A persistent puzzle of quantum mechanics is the emergence of the classical world from its quantum substrate. How is it that the world of everyday experience appears to follow classical rather than quantum laws? This was a mystery to the founders of quantum mechanics, and ultimately they mostly adopted the so called Copenhagen interpretation, which just sweeps the mystery under the rug by declaring that two must be kept separate, pushing all the weirdness into the measurement process.
Over the last three decades an alternative interpretation, known as consistent histories or de-coherent histories, introduced by Robert Griffiths and further developed by Roland Omnes, Murray Gell-Mann, and James Hartle, has attracted widespread interest and support. Its great virtue is that it does away with the Copenhagen division of the world into quantum systems and classical measurement apparatus. All get treated on the same quantum mechanical basis.
The most mysterious quantum phenomena are those associated with non-local behavior and quantum entanglement. A photon fired at two slits somehow manages to go through both of them and interfere with itself in the process of forming an interference pattern on a screen beyond. Quantum systems manage to be in superpositions of quantum states.
These superpositions are very unclassical. Schroedinger was disturbed enough to invent his cat, who paradoxically manages to wind up in a state that is a superposition of dead and alive. The way decoherence deals with the cat is one of its triumphs.
If we were to do the cat experiment in fact, putting the cat in a chamber with a cyanide solution that would release only with the decay of a quantum system, the classical result would be that we would either open the chamber and find a dead cat smelling of bitter almonds or not. In classical Copenhagen, we might, like Schroedinger, imagine that the cat is in a superposition of dead and alive states until we open the chamber and collapse the state function to one of its eigenvalues. So how could we tell the difference? The answer is, that we couldn't, unless we could somehow get the dead and alive states to interfere with each other - somehow showing an interference pattern.
Decoherence says that this can't happen, because even if such a superposition forms, the quantum coherence between the states which could allow such interference is very rapidly destroyed by the random influences of the environment - the heat bath of photons, neutrinos, and gravitons in which we live.
As usual, I am writing about this because I'm trying to understand it, so critique and comment is welcome - as are questions, especially those I haven't thought of.