is the Dirac sea a conventional or controversial model of the vacuum?
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I would describe it as "outdated". It is formally equivalent to the more modern, quantum field-based description, but I think people don't like the way it poses things conceptually.
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What’s the pop-science level distinction between the two interpretations?
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The idea of the Dirac sea was really meant to explain the origin of the positron, which acts just like an electron except it has positive charge. A positron can be created out of nothing so long as an electron is created simultaneously.
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