The mechanism which we think neutrons decay is by the weak force. The interaction between the quarks of a neutron cause one of them to change their flavor to "up". Thus the neutron decays into a proton, an electron and an electron neutrino. And it is known that the boson of the weak force, the W boson acts against the law of conservation of energy(thats why we can not see it). So suppose that we observe the neutron all the time, constantly sending photons. Would it decay or not?
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The quantum Zeno effect doesn't make much difference for neutrons interacting with photons because the energies involved are so tiny relative to the decay energy. In fact, most neutron lifetime measurement experiments (for a recent summary, see the presentations by Dewey and Liu from this recent neutron-physics summer school) have the neutrons decaying in volume of space where the magnetic field is a few tesla, so that the electrons and protons can be guided to a detector. A neutron in a magnetic field is continuously interacting with the virtual photons that make up the field. In that sort of a field, the energy that's exchanged if a neutron's spin reverses is a few hundred nano-eV. This is absurdly smaller than the energy involved in the beta decay, nearly a mega-eV. If there's any effect, it's much smaller than the other uncertainties involved in neutron lifetime measurements.
It may that, in a hand-waving way, you can think of the quantum Zeno effect as one sort of explanation for the stability of neutrons inside of a nucleus. Within the nucleus you have the neutron continuously participating in strong interactions, including charged pion exchange:
In this interaction, the pion field is constantly "checking" whether the neutron has turned into a proton or not, and the neutron lifetime is dramatically different inside of a nucleus than outside. That's not a great example of the quantum Zeno effect, however, because there are nuclei whose lifetime against $\beta^-$ decay is shorter than the free neutron lifetime, as well as nuclei which are $\beta^+$-unstable even though free protons do not decay.
A very interesting example of the quantum Zeno effect in the neutron system is to forbid neutron-antineutron oscillations in regions where there are nuclei or magnetic fields. No one has ever observed a neutron oscillating to an antineutron, but the limit on the lifetime for that process is surprisingly weak because it's hard to really isolate the neutrons.
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