# A Delayed Choice Quantum Eraser Explained by the Transactional Interpretation of Quantum Mechanics

## Abstract

This paper explains the delayed choice quantum eraser of Kim et al. (A delayed choice quantum eraser, 1999) in terms of the transactional interpretation (TI) of quantum mechanics by Cramer (Rev Mod Phys 58:647, 1986, The quantum handshake, entanglement, nonlocality and transactions, 1986). It is kept deliberately mathematically simple to help explain the transactional technique. The emphasis is on a clear understanding of how the instantaneous “collapse” of the wave function due to a measurement at a specific time and place may be reinterpreted as a relativistically well-defined collapse over the entire path of the photon and over the entire transit time from slit to detector. This is made possible by the use of a retarded offer wave, which is thought to travel from the slits (or rather the small region within the parametric crystal where down-conversion takes place) to the detector and an advanced counter wave traveling backward in time from the detector to the slits. The point here is to make clear how simple the transactional picture is and how much more intuitive the collapse of the wave function becomes if viewed in this way. Also, any confusion about possible retro-causal signaling is put to rest. A delayed choice quantum eraser does not require any sort of backward in time communication. This paper makes the point that it is preferable to use the TI over the usual Copenhagen interpretation for a more intuitive understanding of the quantum eraser delayed choice experiment. Both methods give exactly the same end results and can be used interchangeably.

## Keywords

Transactional interpretation Advanced waves Delayed choice Quantum eraser## 1 Complementarity, Which Path Information and Quantum Erasers

*contains nearly all mysteries*of quantum mechanics, namely, wave–particle duality, particle trajectories, collapse of the wave function and non locality. We may see interference, or we may know through which slit the photon passes, but we can never know both at the same time. This is what is commonly referred to as the principle of complementarity. We say two observables are

*complementary*if precise knowledge of one implies that all possible outcomes of measuring the other are equally likely. The fundamental enforcement of complementarity arises from correlations between the detector and the interfering particle in a way that show up in the wave function for the system. It is not, as some undergraduate text books would have you believe, a consequence of the uncertainty principle, although the application of the uncertainty principle makes for an easy calculation when the wave function of the system is difficult to write out. There have been many

*gedanken*(German for thought) experiments over the years to show complementarity. The most famous are the Einstein recoiling slit, Feynman’s light scattering scheme both discussed in Feynman’s lectures on physics [1] and Wheeler’s delayed choice experiment [2]. The TI of Cramer is preferred in this paper, over the usual Copenhagen interpretation (CI), for a more intuitive understanding of the quantum eraser delayed choice experiment. An alternative theory, which also claims to present an alternative perspective to the CI and provides an intuitive understanding of the paradoxes of quantum mechanics, including the quantum eraser and delayed choice, is that by Sohrab [3, 4] which we mention for completeness, but will not discuss further there.

Of particular interest here is the delayed choice quantum eraser *gedanken* experiment by Scully and Druhl [5]. This work described a basic quantum eraser experiment and a delayed choice quantum eraser arrangement. The basic quantum eraser experiment is described using two 3-level \(\Lambda \)-type atoms [6], in the place of two slits. See Fig. 1. The atoms start off in the ground state and then a laser pulse comes in and excites either atom *A* or *B*. The excited atom then decays and emits a signal photon. Interference fringes are sought between these signal photons on a screen some distance away. Let the identical 3-level atoms have one upper level *a* and two lower levels *b* and *c*. The laser excites one of the atoms up to the level *a* but the atom can de-excite to either state *b* or *c*. If both atoms start off in the ground state *c*, there are two possibilities. The excited atom decays and falls back to level *c*, so the excited atom becomes indistinguishable from the other atom which was not excited. In this case we would expect to see an interference pattern since there is no which path information. In the second case, the excited atom drops to level *b* which is distinguishable from level *c*. In this case we have which path information and we would get no interference pattern. That describes the basic quantum eraser.

For a delayed choice quantum eraser [5], the 3-level atoms change to 4-level atoms with levels *a*, *b*, *c*, *d* , with *d* the ground state. See Fig. 2. Instead of one exciting laser pulse there are two closely spaced pulses, which will both go to the same atom. The first laser pulse excites either atom *A* or *B* from the ground state *d* to the upper level *a*. The excited atom then spontaneously decays to *c* emitting the signal photon. The second laser pulse then excites the atom from level *c* to level *b*, which then decays with the emission of a lower energy *idler* photon to the ground state. Now the atoms are inside a cleverly constructed cavity with a trap door separating them. The cavity is transparent to the signal photons and laser light but strongly reflects the idler photons. There is a detector capable of detecting the *idler* photons only near atom *A*. The trap door will prevent the *idler* photon from *B* being detected. Now we have a choice whether to open the trap door or leave it closed. The signal photon detection is now correlated with the idler photon detection. The experiment has become a delayed choice quantum eraser, whether we see interference or not will depend on whether we leave the trap door open or closed. If the trap door is closed and we detect an idler photon, we know that atom *A* was excited. If we do not detect a photon then atom *B* was excited, either way we have which path information that will destroy the fringes. If the trap door is open, then we no longer have which path information since either atom could have emitted the idler photon. In principle the decision, to leave the trap door open or closed, can be made *after* the signal photons have been detected. The *paradox* is, how does the signal photon know which pattern to make, a single slit diffraction pattern or a two-slit interference pattern, if we have not yet decided to leave the trap door open or closed?

