: 422).


### 3.2 Contextuality Theorems



A wealth of theoretical results establish that models and theories
which reproduce the predictions of quantum theory must be in some
sense "contextual". A model or theory is
*noncontextual* when the properties attributed to the system
(e.g., the value of some dynamical variable that the system is found
to have) are independent of the *manner* in which those
properties are measured or observed (the context), beyond the actual
observation itself (for instance, the other properties that are
measured in conjunction with the measurement). In classical mechanics,
the properties that systems are found to have and that are attributed
to them as a consequence of measurement do not in principle depend on
the measurement context, so classical mechanics is noncontextual. The
first contextuality theorems for quantum theory go back to Bell (1966)
and Kochen and Specker (1967). They demonstrate that if one were to
consider quantum measurements as deterministically uncovering the
values of arbitrary pre-existing dynamical variables of quantum
systems, then such a model would have to be contextual (that is,
measured values would have to depend upon the context of measurement).
Thus, no noncontextual deterministic local hidden variable model can
reproduce the observed statistics of quantum theory.



A new way of understanding contextuality operationally was pioneered
by Spekkens (2005). Employing the ontological models framework, an
ontological model of an operational theory is *noncontextual*
when operationally equivalent experimental procedures have equivalent
representations in the ontological model. This understanding of
noncontextuality is then a principle of parsimony akin to
Leibniz' law of Identity of Indiscernibles: no ontological
difference without operational difference. Using this operational
understanding, Spekkens expands the class of ontological models to
which contextuality applies beyond deterministic local hidden variable
models. But the story is essentially the same: hypothesizing
underlying ontic properties for quantum systems that are distinct only
if there are corresponding operational differences cannot account for
the correlations entailed by quantum theory. Thus, no noncontextual
ontological model can reproduce the observed statistics of quantum
theory. (For an extended criticism of Spekkens' operational
understanding of contextuality see Hermens 2011.)



Hypothesizing retrocausal influences alleviates both of these worries.
Retrocausality renders Kochen-Specker-type contextuality potentially
explainable as a form of "causal contextuality" (see the
entry on
 the Kochen-Specker theorem,
 SS5.3). If there is a backward-directed influence of the chosen
measurement setting (and context) on the pre-measurement ontic state,
it is no longer to be expected that the measurement process is simply
uncovering an independently existing definite value for some property
of the system, rather the measurement process can play a causal role
in *bringing about* such values (the measurement process is
retro*causal* rather than retro*dictive*). Indeed, one
might argue contextuality of measured values is just what one might
expect when admitting retrocausal influences. As Wharton (2014: 203)
puts it, "Kochen-Specker contextuality *is* the failure
of the Independence Assumption", i.e., the failure of
measurement independence.



With respect to Spekkens' more general understanding of
contextuality, recall that it is an explicit assumption of the
ontological models framework that the ontic state is independent of
the measurement procedure. Thus, hypothesizing retrocausality is an
unequivocal rejection of the ontological models framework. But one
might wonder whether Spekkens' principle of parsimony might be
recast to apply more generally to retrocausal models.
 SS7.4
 considers a result due to Shrapnel and Costa (2018) that indicates
that retrocausal accounts must indeed accept Spekkens-type
contextuality.


### 3.3 Psi-ontology Theorems



In some models staged in the ontological models framework, an ontic
state \(\lambda\) is only compatible with preparation procedures
associated with one specific quantum state \(\psi\). In that case,
\(\psi\) can be seen as part of \(\lambda\) and, therefore, as an
element of reality itself. Models of this type are referred to as
"\(\psi\)-ontic". In other models, an ontic state
\(\lambda\) can be the result of preparation procedures associated
with different quantum states \(\psi\). In these models, \(\psi\) is
not an aspect of reality and is often more naturally seen as
reflecting an agent's incomplete knowledge about the underlying
ontic state \(\lambda\). Models of this type are referred to as
"\(\psi\)-epistemic". Spekkens (2007) motivates the search
for attractive \(\psi\)-epistemic models by pointing out various
parallels between quantum mechanics and a toy model in which the
analogue of the quantum state is epistemic in that it reflects
incomplete information about an underlying ontic state.



However, so-called "\(\psi\)-ontology theorems" (Pusey,
Barrett, & Rudolph 2012; Hardy 2013; Colbeck & Renner 2012;
see Leifer 2014 for an overview), establish that, given certain
plausible assumptions, only \(\psi\)-ontic models can reproduce the
empirical consequences of quantum theory. The specific assumptions
used to derive this conclusion differ for the various theorems. All of
them, however, rely on the ontological models framework which, as
noted a number of times already, explicitly incorporates an assumption
of measurement independence (Eq. (\ref{eq:independence})). Since, as
shown above, measurement independence implicitly rules out
retrocausality, these theorems apply only inasmuch as retrocausality
is assumed to be absent.



Moreover, the theorem due to Pusey, Barrett, and Rudolph
(2012)--which according to Leifer (2014) uses the least
contestable assumptions of all--relies on an additional
independence assumption according to which the ontic states of two or
more distinct, randomly prepared, systems are uncorrelated before
undergoing joint measurement. Wharton (2014: 203) points out that in a
framework that is open to retrocausality this is a problematic
assumption because one would then expect "past correlations [to]
arise [...] for exactly the same reason that future correlations
arise in entanglement experiments". Exploring the prospects for
retrocausal accounts is therefore one--perhaps
*the*--important open route for developing an attractive
\(\psi\)-epistemic model that may be able to explain the quantum
correlations.


### 3.4 Classical Ontology?



It is worth pausing at this point to consider the metaphysical
motivation for taking a retrocausal approach to quantum theory,
especially in light of circumventing these no-go theorems. The
orthodox reading of the no-go theorems is that, whatever is said about
the ultimate conceptual and ontological framework for understanding
quantum theory, it cannot be completely classical: it must be nonlocal
and/or contextual and/or ascribe reality to indeterminate states. In
short, quantum theory cannot be about local hidden variables. Part of
the appeal of hypothesizing retrocausality in the face of these no-go
theorems is to regain (either partial or complete) classicality in
these senses (albeit, with--perhaps nonclassical--symmetric
causal influences). That is, retrocausality holds the potential to
allow a metaphysical framework for quantum mechanics that contains
causally action-by-contact, noncontextual (or where any contextuality
is underpinned by noncontextual epistemic constraints),
counterfactually definite, determinate (although possibly
indeterministic), spatiotemporally located properties for physical
systems--in other words, a classical ontology.



A nice to way to consider this is in terms of Quine's (1951)
distinction between the "ontology" and the
"ideology" of a scientific theory, where the ideology of a
theory consists of the ideas that can be expressed in that theory.
Ideological economy is then a measure of the economy of primitive
undefined statements employed to reproduce this ideology. The claim
here would then be that the ideology of symmetric causal influences is
more economical than rejecting a classical ontology. Thus in so far as
quantum mechanics is telling us something profound about the
fundamental nature of reality, the hypothesis of retrocausality shows
that the lesson of quantum mechanics is not necessarily that the
quantum ontology can no longer be classical. Some might see this as a
virtue of retrocausal approaches. (Although the Shrapnel-Costa no-go
theorem reviewed in
 SS7.4
 significantly jeopardizes this view.)


