Krein space

Summary

A Krein space is a complex vector space with an inner product that may take negative values on some vectors. It splits as an orthogonal sum of a positive part and a negative part. The positive part is a Hilbert space for the form, and the negative part for its negative. Krein spaces are the natural setting for operators that are self-adjoint for an indefinite form. Such operators occur in relativistic quantum field theory. When the negative part has finite dimension, the space is a Pontryagin space.