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The Bogoliubov inner product (Duhamel two-point function, Bogolyubov inner product, Bogoliubov scalar product, 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.

## Definition

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 ${\displaystyle \langle X,Y\rangle _{A}=\langle Y,X\rangle _{A}}$.

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:

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

## References

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1. D. Petz and G. Toth. The Bogoliubov inner product in quantum statistics, Letters in Mathematical Physics 27, 205-216 (1993).
2. D. P. Sankovich. On the Bose condensation in some model of a nonideal Bose gas, J. Math. Phys. 45, 4288 (2004).