Bogoliubov inner product

Template:Short description 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

$⟨X,Y{⟩}_A=\underset{0}{\overset{1}{\int }}\mathrm{T}\mathrm{r}[{\mathrm{e}}^{xA}{X}^{†}{\mathrm{e}}^{(1-x)A}Y]dx$

The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., $⟨X,X{⟩}_A\ge 0$), and satisfies the symmetry property $⟨X,Y{⟩}_A=(⟨Y,X{⟩}_A{)}^{*}$ 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

$⟨X,Y{⟩}_{\beta H}=\underset{0}{\overset{1}{\int }}⟨{\mathrm{e}}^{x\beta H}{X}^{†}{\mathrm{e}}^{-x\beta H}Y⟩dx$

where $⟨\dots ⟩$ 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:

$⟨X,Y{⟩}_{\beta H}=\frac{{\partial }^2}{\partial t\partial s}\mathrm{T}\mathrm{r}{\mathrm{e}}^{\beta H+tX+sY}{|}_{t=s=0}$