It is pointed out that a slight variation on the Wheeler Delayed Choice Experiment presents the same challenge to orthodox quantum mechanics as Maudlin-type contingent absorber experiments present to the Transactional Interpretation (TI). Therefore, the latter cannot be used as a basis for refutation of TI.

This paper discusses the famous “Delayed Choice Experiment” (DCE) proposed by Wheeler [

TI is a time-symmetric interpretation in which outcomes (and their associated probabilities) supervene on an interaction between an emitter and all absorbers accessible to it. In TI, the emitter emits an “offer wave” (OW) corresponding to the standard (retarded) solution to the Schrodinger Equation, and the absorbers respond to that OW by generating advanced “confirmation waves” (CWs) which correspond to solutions of the complex conjugate Schrodinger equation. The Maudlin experiment can be termed a “Contingent Absorber Experiment” (CAE) in that the placement of at least one absorber is contingent on a previous outcome between the emitter and a nearby absorber. The contingent nature of the more distant absorber placement (together with its various implications) has been seen as a serious challenge to the consistency of the TI approach. It is argued here that the DCE, with a slight variation, presents essentially the same challenge to standard quantum mechanics as Maudlin-type CAEs present to TI.

Let us first review Maudlin’s CAE, illustrated in Figure

The Maudlin experiment.

As illustrated in Figure ^{1}

Maudlin’s challenge has two main features: (1) it seems to involve a situation not amenable to the usual “echoing” picture as given in Cramer [^{2}

The short response to objection (2) is that the weights of incipient transactions are

Objection (1) is the issue more likely to seen as problematic for TI: it seems to thwart the idea of a clearly defined competing set of incipient transactions as described in Cramer [^{3}

Cramer [^{4}. In the Miller experiment, a photon is split by a half-silvered mirror into two beams

Before considering the resolution of these objection (1) issues, let us return to objection (2) to see in more detail how TI’s probabilities are indeed well-defined in CAEs. In the Maudlin experiment, if there is really no other absorber for the OW component heading toward the left, theoretically there may be no incipient transaction on the left, but we may still define the relevant probabilities by taking the total sample space as consisting of the outcomes (Yes, No) for the question “Is the particle detected on the right”? That is, actualization of the incipient transaction between the emitter and

For the more general case, note that we can do a “spectral decomposition” of any observable, that is, we can express it in terms of a sum of the projectors onto its eigenstates. In TI these mathematical projectors represent physical incipient transactions, and the density operator for any emitted OW in terms of the absorbers actually available to it corresponds physically to the set of weighted transactions^{5}^{6}

However, as promised, we return to the issue of an apparent conflict between the “states of the emitter” in the different apparent causal loops presented by such experiments. The way to deal with this issue is by noticing that a similar conundrum already appears in standard, orthodox quantum theory in delayed-choice experiments. Here is where we take note of the observation by Stapp [^{7}

The classic presentation of the DCE is as follows (see Figure

The Delayed Choice Experiment.

Note that the photon has already passed the plane of the slits before the observer has decided whether to measure “which slit” or not. Thus, at a time ^{8}^{9}

The reader may still think that the above indeterminacy of the photon state is just the usual (relatively benign) quantum indeterminacy. However, this is not the case; the delayed choice experiment also presents a “causal loop” problem for standard quantum mechanics, as follows. In the usual language, at ^{10}

Now, one might just conclude that this implies that there is no genuine freedom of choice and the comparison between the two types of experiments (delayed choice versus contingent absorber) ends there. However, we can sharpen the “causal loop” aspect of the delayed choice experiment by proposing that the experimenter does not choose but instead bases removal of

To summarize, in the TI case, the alleged inconsistencies presented by the CAEs are (1) an apparent lack of “fact of the matter” concerning which CWs are present at

Thus we see that the delayed choice experiment (especially when augmented to make the choice dependent on a quantum outcome) seems to raise the same sort of causal loop conundrum that has been used as a basis for criticism of TI. The point here is that

CAEs gain traction as apparent threats to TI based on the idea that there must be a “fact of the matter” about what CWs exist ^{11}

Contingent absorber experiments (CAEs) used as “counterexamples” to the TI picture present essentially the same conceptual challenge as do delayed choice experiments (DCE) for standard quantum mechanics, so the former, therefore, cannot legitimately be used as a basis for rejection of TI. Standard quantum mechanics, not just TI, entails that certain features of the past are indeterminate and can only become determinate based on events occurring in the present. This ontological indeterminacy applies to the photon in the delayed choice experiment just as much as it does to TI’s confirmation waves.

Elitzur ([

TI does not adopt this ontology, since an emitter is not described by a particular CW, but we understand Berkovitz to be attempting to be precise about the circumstances surrounding CW generation, so we will use his terminology in this discussion.

But see Kastner [

Strictly speaking, it could be argued that in a direct action theory (upon which TI was originally based), all photons are “virtual” and can, therefore, have finite (if nearly zero) rest mass. We neglect that aspect here, but it could be a route to saving the “hierarchy” approach.

Specifically, in the expression

The weight is the Born Rule, which in TI is simply the product of the amplitudes of the OW and CW comprising each incipient transaction; see the previous note.

This work does not address the aspect of Stapp [

This is the standard time-symmetric ABL Rule [

In the TI picture of the DCE, the photon’s OW is perfectly well defined; it is only its CW that is not well-defined. This is arguably a simpler way to understand the DCE. That point has previously been made in Kastner [

Although it might be argued that the Copenhagen interpretation would not countenance

This locution obviously implies an “A-series” view of time. This aspect will be more fully addressed in a separate work. For purposes of this presentation, it may be noted that the “B-series” or ‘block universe” view is part of the problem in that it implies causal loops that need not exist in an “A-series” picture. That is, the photon’s status may be indeterminate at