Can decoherence time be shorter than Planck time?

Decoherence times can be estimated and are inverse functions of mass. Since there are no upper bounds on mass, can decoherence time be shorter than Planck time?

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The answer depends on the meaning of time at time scales comparable to the plank time. ::blinks and shakes head at the repetition there:: Which depends on physics we don't know... –  dmckee Jan 16 '11 at 21:59
@dmckee: good point, thanks. –  Humble Jan 16 '11 at 22:18

of course, the decoherence time may be - and quite often is - much shorter than the Planck time. There exists no law saying that "all quantities with the units of seconds have to be longer than the Planck time". The only genuine limitation is that "if you try to measure the timing of an event with the precision better than the Planck time, you will fail." This is an entirely different thing.

The decoherence time is the (approximate) time after which the off-diagonal elements of the density matrix between two macroscopically "different" states of a sufficiently macroscopic system decrease to a negligible fraction of its initial value. The processes that drive decoherence - that make the off-diagonal elements drop - are results of interactions of the macroscopic object with the environment. The more strongly the object interacts with the environment and the higher number of environmental degrees of freedom we are forgetting - i.e. tracing over - the more quickly the decoherence proceeds.

Of course that if you talk about Schrödinger's cat that is emitting lots of photons and other things, the decoherence time - for the "dead" and "alive" states - will be tiny. If it is not shorter than the Planck length, it's surely easy to consider just slightly bigger systems for which it will be shorter than the Planck time. The statement that it is shorter than the Planck time doesn't mean that you have been actually able to "see" events in spacetime with a trans-Planckian resolution, so it's no violation of the laws of quantum gravity.

Much more generally, there is a lot of confusion among the laymen about the statement that "distances and times can't be shorter than the Planck scale". We must refine what distances and times we're talking about. Obviously, the statement may only hold for Lorentz-invariant quantities whose units are either seconds or meters. Not all "lengths" and "times" are equal or the same thing. It depends "length of what" or "time of what" you're talking about. Lengths of different things and times of different things have different properties.

Only when the proper distances between some important points or events decrease beneath the Planck length, one may possibly face some contradictions with quantum gravity - or one produces a black hole that damages the experiment. But for example, the wavelength of an electromagnetic wave (which is the same thing as its periodicity in time, multiplied by $c$) can be much shorter than the Planck length. Arbitrarily short-wavelength photons exist; this is guaranteed by the theory of relativity. An observer may be accelerated to a speed that approaches the speed of light arbitrarily closely, and when it's done, the energy of the original photon may increase arbitrarily high - and the wavelength may become arbitrarily short by the Lorentz contraction. Nothing new is happening to such a photon: it's the same photon that had a low frequency in another inertial system.

There can't exist any restriction on the wavelength of waves (of photons etc.). But when proper distances inside an object (formally) drop beneath the Planck length, you may be sure that the laws of geometry have to be severely modified, to say the least; one should use something else than ordinary geometry to describe what is going on and where it is going on. Similarly for times. However, the density matrix linked to a macroscopic body is not a "real object": it is just a set of numbers that can be used to predict probabilities of various observations of the macroscopic body. Such things can change arbitrarily quickly. And indeed, the more massive body you consider, the shorter the decoherence time will be.

(When people misleadingly talk about the "collapse of the wave function", this "process" may also occur even infinitely quickly. It's no violation of relativity because the wave function is not a "real object": it is a set of numbers encoding the probabilities - and phases - of things we can predict. When one carefully checks whether some genuine information may be transmitted faster than light between two distant EPR entangled particles, the answer is a resounding No. This fact - causality - may be proved easily in quantum field theory. So the EPR particles won't allow you to speed up the communication with other galaxies.)

Best wishes Lubos

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No, Johannes, there is absolutely no Planck scale physics involved in decoherence of large objects - despite the fact that the decoherence time is much shorter than the Planck time. There is no speculation about the fact that wave functions at energy $E$ depend on time as $\exp(Et/i\hbar)$. The periodicity of this function is much shorter than the Planck scale if the energy is greater than the Planck energy - equivalent to mass $10^{-8}$ kg. This doesn't mean that there's any Planck scale physics or quantum gravity here: the frequency is just additive in the mass. Decoherence is similar. –  Luboš Motl Jan 17 '11 at 11:26