Bogoliubov inner product

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 and is named after theoretical physicist Nikolay Bogoliubov.

Definition

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

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

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

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

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

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

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

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