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., $$\langle X,X\rangle_A\ge 0$$), 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:
 * $$ \langle X,Y\rangle_{\beta H}=\frac{\partial^2}{\partial t\partial s}{\rm Tr}\,{\rm e}^{\beta H+tX+sY} \bigg\vert_{t=s=0} $$