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[en] The authors clarify the mathematical difficulty when for bosonic systems (i.e., unbounded operators) they rigorously construct a Liouville space, in which a Liouville operator acts, associated with a Hamiltonian, including unbounded and non-quasi-free cases. Thermo field dynamics are given as an expression equivalent to the Liouville space and operator. The authors investigate the mathematical properties of them, and thus clarify the mathematical properties of the thermal state. 32 refs
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