Englert et al. [7, 8] in 1991 constructed a very nice atom interference *gedanken* experiment that shows the theory in a very straightforward manner, although the experiment would be extremely difficult to perform in practice. Soon afterward in 1993, a polarization experiment by Wineland’s group [9], was the first to demonstrate an actual realization of the Scully–Druhl quantum eraser *gedanken* experiment. They used mercury ions in trap as the two “atoms” and observed linear \(\pi \) and circularly \(\sigma \) polarized light. Choosing to detect linear polarized light, corresponded to the case that the ions in a trap were in the same initial and final state. This implies that there was no which path information and so there was interference. Choosing to observe circular polarized light, corresponded to the situation that the ions were in distinguishable end states after scattering a photon, so which path information was available and hence there was no interference. You could choose to observe interference or not depending on whether you chose to observe linear or circularly polarized light.

There have been many quantum optics experiments involving two photon entangled states and quantum eraser arrangements to demonstrate the complementarity arguments above. Four of the better ones are [10, 11, 12, 13]. One experiment in particular by Zeilinger’s group [14] is worthy of a special note. The arm lengths in their apparatus were very long, between 55 m up to 144 km. They point out that there is no possible communication between one photon and the other in the entangled pair because of the space-like separation between them and they assume no faster-than-light communication is possible.

The most famous real experiment of the delayed choice type is that by Kim et al. [15], using parametric down conversion entangled photons. It has drawn considerably more press than any other experiment of this type and even has a couple of online animations [16]. We choose to present our case for the transactional interpretation (TI) of quantum mechanics using the Kim experiment as our example, but any of the delayed choice quantum erasers would work just as well.

## 2 Introduction to the Transactional Interpretation of Quantum Mechanics

The TI of quantum mechanics was proposed by Cramer [17] in a review article in 1986 and a short overview in 1988 [18]. More recently Cramer has written a book [19] which should become available early in 2016. It is a way to view quantum mechanics that is very intuitive and easily accounts for all the well known quantum paradoxes, Einstein Rosen Podolsky (EPR) experiment [20], which-way detection and quantum eraser experiments, [21, 22]. Unfortunately, it has garnered little support over the years and has fallen off the radar. It deserves a much broader dissemination and part of the motivation to publish this paper was to bring Cramer’s ideas, and the advanced wave concept, to the attention of the younger generation of physicists, who may not have heard of them before. The advanced wave is a standard solution of relativistic wave equation and was utilized by such notable physicists as Dirac, Wheeler, Feynman, Davies, Hoyle and his doctoral student Narlikar. The direct particle interaction theory (which uses advanced waves, traveling backward in time) was used by Wheeler, Feynman, Schwinger, Hoyle and Narlikar. The direct particle interaction does away with the idea of a field, the vacuum field then would be truly empty, with zero energy, as Feynman believed. Frank Wilczek recounts a conversation with Feynman [23].

Around 1982, I had a memorable conversation with Feynman at Santa Barbara. Usually, at least with people he didn’t know well, Feynman was “on” – in performance mode. But after a day of bravura performances he was a little tired and eased up. Alone for a couple of hours, before dinner, we had a wide-ranging discussion about physics. Our conversation inevitably drifted to the most mysterious aspect of our model of the world– both in 1982 and today– the subject of the cosmological constant. (The cosmological constant is, essentially, the energy density of empty space. Anticipating a little, let me just mention that a big puzzle in modern physics is why empty space weighs so little even though there’s so much to it.) I asked Feynman, “Doesn’t it bother you that gravity seems to ignore all we have learned about the complications of the vacuum?” To which he immediately responded, “I once thought I’d solved that one.” Then Feynman became wistful. Ordinarily he would look you right in the eye, and speak slowly but beautifully, in a smooth flow of fully formed sentences or even paragraphs. Now, however, he gazed off into space; he seemed transported for a moment, and said nothing. Gathering himself again, Feynman explained that he had been disappointed with the outcome of his work on quantum electrodynamics. It was a startling thing for him to say, because that brilliant work was what brought Feynman graphs to the world, as well as many of the methods we still use to do difficult calculations in quantum field theory. It was also the work for which he won the Nobel Prize. Feynman told me that when he realized that his theory of photons and electrons is mathematically equivalent to the usual theory, it crushed his deepest hopes. He had hoped that by formulating his theory directly in terms of paths of particles in space–time – Feynman graphs – he would avoid the field concept and construct something essentially new. For a while he thought he had. Why did he want to get rid of fields? “I had a slogan,” he said. Ratcheting up the volume and his Brooklyn accent, he intoned it:

The vacuum doesn’t weigh anything [dramatic pause] because there’s nothing there!