## 4. Motivation II: Time-Symmetry


### 4.1 Deriving Retrocausality from Time-Symmetry



The laws of nature at the most fundamental level at which they are
currently known are combined in the Standard Model of elementary
particle physics. These laws are CPT-invariant, i.e., they remain the
same under the combined operations of charge-reversal *C*
(replacing all particles by their anti-particles), parity *P*
(flipping the signs of all spatial coordinates), and time-reversal
*T*. The asymmetries in time which are pervasive in our everyday
lives are a consequence not of any temporal asymmetry in these laws
but, instead, of the boundary conditions of the universe, notably in
its very early stages. It seems natural to assume that the
time-symmetry of the laws (modulo the combined operation of *C*
and *P*) extends to causal dependences at the fundamental
"ontic" level that underlies the empirical success of
quantum theory. If so, there may be backward-in-time no less than
forward-in-time causal influences at that ontic level.



Price (2012) turns these sketchy considerations into a rigorous
argument. He shows that, when combined with two assumptions concerning
quantum ontology, time-symmetry implies retrocausality (understood
along broadly interventionist lines). The ontological assumptions are
(i) that at least some aspects of the quantum state \(\psi\) are real
(notably, in Price's example, there is a "beable"
encoding photon polarization angle), and (ii) that inputs and outputs
of quantum experiments are *discrete* emission and detection
events. Moreover, it is important to Price's argument that
dynamical time-symmetry (that the dynamical laws of the theory are
time-symmetric) be understood as implying that operational
time-symmetry (that the set of all possible combinations of
preparation and measurement procedures in a theory, with associated
probabilities for outputs given inputs, is closed under interchange of
preparation and measurement) translates into ontic time-symmetry
(operational time-symmetry plus a suitable map between the ontic state
spaces of the symmetric combinations). Given these conditions, any
foundational account that reproduces the empirical verdicts of quantum
theory must be retrocausal.



Leifer and Pusey (2017) (see also Leifer 2017 in
 Other Internet Resources)
 strengthen Price's argument by showing that his assumption
about the reality of (aspects of) the quantum state \(\psi\) can be
relaxed. As they demonstrate, if measurement outcomes depend only on a
system's ontic state \(\lambda\), i.e., if that state completely
mediates any correlations between preparation procedures and
measurement outcomes ("\(\lambda\)-mediation"), this
suffices for operational time-symmetry to entail the existence of
retrocausality. Foundational accounts which like Bohmian mechanics
(Bohm 1952a,b) or GRW-theory (Ghirardi, Rimini, & Weber 1986)
avoid postulating retrocausality do so by violating time-symmetry in
some way. The GRW-theory does so by introducing explicitly
time-asymmetric dynamics. In Bohmian mechanics the dynamics is
time-symmetric, but the theory is applied in a time-asymmetric manner
when assessing which quantum states are actually realized. Notably,
one assumes that the quantum states of a measured system and a
measurement device connected to it are uncorrelated prior to
measurement, whereas they are in general correlated after measurement
(Leifer & Pusey 2017: SSX.).


### 4.2 The Action Duality Principle



The more advanced relativistic quantum theories that underlie modern
particle physics are typically formulated in the Lagrangian, path
integral-based, formalism, where information about the symmetries and
interactions of the theory are encoded in the action, *S*.
Wharton, Miller, and Price (2011) use this formalism as a foundation
to suggest that action symmetries must be reflected as ontological
symmetries. In particular, Wharton et al. claim that for any two
experimental arrangements related by a spacetime transformation (say,
reflection in time) that leaves the action unchanged, the ontologies
must also remain unchanged across the same spacetime transformation.
This principle, referred to by Wharton et al. as *action
duality*, imposes non-trivial constraints on realist accounts.



Notably, Evans, Price, & Wharton (2013) argue that there is such
an action symmetry between a pair of entangled photons passing through
separate polarizers and a single photon passing sequentially through
two polarizers. Thus, if action duality is well-founded, the action
symmetry should be reflected in identical ontologies for the
action-dual experiments. Short of rejecting action duality, this
implies one of two things: either the usual causal explanation for the
behavior of the single photon must be reflected in a causal
explanation for the typically-quantum behavior of the pair of
entangled photons, and so there must be retrocausality at the level of
the ontic state \(\lambda\), or the nonlocality ascribed to the
typically-quantum behavior of the pair of entangled photons must be
similarly ascribed to the behavior of the single photon, and so
nonlocality is even more dramatically widespread than usually assumed,
and indeed is not unique to entangled bipartite quantum systems. In so
far as there is a strong case for the behavior of a single photon
passing sequentially through two polarizers having a perfectly good
causal explanation based on spatiotemporally localized properties, the
argument of Evans et al. is that this amounts to an equally strong
case for retrocausality in quantum theory.


## 5. The Transactional Interpretation



This entry so far has considered the two most significant motivating
arguments in favor of adopting retrocausality as a hypothesis for
dealing with the interpretational challenges of quantum theory. But
these motivations do not by themselves amount to an interpretation or
model of quantum theory.
 SS6
 consists of a survey of a range of retrocausal models, but this
section first considers perhaps the most prominent retrocausal model,
the transactional interpretation. Developed by Cramer in the 1980s
(Cramer 1980, 1986, 1988), the transactional interpretation is heavily
influenced by the framework of the Wheeler-Feynman absorber approach
to electrodynamics (see
 SS1);
 the Wheeler-Feynman schema can be adopted to describe the microscopic
exchange of a single quantum of energy, momentum, *etc.*,
between and within quantum systems.



At the heart of the transactional interpretation is the
"transaction": real physical events are identified with
so-called "handshakes" between forward-evolving quantum
states \(\psi\) and backward-evolving complex-conjugates \(\psi^\*\).
When a quantum emitter (such as a vibrating electron or atom in an
excited state) is to emit a single quantum (a photon, in these cases),
the source produces a radiative field--the "offer"
wave. Analogously to the Wheeler-Feynman description, this field
propagates outwards both forward and backward in time (as well as
across space). When this field encounters an absorber, a new field is
generated--the "confirmation" wave--that
likewise propagates both forward and backward in time, and so is
present as an advanced incident wave at the emitter at the instant of
emission. Both the retarded field produced by the absorber and the
advanced field produced by the emitter exactly cancel with the
retarded field produced by the emitter and advanced field produced by
the absorber for all times before the emission and after the
absorption of the photon; only between the emitter and the absorber is
there a radiative field. Thus the transaction is completed with this
"handshake": a cycle of offer and confirmation waves 



> 
> 
> repeats until the response of the emitter and absorber is sufficient
> to satisfy all of the quantum boundary conditions...at which
> point the transaction is completed. (Cramer 1986: 662) 
> 
> 
> 



Many confirmation waves from potential absorbers may converge on the
emitter at the time of emission but the quantum boundary conditions
can usually only permit a single transaction to form. Any observer who
witnesses this process would perceive only the completed transaction,
which would be interpreted as the passage of a particle (e.g., a
photon) between emitter and absorber.



