Bogoliubov inner product

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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.


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}


  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).