1. Single bosonic mode example
    Applications 

2. Fermionic mode
    Applications 

3. Multimode example

4. References

5. Further reading

In theoretical physics, the Bogoliubov transformation, also known as Bogoliubov-Valatin transformation, were independently developed in 1958 by Nikolay Bogolyubov and John George Valatin for finding solutions of BCS theory in a homogeneous system.^[1][2] The Bogoliubov transformation is an isomorphism of either the canonical commutation relation algebra or canonical anticommutation relation algebra. This induces an autoequivalence on the respective representations. The Bogoliubov transformation is often used to diagonalize Hamiltonians, which yields the stationary solutions of the corresponding Schrödinger equation. The Bogoliubov transformation is also important for understanding the Unruh effect, Hawking radiation, pairing effects in nuclear physics, and many other topics.

The Bogoliubov transformation is often used to diagonalize Hamiltonians, with a corresponding transformation of the state function. Operator eigenvalues calculated with the diagonalized Hamiltonian on the transformed state function thus are the same as before.

Single bosonic mode example

Consider the canonical commutation relation for bosonic creation and annihilation operators in the harmonic basis

Define a new pair of operators

for complex number u and v, where the latter is the Hermitian conjugate of the first.

The Bogoliubov transformation is the canonical transformation mapping the operators and to and . To find the conditions on the constants u and v such that the transformation is canonical, the commutator is evaluated, viz.

It is then evident that is the condition for which the transformation is canonical.

Since the form of this condition is suggestive of the hyperbolic identity

,

the constants {{mvar|u}} and {{mvar|v}} can be readily parametrized as

This is interpreted as a linear symplectic transformation of the phase space. By comparing to the Bloch-Messiah decomposition, the two angles and correspond to the orthogonal symplectic transformations (i.e., rotations) and the squeezing factor corresponds to the diagonal transformation.

Applications

The most prominent application is by Nikolai Bogoliubov himself in the context of superfluidity.^[3][4] Other applications comprise Hamiltonians and excitations in the theory of antiferromagnetism.^[5] 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

,

the same transformation with u and v becomes

To make the transformation canonical, u and v can be parameterized as

Applications

The most prominent application is again by Nikolai Bogoliubov himself, this time for the BCS theory of superconductivity.^[5][6][7][8] The point where the necessity to perform a Bogoliubov transform becomes obvious is that in mean-field 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 -terms, i.e. one must go beyond the usual Hartree–Fock method. In particular, in the mean-field Bogoliubov-de Gennes Hamiltonian formalism with a superconducting pairing term such as , the Bogoliubov transformed operators annihilate and create quasiparticles (each with well-defined energy, momentum and spin but in a quantum superposition of electron and hole state), and have coefficients and given by eigenvectors of the Bogoliubov-de Gennes matrix. Also in nuclear physics, this method is applicable since it may describe the "pairing energy" of nucleons in a heavy element.^[9]

Multimode example

The Hilbert space under consideration is equipped with these operators, and henceforth describes a higher-dimensional quantum harmonic oscillator (usually an infinite-dimensional one).

The ground state of the corresponding Hamiltonian is annihilated by all the annihilation operators:

All excited states are obtained as linear combinations of the ground state excited by some creation operators:

One may redefine the creation and the annihilation operators by a linear redefinition:

where the coefficients must satisfy certain rules to guarantee that the annihilation operators and the creation operators , 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 is different from the original ground state and they can be viewed as the Bogoliubov transformations of one another using the operator-state correspondence. They can also be defined as squeezed coherent states. BCS wave function is an example of squeezed coherent state of fermions.^[10]

References

Further reading

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 Many-Particle Systems, Dover (2003)
• Ch. Kittel: Quantum theory of solids, Wiley (1987)
• M. Wagner, Unitary Transformations in Solid State Physics, Elsevier Science, 1986

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