Experimental observations show that the vacuum energy density is in fact very close to zero. To calculate the vacuum energy in quantum field theory, we must admit that spacetime is probably not a continuum but rather has a discrete nature, at quantum dimensions, and only sum the zero-point energies for vibrational modes having wavelengths larger than, the Planck length (\(10^{-35}\) m) and less than or equal to the size of the universe (diameter approx. \(8.8 \times 10^{26}\) m). This gives a ridiculously large but finite vacuum energy density of about \(10^{111}\) J m\(^{-3}\) or in terms of mass density \( 10^{94}\) kg m\(^{-3}\). Clearly, no where near the experimentally observed value for energy density near zero and mass density, near the critical value of \(10^{-26}\) kg m\(^{-3}\). The quantum field theory vacuum mass density is about 120 orders of magnitude too large—rather embarrassing really.

## 3 Absorber theory and Advanced Waves and Direct Particle Interaction Theory

*direct particle interaction*of absorber theory [26]. Paul Davies later generalized these classical results for the relativistic case of absorber theory [27, 28]. Hoyle and Narlikar also worked on the relativistic absorber theory [29]. There are now three different models for absorbers which have slightly differing advanced wave behavior. Wheeler–Feynman [25], Csonka [30] and Cramer [17]. These models differ with regard to what exactly happens when there is a less than perfect absorber present. They are discussed in the very readable paperback by Nick Herbert [31]. So far, we have a working theory for classical electrodynamics and now for QED. Hoyle and Narlikar have also generalized Einstein’s theory of gravitation by using a direct particle interaction. Their theory reduces to Einstein’s general relativity in the limit of a smooth fluid approximation, in the rest frame of the fluid. This has the benefit of completely incorporating Mach’s principle as a radiative interaction between masses, [32, 33]. Cramer spells out the general quantum version of the theory applicable to all systems not just electrons [34, 35].

## 4 Transactional Interpretation

For an interaction to take place between two particles, emitter and the absorber, Cramer says the emitter must send out an offer wave. This offer wave would be half an advanced and half a retarded wave going out in all directions looking for an absorber, something to interact with. When the retarded offer wave reaches the absorber, that particle sends out a counter wave, also half retarded and half advanced. See Cramer’s wiggle diagram, Fig. 3. The advanced counter wave would travel backward in time, along the exact incident path of the original retarded offer wave (it is the complex conjugate of the retarded offer wave), thus constructive interference takes place along the path between the particles. In the one spatial dimension drawn in Fig. 3, the advanced counter wave reaches the emitter particle at the exact time when the retarded offer wave was emitted. This enables the advanced wave from the absorber to exactly cancel with the advanced wave from the emitter at the location of the emitter. Likewise, the retarded wave from the emitter will cancel the retarded wave from the absorber at the location of the absorber. Only the retarded wave from the emitter and the advanced wave from the absorber along the adjoining path are enhanced by the superposition, they do not cancel out. These waves represent the interaction between the particles.

In three spatial dimensions things are a little more complicated. Advanced and retarded waves travel in all directions not just in the direction of one absorber. Retarded waves carry on into the future and maybe absorbed at some later point in time. An advanced wave travels backward in time to the *big bang*. At this point it is reflected and will move forward in time *as an advanced wave* identical to, and \(\pi \) out of phase with, the incident advanced wave. This will produce a cancellation at every point along the world-line back to the point of emission of the wave. All advanced waves therefore cancel out, [36]. Note that the waves are assumed to travel at speed *c* the speed-of-light in a vacuum, although the advanced wave is traveling backward in time, or with -*t* [34]. Basically, in quantum terms, the regular wave function is the offer wave, (or at least the retarded wave part that does not cancel out) the complex conjugate wave function is the confirmation wave (the advanced part moving between the absorber going back in time to the emitter) and together they give a *handshake* [35], which allows an interaction to take place.

Recently Kastner [37] has expounded the virtues of the transactional method with an additional twist allowing for *free will*. There are many examples of the use of the transactional method in the book and it is well worth a read. In this paper we make no distinction between the original Cramer TI and the Kastner version of possibilist transaction interpretation (PTI). Kastner’s approach [38],

is to consider a growing emergent universe in which the future is not set in stone but is actualized from an underlying substratum of quantum possibilities.

