Before I can talk about interpretations, I've realized I have to address the whole "wave/particle" thing. So I'll see if I can do both in a single post, but I frankly expect to have to do this in one and then talk about interpretations in another.
In our everyday world, we encounter particles all the time; and we encounter waves all the time, too. Normally these are quite distinct types of phenomena from one another. But this is not inherent in the phenomena; it is an effect created by the large number (astronomical, actually) of quanta that make up even the smallest everyday object around us. When you start talking about "particles" that are sufficiently small, they become more "wavy;" and when you talk about quanta, the smallest particles of all, you can't really tell whether they are particles or waves. They exhibit characteristics that are both.
From our point of view, this is paradoxical; how can something be both a particle and a wave? But it turns out that this is the only explanation we have been able to devise that makes sense, and gives the right answers. Remember, science always goes back to the real world; and when you take the answers you get from doing things this way back, they agree with reality with stunning, in fact unprecedented, accuracy and precision. The theory of quantum electrodynamics is the most accurate and precise theory in the history of the world; it predicts the behavior of electrically charged particles with respect to the electromagnetic force to seventeen decimal places, the last time I looked. No other theory of science can even approach this level of accuracy or precision.
But what does this mean? It means that quanta are not particles, nor are they waves, as we generally conceive of particles and waves to be. They are quanta, and they take part in characteristics of both particles and waves, but are neither. And this leads directly to another inherent characteristic of quanta: uncertainty. For a point-like particle, position is a very clear concept; it's right there. But for a wave, it's not so clear; where precisely is a wave? Well, it's over here, and it's over there, and it's all between. So uncertainty is one of the wave-like aspects of quanta. You can't really precisely localize them. In truth, this is just an analogy; there isn't anything you can really say quanta are like. You can only classify their characteristics, at least to the extent that uncertainty lets you do so. But this is close enough for a reasonable understanding.
Well, it looks like I got past that relatively quickly. Let's move right on into interpretations of quantum mechanics.
The first thing to understand is that every interpretation of quantum mechanics must agree with the experimental results, and therefore must substantially agree with actual quantum mechanics as it exists. NONE of these interpretations has any direct effect on the reality we observe, or on the results we predict, at least not so far. No one has devised any experiment that can show a measurable difference between any of these interpretations; they are all on equal footing in that regard. Furthermore, these interpretations are therefore not even hypotheses; we do not foresee the ability to test any of them, so they are and will remain for the foreseeable future nothing but conjectures. None of this is fact, or theory, or law, or even hypothesis. Each is an attempt to explain a highly abstruse mathematical theory in natural language, nothing more.
Before we launch into this, we need to understand where "wave function collapse" comes from.
When a quantum is propagating along ("propagating" is a fancy word for "moving," but since it's kind of hard to think of something that you can't really quite localize "moving," physicists use "propagating" instead), we know how to quantify the various things it might do in the future; we have probabilities for various states its characteristics might have it in when it encounters some situation. A physicist named Erwin Shroedinger made an equation that we still use to describe these probabilities; it is called the "Shroedinger wave equation," wave because it looks very much like the equations of water and sound waves, to such an extent that you wouldn't really think of what it describes as anything but a wave. This is one of those things that when we use it to describe reality, the results agree impressively with what we see.
Now, when the quantum interacts with some other quantum, there will be various probabilities of various outcomes; it might be absorbed, or it might bounce off, or whatnot. After that point, we no longer describe it with the old wave function; we no longer describe probabilities. Something has happened. An outcome has been selected, and now the probabilities are meaningless. We have not probabilities, but an outcome, whose probability is one. This is the "collapse of the wave function."
Because of this feature of the wave function, the first interpretation of quantum mechanics incorporated it as a feature. This interpretation is called the "Copenhagen interpretation," because it was advocated by Niels Bohr, a Danish physicist from, of course, Copenhagen. As Bohr went along, he became more and more convinced that we could never know what was really happening, but the so-called "strict Copenhagen interpretation" asserts to this day that the wave function collapse is a real phenomenon.
