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ResearchCite this article: Elitzur AC, Cohen E. 20161− 1= Counterfactual: on the potency andsignificance of quantum non-events. Phil.Trans. R. Soc. A 374: 20150242.http://dx.doi.org/10.1098/rsta.2015.0242

Accepted: 9 February 2016

One contribution of 14 to a theme issue‘Quantum foundations: information approach’.

Subject Areas:quantum physics

Keywords:quantum oblivion, counterfactuals,non-events, interaction-free measurement,time symmetry

Author for correspondence:A. C. Elitzure-mail: avshalom@iyar.org.il

1− 1= Counterfactual: onthe potency and significance ofquantum non-eventsA. C. Elitzur1 and E. Cohen1,2

1Iyar, The Israeli Institute for Advanced Research, Rehovot, Israel2H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue,Bristol BS8 1TL, UK

ACE, 0000-0002-4578-5986

We study the unique role played in quantummechanics by non-events or ‘counterfactuals’. Ourearlier analysis of ‘quantum oblivion’ has revealedsome subtle stages in the measurement process, whichmay end up in self-cancellation. To these findings, wenow add two insights derived by two time-symmetricinterpretations of quantum mechanics. (i) Like allquantum interactions, the non-event is formed bythe conjunction of forward-plus-backward-evolvingwave functions. (ii) Then, it is another feature of suchdual evolutions, namely the involvement of negativemasses and energies, that enables Nature to makesome events ‘unhappen’ while leaving causal traces.

1. IntroductionSo many, varied and sharp are the differences betweenquantum and classical physics; so persistent the conflictsbetween the quantum formalism and macroscopicreality; and so unsatisfactory have all proposedresolutions remained so far that a unifying hypothesisnaturally emerges: Can all quantum oddities share asubtler common origin? Should that be the case, somenovel ‘method in the madness’ is very likely to emerge,offering hints towards the long sought-for theory thatbetter accounts for these and other oddities.

This is the aim of this paper. We submit that

(i) A promising candidate for a ‘master quantumoddity’ is the unique status of the non-event, i.e. a‘counterfactual’. This is an event which could havehappened but did not. Quantum non-events areunique in that they can exert significant causaleffects just by virtue of this ‘could have’.

2016 The Author(s) Published by the Royal Society. All rights reserved.

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(ii) Earlier [1,2], studying an apparent violation of momentum conservation in a very basicquantum non-event, we showed that, within very brief intervals, momenta have beenexchanged, yet their recurrence ended up in a net zero result. We proposed the term‘quantum oblivion’ (QO) for these self-cancelling interactions.

(iii) Another piece of the puzzle is offered by some time-symmetric interpretations ofquantum mechanics (QM), according to which the quantum interaction is formed by twotime evolutions, going back and forth between past and future.

(iv) Significantly, these time-symmetric models, for independent reasons, have also invokednegative values for otherwise positive parameters, such as negative mass and energy.

(v) The resulting account emerging from all the above is very intuitive. Both positive andnegative values, such as mass and energy, are exchanged between past and futureinteractions of the quantum with macroscopic systems. These give rise to the familiarquantum non-local and non-temporal phenomena, which, despite apparent violations ofspace and time limits, obey conservation laws and the non-signalling principle.

(vi) QO thus turns out to be an event which, in some deeper sense, has occurred and then‘unoccurred’.

(vii) This has implications for several foundational issues in classical physics, bearing in mindthat quantum non-events far outnumber events in our Universe.

This general model has been named ‘quantum hesitation’ [2], honouring de Broglie’s quantum-mechanical rephrasing of Bergson’s definition of time as Nature’s hesitation between possibleoutcomes.

2. The non-event: the odd potency of quantum counterfactualsA simple ordinary quantum event, namely one that has occurred, suffices to elucidate theimportance of its opposite. Consider an excited atom floating in empty space, far from any othermatter. At t1 it emits a photon, which at a much later time t2 is absorbed by a macroscopic systemlocated far away. That system, by irreversibly amplifying the absorption’s effects, constitutesa detector. Assume further that the system can also record the absorption’s precise time. Thusmeasured, the photon’s emission is finalized.