Cramer’s approach means (from the authors view point) that the future is set, the past, present and future may all coexist and we simply have the illusion of flowing through time. To avoid confusion, we quote Cramer on his own interpretation [39];

Let me give an example. When you use your cash card at the grocery store to pay for your purchases, the electronic handshake that occurs between the bank and the cash register insures that money is “conserved” and is neither created nor destroyed, but it does not determine what you elected to purchase. The same is true with quantum transactions, which guarantee the conservation laws but do not determine the future. The real difference between Kastner’s PTI and my TI is that for her, offer and confirmation waves exist as objects only in some multidimensional Hilbert space. In the TI the waves exist in real 3+1 dimensional space. Hilbert space was invented by theorists prone to abstraction because it was the only way they could imagine that quantum waves could be entangled. The TI explains how they can be entangled, because the multi-particle transactions allow only those subset of the waves that satisfy the conservation laws to become real transactions.

Others have considered a Many-Worlds Interpretation, with every possible event happening along parallel realities in order to maintain *free will*. Neither Kastner nor Cramer agree with the many-worlds view [20]. Here, the reader is asked to make up their own mind. This paper is concerned only with; *Does the TI fit the data or not?* It is found that all the usual quantum results hold and the TI is simply an alternative point of view from the CI, and the instantaneously collapsing wave function, way of thinking.

## 5 The Delayed Choice Quantum Eraser by Kim et al.

First we briefly explain the experiment and the observed results. The experimental arrangement can be seen in Fig. 4. An argon laser (\(\lambda _p = 351.1\) nm) is passed through a double slit and illuminates a type II phase matching nonlinear crystal of \(\beta \)-Barium Borate BBO (\(\beta \)-\(Ba\mathrm{{B}}_2\mathrm{{O}}_4\)) The slit *A* allows region *A* of the crystal to be illuminated and slit *B* allows only region *B* of the crystal to be illuminated. This small region is about 0.3 mm long which we take to be the slit width *a*. The separation *d* of the two regions is about 0.7 mm as specified in the paper [15]. So we may discuss regions *A* and *B* of the crystal just as well as the original 2 slits. Parametric down conversion will occur at both sites and from the one pump photon will emerge two photons, a signal and an idler. Note that all possible frequencies are created \(\nu _p = \nu _s + \nu _i\). We are selecting two of the same frequency, or equivalently, twice the pump wavelength \(\lambda _s = 702.2\) nm. The signal and idler photons represent the e-ray and o-ray of the nonlinear crystal.

These photons are momentum entangled and are created essentially at the same time. The probability for a downconversion event is slight, so we may assume that there is only one entangled pair of photons in the system at any given time. Different wavelengths of signal and idler photons exit the crystal at different angles. The required wavelengths are selected by restricting the exit angle. Usually a small range of wavelengths would be selected. For convenience we track only one wavelength, but we should bear in mind that there will be a small bandwidth of wavelengths which will affect the interference pattern of the signal photons and change the visibility of the fringes accordingly. The bandwidth can also be changed using filters in front of the detectors. The detectors will have a less than perfect efficiency which will also affect the fringe visibility. The efficiency of the detectors was not mentioned in the experiment however, and neither was the effective bandwidth.

*f*, (not specified in the paper [15]) and then focussed onto a screen where they can be detected by detector \(D_0\). The detector scans, via stepper motor, along the

*x*-axis to build up a pattern. The lens is used to create the far field condition at the detector so we expect a Fraunhofer type pattern which is built up over time. The idler photons, from region

*A*and

*B*of the crystal, are sent in the direction of a Glen-Thompson prism (a wedge mirror is used in Fig. 4. instead) which separates them into different paths. The idler photons from region

*A*hit BSA and are either reflected or transmitted. The reflected photons will be detected by \(D_3\). The transmitted photons will be reflected by mirror MA and then either transmitted through the beamsplitter BS to detector \(D_2\) or reflected by BS into detector \(D_1\). The idler photons from

*B*hit BSB and are either reflected or transmitted. The reflected photons will be detected by \(D_4\). The transmitted photons will be reflected by mirror MB and then either transmitted through the beamsplitter BS to detector \(D_1\) or reflected by BS into detector \(D_2\).

The time of flight from the crystal to the detector \(D_0\) for the signal photons is 8 ns shorter than for the idler photons which go in the direction of the beamsplitters and were eventually detected by detectors \(D_3, D_4\) or by \( D_1 \) or \(D_2\). The equivalent path length is approximately 2.5 m. We assume that all the detector path lengths, \(D_1\)–\(D_4\), are the same and equal to 2.5 m. This path length will introduce a constant phase shift into each joint detection. It is also assumed that all mirror reflection angles are the same in both paths so that no additional phase shift differences need to be considered. Since all the phase shifts are considered equal they will cancel out and will not effect the overall interference pattern.

All the detectors are linked to a coincidence counter and the interference patterns are recorded. The intensity pattern recorded at \(D_0\) shows no interference when there is a coincidence between \(D_0\) and \(D_3\) or \(D_4\). In these cases, we have which path information, since \(D_3\) only records idler photons from slit *A* and \(D_4\) only records idler photons from slit *B*. Since the signal and idler photons come from the same region of the crystal, we would then know through which path the signal photons came and we expect no interference.