Later formulations of the Copenhagen interpretation, including the most popular today, assert that the wave function does not have real existence, nor does its collapse. These are epiphenomena interjected by our mathematical representation of quanta, not real phenomena in which the quanta participate.
The current "most accepted" approaches all use a concept called "decoherence." Decoherence conjectures that when quanta interact with their environment in a thermodynamically irreversible manner, they imitate "wave function collapse." The "collapse" is therefore "spread out" and dissipated into the environment where its effects become simple causal outcomes.
The "Many Worlds" or Everett interpretation asserts that every quantum event comes out all the different ways; we see only one in our universe, but in nearby universes, each of the alternatives happens. This helps resolve the question of "how does the quantum decide what to do;" in this interpretation, the quantum does everything possible, but in separate realities. Far from being as speculative as it sounds, this is considered a mainstream interpretation of quantum mechanics, of equal stature with decoherent Copenhagen and others I will mention.
Another interpretation is called "Consistent histories," or "decoherent histories." In this interpretation, it is asserted that descriptions of quanta are meaningless when those descriptions violate uncertainty. In other words, to describe the probability of a single parameter as being, for example, spin in X of +1/2 or -1/2, each with a probability of 0.5, has meaning, but to describe the spin of that same particle in, for example, Z, is inconsistent and therefore meaningless. By "meaningless" I mean that they have no connection to reality; given the spin in X is known, the spin in Z literally has no existence. Only descriptions that are consistent with quantum mechanics have meaning, whether they would be meaningful if applied to macroscopic, classical, phenomena or not. It is the interpretation I most favor, although there are others that are also interesting.
David Bohm came up with an interpretation of quantum mechanics that asserts that there is an independent, non-measurable set of what are called hidden variables that cannot be measured, that determine apparently random quantum behavior. In other words, the behavior of quanta is not random, but fully deterministic; we merely cannot ever even theoretically measure the values of the variables that determine this behavior. It therefore appears random to us. There are other hidden variable theories, but they suffer from a defect that Bohm's does not; a combination of factors involving the Aspect realization of the EPR experiment, and something called "Bell's Theorem" eliminate these theories (but not Bohm's) from possibility.
John Cramer has proposed an interpretation called the "Transactional Interpretation." The assertion here is that both the so-called "advanced" and "retarded" solutions to Maxwell's equations represent real phenomena. The retarded solution is our familiar picture of waves propagating forward in time; but the advanced solution proposes that waves can propagate backward in time and affect the outcome of an experiment prior to the determination of the conditions that it began with. This is very odd, but because of the EPR experiment, it is not inconsistent with reality. Dr. Cramer actually teaches classes in physics using this interpretation at the University of Washington in Seattle. It is supported by a paper written in the 1950s by John Wheeler and Richard Feynman, advocating a theory called "Wheeler-Feynman absorber theory."
A very interesting interpretation is Backward Causation. This interpretation asserts that events can have an effect on events backward in time. It is functionally equivalent to Cramer's TI, but does not propose a mechanism. It asserts that backward causation happens only in the realm where Planck's Constant is not of appreciable effect; Planck's Constant is the constant that relates the wavelength of a quantum's wave-like characteristics to its individual energy content.
There are others. What is most important to remember is that essentially, none of these interpretations says there is anything wrong with the mathematical theories of QM. Some of them claim that there is something missing, but in most if not all cases, they also claim that this "something" can never be directly measured.
And there you have it.
If we're going to discuss why we have all these different interpretations, I suggest that is another thread, and it involves the EPR experiment and Bell's Theorem and Aspect. From these experiments grows another that I consider to be one of the most revealing ever conducted: the Delayed Choice Quantum Eraser. If folks would like some website references to look over the various interpretations, I have several that you will find informative. But I also suggest that you want to find out about Bell's Theorem and EPR before you go there, and properly understand the DCQM, because otherwise you'll be left with a spinning head and nothing to really hang any of these interpretations on to differentiate them.
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