And FINALLY I have time to walk through what's really happening here. This post must come before the one on Afshar's experiment, so that it is possible to understand what happens in Afshar's experiment.
The first thing to understand is what "entangled" means. A lot of wild talk has been floated around about what the meaning of this term might be; things like some mysterious connection between particles at random locations, and so forth. The truth is much more mundane. Entangled particles simply have parameters that have values that are predictable from values on the other particle, using the laws of physics. In most cases, the laws involved are the simplest possible: conservation laws. In the explanations of these experiments that I have been using, specifically the conservation of angular momentum, which for a quantum is quantized as spin angular momentum, or spin.
Since angular momentum is conserved, if a spinless system emits two particles that have spin, then those particles must have equal and opposite spins. Equally, if a system that has spin still has the same spin after emitting the particles, the same must apply- spin can be neither created nor destroyed, that is the meaning of "conservation," and again, the particles must have equal and opposite spins.
Now spin is not the only thing that is conserved; for example, if two particles are emitted from an uncharged source, and one is measured to have a charge, the other must have an equal and opposite charge. If two particles emerge from a motionless source (and in their own frames of reference, all sources can be considered motionless), then if the source is still motionless in that frame after they are no longer in contact with it, then their momenta must be equal and opposite, and the vectors of their positions must also be in that frame, at any arbitrary time, unless they interact with something else.
But spin is special; and the reason is that it is both discrete, in other words it takes on small integer quantized values in all known elementary particles and for any given type of particle can only have one of two values on any given axis, and different on different axes of the same particle; but in addition to all of that, its value on one axis, while not determined by the value on another, is influenced by it. In other words, if you sort a large number of particles based on their spins on a particular axis, and then measure their spins on another axis, consistently, that is, always on the same second axis, you will find a particular probability that the spin measured on the second axis is the same as the spin on the first; and that probability will depend on the angle between the two axes. And if you take the other half (assuming a random distribution) of the particles from the first measurement, and make a second measurement at the same second angle, you will find precisely the opposite distribution of spins.
This probability distribution is a quantum mechanical property of photons. It might seem abstruse, but in fact, it is very real and concrete. This is the reason for a very interesting phenomenon in optics, that of polarization. And polarization is also dependent upon another quantum mechanical phenomenon: scattering. I will begin by explaining scattering, then move to polarization, and finally show how polarization is used in all of these experiments. Incidentally, it is worth noting that the two characteristics of spin, that it is discrete, and that a single particle can have two different values on different axes (though apparently not at the same time under uncertainty), when combined with the fact that it is a conserved quantity, make spin a particularly facile property to use in tests of EPR; that it is relatively easy to measure makes the picture complete, and this is why polarization and spin are so important in these experiments. Theoretically, we could use position and momentum, or time and energy, in the same way; but these are much more difficult to measure accurately than spin, they are continuous rather than discrete, and their expectation values under quantum and classical assumptions are more difficult to distinguish from one another.
Scattering is what happens when two particles interact in such a way that they only exchange momentum. The pool table is the classical description of scattering; the balls bounce off one another, with their trajectories modified at each bounce, but remain pool balls (unless one uses a hammer and breaks one). A great deal of work in physics has been done on scattering; there is elastic scattering, inelastic scattering, deep scattering both elastic and inelastic, and very much more. The math involved is abstruse, and when quantum mechanical representations of action, position, momentum, and spin are involved, this is one of the more complex areas of all of physics. So we must accept in advance that we will have to simplify and condense, and even gloss over some details of how these processes work if we are to gain an understanding of what is involved. I'll do my best to indicate where I am speaking of these matters in a manner that is less than precisely physically accurate, in order to illuminate them in a fashion that permits deeper understanding without having to know the math; apologies in advance if I slip up here or there.
When a photon scatters off an atom, it generally scatters off the electron shells, and in the case of a visible light photon, almost always off the outermost shell. Visible light photons don't "pack enough punch" to get past these outermost electrons and scatter from inner shells. Of course, by random chance, a few do; but they are rare and can be ignored for the purpose of understanding what happens here. Remembering a prior post on electron shells, most readers will also understand that this scattering could take place some distance from the atom; it is a very low probability, but they could encounter an electron there, due to uncertainty. In general, though, the photon must come close enough to the atom to be within a high-probability position for one or more of the electrons in the outermost shell; and then one of those electrons must actually be there for the photon to interact with. Finally, the photon must be of an energy that does not represent the correct amount for the electron to absorb it and be raised to a higher energy level; that is not scattering, it is absorption.