Apparently, the detector has recorded a single event, namely the photon’s absorption. ByQM, however, there have been numerous events prior to t2: every instance in which the detectorremained silent is also a measurement—hence a quantum event. Logically, the non-event is subjectto the counterfactual statement ‘it could have happened’, quantum indeterminism being the onlyreason why, just by chance, it did not. What makes this counterfactual intriguing, far beyond logic,is the non-event’s causal efficacy: another physical event can occur just by virtue of this ‘could have’.

Consider first the time–energy uncertainty ruling the photon’s emission by the atom:

�E�t ≥ h̄2

, (2.1)

making the atom, respectively, superposed:

|ψ〉 = 1√2

(|e〉 + |g〉), (2.2)

where, as usual, e and g stand for excited and ground states, respectively. Therefore, the distantdetector, wherever positioned, is constantly measuring the atom, even when the measurementends with no click (which is the more frequent occurrence), making the atom undergo a subtle‘collapse’

|ψ〉 = 1√2

(|e〉 + |g〉) → |e〉. (2.3)

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This shift from superposition to a definite state repeats itself periodically.1 Eventually, the cyclecomes to an end by detection:

|ψ〉 =[

1√2

(|e〉 + |g〉) ↔ |e〉]

→ |g〉. (2.4)

It is the quantum non-event’s causal efficacy, sometimes equalling that of a positive event, forwhich interaction-free measurement (IFM) [3] provides a lucid example: a detector’s otherwise-impossible click sometimes does occur as a result of another no-click. But then, as the Universecontains countless potential detectors for every excited atom located anywhere, the actual numberof non-events turns out to be astronomically huge. A plethora of even more intriguing phenomenacan therefore be revealed by studying quantum non-events, in both experiment and theory.

(a) The information conveyed by the intervals between Feynman’s clicksRichard Feynman’s genius as a physicist was matched by his inspirational teaching. He famouslyargued that the double-slit experiment ‘has in it the heart of quantum mechanics. In reality, itcontains the only mystery’ [4]. His lecture on this topic employs a stream of single electronsundergoing interference, with two sensitive detectors, each placed next to one of the slits, toindicate which path the electron has taken. His vivid description, best quoted verbatim, is verysuitable for our study of non-events:

The first thing we notice with our electron experiment is that we hear sharp ‘clicks’ fromthe detector (that is, from the loudspeaker). And all ‘clicks’ are the same. There are no ‘half-clicks’.

We would also notice that the ‘clicks’ come very erratically. Something like: click . . . .. click-click . . . click . . . . . . .. click . . . . click-click . . . . . . click . . . , etc., just as you have, no doubt, hearda Geiger counter operating. If we count the clicks which arrive in a sufficiently long time—say for many minutes—and then count again for another equal period, we find that the twonumbers are very nearly the same. So we can speak of the average rate at which the clicksare heard (so-and-so-many clicks per minute on the average). [4, pp. 1–5]

These clicks are, by nature, indeterministic. But what about the intermediate non-clicks? This isactually how IFM [3] was conceived: omit one detector, and, lo and behold, interference vanishesjust as before.

Non-events, then, not only far outnumber the events in our Universe, they contain a significantamount of quantum information, and take a major causal portion in every physical process.

(b) The simplest non-event offering a new insightEinstein’s advice ‘Make things as simple as possible, but not simpler’ may offer further help. Thefollowing experiment provides the simplest quantum interaction that can shed new light on thenature of non-events.

Consider an electron and a positron, both with spin state |z+〉 = (1/√

2)(|x+〉 + |x−〉), split bytwo different Stern–Gerlach magnets positioned at (t0, xe− , y0) and (t0, xe+ , y0), respectively, toenable the asymmetric interaction in figure 1. The magnets split the particles’ paths accordingto their spins in the x-direction:

|ψe−〉 = 1√2

(∣∣1′e−

⟩ + ∣∣1′′e−

⟩)and |ψe+〉 = 1√

2

(∣∣2′e+

⟩ + ∣∣2′′e+

⟩). (2.5)

Let care be taken to ensure that, should the particles turn out to reside in the intersecting paths,they would mutually annihilate.

