Bogoliubov inner product

Special inner product in the space of operators

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.

## Definition[edit]

Let  A {\displaystyle A} be a 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 ⟨ e x β H X † e − x β H Y ⟩ d x {\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  H {\displaystyle H} and inverse temperature  β {\displaystyle \beta } .

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}}

## References[edit]

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).