When the coincidence counts are between \(D_0\) and \(D_1\) there is an interference pattern. The beamsplitter BS mixes the idler photons from both regions and we have now *erased* the which path information. There is also an interference pattern when there is a coincidence between \(D_0\) and \(D_2\) but this pattern differs from the previous one by a phase shift of \(\pi \). In other words if one pattern shows a co-sinusoidal interference the other will be sinusoidal. The experiment is considered a delayed choice quantum eraser since the signal photons path length is shorter than the idler photons. It would seem that the signal photons are detected first, then we make a selection of which coincidence detections to look at, and depending on that choice we see or do not see interference of the signal photons. The paradox being, how can you influence the signal photon, basically tell it to interfere or not, by making a choice of detector \(D_1\)–\(D_4\), *8 ns after* the signal photon has already been detected by \(D_0\). This however is the wrong way to think about this problem. If looked at in the correct way there is no paradox.

These observations can easily be explained in terms of the TI of quantum mechanics as follows. A brief account of this experiment is given in the book by Kastner [37], we give a bit more detail here.

## 6 Transactional Interpretation Derivation

An offer wave will go out from the slits and get absorbed by a detector. The detector will then send back an advanced wave (backwards in time) along the same path as the incident wave to the slits to *handshake* and confirm the interaction. Only then does the photon actually leave the slit region. The offer wave is a momentum entangled two-photon state (or bi-photon). The possible transactions will depend on the detector configuration which generates the counter wave. We will go through the process step by step.

*p*stands for pump. The \(\alpha \) is the single slit diffraction pattern, a sinc function of the usual kind.

*A*and

*B*stand for the photon wave functions from the two slits, of plane wave type. Parametric downconversion inside the \(\beta \)-barium borate (BBO) crystal duplicates each pump photon into a signal and an idler photon. For type I parametric down conversion, the signal and idler have the same polarization, for type II the signal and idler polarizations are perpendicular. This is of no importance here since the signal photons from regions

*A*and

*B*interfere at detector \(D_0\) and both idler photons interfere at one of the four detectors \(D_1\)–\(D_4\). The offer wave from the 2 slits and crystal then becomes,

The time dependent, correlation function calculation can be found in Appendix 1. This is for comparison with the TI approach taken below. We skip the details of the parametric downconversion process in what follows, but they can be found in [6, 40, 41, 42] and these results are used in the Appendix 1 calculation. The first reference refers to 5 basic quantum experiments and has simple theory accessible to undergraduates [40]. The second reference has more theory but still some experiment, and is geared more for graduates and researchers [41] and the last two reference is a theory paper and a text book [6, 42].

*x*on the screen, then our offer wave becomes [37]

*a*is the slit width and \(k_x = k \sin \theta \) and the angle \(\theta \) is the angular displacement from the center of the slits to the position

*x*on the screen. For the paraxial ray approximation this would be

*f*is the focal length of the lens which is taken to be roughly the slit screen distance and \(\lambda \) is the wavelength of the signal photons and we have used \(k = 2 \pi /\lambda \). Hence

*A*and

*B*to the screen at position

*x*. Also we assume that the slit separation can be given by \(d = d_A - d_B \). The offer wave can now be written as,

**Case 1**

*r*and

*t*of the lossless beamsplitters and

Using our result Eq. (16) it is easy to generalize to a small spread of wavelengths (bandwidth=\(\Delta \lambda \)) by using a computer code to plot the equation and summing the interference patterns for \(\lambda \), \(\lambda \pm \Delta \lambda \), \( \lambda \pm \Delta \lambda /2 \) and \( \lambda \pm \Delta \lambda /4 \). This will give a quite accurate interference pattern which will match the experimental data very well. If you also include the detector efficiency \(\eta _1\) then you can match the experimental fringe visibility almost exactly. This is easy to do with a symbolic manipulation code like *Mathematica*, which also plots the results for you.

**Case 2**

**Case 3**

## 7 Discussion

The TI is related to the direct particle interaction theory of Wheeler–Feynman and Hoyle–Narlikar and involves advanced waves as well as the usual retarded waves. The advanced waves are natural solutions to the relativistic wave equation and are required to conserve momentum in direct particle interactions. This paper has briefly considered the pros and cons of direct particle interactions verse conventional field theory methods. In terms of vacuum energy density the direct particle approach tells us there is no vacuum field and thus its energy is identically zero, close in fact to the observed value. Quantum field theory tells us that the vacuum energy density is huge and gives a value 120 times too large. Direct particle or source theory does away with self interaction and subtracting infinities is only needed for charge renormalization. Charge renormalization follows in the same manner as in the field theory case when you introduce a size cutoff (no point particles) of the Schwarzschild radius of the particle. There is also a size limit to the universe to prevent a divergent advanced wave integral due to the Rindler horizon for an accelerating expansion of the universe [43]. Advanced waves have never been detected in practice and this lack of experimental evidence is enough for some to rule them out altogether.