1This is analogous to a photon’s wave function undergoing position measurement upon passing a slit and immediatelyspreading again by diffraction, until the final detection of the photon.

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|CLICKÒ1 |READYÒ1

|READYÒ2 |CLICKÒ2

1¢ 1≤ 2¢ 2≤(a)

(b) (c)

(d)

e+e–

2¢1≤2¢1¢

e+e+

e+e–

e–e–

2≤

Figure 1. Possible electron–positron interactions and their possible outcomes. (a) The setting, (b,c) annihilation and (d)oblivion. (Online version in colour.)

Let us follow the time evolution of these two wave functions plus two nearby detectors|READY〉1, |READY〉2, set to measure the photons emitted upon pair annihilation, which wouldchange their states to |CLICK〉1 or |CLICK〉2.

Initially, the total wave function is the separable state:

|ψ〉 = 12

(∣∣1′e−

⟩ + ∣∣1′′e−

⟩) (∣∣2′e+

⟩ + ∣∣2′′e+

⟩) |READY〉1|READY〉2. (2.6)

The particles, depending on their positions at t1 or t2, may (not) annihilate and consequently (not)release a pair of photons, which would in turn (not) trigger one of the detectors.

At t0 ≤ t< t1, then, the superposition is still as in equation (2.6). But at t1 < t< t2,either a photon pair is emitted (figure 1b), indicating that the system ended up in∣∣1′′

e−⟩ ∣∣2′

e+⟩ |CLICK〉1|READY〉2, or not, thereby

|ψ〉 = 1√3

[(∣∣1′e−

⟩ + ∣∣1′′e−

⟩) ∣∣2′′e+

⟩ + ∣∣1′e−

⟩ ∣∣2′e+

⟩] |READY〉1|READY〉2, (2.7)

which is a superposition of an interesting type: one component of it is a superposition in itself.Similarly, at t> t2, if a photon pair is emitted (figure 1c), we know that the particles ended

up in paths 1′ and 2′ :∣∣1′

e−⟩ ∣∣2′

e+⟩ |READY〉1|CLICK〉2. Otherwise, however (figure 1d), we find the

product state:

|ψ〉 = 1√2

(∣∣1′e−

⟩ + ∣∣1′′e−

⟩) ∣∣2′′e+

⟩ |READY〉1|READY〉2, (2.8)

which is peculiar. The positron is observably affected: its momentum has thus changed, because ifwe time-reverse its splitting, it may fail to return to its source. Not so with the electron: it remainssuperposed, hence its time-reversibility remains intact (figure 2).

This is QO [1], where one party of the interaction ‘remembers’ it through momentum change,whereas the other remains unaffected, in apparent defiance of momentum conservation.

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e+

e+

e–

e–

Figure 2. The peculiar momentum exchange due to outcome (d) in figure 1: the positron’s interference is changed while thatof the electron remains intact. (Online version in colour.)

(c) The critical interval: the non-event was not that silent after allIt is obviously the intermediate time interval t1 < t< t2 that conceals the momentum conservationin QO. The details, however, are not trivial.

The two particles, during this interval, become partly entangled in both positions andmomenta. Should one be found on the left path, the other will be on the right. The momentumentanglement is even more interesting: suppose that within the interval we reunite the two halvesof each particle’s wave function through the original beam splitter, to see whether they return totheir source. Either one of the particles may fail to do that; in which case the other must not. Thisentanglement is identical to that of the electron–positron pair in Hardy’s experiment [5]:

|ψ〉 = 1√3

[(∣∣1′e−

⟩ + ∣∣1′′e−

⟩) ∣∣2′′e+

⟩ + ∣∣1′e−

⟩ ∣∣2′e+

⟩]. (2.9)

