Bogoliubov inner product

Short description: 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

Let [math]\displaystyle{ A }[/math] be a self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as

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

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

[math]\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 }[/math]

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:

[math]\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} }[/math]

## 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).

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