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. 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.
The most prominent application is by Nikolai Bogoliubov himself in the context of superfluidity. Other applications comprise Hamiltonians and excitations in the theory ofantiferromagnetism. 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. Bogoliubov transforms are also used extensively in quantum optics, particularly when working with gaussian unitaries.
Fermionic mode
For the anticommutation relations the Bogoliubov transformation can only satisfy the first of these anticommutation relations when Therefore, the only non-trivial possibility is corresponding to particle-antiparticle interchange. Thus, for a single particle, the transformation can only be implemented for a Dirac fermion where particle and antiparticle are distinct or or for multi-fermionic systems, in which there is more than one type of fermion.
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 -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, 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.
Multimode example
The Hilbert space under consideration is equipped with these operators, and henceforth describes a higher-dimensional quantum harmonic oscillator. 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.
