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 mechanicsD. P. Sankovich
On the Bose condensation in some model of a nonideal Bose gas
'' J. Math. Phys.'' 45, 4288 (2004). 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 ^ X^\dagger^Yx

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 $\langle X,Y\rangle_A=(\langle Y,X\rangle_A)^*$ where $\alpha^*$ is the complex conjugate of $\alpha$.

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
:$\langle X,Y\rangle_= \int\limits_0^1 \langle^ X^\dagger^Y\rangle dx$
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:
:$\langle X,Y\rangle_=\frac\,^ \bigg\vert_$

References

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Quantum mechanics
Statistical mechanics