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The '''Bogoliubov inner product''' (''Duhamel two-point function'', ''Bogolyubov inner product'', ''Bogoliubov scalar product'', ''[[Ryogo Kubo|Kubo]]-Mori-Bogoliubov inner product'') is a special [[inner product]] in the space of [[operator (mathematics)|operator]]s. The Bogoliubov inner product appears in [[quantum statistical mechanics]]<ref>D. Petz and G. Toth. [http://dx.doi.org/10.1007/BF00739578 The Bogoliubov inner product in quantum statistics], ''Letters in Mathematical Physics'' '''27''', 205-216 (1993).</ref><ref>D. P. Sankovich. [http://dx.doi.org/10.1063/1.1795971 On the Bose condensation in some model of a nonideal Bose gas], ''[[Journal of Mathematical Physics|J. Math. Phys.]]'' '''45''', 4288 (2004).</ref> and is named after theoretical physicist [[Nikolay Bogoliubov]].

==Definition==
Let <math>A</math> be a [[self-adjoint operator]]. The Bogoliubov inner product of any two operators X and Y is defined as
:<math> \langle X,Y\rangle_A=\int\limits_0^1 {\rm Tr}[ {\rm e}^{xA} X^\dagger{\rm e}^{(1-x)A}Y]dx</math>

The Bogoliubov inner product satisfies all the axioms of the inner product: it is [[Sesquilinear form|sesquilinear]], positive semidefinite (i.e., <math>\langle X,X\rangle_A\ge 0</math>), and satisfies the symmetry property <math>\langle X,Y\rangle_A=\langle Y,X\rangle_A</math>.

In applications to [[quantum statistical mechanics]], the operator <math>A</math> has the form <math>A=\beta H</math>, where <math>H</math> is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the quantum system and <math>\beta </math> is the [[inverse temperature]]. With these notations, the Bogoliubov inner product takes the form
:<math> \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</math>
where <math>\langle \dots \rangle</math> denotes the thermal average with respect to the Hamiltonian <math> H </math> and inverse temperature <math> \beta </math>.

In quantum statistical mechanics, the Bogoliubov inner product appears as the second order term in the expansion of the statistical sum:
:<math> \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} </math>

==References==
{{reflist}}

[[Category:Quantum mechanics]]
[[Category:Statistical mechanics]]

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en>Christian75: Added {{unreferenced}} tag to article using TW