Bogoliubov inner product

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:

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