The transactional interpretation takes the wave function to be a real
physical wave with spatial extent. The wave function of the quantum
mechanical formalism is identical with the initial offer wave of the
transaction mechanism and the collapsed wave function is identical
with the completed transaction. Quantum particles are thus not to be
thought of as represented by the wave function but rather by the
completed transaction, of which the wave function is only the initial
phase. As Cramer explains:



> 
> 
> The transaction may involve a single emitter and absorber or multiple
> emitters and absorbers, but it is only complete when appropriate
> boundary conditions are satisfied at all loci of emission and
> absorption. Particles transferred have no separate identity
> independent from the satisfaction of these boundary conditions. (1986:
> 666)
> 
> 
> 



The amplitude of the confirmation wave which is produced by the
absorber is proportional to the local amplitude of the incident wave
that stimulated it and this, in turn, is dependent on the attenuation
it received as it propagated from the source. Thus, the total
amplitude of the confirmation wave is just the absolute square of the
initial offer wave (evaluated at the absorber), which yields the Born
rule. Since the Born rule arises as a product of the transaction
mechanism, there is no special significance attached to the role of
the observer in the act of measurement. The "collapse of the
wave function" is interpreted as the completion of the
transaction.



The transactional interpretation explicitly interprets the quantum
state \(\psi\) as real, and so does not constitute an attempt to
exploit the retrocausality loopholes to the theorems that rule out
\(\psi\)-epistemic accounts. Additionally, the transactional
interpretation subverts the dilemma at the core of the EPR argument
(Einstein, et al. 1935) by permitting incompatible observables to take
on definite values simultaneously: the wavefunction, according to the
transactional interpretation,



> 
> 
> brings to each potential absorber the full range of possible outcomes,
> and all have "simultaneous reality" in the EPR sense. The
> absorber interacts so as to cause one of these outcomes to emerge in
> the transaction, so that the collapsed [wavefunction] manifests only
> one of these outcomes. (Cramer 1986: 668).
> 
> 
> 



Most importantly, however, the transactional interpretation employs
both retarded and advanced waves, and in doing so admits the
possibility of providing a "zigzag" explanation of the
nonlocality associated with entangled quantum systems.



Before turning to one of the more significant objections to the
transactional interpretation, and to retrocausality in general, it is
instructive to tease apart here two complementary descriptions of this
transaction process. On the one hand there is a description of the
real physical process, consisting of the passage of a particle between
emitter and absorber, that a temporally bound experimenter would
observe; and on the other hand there is a description of a dynamical
process of offer and confirmation waves that is instrumental in
establishing the transaction. This latter process simply cannot occur
in an ordinary time sequence, not least because any temporally bound
observer by construction cannot detect any offer or confirmation
waves. Cramer suggests that the "dynamical process" be
understood as occurring in a "pseudotime" sequence:



> 
> 
> The account of an emitter-absorber transaction presented here employs
> the semantic device of describing a process extending across a
> lightlike or a timelike interval of space-time as if it occurred in a
> time sequence external to the process. The reader is reminded that
> this is only a pedagogical convention for the purposes of description.
> The process is atemporal and the only observables come from the
> superposition of all "steps" to form the final
> transaction. (Cramer 1986: 661, fn.14)
> 
> 
> 



These steps are of course the cyclically repeated exchange of offer
and confirmation waves which continue "until the net exchange of
energy and other conserved quantities satisfies the quantum boundary
conditions of the system" (1986: 662). There is a strong sense
here that any process described as occurring in pseudotime is *not
a process at all* but, as Cramer reminds, merely a
"pedagogical convention for the purposes of description".
Whether it is best to understand causality according to the
transactional interpretation in terms of processes underscored by
conserved quantities is closely tied to how one should best understand
this pseudotemporal process.



Maudlin (2011) outlines a selection of problems that arise in
Cramer's theory as a result of the pseudotemporal account of the
transaction mechanism: processes important to the completion of a
transaction take place in pseudotime only (rather than in real time)
and thus cannot be said to have taken place at all. Since a temporally
bound observer can only ever perceive a completed transaction, i.e., a
collapsed wavefunction, the uncollapsed wavefunction never actually
exists. Since the initial offer wave is identical to the wavefunction
of the quantum formalism, any ensuing exchange of advanced and
retarded waves required to provide the quantum mechanical
probabilities, according to Maudlin, also do not exist. Moreover,
Cramer's exposition of the transaction mechanism seems to
suggest that the stimulation of sequential offer and confirmation
waves occurs deterministically, leaving a gaping hole in any
explanation the transactional interpretation might provide of the
stochastic nature of quantum mechanics. Although these problems are
significant, Maudlin admits that they may indeed be peculiar to
Cramer's theory. Maudlin also sets out a more general objection
to retrocausal models of quantum mechanics which he claims to pose a
problem for "any theory in which both backwards and forwards
influences conspire to shape events" (2011: 184).



Maudlin's main objection to the transactional interpretation
hinges upon the fact that the transaction process depends crucially on
the fixity of the absorbers "just sitting out there in the
future, waiting to absorb" (2011: 182); one cannot presume that
present events are unable to influence the future disposition of the
absorbers. Maudlin offers a thought experiment to illustrate this
objection. A radioactive source is constrained to emit a
\(\beta\)-particle either to the left or to the right. To the right
sits absorber *A* at a distance of 1 unit. Absorber *B* is
also located to the right but at a distance of 2 units and is built on
pivots so that it can be swung around to the left on command. A
\(\beta\)-particle emitted at time \(t\_{0}\) to the right will be
absorbed by absorber *A* at time \(t\_{1}\). If after time
\(t\_{1}\) the \(\beta\)-particle is not detected at absorber *A*,
absorber *B* is quickly swung around to the left to detect the
\(\beta\)-particle after time \(2t\_{1}\).



According to the transactional interpretation, since there are two
possible outcomes (detection at absorber *A* or detection at
absorber *B*), there will be two confirmation waves sent back
from the future, one for each absorber. Furthermore, since it is
equally probable that the \(\beta\)-particle be detected at either
absorber, the amplitudes of these confirmation waves should be equal.
However, a confirmation wave from absorber *B* can only be sent
back to the emitter if absorber *B* is located on the left. For
this to be the case, absorber *A* must not have detected the
\(\beta\)-particle and thus the outcome of the experiment must already
have been decided. The incidence of a confirmation wave from absorber
*B* at the emitter *ensures* that the \(\beta\)-particle
is to be sent to the left, even though the amplitude of this wave
implies a probability of a half of this being the case. As Maudlin
(2011: 184) states so succinctly, "Cramer's theory
collapses".



The key challenge from Maudlin is that any retrocausal mechanism must
ensure that the future behavior of the system transpires consistently
with the spatiotemporal structure dictated by any potential future
causes: "stochastic outcomes at a particular point in time may
influence the future, but that future itself is supposed to play a
role in producing the outcomes" (2011: 181). In the
transactional interpretation the existence of the confirmation wave
itself presupposes some determined future state of the system with
retrocausal influence. However, with standard (i.e., forward-in-time)
stochastic causal influences affecting the future from the present, a
determined future *may not necessarily be guaranteed* in every
such case, as shown by Maudlin's experiment.