It only takes one experimental observation to refute a theory. Cramer and Herbert [44] considered several experimental possibilities of nonlocal quantum signaling (retrocausal signals) involving path entangled systems and in all cases found that the complementarity between two-photon interference and one-photon interference blocks any potential nonlocal signal [45]. The traditional way of thinking about an instantaneous wave function collapse, at a certain time at a certain place, which is clearly in conflict with relativity, is superseded in the transactional picture. The wave function collapse is among the most confusing aspects of quantum mechanics (as a component of the measurement problem) and is simply resolved using the TI method of Cramer, or PTI of Kastner. Indeed the Copenhagen approach actually evades the entire issue by taking the wave function and its collapse as epistemic—a measure of knowledge rather than a physical entity. This approach is observer-dependent; it is subject to the ‘Heisenberg Cut’ in which there is no physically grounded and non-arbitrary account of what constitutes an ‘observer’. In the transactional approach, there is no observer-dependence: it is absorbers that provide the missing ingredient that defines when a measurement and attendant collapse occurs.

Advanced waves are natural solutions to relativistic wave equations. In order to use this theory for the nonrelativistic case it is necessary to think of two Schrödinger equations: one Schrödinger equation for the wave function \(\psi \) and one for its complex conjugate \(\psi ^*\), which becomes the advanced wave. This makes sense if we think of the Schrödinger equation as a square root version of the relativistic Klein Gordon equation.

Furthermore, work by Hogarth [46] and Hoyle and Narlikar (HN) [32, 33, 47, 48, 49] has paved the way to a new version of direct particle interaction gravitational theory, which is fully Machian, incorporates advanced waves and has Einstein’s theory as a special case. The HN theory may be quantized as in their book [32, 33] using the path integral technique pioneered by Feynman [26].

It is interesting to note that the mass field *m*(*x*) in HN theory looks similar to the source field *S*(*x*) introduced by Schwinger [50, 51]. Wheeler never gave up on the absorber theory, which is a direct particle interaction (action-at-a-distance) theory. It simply wasn’t popular at the time and dropped off the radar. Gerard t’Hooft found a way to renormalize Yang Mills field theories in a way similar to QED and most physicists took that path. We believe the works of Cramer, Wheeler–Feynman, Hoyle–Narlikar, and Schwinger’s source theory, are all direct particle interactions. How source theory is related to the Feynman path integrals is explained by Schweber [52]. It should be noted that Schwinger was able to derive the Casimir force using the source field method in which there are no nontrivial vacuum fields [53, 54]. The action at a distance theories are well worth study and may lead to a consistent picture of quantum gravity. Radiation reaction can be dealt with using the half retarded half advanced absorber picture. Many QED results thought to be vacuum fluctuation related can in fact be derived by considering source fields instead, including the Lamb shift and particle self energy [53].

## 8 Conclusions

The main aim of this paper is to draw attention to the fact that the TI of quantum mechanics by John Cramer is perfectly viable and legitimate, and should be given due consideration by the physics community, which has not been the case thus far. The TI by Cramer [17], gives a simple and intuitive picture for wave function collapse distributed over the entire path of the interacting system (in Kastner’s approach, the collapse is what establishes that path). In the case of the Kim experiment [15], the wave function would collapse along the entire path between the slits (or the regions *A* and *B* of the down converting crystal) and the detectors and it would happen in a way distributed over time, not in an instant. The TI picture rules out the possibility of any backward in time signals using quantum delayed choice experiments. In fact it makes clear the idea is nonsense since the advanced counter wave from the detector must travel the entire distance back to the slit in order for the photon (from the slit) to make the trip in the first place. The choice is really no longer delayed since the photon *knows* where it will end up because of the advanced wave coming backwards in time to confirm the interaction or *handshake*, as Cramer puts it. The alternative way of avoiding wave function collapse is to use the correlation functions as in Appendix 1. The calculations are far more long winded, than the fairly quick and easy calculation in the main paper, and in the opinion of the author the correlation function method masks what is really going on and thus leaves room for misinterpretation.

## Notes

### Acknowledgments

HF thanks both J. G. Cramer and R. E. Kastner for reviewing the paper and for suggesting great improvements. Thanks also go to K. Wanser and P. W. Milonni for helpful comments.

