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We discuss self-adjoint operators given formally by expressions quadratic in bosonic creation and annihilation operators. We give conditions when they can be defined as self-adjoint operators, possibly after an infinite renormalization. We also discuss explicit formulas for their infimum. Our main motivation comes from local quantum field theory, which furnishes interesting examples of bosonic quadratic Hamiltonians that require an infinite renormalization.
Jan Dereziński (Fri,) studied this question.