For a material to be transparent, it must not absorb visible light photons very often. In addition, it also must either not scatter them very often, or the scattering must happen in a fashion that does not significantly alter their trajectories; if they are scattered much, then the material will not be transparent, but translucent. In most transparent materials, the second is the case; although the photons are scattered, they are by and large scattered back by the same amount later, so there is no great difference between their trajectory when they enter the material, and when they exit it. Scattering, however, does reduce their signal velocity, and as a result, we perceive the speed of light in transparent media to be lower than the speed of light in a vacuum. Note carefully that a vacuum is a medium in which scattering does not occur, because there are no atoms in it. That is why the speed of light in a vacuum is the yardstick against which we measure all other speeds.
Most transparent materials have atoms (or more properly, in most cases, molecules) at random orientations; water and glass are good examples. Thus, the angles at which photons scatter must be relatively consistent, and relatively small; that they will scatter has to be obvious from the fact that the atoms in a liquid or a solid are in very close proximity, leading to surface tension in liquids and solidity in a solid, due to their Van der Waals interactions (these are electromagnetic interactions between the electrons and nuclei of adjacent atoms). There is thus little chance for a photon to traverse a significant distance in such a material without being scattered. There are, however, transparent materials that have their atoms or molecules arranged in a regular formation. These are called crystals. And some very interesting things can happen in such materials, particularly if they consist of atoms or molecules that are asymmetric. This is because the scattering angle of a photon depends not only on its vector and that of the electron it scatters from, but also on their relative spins. In an asymmetric molecule or atom, the scattering angle can therefore differentiate between photons with one spin or the other along the axis of the asymmetry of the atom or molecule, and if all the atoms or molecules are lined up, then all the photons with one spin will leave at one angle, and all the photons with the other spin will leave at another.
An inexact analogy would be the behavior of the cueball after striking the object ball in pool, based on whether the cueball has forespin or backspin (many people confuse backspin and forespin with "english;" true english in pool is sideways transverse spin). A cueball with forespin or topspin will pause slightly after hitting the object ball, and then "follow" it; a cueball with backspin will pause slightly and then "come back" toward the player who hit it. The big differences that make this an inexact analogy have mostly to do with the nature of photon spin which cannot be transverse but must be longitudinal, and with the differences between classical spin of a pool ball and quantum mechanical spin of a photon; however, the analogy serves to give one a proof of the possible effects of spin on post-scattering trajectories.
What is longitudinal spin? It is spin around an axis that is coincident with the direction of movement. In other words, like a corkscrew. When you think about the likely effects of this type of spin upon the trajectory of a photon that scatters from a "bump" on an asymmetric molecule or atom in a crystal, and the effects on all the photons over all the aligned "bumps" on all the atoms in the lattice, you will see how polarization can happen; the photons spinning this way will go this way, and the photons spinning that way will go that way. This is another inexact analogy, but much more subtly so; I have misrepresented the quantum mechanical spin of a photon like a classical spin. But you really do get the flavor of what the equations that describe this type of scattering say from this analogy, and I don't know of another that is more exact that is simple enough to understand without math, so I'll stop there and let this one stand; remember, however, that there are parts of this analogy that are not accurate, and I will have occasion to note some of them later.
So now we have a good analogy that gives us a pretty good idea of what happens inside a crystal that can cause the emergence of two beams of light where one went in, with the photons in each beam sorted by their measured (yes, that scattering this way or that way was indeed a measurement) spins, UP or DOWN with reference to the optical axis of the crystal lattice. The optical axis, of course, is the direction along the crystal lattice in which the "bumps" "stick out" and can affect the trajectories of the photons based on their spins as I have described. I quoted those terms because, again, we are in an area where analogy can give us the flavor of what's happening, but is not precisely accurate. Now, note that I said "measurement" in there; that's very important, and it's not in any way an analogy. This is a real, quantum mechanical, exact, precise measurement in every sense of the word. We have specifically measured the spins of the photons along the optical axis of the crystal, and that also is an exact, precise description of the situation, no analogy at all. This should help make it clearer where this analogy breaks down, and make it clear that that breakdown cannot affect the conclusions I will draw from this description.
OK, now what about the whole wave description of polarization, you know, the right-handed/left-handed circular polarization, and the plane polarization, and elliptical polarization and all that stuff? Well, we could get into that- but if we do, it's a complete other discussion. But there's something more important to understand here, and it is the key to Afshar: if I use that description, I will get the same results as if I use the particle description I have above. This is called the principle of complementarity. Many people misstate that principle as "you can't measure wave and particle behavior at the same time." While nominally true, that ignores a more important fact: both measurements are equally valid, and will yield the same result. And further misunderstanding is possible, based ONLY on the more egregious misstatement that "you can't see both wave and particle behavior in the same experiment." This last is simply not true. What is true is that you can analyze a single measurement, i.e. a single interaction between a photon and an atom, either in terms of wave mechanics, or in terms of matrix mechanics, but you can't switch methods in the middle of the analysis of that measurement and expect to get a consistent answer. However, no matter whether you use wave mechanics or matrices, you will get the SAME answer. Afshar makes two measurements, and claims to have "measured wave and particle behavior in the same experiment," which is true, but then claims this "violates complementarity," which is not.
