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 mechanics[1][2] and is named after theoretical physicist Nikolay Bogoliubov.

Definition

edit

Let   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.,  ), and satisfies the symmetry property   where   is the complex conjugate of  .

In applications to quantum statistical mechanics, the operator   has the form  , where   is the Hamiltonian of the quantum system and   is the inverse temperature. With these notations, the Bogoliubov inner product takes the form

 

where   denotes the thermal average with respect to the Hamiltonian   and inverse temperature  .

In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum:

 

References

edit
  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).