Decoherence is the process that occurs as a system transitions from quantum to classical mechanical behavior.

It is not possible to observe decoherence at normal human scales because the transition occurs at timescales that are shorter than we are currently able to measure. It is possible for a particle at the quantum level to be in two places at the same time (and sometimes I wonder if they can be in two times at the same place).

Decoherence has now been experimentally observed by David Wineland and colleagues at the National Institute of Standards and Technology in Boulder, Colorado, where they used a single beryllium ion confined in a magnetic trap and cooled so that quantum effects dominate its behavior. When the magnetic trap was disturbed the ion decohered.

Applications of decoherence include quantum computers that use quantum superposition.

Source:
Schroedinger's Cat in an Atomic Cage http://physics.nist.gov/News/Update/960528.html 5/25/2001

Quantum mechanics is all about an object being observed, and that the object is really in a superposition until it is seen to be otherwise; Decoherence is the idea that the wavefunctions of those superpositions can interfere with each other, causing those wavefunctions to collapse, making those objects behave classically.

This takes the ambiguity out of the idea of when a particle is observed; it's observed when the light hits it, interfering with and collapsing the wavefunction.

So, using decoherence, we can refute the common sense based attacks on Quantum Mechanics; some people pose it this way, "So you're saying the moon isn't there unless someone's looking at it?" If someone's really strictly following quantum mechanical dogma, they will answer, "Yes, we cannot be sure it's there unless we're looking." Decoherence comes in and says, "The moon will be there because all the individual particles of the moon are interacting, plus it is being hit with light from the sun, so their wavefunctions have collapsed to an eigenvalue of the superposition; it will act classically."

To quote Claus Kiefer and Erich Joos from their paper here: http://xxx.lanl.gov/abs/quant-ph/9803052,

What are the achievements of the decoherence mechanism? Decoherence can certainly explain why and how within quantum theory certain objects (including fields) appear classical to "local" observers... It is important to emphasise that classical properties are not an a priori attribute of objects, but only come into being through the interaction with the environment.

So, to summarize, decoherence describes how the environment collapses the wavefunction of a given object, describing the transition between quantum mechanical behavior and classical physical behavior. This idea that interaction between nearby wavefunctions could collapse them both and cause them both to act classically makes quantum theory a lot less objectionable, in my view, by describing why things we can see act classically.

I highly recommend that paper I referenced above, it's full of good information. Don't get too hung up on the big equations.

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