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 mechanics^[1][2] and is named after theoretical physicist Nikolay Bogoliubov.

Let ${\displaystyle A}$ be a self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as

${\displaystyle \langle X,Y\rangle _{A}=\int \limits _{0}^{1}{\rm {Tr))[{\rm {e))^{xA}X^{\dagger }{\rm {e))^{(1-x)A}Y]dx}$

The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., ${\displaystyle \langle X,X\rangle _{A}\geq 0}$), and satisfies the symmetry property ${\displaystyle \langle X,Y\rangle _{A}=(\langle Y,X\rangle _{A})^{*))$ where ${\displaystyle \alpha ^{*))$ is the complex conjugate of ${\displaystyle \alpha }$.

In applications to quantum statistical mechanics, the operator ${\displaystyle A}$ has the form ${\displaystyle A=\beta H}$, where ${\displaystyle H}$ is the Hamiltonian of the quantum system and ${\displaystyle \beta }$ is the inverse temperature. With these notations, the Bogoliubov inner product takes the form

${\displaystyle \langle X,Y\rangle _{\beta H}=\int \limits _{0}^{1}\langle {\rm {e))^{x\beta H}X^{\dagger }{\rm {e))^{-x\beta H}Y\rangle dx}$

where ${\displaystyle \langle \dots \rangle }$ denotes the thermal average with respect to the Hamiltonian ${\displaystyle H}$ and inverse temperature ${\displaystyle \beta }$.

In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum:

${\displaystyle \langle X,Y\rangle _{\beta H}={\frac {\partial ^{2)){\partial t\partial s)){\rm {Tr))\,{\rm {e))^{\beta H+tX+sY}{\bigg \vert }_{t=s=0))$