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.
Let [math]\displaystyle{ A }[/math] be a self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as
The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., [math]\displaystyle{ \langle X,X\rangle_A\ge 0 }[/math]), and satisfies the symmetry property [math]\displaystyle{ \langle X,Y\rangle_A=(\langle Y,X\rangle_A)^* }[/math] where [math]\displaystyle{ \alpha^* }[/math] is the complex conjugate of [math]\displaystyle{ \alpha }[/math].
In applications to quantum statistical mechanics, the operator [math]\displaystyle{ A }[/math] has the form [math]\displaystyle{ A=\beta H }[/math], where [math]\displaystyle{ H }[/math] is the Hamiltonian of the quantum system and [math]\displaystyle{ \beta }[/math] is the inverse temperature. With these notations, the Bogoliubov inner product takes the form
where [math]\displaystyle{ \langle \dots \rangle }[/math] denotes the thermal average with respect to the Hamiltonian [math]\displaystyle{ H }[/math] and inverse temperature [math]\displaystyle{ \beta }[/math].
In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum:
Original source: https://en.wikipedia.org/wiki/Bogoliubov inner product.
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