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Is it possible to increase indefinitely the lifetime of unstable particles by applying the quantum Zeno effect? Is there a bound from theoretical principles about the maximum extension one can get in a unstable energy level in general from continuous measurement alone?

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Yes, it is possible, but this effect is hard to calculate precisely, and has no chance of experimental realization. You can see this happen in the Unruh effect, when you accelerate a particle fast enough, the lifetime increases, due to the constant measurement by the thermal bath (or in the rest frame, due to the constant particle emission).

Exponential decay which is interrupted by a measurement is still exponential. If you have a particle which is decaying according to Fermi's golden rule, the probability of finding decay products at time t is exponentially growing in t, with a rate proportional to the matrix element squared integrated over the final states. This means that if you are working on time-scales where Fermi's rule is applicable, measuring whether that particle has decayed or not will do nothing to the decay rate.

The way Fermi's golden rule is derived, it is from different states cancelling in amplitude, over a time-scale comparable to the inverse amount of energy violation in the state. So when $\Delta t \gg {1\over \Delta E}$, Fermi's rule applies, and you get exponential decay.

Quantum zeno effect happens when you muck up the states in intermediate times, before Fermi's golden rule has a chance to become valid. So the time between interactions must be comparable to the inverse energy difference, not to the decay time. These are very different when you have a small coupling, so it is easy to mislead yourself. For example, if you want to stabilize the neutron, you need to measure it on inverse MeV timescales, which is the time it takes to cross 1000 nuclear diameters, or once every $10^{-20}s$, not once every seven minutes. This rate of interaction is experimentally hopeless (let alone keeping it up for seven minutes).

Similarly, for a muon decay, you need to measure once every inverse 100 MeV's. Likewise for pion decay. For proton decay, same thing, so we aren't wrecking proton decay by zeno effect by putting it in a vat of water.

There aren't any pairs of unstable particles which are separated in energy by less than the eletron mass, so for realistic zeno effect experiments, you are limited to atomic or perhaps nuclear energy levels.

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what if the interaction/measurement is continuous, why have the abitrary timeslicing? or you need to deliver a certain amount of measurement power (energy over time) in order for the effect to work? –  lurscher Aug 1 '12 at 7:10
    
@lurscher: You need one quantum to interact to carry information out. –  Ron Maimon Aug 1 '12 at 7:30
    
duh, of course! –  lurscher Aug 1 '12 at 7:32
    
"when you accelerate a particle fast enough, the lifetime increases, due to the constant measurement by the thermal bath" - do you presume that a particle on a planet's surface also experience a thermal bathe because it gets prolonged lifetime? What is the origin of such thermal bath? Hawking radiation? –  Anixx Dec 18 '12 at 1:52
    
"You need one quantum to interact to carry information out." - what is the box is a passive detector of the decay products? –  Anixx Dec 18 '12 at 2:01
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