The Bogoliubov inner product (also known as the Duhamel two-point function, Bogolyubov inner product, Bogoliubov scalar product, or Kubo–Mori–Bogoliubov inner product) is a special inner product in the space of operators. The Bogoliubov inner product appears in quantum statistical mechanicsD. Petz and G. Toth. The Bogoliubov inner product in quantum statistics, Letters in Mathematical Physics 27, 205-216 (1993).D. P. Sankovich. On the Bose condensation in some model of a nonideal Bose gas, J. Math. Phys. 45, 4288 (2004). and is named after theoretical physicist Nikolay Bogoliubov.
Definition Let be a nonnegative 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 where is the complex conjugate of .
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 choices of notation, the Bogoliubov inner product takes the form
resembling a thermal average , where . In terms of the partition function, or statistical sum, this average is
where represents a perturbation of by , and .
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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