Maudlin's challenge to the transactional interpretation has been
met with a range of responses (see P. Lewis 2013 and the entry on
 action at a distance in quantum mechanics
 for more discussion of possible responses). The responses generally
fall into two types (P. Lewis 2013). The first type of response
attempts to accommodate Maudlin's example within the
transactional interpretation. Berkovitz (2002) defends the
transactional interpretation by showing that causal loops of the type
found in Maudlin's experiment need not obey the assumptions
about probabilities that are common in linear causal situations.
Marchildon (2006) proposes considering the absorption properties of
the long distance boundary conditions: if the universe is a perfect
absorber of all radiation then a confirmation wave from the left will
always be received by the radioactive source at the time of emission
and it will encode the correct probabilistic information. Kastner
(2006) proposes differentiating between competing initial states of
the radioactive source, corresponding to the two emission
possibilities, that together characterize an unstable bifurcation
point between distinct worlds, where the seemingly problematic
probabilities reflect a probabilistic structure across both possible
worlds.



The second type of response is to modify the transactional
interpretation. For instance, Cramer (2016) introduces the idea of a
hierarchy of advanced-wave echoes dependent upon the magnitude of the
spatiotemporal interval from which they originate. Kastner (2013)
surmises that the source of the problem that Maudlin has exposed,
however, is the idea that quantum processes take place in the
"block world", and rejects this conception of processes in
her own development of the transactional interpretation. According to
her "possibilist" transactional interpretation, all the
potential transactions exist in a real space of possibilities, which
amounts at once to a kind of modal realism and an indeterminacy
regarding future states of the system (hence Kastner's rejection
of the block universe view). The possibilist transactional
interpretation arguably handles multi-particle scenarios more
naturally, and presents the most modern sustained development of the
transactional interpretation (although see P. Lewis 2013 for
criticisms of the possibilist transactional interpretation specific to
Maudlin's challenge).


## 6. Developments Towards a Retrocausal Model



The transactional interpretation might be seen as the most
prominent--and historically significant--retrocausal model
on the market, but it is not the only one. While retrocausality in
quantum mechanics has been the subject of considerable analysis and
critique over the years (Rietdijk 1978; Sutherland 1983, 1985, 1989;
Price 1984, 2001; Hokkyo 1988, 2008; Miller 1996, 1997; Atkinson 2000;
Argaman 2010; Evans 2015), the focus of this section is a review of
some of the more concrete proposals for retrocausal models.


### 6.1 The Two-State Vector Formalism



What is now known as the two-state vector formalism was first proposed
by Watanabe (1955), and then rediscovered by Aharonov, Bergmann, and
Lebowitz (1964). The proposal is that the usual forward evolving
quantum state contains insufficient information to completely specify
a quantum system; rather the forward evolving state must be
supplemented by a backward evolving state to provide a complete
specification. Thus, according to the two-state vector formalism, the
complete quantum description contains a state vector that evolves
forward in time from the initial boundary condition towards the
future, and a state vector that evolves backward in time from the
future boundary condition towards the past. It is only a combination
of complete measurements at both initial and final boundaries that can
provide a complete specification of a quantum system. The two-state
vector formalism is empirically equivalent to standard quantum
mechanics (Aharonov & Vaidman 1991, 2008).



The emphasis of the two-state vector formalism is on the operational
elements of the theory, and there are very few ontological
prescriptions, including how best to understand causality. It is in
principle compatible with a variety of supplemented retrocausal
ontologies, e.g., the causally symmetric Bohm model
 (SS6.3).


### 6.2 Toy Models



Toy models have been employed to illustrate how retrocausality could
be possibly realized. Price (2008) suggests a simple toy model
featuring linked nodes that can assume different values, which he dubs
the "Helsinki model". If one interprets the nodes as
partially ordered in time and the values of the exogenous boundary
nodes as controllable, the dynamics specified by Price entails causal
bi-directionality, i.e., both ordinary forward causality and
retrocausality, understood in an interventionist sense. Although the
model does indeed display particular behavior that can be naturally
interpreted as retrocausal, a full-blown analogy with standard quantum
mechanics is limited.



The Helsinki model consists of three primitive endogenous nodes, each
node comprising a meeting point of three edges, with two
"internal" edges linking the three nodes and five
"external" edges. Each edge has one of three
"flavors", *A*, *B*, or *C*. The system is
temporally oriented with three exogenous nodes, each joined to a
single edge, representing system "inputs"
(preparations/interventions), and two system "outputs"
(measurement outcomes/observations) joined to the remaining two edges.
The two internal edges represent hidden flavors that cannot be
directly controlled or observed. There are two basic rules that govern
the dynamics of this toy system: (i) each node must be strictly
inhomogeneous--i.e., comprising three edges of different
flavors--or strictly homogeneous--i.e., three edges of the
same flavor, and (ii) successive homogeneous nodes are prohibited.



The retrocausal behavior of the model arises as a result of the heavy
constraints that the two basic rules place on the possible flavors of
the hidden edges, which establishes correlations between the input
states and the hidden states. Thus interventions on the left or right
exogenous variables influence the complete set of possible hidden
states of the system. Assuming that the hidden states are in the past
of these variable choices, the hidden state depends
"retrocausally" on the left and right input settings.
Moreover, under specific node variables, some interventions on the
left or right exogenous variables amount to interventions on the
distant output variable, displaying a kind of nonlocality (but
violating no-signaling).



More ambitiously, Corry (2015) proposes three toy models, also using
nodes and links, which exhibit Bell-type correlations and account for
them in a purely local manner in that the constraints governing nodes
make reference only to information available at the respective nodes.
The models do not in general specify values for unobserved nodes,
however. As a way out, Corry suggests, one may either postulate such
values at the price of accepting "retrofinkish
dispositions" (dispositions which, if they were triggered, would
not have been there in the first place) or simply deny the existence
of such values outright.


### 6.3 The Causally Symmetric Bohm Model



A model that hypothesizes retrocausality and reproduces the
consequences of standard quantum theory without violating Lorentz
invariance is Sutherland's (2008, 2015, 2017) causally symmetric
version of Bohmian mechanics. This model adds to the standard quantum
state \(\psi\_i\) of ordinary Bohmian mechanics, which is fixed by an
initial boundary condition, an additional quantum state \(\psi\_f\),
which is fixed by some final boundary condition. An analogue to the
"guiding equation" for particle motion in ordinary Bohmian
mechanics is derived by Sutherland that is symmetric with respect to
these states. The zero-component of the probability current, which is
directly related to the probability density, is computed as 



\[\tag{2}
\rho(\mathbf x, t)={\rm Re}\left(\psi\frac{\psi^\*\_f\psi\_i}{a}\right)\,\label{eq:probdens}
\]


where 

 
\[\tag{3}
a=\int\psi\_f^\*(\mathbf x,t)\psi\_i(\mathbf x,t)d^3x\,.
\]


Finally, for consistency, the model also requires that the conditional
probability of the final state being \(\psi\_f\), given the initial
state \(\psi\_i\), is 

 
\[\tag{4}
\rho(\psi\_f\mid \psi\_i)=\lvert a\rvert^2\,\label{eq:addass}\,.
\]


Berkovitz (2008) criticizes the use of the additional assumption Eq.
(\ref{eq:addass}) and the need for it in Sutherland's model,
arguing that it leads to an undesirable form of causal loops and has
an *ad hoc* character because there is no independent reason to
think that \(\psi\_i\) and \(\psi\_f\) should be correlated in this
way.