## References

- 1.Feynman, R.P., Leighton, R., Sands, M.: The Feynman Lectures of Physics, vol. III. Addison Wesley, Reading (1965)Google Scholar
- 2.Wheeler, J.A.: Delayed choice. In: Wheeler, J.A., Zurek, W.H. (eds.) Quantum Theory and Measurement. Princeton University Press, Princeton (1983). See also experimental verification by Jacques et al. arXiv:610241 [quant-ph] (2006)
- 3.Sohrab, S.H.: Quantum theory of fields from Planck to Cosmic scales. WSEAS Trans. Math.
**9**(9), 734–756 (2010)MathSciNetGoogle Scholar - 4.Sohrab, S.H.: Implications of a scale invariant model of statistical mechanics to nonstandard analysis and the wave equation. WSEAS Trans. Math.
**5**(3), 93–103 (2008)Google Scholar - 5.Scully, M.O., Druhl, K.: Quantum eraser: a proposed quantum correlation experiment concerning observation and ‘delayed choice’ in quantum mechanics. Phys. Rev. A
**25**, 2208 (1982)ADSCrossRefGoogle Scholar - 6.Scully, M.O., Zubairy, M.S.: Quantum Optics, Cambridge University Press (1997). See p. 223 for dark states and coherent trapping. See p. 583 chap. 21 for spontaneous emission ad the Scully Druhl paper derivation. See chap. 16, p. 461 for parametric down conversionGoogle Scholar
- 7.Scully, M.O., Englert, B.G., Walther, H.: Quantum optical tests of complementarity. Nature
**351**, 6322 (1991)CrossRefGoogle Scholar - 8.Englert, B.-G., Walther, H., Scully, M.O.: Quantum optical Ramsey fringes and complementarity’. Appl. Phys. B
**54**, 366–368 (1992)ADSCrossRefGoogle Scholar - 9.Eichmann, U., Bergquist, J.C., Bollinger, J.J., et al.: Young’s interference experiment with light scattered from two atoms. Phys. Rev. Lett.
**70**(16), p2359–2362 (1993)ADSCrossRefGoogle Scholar - 10.Holladay, W.: A simple quantum eraser. Phys. Letts.
**A183**(4), 280–282 (1993)ADSCrossRefGoogle Scholar - 11.Devereux, M.: Testing quantum eraser and reversible quantum measurement with Holladay’s simple experiment. J. Phys Conf. Ser.
**462**, 012010 (2013)CrossRefGoogle Scholar - 12.Herzog, T.J., Kwiat, P.G., Weinfurter, H., Zeilinger, A.: Complementarity and the quantum eraser. Phys. Rev. Lett.
**75**(17), 3034 (1995)ADSCrossRefGoogle Scholar - 13.Kwiat, P.G., Steinberg, A.M., Chiao, R.Y.: Observation of a quantum eraser: a revival of coherence in a two-photon interference experiment. Phys. Rev. A
**45**, 7729 (1992)ADSCrossRefGoogle Scholar - 14.Ma, X.-S.: Quantum erasure with causally disconnected choice. PNAS
**110**(4), 1221–1226 (2013)ADSCrossRefGoogle Scholar - 15.Kim, Y.-H., Yu, R., Kulik, S.P., Shih, Y. H.: A delayed choice quantum eraser. arXiv:9903047 [quant-ph] (1999)
- 16.See for example http://www.youtube.com/watch?v=u9bXolOFAB8. There are other animations discussing the philosophy of the experiment online
- 17.Cramer, J.G.: The transactional Interpretation of quantum mechanics. Rev. Mod. Phys.
**58**, 647 (1986)ADSMathSciNetCrossRefGoogle Scholar - 18.Cramer, J.G.: An overview of the transactional interpretation of quantum mechanics. Int. J. Theor. Phys.
**27**, 227 (1988)MathSciNetCrossRefGoogle Scholar - 19.Cramer, J.G.: The Quantum Handshake, Entanglement, Nonlocality and Transactions. Springer, Berlin (2015)Google Scholar
- 20.Kastner, R.E., Cramer, J.G.: Why Everettians should appreciate the transactional interpretation. arXiv:1001.2867 [quant-ph]
- 21.Cramer, J.G.: A transactional analysis of interaction free measurements. Found. Phys. Lett.
**19**, 63–73 (2006), also arXiv:0508102 [quant-ph] - 22.Cramer, J.G.: The quantum Eraser. In: Analog Science Fiction/Fact Magazine (1998). http://www.npl.washington.edu/AV/altvw90.html
- 23.Wilczek, F.: The Lightness of Being. Basic Books, New York (2008)Google Scholar
- 24.Dirac, P.A.M.: Classical theory of radiating electrons. Proc. R. Soc. Lond.
