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^[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

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

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

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

where ${\displaystyle ⟨\dots ⟩}$ 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:

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

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