Sutherland's model is explicitly retrocausal in that particle
dynamics are influenced by \(\psi\_f\), which in turn depends on the
future measurement setting. It also contains action-by-contact causal
influences in that Bell-type correlations are accounted for in the
"zigzag" manner envisaged by Costa de Beauregard, as
outlined in
 SS1.


### 6.4 The Relational Blockworld Interpretation



Silberstein, Stuckey, and McDevitt (2018), based on earlier work
together with Cifone (Stuckey, Silberstein, & Cifone 2008;
Silberstein, Cifone, & Stuckey. 2008), suggest a realist
interpretation that they call the "relational blockworld
view". The ontology of this interpretation consists of so-called
"spacetimesource elements", which are characterized as
"amalgams of space, time, and sources" (Silberstein,
Stuckey, & McDevitt 2018: 153).



Technically, the relational blockworld approach is set up as a
modification of lattice gauge theory, with the Feynman path integral
functioning as an "adynamical global constraint". While
adynamical and acausal rather than retrocausal in spirit, on the
authors' view this interpretation exploits the retrocausality
loopholes of the no-go theorems. They conceive of the relational
blockworld view as \(\psi\)-epistemic. In more recent work (2018: Ch.
6), Silberstein, Stuckey and McDevitt have developed this approach
into an ambitious overarching programme in fundamental physics, aiming
at field-theoretic unification as well as a novel account of quantum
gravity.


### 6.5 The Two-Time Boundary Value Model



Schulman (1997, 2012) proposes a solution to the quantum measurement
problem based on the possibility of future conditioning, which
originates from his analysis of the thermodynamic arrow of time.
Beginning with Loschmidt's challenge to Boltzmann--that it
should not be possible to derive irreversible dynamical behavior from
time-symmetric dynamics (see the entry on
 thermodynamic asymmetry in time)--Schulman
 notes that successful "retrodiction" of past macroscopic
states of some thermodynamic system is indeed asymmetric to successful
"prediction" of future macroscopic states. For successful
retrodiction, rather than evolving each of the microscopic states of
some macroscopic state according to the dynamical laws (which is
identical to the process of prediction, and is the basis of
Loschmidt's challenge), one instead hypothesizes a prior
macroscopic state to which the prediction process can be applied such
that the current macroscopic state of the system obtains with high
likelihood given the dynamical evolution of the prior microscopic
states. With respect to the set of possible microscopic states for the
current macroscopic state, since the evolution of the vast majority
(on Liouville measure) of these states according to the dynamical laws
result in trajectories that conflict with the retrodiction hypothesis,
these states are effectively rejected in the process of retrodiction
in favor of those special few microscopic states that correspond to
the dynamical evolution of acceptable hypothetical initial
conditions.



Schulman's proposal in response to this asymmetry is that, just
as the set of possible final microscopic states must be restricted for
successful retrodiction, so should the permissible initial microscopic
states be restricted for the purposes of prediction. Thus, since final
microscopic states are subject to conditioning from the past state of
the system, the initial microscopic states will be subject to
conditioning from the future state of the system--a conditioning
that is hidden in thermodynamic processes where the microscopic states
of the system are indistinguishable macroscopically. This future
conditioning is a central feature of Schulman's two-time
boundary condition proposal and the hidden nature of this conditioning
Schulman calls "cryptic constraints".



Regarding quantum theory, Schulman claims that quantum superpositions
of macroscopically distinct states generated by quantum evolution,
which are at the heart of many of the nonclassical elements of quantum
theory, are what he calls "grotesque" states, and proposes
that these states are avoided in quantum systems due to future
conditioning and cryptic constraints. Just like there are special
microscopic states of thermodynamical systems that evolve
"against" the second law of thermodynamics (which are
identified in the process of retrodiction), so too must there be
"special" microscopic states of a quantum system which do
not lead to grotesque states when evolved according to the quantum
dynamical laws, rather these states will lead to one particular
definite state of the superposition. Schulman's solution to the
quantum measurement problem is that, in every performed experiment,
the initial conditions of the quantum system are among these special
states which, through pure unitary quantum evolution, yield a single
outcome to the experiment (Schulman 1997: 214). Grotesque states, the
problematic states of the quantum measurement problem, are thus
avoided.



Schulman's proposal implicitly assumes that, in the preparation
of an experiment, it is only the macroscopic state of the system which
is under the control of the experimenter; it is impossible to control
the precise microscopic state. It is this initial microscopic state
that Schulman suggests is always a special state. Schulman envisages
the special-state constraint as correlated with future conditions and,
since these constraints are not apparent to the experimenter until the
future conditions are "measured" at the end of the
experiment, these constraints are cryptic. As a result of the future
conditioning of initial states, Schulman's proposal is a kind of
retrocausal mechanism, understood in an interventionist sense. (For an
extension of Schulman's model into the Lagrangian schema
 (SS6.6)
 see Almada et al. 2016 and Wharton 2014, 2018.)


### 6.6 "All-at-once" Lagrangian Models



Wharton (2010a; see also Wharton 2007, 2010b, 2013, 2016, 2018;
Wharton, Miller, & Price 2011; Evans, Price, & Wharton 2013)
proposes a "novel interpretation of the Klein-Gordon
equation" for a neutral, spinless relativistic particle. The
account is a retrocausal picture based on Hamilton's principle
and the symmetric constraint of both initial and final boundary
conditions to construct equations of motion from a Lagrangian, and is
a natural setting for a perspectival interventionist account of
causality. Wharton treats external measurements as physical
constraints imposed on a system in the same way that boundary
constraints are imposed on the action integral of Hamilton's
principle; the final measurement does not simply reveal preexisting
values of the parameters, but constrains those values (just as the
initial boundary condition would). Wharton's model has been
described as an "all-at-once" approach, since the dynamics
of physical systems between an initial and final boundary emerges
*en bloc* as the solution to a two-time boundary value
problem.



On this interpretation, one considers reality exclusively between two
temporal boundaries as being described by a classical field \(\phi\)
that is a solution to the Klein-Gordon equation: specification of
field values at both an initial and final boundary (as opposed to
field values and their rate of change at only the initial boundary)
constrains the field solutions between the boundaries. Wharton argues
that constraining such fields at both an initial and a final boundary
(or a closed hypersurface in spacetime) generates two strikingly
quantum features: quantization of certain field properties and
contextuality of the unknown parameters characterizing the field
between the boundaries. (That there are unknown parameters before the
imposition of the final condition is ensured due to the
underdetermination of the classical field by specifying only the field
values, and not their rate of change, in the initial data.)