The state underlying this entanglement is a higher order superposition in that it is composedof a superposed plus a non-superposed state. Equally interesting is this state’s sequel,which is unentanglement. The difference from the familiar ‘disentanglement’ is fundamental:disentanglement is the natural consequence of entanglement, e.g. the Einstein–Podolsky–Rosen(EPR) correlations following spin measurements. Unentanglement, in contrast, creates thedeceptive impression that entanglement never took place. Rephrased in more familiar terms,oblivion is the process where the wave function undergoes momentary decoherence followedby ‘recoherence’. Notice that, in contrast to the familiar decoherence induced by the macroscopicenvironment, which is usually believed to be irreversible, here decoherence temporarily ensuesby the interaction’s mere potential to become macroscopic. Nature, so to speak, takes back itsassertion.

(d) Oblivion’s ubiquity: every detector’s pointer must be superposed in the conjugatevariable

We now submit our main argument. Rather than a curious interaction, Oblivion is part and parcelof every routine quantum measurement. Its elucidation can therefore shed new light on the nature ofmeasurement, further enabling some novel varieties of it.

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A1 A2

1¢ 1≤(a)2¢ 2≤

A2A1

(c) 1¢ 1≤2¢ 2≤

A2A1

2≤≤1≤≤

1≤¢ 2≤¢(b) 2¢ 2≤

Figure 3. Similar set-up to that in figure 1 butwith twoatoms that donot annihilate upon interactionbut (a)merely formelasticcollision. In this version, the critical interval is the detector’s long exposure time which does not allow the precise detectiontime. (b) Measurement ending up with collision, where the nearby detectors’ widths signify that A1’s and A2’s positions remainsuperposed. (c) IFM. (Online version in colour.)

Recall first the trivial preparation required by quantum measurement, e.g. in the commonposition measurement.2 The detector’s pointer, positioned at a specific location, reveals theparticle’s presence in that precise place by receiving momentum from it. This naturally obligesthe pointer to have precise (preferably 0) momentum. In return, however, the pointer’s positionmust be highly uncertain.

Let this trade-off be studied with the aid of our first experiment (figure 1) with onemodification. In the original version, the experiment’s two possible interactions are annihilations,which are mutually exclusive events. So let us replace annihilation by mere collision (figure 3):two superposed atoms A1 and A2 undergo splitting and interactions exactly like the electron andpositron in figure 1. These atoms, unable to annihilate, can only collide, which can now happen onboth possible occasions at t1 and t2, namely at the two locations, where A2 can turn out to reside.Instead of photon detectors, therefore, we now detect the atoms themselves.

This experiment’s version has greater sensitivity. Suppose that the atom detector on path2′′ (positioned closer to the beam splitter; figure 3b) remains silent: we thereby know that the

2It is also worth noting that position measurement is part of nearly all standard quantum measurements, e.g. spinmeasurement necessitating two detectors placed within the Stern–Gerlach magnet.

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two atoms have collided, yet remain oblivious about this collision’s location. What we thusmeasure is the ordinary momentum exchange: both atoms’ momenta have been reversed alongthe horizontal axis.

Here, oblivion is negligible, affecting only the two atoms’ positions which undergo only littlechange. Yet, because A2 has vanished from the macroscopically distant location on 2′′, the finaloutcome is a position measurement of A2.

Conversely, we may fail to detect A1 and A2 in paths 1′′′ 1′′′′ and 2′′′ 2′′′′, to which they wouldhave been diverted in case of collision. We are now certain that atom A2 resides on path 2′′, whileA1 has returned to its initial superposition over 1′ and 1′′ (figure 3c). In other words, A2 undergoesmomentum oblivion, which is again an ordinary position measurement of A1, but this time themeasurement is interaction free—a point studied in greater detail below.

To summarize: we have studied an asymmetric interaction between two atoms, where twohalves of A1’s wave function interact with one half of A2, under great time and positionresolution. Two momentum exchanges can occur between the two atoms:

(i) A2 turns out to have collided with A1. This amounts to A2 undergoing positionmeasurement. The price exacted by the uncertainty principle is a minor position oblivionof both A1 and A2.