OK, now back to the main thread; we'll get to an analysis of Afshar when we've disposed of the impossibilities in my previous description of the DCQE (yes, I exaggerated for effect, and yes, I always intended it this way, I think it makes the point in a very forceful, anti-woo-woo way, on a subject that lends itself to a great deal of woo).
Now that we understand polarization, let's talk a little more about spin, and make sure we understand the implications of it. But first, let's talk again about phase.
Phase is the precise position in spacetime at which a wave has a certain value; for example, precisely where and when the value of the parameter that is "waving" achieves its maximum value. Because waves repeat themselves, this is not a single point, but a collection of points. Note that two waves that are 180° out of phase, that is, when the maximum value of one wave coincides with the minimum value of another, if the waves have the same magnitude, that is, the same maximum and minimum values, they cancel, and we measure nothing.
When we talk about the phase of something that is rotating, it quickly becomes clear that this is a wave phenomenon because it repeats; and that means it must have phase. Now, if that something has conserved and quantized angular momentum as quanta do, then the phase must reinforce itself, because if it did not, then the angular momentum would disappear and conservation would be violated. It turns out when you analyze this situation using topology, that branch of geometry which is concerned with the shapes and relationships of things rather than their exact sizes and positions, there are two possibilities. Either the spin reinforces itself on every rotation, or it reinforces itself on every other rotation. Note that this means that when two particles encounter one another, in the first case, the two spin phases coincide, and double the magnitude; in the second case, and I'll state right now that this is impossible because the particles would cease to exist and thereby violate all the conservation laws for their parameters, the two spin phases anti-coincide, and cancel. This observation is the basis of the spin-statistics theorem, which I often call "the Laws of Spin and Statistics," after one of my favorite expositions on QM, The Force of Symmetry, by Vincent Icke. Icke himself notes that the discussion here of spin and phase is another inexact analogy, and I repeat that warning; he and I are encouraging you to visualize things in a way that has some differences from the exact mathematics, but it is a very illuminating analogy, and without actually using the math, I know again of none better. This will give us the idea, without overwhelming us in abstruse math.
Now, what are these "waves" that are involved with spin? They are, of course, Schroedinger's probability function. And so what we see is that if two particles with spins that reinforce on successive rotations encounter one another, the probability that they will remain together in the same position and quantum state is doubled, all other factors (primarily momentum) being in proper alignment; however, if two particles that reinforce only after two rotations encounter one another, and their spins are out of phase, then the probability that they will be in the same quantum state is canceled; that means it's zero. And we all know what "zero probability" means.
Physicists call particles whose rotational phase reinforces on successive rotations "integer spin" particles, or "bosons," because their statistics were first described by Satyendra Nath Bose and Albert Einstein, and are therefore called the "Bose-Einstein statistics." These particles are the quanta of forces and energy: the photon, the gluon, and the W and Z bosons, and perhaps the graviton ("perhaps" because no consistent quantum mechanical description of the graviton exists currently). As many bosons as might be desired (or can be gathered) can occupy the same quantum state; they are in fact more likely to do so, and this leads to interesting phenomena like the coherence of light in a laser beam and the anomalous behaviors of Bose-Einstein condensates, for example superfluidity. It also leads to them building up together and exhibiting what we call "forces" and "fields" as they transfer momentum between fermions. These particles can also be characterized as "energy," though traditionally only the photon is so described. All but the W and Z bosons are massless, and move forever at the speed of light.
Physicists call particles whose rotational phase reinforces on every other rotation "half-integer spin" particles, or "fermions," because their statistics were first described by Enrico Fermi and Paul Dirac, and are therefore called the "Fermi-Dirac statistics." These particles are the quanta of matter: quarks, electrons, muons, tauons, and neutrinos. Only two identical fermions can occupy a single quantum state, and only if they are of opposite spin. This is called the "Pauli exclusion principle," after Wolfgang Pauli who discovered it, and is responsible for the fact that fermions build up into finite-sized aggregates that we call "matter." These particles can cancel if all of their properties other than the one called "mass" are equal and opposite; particles with these opposite properties from the ones we commonly encounter in everyday life are called "antimatter." The mass is also conserved, but can be interconverted with the energy of bosons; and since the antiparticle of the photon is itself (a curious requirement of the conservation of energy), when this annihilation occurs, the mass of the disappearing particles can be carried away by photons as energy, while their other parameters (charge, spin, and so forth) cancel each other out. These particles must always move at less than the speed of light because of their possession of mass.
So now we understand spin, and polarization; and phase. I will pick up the thread in the next post, and we will see precisely how these properties interact in these experiments.
LOL
