in theoretical physics, the bogoliubov transformation, named after nikolay bogolyubov, is a unitary transformation  from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an isomorphism of the commutation relation algebra. the bogoliubov transformation is often used to diagonalize hamiltonians, which yields the steady-state solutions of the corresponding schrÃ¶dinger equation. the solutions of bcs theory in a homogeneous system, for example, are found using a bogoliubov transformation. the bogoliubov transformation is also important for understanding the unruh effect, hawking radiation and many other topics. == single bosonic mode example == consider the canonical commutation relation for bosonic creation and annihilation operators in the harmonic basis :\left [ \hat{a}, \hat{a}^\dagger \right ] = 1~. define a new pair of operators :\hat{b} = u \hat{a} + v \hat{a}^\dagger :\hat{b}^\dagger = u^* \hat{a}^\dagger + v^* \hat{a}~, where the latter is the hermitian conjugate of the first. the bogoliubov transformation is a canonical transformation of these operators. to find the conditions on the constants u and v such that the transformation is canonical, the commutator is evaluated, viz. :\left [ \hat{b}, \hat{b}^\dagger \right ] = \left [ u \hat{a} + v \hat{a}^\dagger , u^* \hat{a}^\dagger + v^* \hat{a} \right ] = \cdots = \left ( |u|^2 - |v|^2 \right ) \left [ \hat{a}, \hat{a}^\dagger \right ]. it is then evident that \,|u|^2 - |v|^2 = 1 is the condition for which the transformation is canonical. since the form of this condition is suggestive of the hyperbolic identity :\cosh^2 x - \sinh^2 x = 1, the constants  and  can be readily parametrized as :u = e^{i \theta_1} \cosh r :v = e^{i \theta_2} \sinh r ~ . ===applications=== the most prominent application is by nikolai bogoliubov himself in the context of superfluidity.nikolai bogoliubov: on the theory of superfluidity, j. phys. (ussr), 11, p. 23 (1947) other applications comprise hamiltonians and excitations in the theory of antiferromagnetism.see e.g. the textbook by charles kittel: quantum theory of solids, new york, wiley 1987. when calculating quantum field theory in curved space-times the definition of the vacuum changes and a bogoliubov transformation between these different vacua is possible, this is used in the derivation of hawking radiation. == fermionic mode == for the anticommutation relation :\left\{ \hat{a}, \hat{a}^\dagger \right\} = 1, the same transformation with u and v becomes :\left\{ \hat{b}, \hat{b}^\dagger \right\} = (|u|^2 + |v|^2) \left\{ \hat{a}, \hat{a}^\dagger \right\} to make the
Quantum_field_theory