(ii) A2 has not collided with A1. This again amounts to A2 undergoing positionmeasurement—‘collapsing’ to the remote 2′′ path. Here, however, a macroscopicmomentum oblivion plagues A1, making the measurement interaction free.

Both (i) and (ii) occur under an unusually high space and time resolution, enabling a novel studyof the critical interval. During this interval, entanglement between the two atoms has ensued, asthey have assumed new possible locations

|ψ〉 = 12

[|1′′′〉|2′′′′〉 + |1′′′′〉|2′′′′〉 + (|1′′′〉 + |1′′′′〉)|2′′〉] , (2.10)

which has remained undistinguished until the macroscopic detection which finalized theinteraction and sealed the oblivion.

The generalization is natural. During every quantum measurement, the detector’s pointerinteracts with a particle in the same manner as atoms 1 and 2 in the above setting: both themeasured particle and the measuring pointer are superposed, but their wave functions interactasymmetrically: the particle’s wave function’s part interacts with the pointer’s wave function’swhole. To make the analogy more realistic, recall that in reality the pointer’s superposition iscontinuous, smeared over an array of locations, which is much wider than the two discretelocations in the former case. Momentum measurement thus becomes much more precise. Thistransition from discrete to continuous superposition also opens the door for several interestinginterventions [1,2].

3. The clue of Escher’s self-drawing hands: use time to bypassspatial restrictions

The above account of QO is a purely orthodox one, based on standard quantum theory alone.It thus says nothing about the ‘decision’ of which of all possible particle–pointer interactionswill eventually be materialized, annihilating all its competitors. Such an explanation lies, ofcourse, outside of quantum formalism, which is the very reason for the conception of severalinterpretations of QM. In our case, it turns out that two of the most radical interpretations offerthe best clues for a deeper account of quantum measurement in general and oblivion in particular.

Taking an example from M. C. Escher may help us to elucidate our proposal. This artist hasearned his reputation by paintings of impossible causal loops, such as a pair of hands drawingone another, or a waterfall where the water is enclosed into an eternal circle, etc. These picturesdo not challenge our physical intuition because we know that the artist has proceeded through

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leftpolarization

measurement

rightpolarization

measurement

offerwave yR

offerwave yL

confirmationwave yL*

confirmationwave yR*

polarization-entangledtwo-photon source

x

t

Figure 4. Transactions between past and future events in an EPR experiment. (Online version in colour.)

normal causality to create the paradoxical causal chain. In other words, the apparently consecutivestages that we see in the picture were drawn in a different order in time.

Is it possible that the causal anomalies manifested by quantum uncertainty and non-localityowe their existence to a somewhat similar creation? This is the spirit of the time-symmetricinterpretations considered below.

(a) The transactional interpretation: each quantum interaction is affected by the past aswell as the future

The following is a brief introduction of Cramer’s transactional interpretation of QM [6,7]. Anyquantum event that involves the exchange of conserved quantities (energy, momentum, angularmomentum, etc.), and that can be represented by a matrix element, is considered to have beenformed in three stages:

1. An ‘offer wave’ (the usual retarded wave function ψ(x) or a ket |ψ〉) originates from the‘source’ (the object supplying the quantities transferred) and spreads through space–timeuntil it encounters the ‘absorber’ (the object receiving the conserved quantities).

2. The absorber responds by producing an advanced ‘confirmation wave’ (the complexconjugate wave function ψ∗(x) or a bra 〈ψ |) which travels in the reverse-time directionback to the source, giving rise to the density ψ∗ψ .

3. The source chooses between the possible transactions {i} based on the strengths of theψi

∗ψi echoes it receives, and reinforces the selected transaction repeatedly until theconserved quantities are transferred and the potential quantum event becomes real.

