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. Theoretical physics is physics that employs mathematical models and abstractions rather than experimental processes. ... Nikolai Nikolaevich Bogoliubov (21 August 1909 – 13 February 1992) was a Russian-Ukrainian mathematician and theoretical physicist known for his work in statistical field theory and dynamical systems. ... A unitary transformation is an isomorphism (but not an antiisomorphism; that corresponds to an antiunitary transformation) between two Hilbert spaces or an automorphism of a single Hilbert space. ... In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in case G is a locally compact (Hausdorff) topological group... In quantum field theory, if V is a real vector space equipped with a nonsingular real antisymmetric bilinear form (,) (i. ... In quantum field theory, if V is a real vector space equipped with a nonsingular real antisymmetric bilinear form (,) (i. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

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 quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. ...

The ground state of the corresponding Hamiltonian is annihilated by all the annihilation operators: In physics, the ground state of a quantum mechanical system is its lowest-energy state. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ...

All excited states are obtained as linear combinations of the ground state excited by some creation operators: In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...

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

where the coefficients u_ij,v_ij must satisfy certain rules to guarantee that the annihilation operators and the creation operators , defined by the Hermitean conjugate equation, have the same commutators. In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. ...

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 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, a squeezed coherent state is every state in the Hilbert space of quantum mechanics that saturates the uncertainty principle that is the product of the corresponding two operators takes on its minimum value: The simplest such state is the ground state of the quantum harmonic oscillator. ...

In physics, the Bogoliubov transformation is important for understanding of the Unruh effect and Hawking radiation, among many other things. A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... The Unruh effect, discovered in 1976 by Bill Unruh of the University of British Columbia, is the prediction that an accelerating observer will observe black-body radiation where an inertial observer would observe none, that is, the accelerating observer will find themselves in a warm background. ... In physics, Hawking radiation is thermal radiation thought to be emitted by black holes due to quantum effects. ...