From within Wharton's picture, an invariant joint probability
distribution associated with each possible pair of initial and final
conditions can be constructed, and the usual conditional probabilities
can be formed by conditioning on any chosen portion of the boundary
(Wharton 2010a,b). Probability is then a manifestation of our
ignorance: if one knew only the initial boundary, one would only be
able to describe the subsequent field probabilistically (due to the
future constraint); given the final boundary as well, one would then
be able to retrodict the field values between the two boundaries. (See
Evans, Gryb, and Thebault 2016 for a proposed extension of this
schema to the cosmological context.)



In more recent work (Wharton 2016), Wharton explores the prospects for
a realist retrocausal interpretation of quantum theory based on the
Feynman path integral formalism (Feynman 1942). In that formalism as
applied to a single particle undergoing position measurements at
spacetime points \((\mathbf{x\_0},t\_0)\) and \((\mathbf{x\_1},t\_1)\),
the joint probability distribution for all pairs of points and given
times \(t\_0\) and \(t\_1\) is given by 

 
\[\tag{5}
P(\mathbf{x\_0},\mathbf{x\_1})=\left|\sum\_{\mathbf{x\_0}\mapsto\mathbf{x\_1}}\exp(iS/\hbar)\right|^2\,.\label{eq:feynman}
\]


The sum in this formula is to be understood as the infinitesimal limit
of a discretized set of spacetime trajectories connecting
\((\mathbf{x\_0},t\_0)\) and \((\mathbf{x\_1},t\_1)\). *S* is the
classical action of the particle along the respective trajectory.



A straightforward but naive interpretation of this equation according
to which the probability reflects ignorance concerning the path taken
does not work because the right hand side of Eq. (\ref{eq:feynman}) is
not a sum of positive numbers (interpretable as probabilities of
trajectories) but rather a positive number obtained by taking the
modulus squared of a sum of complex numbers. Wharton explores the
prospects for giving an ignorance interpretation of the path integral
along less straightforward lines, noting that Eq. (\ref{eq:feynman})
can be brought into the form 

 
\[\tag{6}
P(\mathbf{x\_0},\mathbf{x\_1})=\sum\_{c\_i}\left|\sum\_{A\in c\_i}\exp(iS\_A/\hbar)\right|^2\,,\label{eq:wharton}
\]


where the \(\left|c\_i\right|\) are distinct *groups of
trajectories* *A*. The form Eq. (\ref{eq:wharton}), according
to Wharton, invites an interpretation of the probability in terms of
ignorance concerning the actual group of trajectories \(c\_i\), and he
tentatively proposes a field interpretation of the \(c\_i\). The
interpretation is retrocausal because the probabilities over groups of
trajectories are influenced by the future measurement time and
setting. Open questions for this approach concern the grouping of
trajectories, which is so far inherently ambiguous, the details of the
suggested field interpretation, as well as generalizations to
many-particle and other more general settings.


### 6.7 The Q-based interpretation



The Husimi Q-function (Husimi 1940) is a non-negative distribution
function on phase space with a wide applicability, e.g. in quantum
optics. It is obtained from the better known Wigner function (Wigner
1932) by smoothening with a Gaussian filter. The Q-based
interpretation (Drumond and Reid 2020; Friederich 2021) assumes that
any quantum system has a precise phase space location, and that phase
space locations are distributed according to the Q-function. As a
consequence, all dynamical variables of all quantum systems have
definite values in this interpretation, and the measurement problem is
avoided.



Drummond and Reid (2020) argue that this interpretation avoids the
no-go theorems discussed in
 SS3
 via the retrocausality loophole. In support of this claim, they point
to a result by Drummond (2021), according to which, for a wide class
of quantum field theories, the Q-function evolves via diffusion
backward in time for half of all degrees of freedom (and forward for
the other half). Friederich (2021) suggests that failure of temporal
locality, independently motivated by Adlam (2018, 2022), could also
make those theorems inapplicable to the Q-based interpretation.



It is not immediately obvious that the Q-based interpretation is
empirically adequate since it does not exactly reproduce the quantum
probabilities derived from the Born rule. However, as argued by
Drummond and Reid (2020), for various measurement setups and at the
level of macroscopic measurement devices, the discrepancy with
standard quantum mechanics based on the Born rule may not be
detectable. Friederich (2021) provides a general argument that the
Q-based interpretation is practically indistinguishable from standard
quantum mechanics at the level of measurement devices, parallelling
the standard argument for the empirical adequacy of Bohmian mechanics,
as sketched in the entry on
 Bohmian mechanics,
 SS10. Interestingly, a version of Bohmian mechanics that also
treats the Husimi function as the actual phase space probability
distribution was earlier suggested by de Polavieja (1996).


## 7. Objections Against Retrocausality in Quantum Mechanics



This final section reviews some of the most common, and then two of
the most significant, objections against the proposal of
retrocausality in quantum mechanics.


### 7.1 General Arguments Against Retrocausality



There is a tradition in philosophy for regarding the very idea of
retrocausality as incoherent. The most prominent worry (Flew 1954,
Black 1956), is the so-called "bilking argument" (see the
entry on
 time travel).
 Imagine a pair of events, a cause, *C*, and an effect, *E*,
which we believe to be retrocausally connected (*E* occurs
earlier in time than *C*). It seems possible to devise an
experiment which could confirm whether our belief in the retrocausal
connection is correct or not. Namely, once we had observed that
*E* had occurred, we could then set about ensuring that *C*
does not occur, thereby breaking any retrocausal connection that could
have existed between them. If we were successful in doing this, then
the effect would have been "bilked" of its cause.



The bilking argument drives one towards the claim that any belief an
agent might hold in the positive retrocausal correlation between event
*C* and event *E* is simply false. However, Dummett (1964)
disputes that giving up this belief is the only solution to the
bilking argument. Rather, according to Dummett, what the bilking
argument actually shows is that a set of three conditions concerning
the two events, and the agent's relationship to them, is
incoherent:


1. There exists a positive correlation between an event *C* and
an event *E*.
2. Event *C* is within the power of an agent to perform.
3. The agent has epistemic access to the occurrence of event *E*
independently of any intention to bring it about.



It is interesting to note that these conditions do not specify in
which order events *C* and *E* occur. On simple reflection,
there is a perfectly natural reason why it is not possible to bilk
*future* effects of their causes, since condition (iii) fails
to hold for future events: we simply have no access to which future
events occur independently of the role we play as causal agents to
bring the events about. When we lack that epistemic access to past
events, the same route out of the bilking argument becomes
available.



Dummett's defense against the bilking argument is especially
relevant to quantum mechanics. In fact, once a suitable specification
is made of how condition (iii) can be violated, we find that there
exists a strong parallel between the conditions which need to hold to
justify a belief in bringing about the past and the structure of
quantum mechanics. Price (1996: 174) points out that bilking is
impossible in the following circumstances: rather than suppose that a
violation of condition (iii) entails that the relevant agent has
*no* epistemic access to the relevant past events independently
of any intention to bring them about, suppose that the means by which
knowledge of these past events is gathered breaks the claimed
correlation between the agent's action and those past events.
Such a condition can be stated as follows:


4. The agent can gain epistemic access to the occurrence of event
*E* independently of any intention to bring it about and without
altering event *E* from what it would have been had no epistemic
access been gained.