It is quantum non-locality where the transactional interpretation offers its best reward. It takesadvantage of the trivial effect that entangled systems, though spacelike-separated, have beenlocally connected in the past. The quantum transaction, along the resulting space–time zigzag,thus makes the effect natural, as in Cramer’s own illustration for the EPR experiment (figure 4).An interesting variant of the transactional interpretation (TI) is Kastner’s possibilist transactionalinterpretation [8], which deals with several consistency and relativistic issues left open in theoriginal version. Indeed, Cramer’s own revised version [7] seems to respond to these challenges.

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(b) The two-state-vector formalism: the combined causal effect of past and future statescan yield odd values

Despite several important differences, the time-symmetric idea of TI appears also in Aharonov’stwo-state-vector formalism (TSVF). Its basic assumption was proposed by Aharonov et al. [9]:every quantum system is determined by two wave functions. One (also known as the pre-selectedwave function) evolves forward in time while the other (post-selected) evolves backward. Theforward- and backward-evolving wave functions, |ψ〉 and 〈φ|, respectively, define the so-calledtwo-state vector 〈|φ|ψ〉. Both are equally important for describing the quantum system in betweenthe pre- and post-selection via the weak value of any operator A defined by

〈A〉w = 〈φ|A|ψ〉〈φ|ψ〉 . (3.1)

Here a logical catch ensues: ‘the state between two measurements’ cannot, by the very nature ofquantum measurement, be revealed by measurement. Weak measurement [10,11] was conceivedin order to bypass this obstacle as well as to test other TSVF predictions. This led to numerousintriguing works, both theoretical and experimental. In what follows we demonstrate its valuefor the understanding of non-events.

(c) Why time-symmetric causality?Our reasons for opting for these two time-symmetric interpretations have been explained inseveral earlier works. Although their novel pictures of quantum reality are still incomplete, attimes even raising new questions, they offer very elegant explanations for well-known quantumparadoxes, as well as some novel ones, presented also in earlier works of ours [12].

Such is the present case. Our analysis of QO has revealed a subtle quantum interactionthat rigorously applies for quantum measurement in general, with a particular insight into theintriguing causal efficacy of IFM: the interaction between the quantum and the macroscopic pointerturns out to consist of a feeble occurrence of the event, followed by its ‘unoccurrence’. It is therefore onlynatural to ask whether these antagonistic stages occur during the above back-and-forth exchangesbetween future and past.

4. The complementary clue: erasure must be added for makingthe deception perfect

But it was not only for its elegance that we have found time-symmetric quantum causality tonaturally integrate with our findings. Both TI and TSVF, for their own reasons, had to invokeunique values such as negative energy and negative momentum as parts of the interactions acrosstime. This bring us back to the visual analogy we employed earlier: when Escher was drawing hismasterpieces, he was not only going back-and-forth in terms of the paradoxical causality he waspresenting; without doubt, he also used a great deal of erasure in the process. With the appropriatecaution needed when following analogies, let us follow this one and ask whether Nature is alsorefining her quantum feats this way.

(a) Negative energy exchange in transactional interpretationCramer [6] drew his initial inspiration from the Wheeler–Feynman (W-F) ‘absorber theory’ ofelectromagnetism. They invoked retarded and advanced waves going between the source and theabsorber, such that, due to destructive interference, all the intermediate anomalies of advancedwaves eventually disappear. Cramer found it natural to follow this reasoning into the quantumrealm:

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The emitter can be considered to produce an ‘offer’ wave F1 which travels to the absorber.The absorber then returns a ‘confirmation’ wave to the emitter and the transaction iscompleted with a ‘handshake’ across space-time. To an observer who had not viewedthe process in the pseudo-time sequence employed in the above discussion, there is noradiation before T1 or after T2 but a wave travelling from emitter to absorber. This wave canbe reinterpreted as a purely retarded wave because its advanced component G2, a negativeenergy wave travelling backwards in time from absorber to emitter, can be reinterpreted asa positive energy wave travelling forward in time from emitter to absorber, in one-to-onecorrespondence with the usual description.Thus the W-F time symmetric description of electrodynamic processes is completelyequivalent in all observables to the conventional electrodynamic description. Time-symmetric electrodynamics, in both its classical and quantum mechanical forms, leads topredictions identical with those of conventional electrodynamics. [6, p. 661]

