- Great 'paradox'!
When an electron neutrino is formed, it's a superposition of the three neutrino mass eigenstates. Each comes with a slightly different proton-and-electron recoil. So when beta decay produces an electron neutrino, the proton and electron don't have a definite total energy-momentum; they're a superposition of three possibilities.
In short, the neutrino's mass and its partners' momenta are linked - entangled, in fact. That's why you don't need to touch the neutrino to 'collapse' it. If you *could* measure the proton and electron precisely enough to pin down the neutrino's mass precisely, you'd be measuring the neutrino's mass - and you'd find it was one mass eigenstate, no longer a flavor state, no longer oscillating.
In reality nobody can measure the recoil anywhere near finely enough (the mass differences are absurdly tiny), so the three mass eigenstates stay superposed. The "mass" you'd extract from a realistic experiment along this line isn't one of the eigenvalues - it's a fuzzy number with an uncertainty far bigger than the gaps between the masses. The superposition survives, and what you see is an electron neutrino.
[@j_bertolotti](https://mathstodon.xyz/@j_bertolotti )
John Carlos Baez
npub1nf…3nqe4
2026-06-13 20:56:51 UTC
in reply to nevent1q…0v2u
Author Public Key
npub1nf4p4rh06z6n6lsvje4txk7eqs23y3hs8vd7nraq6tgwady5qvsqy3nqe4Published at
2026-06-13 20:56:51 UTCEvent JSON
{
"id": "2294195895119246086e2d7683308cf7becf6e147a384b9f27e82a7afc31d757",
"pubkey": "9a6a1a8eefd0b53d7e0c966ab35bd904151246f03b1be98fa0d2d0eeb4940320",
"created_at": 1781384211,
"kind": 1,
"tags": [
[
"e",
"f3926da9e12cb74d1516b355d3da94ae4c53c067cfa9887fe4ddf7ec4514e1f9",
"",
"reply",
"47d069b9227182c947fa6974c32d7cdf8764f5916ef0e6d88011720b2cf1db53"
],
[
"p",
"47d069b9227182c947fa6974c32d7cdf8764f5916ef0e6d88011720b2cf1db53"
],
[
"proxy",
"https://mathstodon.xyz/@johncarlosbaez/116744795693019973",
"web"
],
[
"p",
"9a6a1a8eefd0b53d7e0c966ab35bd904151246f03b1be98fa0d2d0eeb4940320"
],
[
"p",
"e04ab873a2a16d381911efeaa323e063c07286adf850d765122d2723abde0122"
],
[
"e",
"e551e4de9716331c64da8c8c182c83a1e818de93b4d7fc979fe0c26d9a2736f0",
"",
"root",
"9a6a1a8eefd0b53d7e0c966ab35bd904151246f03b1be98fa0d2d0eeb4940320"
],
[
"proxy",
"https://mathstodon.xyz/users/johncarlosbaez/statuses/116744795693019973",
"activitypub"
],
[
"L",
"pink.momostr"
],
[
"l",
"pink.momostr.activitypub:https://mathstodon.xyz/users/johncarlosbaez/statuses/116744795693019973",
"pink.momostr"
],
[
"-"
]
],
"content": "- Great 'paradox'! \n\nWhen an electron neutrino is formed, it's a superposition of the three neutrino mass eigenstates. Each comes with a slightly different proton-and-electron recoil. So when beta decay produces an electron neutrino, the proton and electron don't have a definite total energy-momentum; they're a superposition of three possibilities. \n\nIn short, the neutrino's mass and its partners' momenta are linked - entangled, in fact. That's why you don't need to touch the neutrino to 'collapse' it. If you *could* measure the proton and electron precisely enough to pin down the neutrino's mass precisely, you'd be measuring the neutrino's mass - and you'd find it was one mass eigenstate, no longer a flavor state, no longer oscillating. \n\nIn reality nobody can measure the recoil anywhere near finely enough (the mass differences are absurdly tiny), so the three mass eigenstates stay superposed. The \"mass\" you'd extract from a realistic experiment along this line isn't one of the eigenvalues - it's a fuzzy number with an uncertainty far bigger than the gaps between the masses. The superposition survives, and what you see is an electron neutrino. \n\n[@j_bertolotti](https://mathstodon.xyz/@j_bertolotti )",
"sig": "f3e28cd7d23402616a6367ba5e0937599a95760c6b42465dfea67c25b61b7ae557676b6c6cd8086894999611d5afdb915f84d6601e69bd808506d49f23b28578"
}