Bogoliubov inner product explained

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

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} ^{1} ^{0} {\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)} 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 _{\beta} = \int\limits ^{1} _{0} \langle{\rm e} ^{x\beta} X ^{\dagger{\rm} e} ^{-x\beta} Y\rangle dx where \langle ... \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} = (\partial ^{2})/(\partial t\partial s) {\rm Tr}{\rm e} ^{\beta} \ vert _{t=s=0}

Notes and 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).