(b) Weak quantum values can be odd, even negativeNext comes what may be TSVF’s most intriguing prediction. Consider the valuesstraightforwardly predicted for mass and momentum in the following experiment. The threeboxes paradox [13] is a by-now familiar surprise yielded by TSVF. A particle is prepared withequal probability to reside in one out of three boxes:

|ψ〉 = 1√3

(|1〉 + |2〉 + |3〉). (4.1)

Later, it is post-selected in the state

〈φ| = 1√3

(〈1| + 〈2| − 〈3|). (4.2)

What is the particle’s state between these two strong measurements? By definition, anothermeasurement of this kind is unsuitable for answering this question, as it would reveal the state uponthe intermediate measurement. It is weak measurement, again, that comes to help. TSVF predictsthe following weak values of the projection operators, Πi ≡ |i〉〈i| for i = 1, 2, 3:

〈Π1〉w ≡ 1,

〈Π2〉w ≡ 1

and 〈Π3〉w ≡ −1.

⎫⎪⎪⎬⎪⎪⎭

(4.3)

Therefore, the total number of particles is 1, as it should be, but in this case 1 is a sum of 2 ordinaryparticles plus 1 odd. That is, the first two boxes contain a particle with certainty, while the third boxcontains a particle with an apparent ‘negative probability’. Can this odd mathematical value begiven physical meaning? If we choose to trust the formalism,3 we can assign the negative valueto mass. In this case, the last equation denotes a negative weak value for the particle’s very existence inthe third box. In order to fully grasp the paradoxical nature of this term, let us consider within thecontext of its standard versions:

IF ‘P = 1’ means ‘the particle certainly resides within this box’;‘P = 0’ means ‘the particle has never resided within this box’;

THEN ‘P = −1’ means ‘the particle certainly unresides within this box’.

As absurd as the latter expression may sound, we find it mathematically simple and physicallysignificant (i.e. weak interactions with the third box would flip their sign in this specific pre- andpost-selections), the alternative being dismissing it as having ontological meaning, by relying

3Admittedly, there is no consensus on this point. We, for example, support the more physical view on weak valuesrepresenting real properties of single particles in pre-/post-selected ensembles, while Kastner [14], for instance, suggeststhat weak values represent mere amplitudes and should be applied to statistical ensembles rather than to single particles.

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on forward evolving interference and retrodiction.4 The choice to trust the mathematics has ledAharonov et al. [15] to assign a negative sign to every interaction involving the third box, as longas it is weak enough. Obviously, this cannot be done to the particle’s charge, as charge plays norole in this case. The remaining choice therefore is mass. The simplest way to prove this predictionis through the particle’s momentum: a collision with another particle must give the latter a ‘pull’rather than a ‘push’, even though their initial velocities were opposite. Moreover, since weakvalues are additive, if one is interested to know, for instance, what is the net interaction with thesecond and third boxes, one will find the 1 − 1 = 0 net interaction which motivated the title of thispaper. A preliminary empirical test of this prediction has already yielded promising results [16].

In passing, it is worth comparing this step of the TSVF with Dirac’s choice to trust themathematics upon encountering the negative value for the electron’s charge, following the four-spinor solution of his famous equation. That choice later led to the discovery of the positron. Thismay be the case with the present choice as well.

(c) Weak, odd and negative: but not necessarily rareTo the extent that the above weak values are indeed part of quantum reality rather than amere mathematical curiosity, one cannot help doubting whether their existence is limited to theprobabilistic fringe that allows them, namely under very unique pre- and post-selections. If wedo not observe such values under ‘ordinary’ quantum interactions, it may be because they do notexist there, but there is also another possibility which merits consideration: perhaps even a normalquantum value is the sum of several odd quantum values.