The significance of this weakened violation of condition (iii) is that
it is just the sort of condition one would expect to hold if the
system in question were a quantum system. The very nature of quantum
mechanics ensures that any claimed positive correlation between the
future measurement settings and the hidden variables characterizing a
quantum system cannot possibly be bilked of their causes because
condition (iv) is perennially violated. Moreover, so long as we
subscribe to the epistemic interpretation of the wavefunction, we lack
epistemic access to the "hidden" variables of the system
and we lack this access *in principle* as a result of the
structure of quantum theory.



Another prominent challenge against the very idea of retrocausality is
that it inevitably would give rise to vicious causal loops (Mellor
1998). (See Faye 1994 for a response and the entry on
 backward causation
 for a more detailed review of the objections raised against the idea
of retrocausality.)


### 7.2 Retrocausality Just Is Superdeterminism



Recall
 (SS3.1)
 that retrocausality can be motivated as an explicit violation of the
assumption of measurement independence (Eq. (\ref{eq:independence}))
as a means of circumventing Bell's theorem. But there is another
possible mechanism that might underpin violations of measurement
independence, namely, superdeterminism. This is the idea that, in the
terminology of
 SS3,
 the joint past of (i) the measurement setting \(\alpha\) and (ii) the
hidden state of the measured system \(\lambda\) determines both of
them completely and induces the relevant quantum correlations between
them (see the entry on
 Bell's Theorem,
 SS8.1.1, and references therein). Hossenfelder (2020) argues
against referring to violations of measurement independence as
'retrocausality', preferring to consider such violations
as exclusively instances of 'superdeterminism', since
"the idea of a cause propagating back in time is
meaningless" (2020:10; see also Hossenfelder and Palmer (2020)).
The essential point of contention here, and Hossenfelder points this
out explicitly, is that each of retrocausality and superdeterminism
invokes a different understanding of the nature of causality.



The characterisation of causality that Hossenfelder (2020: 10) takes
to undermine the proposal of retrocausality is that *A* is a
cause of *B* if and only if *A* and *B* are correlated
and *B* is in the forward lightcone of *A*: "If you
think you have a situation where B "retrocauses" *A*,
then this merely means *A* causes *B*". In this way,
such correlations between events should never be understood as
retrocausal, and should always be understood as superdeterministic.
This characterization of causality is reminiscent of Hume's
conventionalism about cause and effect, according to which it is
simply a matter of terminology that causes must precede their effects
(see the entry on
 Backward Causation,
 SS2.3). As noted in the metaphysical preliminaries set out in
 SS2.1,
 the conception of causality that most naturally coheres with the
proposal of retrocausality is an interventionist one. Hossenfelder
acknowledges this, and argues that the reason the interventionist
account of causation is inappropriate as a proposal in the quantum
context "is that "agents" and their
"interventions" are macroscopic terms that do not appear
in quantum mechanics".



What Hossenfelder's critique emphasizes is that retrocausality
requires a certain package of metaphysical views to be a viable
proposal. Hossenfelder's analysis relies on a conception of
causality that is at odds with the interventionist account, especially
as understood along perspectival lines, as outlined in
 SS2.1.
 As a result, a defence of retrocausality against Hossenfelder's
objection would need to defend the fruitfulness of this account of
causality.


### 7.3 Retrocausality Requires Fine Tuning



Causal modeling (Spirtes, Glymour, & Scheines 2000; Pearl 2009) is
a practice that has arisen from the field of machine learning that
consists in the development of algorithms that can automate the
discovery of causes from correlations in large data sets. The causal
discovery algorithms permit an inference from given statistical
dependences and independences between distinct measurable elements of
some system to a causal model for that system. As part of the
algorithms, a series of constraints must be placed on the resulting
models that capture general features that we take to be characteristic
of causality. Two of the more significant assumptions are (i) the
causal Markov condition, which ensures that every statistical
dependence in the data results in a causal dependence in the
model--essentially a formalization of Reichenbach's common
cause principle--and (ii) faithfulness, which ensures that every
statistical independence implies a causal independence, or no causal
independence is the result of a fine-tuning of the model.



It has long been recognized (Butterfield 1992; Hausman 1999; Hausman
& Woodward 1999) that quantum correlations force one to give up at
least one of the assumptions usually made in the causal modeling
framework. Wood and Spekkens (2015) argue that any causal model
purporting to causally *explain* the observed quantum
correlations must be fine-tuned (i.e., must violate the faithfulness
assumption). More precisely, according to them, since the observed
statistical independences in an entangled bipartite quantum system
imply no signaling between the parties, when it is then assumed that
every statistical independence implies a causal independence (which is
what faithfulness dictates), it must be inferred that there can be no
(direct or mediated) causal link between the parties. Since there is
an observed statistical dependence between the *outcomes* of
measurements on the bipartite system, we can no longer account for
this dependence with a causal link unless this link is fine tuned to
ensure that the no-signaling independences still hold. There is thus a
fundamental tension between the observed quantum correlations and the
no-signaling requirement, the faithfulness assumption and the
possibility of a causal explanation.



Formally, Wood and Spekkens argue that the following three assumptions
form an inconsistent set: (i) the predictions of quantum theory
concerning the observed statistical dependences and independences are
correct; (ii) the observed statistical dependences and independences
can be given a causal explanation; (iii) the faithfulness assumption
holds. Wood and Spekkens conclude that, since the faithfulness
assumption is an indispensable element of causal discovery, the second
assumption must yield. The contrapositive of this is that *any*
purported causal explanation of the observed correlations in an
entangled bipartite quantum system falls afoul of the tension between
the no-signaling constraint and no fine tuning and, thus, must violate
the assumption of faithfulness. Such causal explanations, so the
argument goes, including retrocausal explanations, should therefore be
ruled out as viable explanations.



As a brief aside, this fine-tuning worry for retrocausality in the
quantum context arises in a more straightforward way. There is no good
evidence to suggest that signaling towards the past is possible; that
is, there is no retrocausality at the operational level. (Pegg 2006,
2008 argues that this can be explained formally as a result of the
completeness condition on the measurement operators, introducing an
asymmetry in normalization conditions for preparation and
measurement.) Yet, despite there being no signaling towards the past,
retrocausal accounts assume causal influences towards past. That these
causal influences do not show up as statistical dependences
exploitable for signaling purposes raises exactly the same fine-tuning
worry as Wood and Spekkens raise.



