What is the meaning of Wheeler's delayed choice experiment? Wheeler's delayed choice experiment is a variant of the classic double slit experiment for photons in which the detecting screen may or may not be removed after the photons had passed through the slits. If removed, there are lenses behind the screen refocusing the optics to reveal which slit the photon passed through sharply. How must this experiment be interpreted?


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*Does the photon only acquire wave/particle properties only at the moment of measurement, no matter how delayed it is?

*Can measurements affect the past retrocausally?

*What was the history of the photon before measurement?

*What are the beables before the decision was made?

 A: The actual meaning of the colloquial phrase of photons "acquiring particle properties" or "acting like a particle" is really nothing more than saying that photons interact locally and in discrete packages, despite being described much of the time by a spatially distributed wave-function.
Photons, when they are left to travel freely, travel as waves. (The same is true of electrons and other matter/antimatter particles.) But photons can be absorbed by electrons, such as those in light detectors or photoplates; and despite the fact that the wave-function of the photon may be distributed across more than one such detector or more than one cell of the plate, we find that the photon is always absorbed at only one location.
In the old days of quantum mechanics, one would say that the photon "acted like a wave through the slit, and like a particle at the plate". What one would say nowadays is that the photon evolved according to the Schrödinger equation until


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*it is interrupted by a measurement device, at which point the wavefunction collapses and gives a definite outcome for whatever that measurement device is measuring; or