**A167**, 148 (1938)ADSCrossRefGoogle Scholar - 25.Wheeler, J.A., Feynman, R.P.: Interaction with the absorber as a mechanism of radiation. Rev. Mod. Phys.
**17**, 157 (1945)ADSCrossRefGoogle Scholar - 26.Feynman, R.P.: Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys.
**20**(2), 367–387 (1948)ADSMathSciNetCrossRefGoogle Scholar - 27.Davies, P.C.W.: Extension of Wheeler–Feynman quantum theory to the relativistic domain I. Scattering processes. J. Phys. A
**4**, 836–845 (1971)ADSCrossRefGoogle Scholar - 28.Davies, P.C.W.: Extension of Wheeler–Feynman quantum theory to the relativistic domain II. Emission processes. J. Phys. A
**5**, 1025–1036 (1972)ADSCrossRefGoogle Scholar - 29.Hoyle, F., Narlikar, J.V.: Electrodynamics of direct interparticle action. I. The quantum mechanical response of the universe. Ann. Phys.
**54**, 207 (1969) and ibid**62**, 44 (1971)Google Scholar - 30.Csonka, P.L.: Advanced effects in particle physics. Phys. Rev.
**180**, 1266–1281 (1969)ADSCrossRefGoogle Scholar - 31.Herbert, N.: Faster than Light: Superluminal Loopholes in Physics, A Plume book, pp. 77–97. Penguin Group, New York (1989)Google Scholar
- 32.Hoyle, F., Narlikar, J.V.: Action at a Distance in Physics and Cosmology. W. H. Freeman and Company, San Francisco (1974)Google Scholar
- 33.Hoyle, F., Narlikar, J.V.: Lectures on Cosmology and Action at a Distance Electrodynamics. World Scientific, Singapore (1996)Google Scholar
- 34.Cramer, J.G.: The plane of the present and the new transactional paradigm of time, chapter 9. In: Drurie, R. (Ed.) Time and the Instant. Clinamen Press, Manchester (2001). Also arXiv:0507089 [quant-ph]
- 35.Cramer, J.G.: The quantum Handshake. In: Analog Science Fiction/Fact Magazine (1986). http://www.npl.washington.edu/AV/altvw16.html
- 36.Cramer, J.G.: The arrow of electromagnetic time and the generalized absorber theory. Found. Phys.
**13**(9), 887–902 (1983)ADSCrossRefGoogle Scholar - 37.Kastner, R.E.: The Transactional Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
- 38.Kastner, R.E.: The Possibilist Transactional Interpretation and Relativity. arXiv:1204.5227 [quant-ph]. See Kastner’s webpage for more papers and interesting details. http://transactionalinterpretation.org
- 39.Cramer, J.G.: Private communicationGoogle Scholar
- 40.Galvez, E.J., et al.: Interference with correlated photons: five quantum mechanics experiments for undergraduates. Am. J. Phys.
**73**(2), 127–140 (2005)ADSCrossRefGoogle Scholar - 41.Di Giuseppe, G., et al.: Entangled-photon generation from parametric down-conversion in media with inhomogeneous nonlinearity. Phys. Rev.
**A66**, 013801 (2002)ADSCrossRefGoogle Scholar - 42.Milonni, P.W., Fearn, H., Zeilinger, A.: Theory of two-photon down-conversion in the presence of mirrors. Phys. Rev. A
**53**(6), 4556–4566 (1996)ADSCrossRefGoogle Scholar - 43.Fearn, H.: Mach’s principle, Action at a Distance and Cosmology. J. Mod. Phys.
**6**, 260–272 (2015)CrossRefGoogle Scholar - 44.Cramer, J.G., Herbert, N.: An Inquiry into the Possibility of Nonlocal Quantum Communication. arXiv:1409.5098 [quant-ph]
- 45.Jaeger, G., Horne, M.A., Shimony, A.: Complementarity of one-particle and two-particle interference. Phys. Rev. A
**48**, 1023–1027 (1993)ADSCrossRefGoogle Scholar - 46.Hogarth, J.E.: Considerations of the absorber theory of radiation. Proc. R. Soc.
**A267**, 365–383 (1962)ADSMathSciNetCrossRefGoogle Scholar - 47.Hoyle, F., Narlikar, J.V.: A new theory of gravitation. Proc. R. Soc. Lon.
**A282**, 191 (1964)ADSMathSciNetCrossRefGoogle Scholar - 48.Hoyle, F., Narlikar, J.V.: On the gravitational influence of direct particle fields. Proc. R. Soc. Lond.
**A282**, 184 (1964)ADSMathSciNetCrossRefGoogle Scholar - 49.Hoyle, F., Narlikar, J.V.: A conformal theory of gravitation. Proc. R. Soc. Lond.
**A294**, 138 (1966)ADSCrossRefGoogle Scholar - 50.Schwinger, J.: Particles and sources. Phys. Rev.
**152**(4), 1219–1226 (1966)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 51.Schwinger, J.: Particles, Sources and Fields. Advanced Books Program, vol. I–III. Perseus Books, Reading (1998)Google Scholar
- 52.Schweber, S.S.: The sources of Schwinger’s Green’s functions. PNAS
**102**(22), 7783–7788 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar - 53.Milonni, P.W.: The Quantum Vacuum An Introduction to Quantum Electrodynamics. Academic Press, New York (1994). See p. 474Google Scholar
- 54.Schwinger, J., DeRaad Jr, L.L., Milton, K.A.: Casimir effect in dielectrics. Ann. Phys.
**115**, 1 (1978)ADSMathSciNetCrossRefGoogle Scholar - 55.Collett, M.J., Loudon, R.: Output properties of parametric amplifiers in cavities. J. Opt. Soc. Am.
**B4**(10), 1525 (1987)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.