Кванттық оптикада және физиканың басқа да салаларында Боголюбов түрленуі

Mathematical operation in quantum optics, general relativity and other areas of physics

In theoretical physics, the Bogoliubov transformation, also known as the Bogoliubov–Valatin transformation, was 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, Davies Fulling radiation (moving mirror model), 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 of antiferromagnetism. 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. Bogoliubov transforms are also used extensively in quantum optics, particularly when working with gaussian unitaries (such as beamsplitters, phase shifters, and squeezing operations).

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 (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.