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 {\displaystyle A} be a nonnegative self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as
⟨ X , Y ⟩ A = ∫ 0 1 T r [ e − x A X † e − ( 1 − x ) A Y ] d x {\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., ⟨ X , X ⟩ A ≥ 0 {\displaystyle \langle X,X\rangle _{A}\geq 0} ), and satisfies the symmetry property ⟨ X , Y ⟩ A = ( ⟨ Y , X ⟩ A ) ∗ {\displaystyle \langle X,Y\rangle _{A}=(\langle Y,X\rangle _{A})^{*}} where α ∗ {\displaystyle \alpha ^{*}} is the complex conjugate of α {\displaystyle \alpha } . In applications to quantum statistical mechanics, the operator A {\displaystyle A} has the form A = β H {\displaystyle A=\beta H} , where H {\displaystyle H} is the Hamiltonian of the quantum system and β {\displaystyle \beta } is the inverse temperature. With these notations, the Bogoliubov inner product takes the form
⟨ X , Y ⟩ β H = ∫ 0 1 T r [ e − x β H X † e − ( 1 − x ) β H Y ] d x {\displaystyle \langle X,Y\rangle _{\beta H}=\int \limits _{0}^{1}{\rm {Tr}}[{\rm {e}}^{-x\beta H}X^{\dagger }{\rm {e}}^{-(1-x)\beta H}Y]dx}
In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum:
⟨ X , Y ⟩ β H = ∂ 2 ∂ t ∂ s T r e β H + t X + s Y | t = s = 0 {\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}}
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