Bogoliubov transformation

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In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, 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_^n a_^\dagger |0\rangle]

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

[a'_i = \sum_j (u_ a_j + v_ a^\dagger_j)]

where the coefficients [u_,v_] must satisfy certain rules to guarantee that the annihilation operators and the creation operators [a^_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'_] 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.

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