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 A be a self-adjoint operator. The Bogoliubov inner product of any two operators X and Y is defined as

⟨X,Y⟩A=∫01Tr[exAX†e(1−x)AY]dx

The Bogoliubov inner product satisfies all the axioms of the inner product: it is sesquilinear, positive semidefinite (i.e., ⟨X,X⟩A≥0), and satisfies the symmetry property ⟨X,Y⟩A=(⟨Y,X⟩A)* where α* is the complex conjugate of α.

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

⟨X,Y⟩βH=∫01⟨exβHX†e−xβHY⟩dx

where ⟨…⟩ denotes the thermal average with respect to the Hamiltonian H and inverse temperature β.

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

⟨X,Y⟩βH=∂2∂t∂sTreβH+tX+sY|t=s=0

## 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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