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### Is there any theorem that suggests that QM+SR has to be an operator theory?

UPDATE To make my question more precise, I'll define what I mean by an operator theory: An operator theory is a theory in which the dynamical objects are operators, i.e., the equations of motion are ...
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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.