Bogoliubov inner product

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 $A$ be a self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as

$\langle X,Y\rangle_A=\int\limits_0^1 {\rm Tr}[ {\rm e}^{xA} X^\dagger{\rm e}^{(1-x)A}Y]dx$

The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., $\langle X,X\rangle_A\ge 0$), and satisfies the symmetry property $\langle X,Y\rangle_A=\langle Y,X\rangle_A$.

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

$\langle X,Y\rangle_{\beta H}= \int\limits_0^1 \langle{\rm e}^{x\beta H} X^\dagger{\rm e}^{-x\beta H}Y\rangle dx$

where $\langle \dots \rangle$ 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:

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

References

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).