*until it has interacted in an uncontrolled (but consistent) manner with enough of the world around it that it decoheres, in which case it ends up being in a probabilistic mixture of states which are stable under that interaction.
This may sound quite similar to the "wave/particle duality" way of saying things, but in practice it gives you a much better shot at understanding how a photon or electron will actually behave when you get your hands on the mathematics.
(Incidentally, the question of "when something counts as a measurement device" is one which is still an open topic at a fundamental level, even though in practical terms we know enough to predict the outcomes of most experiments. A great number of physicists also believe that measurement is in some sense a special case of decoherence. This is all part of understanding the Measurement Problem of quantum mechanics.)
As for the "history" and the "beables" prior to measurement, or prior to the decision of whether to make the measurement or not, these are questions of the interpretation of quantum mechanics. There is no commonly-agreed-upon answer. But the short story is that — no matter how long it took for you to decide whether or not to measure — if you don't measure, the trajectory of the photon is still described by the Schrödinger equation, and you can still cause different "possible paths" to interfere with one another (e.g. in a sum-over-histories description of the evolution of the particle).
A: The concept of "evolution relative to the Schroedinger equation" is an insightful means of considering your questions via a holistic interpretation of the reality to which most of modern physics seems to point.  One should recognize that interacting with a measurement device is another aspect of interacting with "the world".  This concept of the photon as a wave "interacting with the world" over "many paths" simultaneously is a much more significant element of determining the final outcome of all the interactions in which what we are taught to think of as a photon is involved than a semi-classical interpretation of the photon as a particle some of the time and a wave some of the time might suggest.  (What we call a photon is, fundamentally, no more than our perception of the localization of a collection of properties associated with specific, quantum fields via a mode of "bundling" that interact in a pre-defined manner with particles with specific properties.)
"The world" exists in a multi-dimensional framework that includes time.  "The world" evolves in time, as must all experiments performed in "the world" that we detect as beings made of matter.  The perception of a wave function as being uniquely linked to one photon that is somehow in one position in some temporal interval associated with measurement in a detector may be one cause of the questions that have been posed.  Stop thinking of photons as highly localized in space and time in the same manner that one might think of a little ball as being highly localized in space and time when what is called a photon is, in fact, better conceived as a disperse, wave-like field that interacts with localized objects called atoms that we have learned to use to make what we call "particle detectors" (using a concept based on classical thinking).  
The stuff that we use to interact with the photon in a detector is in a complex form that we call "an atom".  Because it is in this complex, atomic system, it is bound by rules that are defined by quantum physics relative to energy states and other particle properties.  The photon in free space (in the classic, quantum description of what physics calls a "photon") is a free agent until it "gets mixed up" with the particle "crowd" that comprises the atom.  The atom has a great deal of mass relative to the "photon", and it has a great deal of power to produce what we perceive as localized phenomenon in time and space, because the atom is, due to its mass (and, formally, momentum), a relatively localized phenomenon.
Re-think the photon crudely as electro-magnetic energy in the environment (manifested in quantum fields) that is bundled by an atom due to the atom's relatively high momentum, which forces its probability wave to be relatively localized.  Think of the electro-magnetic energy of what we call a "photon" as being highly interactive with "the world" until it has sufficiently interacted with the particle detector's energy "bundling" atoms to produce a result.  At that point, we get some output data from the machine.  
Because of the highly interactive nature of the photon with the world around it given its "many paths" aspect, we may find the results a bit surprising if we are too used to digging a rut in the same logical path by forming what one author described as "cog-webs" that define the photon as a particle some of the time.  If we think of detection of a photon by a measurement device as a means of localizing ("bundling") electro-magnetic energy that is disperse in the environment (that follows "many paths" that overlap with other photons' "many paths"), and think of the speed of light as the rate at which electro-magnetic energy can be localized by atoms (with mass) to produce quantized changes in energy in things called atoms that we can use to detect the electro-magnetic energy in the environment, then it might come as no surprise that we begin to gather data about the environment that exceeds our expectations in some three-dimensional, directional sense in a given, laboratory experiment.  
If we use an electro-magnetic energy source in a particular position sending energy in a more or less directional manner to provide most of that electro-magnetic energy being put into a given environment in which an experiment is occurring, it should come as no surprise that we detect information that is biased relative to a specific directional thumb-print in space and time, because most of the energy we gather carries a certain amount of information due to its point of origin, and objects in the related path.  Because we are gathering electro-magnetic energy from the environment, should we expect all of the information reflected in what we call "a signal" to originate from one direction?  If electro-magnetic energy bundling has a speed associated with it, any changes occurring during the "bundling" process that manipulate the directions from which energy can be gathered into a "bundle" will, most likely, be reflected in the results produced by the "bundling" process.
Don't trap your mind in a logical framework around a specific, laboratory experimental set-up that amounts to a temporal, Rube Goldberg machine that may be more likely to hide the reality that we are observing than to reveal it if it manifests an anthropic inclination to localize source and detector (and thus disperse energy) in a directional sense based on our inclination to perceive cause and effect associated with spatial momentum in interactions involving large chunks of matter due to our experience with a universe in which entropy generally increases as time passes by biasing an experiment to investigate photons as particles, and, what a shock, that generates results that suggest that a photon is not just a wave, but a form of energy that can be localized by atoms after following many paths.  Be VERY CAREFUL how you link relativity to quantum physics with what are still defined by many as mass-less particles.  The "speed of light" is an extremely classical term premised on ancient, anthropic perspective.  (Feynman even dared to conceive of instantaneous action at a distance relative to electro-magnetic theory, a concept which, in his original formulation, he later described in negative terms in his Nobel lecture.)
What if Galileo had stepped onto a hillside that was distant from another on which stood a friend with a lantern and a shielding cloak, and conceptualized his experiment as an attempt to measure the rate at which electro-magnetic energy seeped from the environment into his eyes (comprised of atoms with mass and momentum) to form a bundled "quantum" of light energy that would cause the measuring neurons in his retina to fire and transmit a signal to the optical center of his brain localizing the bundle of energy in a specific area of his visual field after he gave the high sign to his distant buddy to remove the shield that blocked the lantern's light?  Would we still think that there was something called a photon with a specific speed as it flew through space-time, or would we perceive what we call a "photon" as a bundle of properties associated with energy that is localized by atoms with mass (and momentum), that we once found it easy, given science's classical pedigree, to think of as a little ball flying through space?  
If a particle is free in space and not interacting with others in an atom, it is not required to be "quantized".  Quantum physic's particle properties are associated with unique quantum fields.  The concept of a field was created to explain transmission of energy between objects lacking a physical connection.  Fully embrace the concept of waves and fields, and by-pass the "wave-particle duality" perspective relative to light as something that is sometimes one and sometimes the other and the strong, directional perspectives that accompany it.  Your other questions should fade in the process.