An obvious response to the challenge set by Wood and Spekkens is to
simply reject the assumption of faithfulness. But this should not be
taken lightly; the intuition behind the faithfulness assumption is
basic and compelling. When no statistical correlation exists between
the occurrences of a pair of events, there is no reason for supposing
there to be a causal connection between them. Conversely, if we were
to allow the possibility of a causal connection between statistically
uncorrelated events, we would have a particularly hard task
determining which of these uncorrelated sets could be harboring a
conspiratorial causal connection that hides the correlation. The
faithfulness assumption is thus a principle of parsimony--the
simplest explanation for a pair of statistically uncorrelated events
is that they are causally independent--much the same way that
Spekkens' (2005) definition of contextuality is, too (see
 SS3.2);
 indeed, Cavalcanti (2018) argues that contextuality can be construed
as a form of fine-tuning.



There are, however, well-known examples of systems that potentially
show a misapplication of the faithfulness assumption. One such
example, originating in Hesslow (1976), involves a contraceptive pill
that can cause thrombosis while simultaneously lowering the chance of
pregnancy, which can also cause thrombosis. As Cartwright (2001: 246)
points out, given the right weight for these process, it is
conceivable that the net effect of the pills on the frequency of
thrombosis be zero. This is a case of "cancelling paths",
where the effect of two or more causal routes between a pair of
variables cancels to achieve statistical independence. In a case such
as this, since we can have independent knowledge of the separate
causal mechanisms involved here, there are grounds for arguing that
there really is a causal connection between the variables despite
their statistical independence. Thus, it is certainly possible to
imagine a scenario in which the faithfulness assumption could lead us
astray. However, in defense of the general principle, an example such
as this clearly contains what Wood and Spekkens refer to as fine
tuning; the specific weights for these processes would need to match
precisely to erase the statistical dependence, and such a balance
would generally be thought as unstable (any change in background
conditions, etc. would reveal the causal connection in the form of a
statistical dependence).



Nager (2016) raises the possibility that unfaithfulness can occur
without conspiratorial fine tuning if the unfaithfulness arises in a
stable way. In the quantum context, Nager suggests that the
fine-tuning mechanism is what he calls "internal cancelling
paths". This mechanism is analogous to the usual cancelling
paths scenario, but the path-cancelling mechanism does not manifest at
the level of variables, but at the level of values. On this view, such
fine tuning would occur as a result of the particular causal and/or
nomological process that governs the system, and it is in this sense
that the cancelling paths mechanism is internal, and it is the fact
that the mechanism is internal that renders the associated fine tuning
stable to external disturbances. Thus 



> 
> 
> if the laws of nature are such that disturbances always alter the
> different paths in a balanced way, then it is physically impossible to
> unbalance the paths. (Nager 2016: 26) 
> 
> 
> 



The possibility raised by Nager would circumvent the problem that
violations of faithfulness ultimately undermine our ability to make
suitable inferences of causal independence based on statistical
independence by allowing only a specific kind of
unfaithfulness--a principled or law-based unfaithfulness that is
"internal" and is thus stable to background
conditions--which is much less conspiratorial, as the fine-tuning
is a function of the specific process involved. Evans (2018) argues
that a basic retrocausal model of the sort envisaged by Costa de
Beauregard (see
 SS1)
 employs just such an internal cancelling paths explanation to account
for the unfaithful (no signaling) causal channels. See also Almada et
al. (2016) for an argument that fine tuning in the quantum context is
robust and arises as a result of symmetry considerations. Furthermore,
Evans (2021) argues that, to the letter of Wood and Spekkens'
analysis, cases of fine tuning may be more ubiquitous than we suspect,
and so we should not be so worried about such quantum cases.


### 7.4 Contextuality for Exotic Causal Structures



Recall
 (SS3.2)
 that Spekkens' (2005) claim that no noncontextual ontological
model can reproduce the observed statistics of quantum theory based on
his principle of parsimony (that there can be no ontological
difference without operational difference) was sidestepped by
retrocausal approaches due to the explicit assumption of the
ontological models framework that the ontic state is independent of
the measurement procedure (i.e., that there is no retrocausality). It
was noted there the possibility that Spekkens' principle of
parsimony might be recast to apply more generally to retrocausal
models. Shrapnel and Costa (2018) achieve just this in a no-go theorem
that applies to any exotic causal structure used to sidestep the
ontological models framework, including retrocausal accounts,
rendering such models contextual after all.



Shrapnel and Costa's result is based on a generalization of the
ontological models framework which replaces the operational
preparation, transformation, and measurement procedures with the
temporally and causally neutral notions of local controllables and
environmental processes that mediate correlations between different
local systems, and generate the joint statistics for a set of events.
"These include any global properties, initial states, connecting
mechanisms, causal influence, or global dynamics" (2018: 5).
Furthermore, they replace the ontic state \(\lambda\) with the ontic
"process" \(\omega\):



> 
> 
> our ontic process captures the physical properties of the world that
> remain invariant under our local operations. That is, although we
> allow local properties to *change* under specific operations,
> we wish our *ontic process* to capture those aspects of reality
> that are independent of this probing. (2018: 8)
> 
> 
> 



As a result, the notion of \(\lambda\)-mediation (encountered in
 SS4.1)
 is replaced by the notion of \(\omega\)-mediation, in which the ontic
process \(\omega\) completely specifies the properties of the
environment that mediate correlations between regions, and screens off
outcomes produced by local controllables from the rest of the
environment. Shrapnel and Costa (2018: 9) define the notion of
"instrument noncontextuality" as a law of parsimony (along
the lines of Spekkens' own definition of noncontextuality):
"Operationally indistinguishable pairs of outcomes and local
controllables should remain indistinguishable at the ontological
level". They then show that no instrument noncontextual model
can reproduce the quantum statistical predictions.



> 
> 
> Crucially, what is contextual is not just the traditional notion of
> "state", but any supposedly objective feature of the
> theory, such as a dynamical law or boundary condition. (2018: 2)
> 
> 
> 



Since preparations, transformations, and measurements have been
replaced by local controllables, there is no extra assumption in
Shrapnel and Costa's framework that \(\omega\) is correlated
with some controllables but independent of others. Thus the usual
route out of the ontological models framework, and so the no-go
theorems of
 SS3,
 open to retrocausal approaches--that the framework assumes no
retrocausality--is closed off in the Shrapnel-Costa theorem,
rendering retrocausal approaches contextual along with the rest of the
models captured by the ontological models framework.



This presents a significant worry for retrocausal approaches to
quantum theory. If the main motivation for pursing the hypothesis of
retrocausality is to recapture in some sense a classical ontology for
quantum theory (see
 SS3.4),
 then the Shrapnel-Costa theorem has made this task either impossible,
or beholden to the possibility of some further story explaining how
the contextual features of the model arise from some noncontextual
footing. On this latter point, it is difficult to see how this story
might be told without significantly reducing the ideological economy
of the conceptual framework of retrocausality, again jeopardizing a
potential virtue of retrocausality (see Evans 2020 for further
discussion of this point).



As mentioned above
 (SS7.3),
 contextuality can be construed as a form of fine tuning (Cavalcanti
2018; Adlam 2021), especially when the demand for noncontextuality is
understood as a requirement of parsimony, as above. The worries raised
in this section and the last underline the fact that the challenge to
account for various types of fine tuning is the most serious
principled obstacle that retrocausal accounts continue to face.