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  "body": "\n## Summary\n\nA Krein space is a complex vector space with an inner product that may take negative values on some vectors.\nIt splits as an orthogonal sum of a positive part and a negative part.\nThe positive part is a Hilbert space for the form, and the negative part for its negative.\nKrein spaces are the natural setting for operators that are self-adjoint for an indefinite form.\nSuch operators occur in relativistic quantum field theory.\nWhen the negative part has finite dimension, the space is a Pontryagin space.\n\n## Explanations\n\n### L1 human\n\nIn ordinary geometry every nonzero vector has positive squared length.\nIn a Krein space some vectors have negative or zero \"squared length\", as in the space-time of special relativity.\nThe space still breaks into a part where lengths are positive and a part where they are negative, and each part looks like a Hilbert space.\nThe simplest example is $\\C^2$ with $[x, y] = x_1 \\overline{y_1} - x_2 \\overline{y_2}$.\n",
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