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Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
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Positive Definiteness of Killing Form in Gauge Theory

This question is related to requirement that the gauge group of a gauge theory be a direct product of compact simple groups and $U(1)$ factors but is not the same as, for example, this question (...
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What does the identity operator look like in QFT, written in momentum eigenstates? [duplicate]

This is a followup to a question I read recently: What does the identity operator look like in Quantum Field Theory? Out of curiosity, I was writing down what I figured to be the momentum basis ...