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 steadystate solutions of the corresponding Schrödinger 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.
Contents

Single bosonic mode example 1

Fermionic mode 2

Multimode example 3

References 4

Literature 5

External links 6
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 u and v 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.^{[1]} Other applications comprise Hamiltonians and excitations in the theory of antiferromagnetism.^{[2]} When calculating quantum field theory in curved spacetimes 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 transformation canonical, u and v can be parameterized as

u = e^{i \theta_1} \cos r \,\!

v = e^{i \theta_2} \sin r \,\! .
Applications
The most prominent application is again by Nikolai Bogoliubov himself, this time for the BCS theory of superconductivity .^{[2]} The point where the necessity to perform a Bogoliubov transform becomes obvious is that in meanfield approximation the Hamiltonian of the system can be written in both cases as a sum of bilinear terms in the original creation and destruction operators, involving finite \,\langle a_i^+a_j^+\rangleterms, i.e. one must go beyond the usual Hartree–Fock method (> Hartree–FockBogoliubov method). Also in nuclear physics this method is applicable since it may describe the "pairing energy" of nucleons in a heavy element.^{[3]}
Multimode example
The Hilbert space under consideration is equipped with these operators, and henceforth describes a higherdimensional quantum harmonic oscillator (usually an infinitedimensional one).
The ground state of the corresponding Hamiltonian is annihilated by all the annihilation operators:

\forall i \qquad a_i 0\rangle = 0
All excited states are obtained as linear combinations of the ground state excited by some creation operators:

\prod_{k=1}^n a_{i_k}^\dagger 0\rangle
One may redefine the creation and the annihilation operators by a linear redefinition:

a'_i = \sum_j (u_{ij} a_j + v_{ij} a^\dagger_j)
where the coefficients \,u_{ij},v_{ij} must satisfy certain rules to guarantee that the annihilation operators and the creation operators a^{\prime\dagger}_i, defined by the Hermitian conjugate equation, have the same commutators for bosons and anticommutators for fermions.
The equation above defines the Bogoliubov transformation of the operators.
The ground state annihilated by all a'_{i} is different from the original ground state 0\rangle and they can be viewed as the Bogoliubov transformations of one another using the operatorstate correspondence. They can also be defined as squeezed coherent states. BCS wave function is an example of squeezed coherent state of fermions.^{[4]}
References

^ Nikolai Bogoliubov: On the theory of superfluidity, J. Phys. (USSR), 11, p. 23 (1947)

^ ^{a} ^{b} See e.g. the textbook by Charles Kittel: Quantum theory of solids, New York, Wiley 1987.

^ Vilen Mitrovanovich Strutinsky: Shell effects in nuclear physics and deformation energies, Nuclear Physics A, Vol. 95, p. 420442 (1967), [1] .

^ Svozil, K. (1990), "Squeezed Fermion states", Phys. Rev. Lett. 65, 33413343. doi:10.1103/PhysRevLett.65.3341
Literature
The whole topic, and a lot of definite applications, are treated in the following textbooks:

J.P. Blaizot and G. Ripka: Quantum Theory of Finite Systems, MIT Press (1985)

A. Fetter and J. Walecka: Quantum Theory of ManyParticle Systems, Dover (2003)

Ch. Kittel: Quantum theory of solids, Wiley (1987)
External links
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