It helps in the diagonalization of the many body interaction Hamiltonian (such as in BEC). However, I am interested in the physical significance of the coefficients of the operators in this transformation and their indices.
Since Hamiltonians are Hermitian operators, the transformation which diagonalize them must be Unitary, therefore it can be formulated as a rotation. Hence, the Bogoliubov transformation is merely a rotation of the phase space, the coefficients can be expressed in terms of trigonometric (for Fermions) or hypertrignometric functions (for Bosons). Recall that the fundamental rotation matrix is(in matlab matrix notation)
[cos(theta), -sin(theta); sin(theta), cos(theta], and you'll notice that your solution for the coefficients is such that they obey the trigonometric (resp. hypertrigonometric) identities: cos(theta)^2+sin(theta)^2=1 (resp. cosh(theta)^2-sinh(theta)^2=1) due to the constraints on the commutation relations of Fermionic (resp. Bosonic) operators.
Hope this was helpful,
Ofer
Superalgebraic representation of Bogoliubov transformations for fermions shows that they are really Lorentz transformations - see my article
https://arxiv.org/abs/1707.03687
Bogoliubov transformation arises in the problem of two coupled harmonic oscillators.
In this problem of two coupled oscillators described by coordinates (x1,p1,x2,p2) if we introduce a canonical transformation which will decouple two oscillators then transformation matrices satisfy a set of commutation relations identical to Lorentz transformation rules (Lie algebra) in 2+1 dimensions.
ref: https://arxiv.org/pdf/cond-mat/9611082.pdf
As already said by other users, Bogoliubov transformations can be visualized as rotations. From the energy-spectrum standpoint, such rotations allow you to perform a change of basis from a basis where your Hamiltonian is not diagonal to a basis where your Hamiltonian is indeed diagonal. From the eigenstates standpoint, the coefficients of the inverse rotations are the weights of the linear combinations which give you new creation and destruction operators. Basically these new operators tell you the structure or the "shape" of the elementary excitations in the system, very much like one says that A_j^+ creates a particle in site j or that a_k^+ creates a particle having momentum k. I hope this answers your question. For a more detailed explanation, please take a look to
https://arxiv.org/abs/1705.02115
and to
Article Two-species boson mixture on a ring: A group theoretic appro...
where we discuss the importance of Bogoliubov transformations in relation to systems of interacting bosons.
It is necessary to get a compact for of an energy values and eigenstates as well.
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