BIGpedia - Bogoliubov transformation - Encyclopedia and Dictionary Online

In theoretical physics, the Bogoliubov transformation 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 CCR/CAR algebra.

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:

$\forall i \qquad a_i |0\rangle = 0$

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 $u i j, v i j$ must satisfy certain rules to guarantee that the annihilation operators and the creation operators $a^{\prime\dagger}_i$ , defined by the Hermitean conjugate equation, have the same commutators.

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 operator-state correspondence . They can also be defined as squeezed coherent states.

In physics, the Bogoliubov transformation is important for understanding of the Unruh effect and Hawking radiation, among many other things.

Categories: Quantum field theory