We are well aware of the extent of speculation in which we are now indulging towards the endof this work. Not only do we favour the highly unorthodox account of time-symmetric quantumcausality, but we are willing to ascribe it exchanges of negative momentum and mass to accountfor the fact that, despite its spatial and temporal oddities, the quantum interaction does obeyconservation laws.

5. From information to ontologyHow, then, can non-events play a part in the quantum process? The challenge’s acuity is reflectedin the various radical moves it has elicited from the rival interpretations of QM. Among these,two famous schools have resorted to the following opposite extremes:

(i) Abandonment of ontology: Copenhagen. Since counterfactuals are facts of our knowledge,just like actual facts, let us define QM as dealing only with knowledge, information, etc.,rather than with objective reality.

(ii) Excess ontology: Many Worlds. The counterfactual does occur, but in a different world, splitfrom ours at the instant of measurement.

Our option is moderate ontology. It does not seek shelter in strict positivism, which can seldombe falsified yet is equally barren. It rather indulges into a modicum of speculation with itsinherent risk of being proven wrong. Our inspiration comes from Wigner’s seminal essay ‘Theunreasonable effectiveness of mathematics in the natural sciences’ [17]. Following a long line ofthinkers ever since Plato, Wigner mused that ‘The miracle of the appropriateness of the languageof mathematics for the formulation of the laws of physics is a wonderful gift which we neitherunderstand nor deserve’. What, after all, is so unique about science? It is the fact that mereinformation, especially mathematical, can yield, with appropriate manipulation of its symbols,additional information, which may strike us as odd or even wrong. Yet, if we take it seriously, wemay find that Nature indeed possesses what this new information conveys!

4This perfectly reasonable approach was recently suggested to us via private communication from W. M. Stuckey (2015).

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Our moderate ontology concerns the nature of time. It is from the present physical andmathematical attributes of time that we seek to derive a new property. Our first derivation(which is, let it be stressed again, a rigorous one) is QO [1], whereby quantum events happenand sometimes also unhappen. This ontology offers, paradoxically, a demystifying account of thequantum world [2]. We now dare to speculate further into the nature of time and endorse the ideaof Becoming [18,19]; namely, the genuine emergence of events together with their surrounding space–time [18,20]. It is within this Becoming, our speculation goes, that happening and unhappeningcan most naturally occur.

Becoming, however, is notoriously hard to reconcile with general relativity, which seems toportray a block universe where all events—past, present and future—have the same degree ofexistence, just like events in different locations. It is therefore significant that even ardent blockuniverse adherents, such as Stuckey et al. [21], resort to their own moderate ontology. They invoke‘space–timesource elements’ which are presented as more fundamental than space–time. This, webelieve, is not so remote from Becoming, especially its recent moderate variants in the works ofCramer [6,7] and Kastner [8]. Something that is as yet unknown about time may be waiting to berevealed.

6. SummaryHow, in conclusion, can an event which has never occurred but just could have exert observableeffects? Our analysis of QO has revealed some elusive intermediate stages in the informationyielded by measurement, which can sometimes end up in nearly perfect self-cancellation,still leaving some traces in neighbouring systems. We then found it natural to follow thetransactional interpretation and the two-state-vector formalism of QM in search of a morecomprehensive account of this process. Allowing ourselves one more unorthodox hypothesis,we suggest that Becoming may be the fundamental ingredient of physical reality which makessuch phenomena possible. Several insights emerging from these works, and moreover someexperimental predictions of the TSVF, converged into a picture in which both advanced andretarded actions form the quantum event, where even physical values known so far to be onlypositive, such as energy and mass, can also change their sign.

Whether the luring analogies which have guided us were genuine, the burden of proof remainswith us, to be addressed in works to follow.

Authors’ contributions. Both authors contributed substantially to the ideas, contents and presentation of this work.Both authors gave their final approval for publication.Competing interests. We do not have any competing interests with regard to this manuscript.Funding. E.C. acknowledges support from ERC-AD NLST.Acknowledgements. We are grateful to Yakir Aharonov, William Mark Stuckey and Ruth Elinor Kastner forinspiring discussions and helpful comments.

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