Bogoliubov inner product - Alchetron, the free social encyclopedia

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 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 = ∫ 0 1 T r [ e x A X † e ( 1 − x ) A Y ] d x

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 .

In applications to quantum statistical mechanics, the operator A has the form A = β H , 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 ⟩ β H = ∫ 0 1 ⟨ e x β H X † e − x β H Y ⟩ d x

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 ⟩ β H = ∂ 2 ∂ t ∂ s T r e β H + t X + s Y | t = s = 0

References

Bogoliubov inner product Wikipedia

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