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. 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.
Contents
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
where the latter is the hermitian conjugate of the first.
The Bogoliubov transformation is the canonical transformation mapping the operators
It is then evident that
Since the form of this condition is suggestive of the hyperbolic identity
the constants u and v can be readily parametrized as
Applications
The most prominent application is by Nikolai Bogoliubov himself in the context of superfluidity. Other applications comprise Hamiltonians and excitations in the theory of antiferromagnetism. 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 . 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
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
The equation above defines the Bogoliubov transformation of the operators.
The ground state annihilated by all
Literature
The whole topic, and a lot of definite applications, are treated in the following textbooks: