Bogoliubov inner product
The Bogoliubov inner product (Duhamel two-point function, Bogolyubov inner product, Bogoliubov scalar product, Kubo-Mori-Bogoliubov inner product) is a special inner product in the space of operators. The Bogoliubov inner product appears in quantum statistical mechanics and is named after theoretical physicist Nikolay Bogoliubov.
Let be a self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as
The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., ), and satisfies the symmetry property .
In applications to quantum statistical mechanics, the operator has the form , where is the Hamiltonian of the quantum system and is the inverse temperature. With these notations, the Bogoliubov inner product takes the form
where denotes the thermal average with respect to the Hamiltonian and inverse temperature .
In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum:
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