Are elementary particles stable, like many atoms or molecules are stable if there are no reaction partners nearby? Or can they be observed just for very short instances of time?
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How elementary are you talking. All mesons are unstable, a proton is possibly the only stable baryon – its half life is at least $10^{32}$ years. If you consider quarks, it's a tricky answer. Strange and heavier quarks decay via the Weak interaction –but so do down quarks, not in a proton, but certainly in a neutron ($n\rightarrow p + e^- + \bar{\nu}_e$). I suppose the up quark is as stable as the proton.
The electron (positron) is the only stable charged lepton – the heavier mu and tau decay.
Neutrinos (and anti neutrinos) are tricky, since they oscillate. It's not decay, but it's not exactly stability either.
On the gauge boson side of things, I would say the photon is stable – it moves at the speed of light, so it can't decay – its time dilation is infinite. Nevertheless, it is its own antiparticle and can just be absorbed (so it disappears, but it doesn't decay).
The W and Z bosons decay. (Likewise the Higgs).
Regarding the strong interaction: the gluon, it's confined – we never see them. They are kicked out in jets (see 3-Jet Events) and hadronize, but that's not decay either.
For a complete listing, see the legendary PDB.
