Is Bogoliubov transformation simply a basis transformation?

Question

I encounter Bogoliubov transformation when I'm learning 1D transverse field Ising Model.But I think it's just a simple basis transformation, I don't understand why people give it a special name?

Answer 1

Any invertible transformation can be understood as a change of basis, but that doesn’t make it trivial (at least from a conceptual point of view). Some such transformations are so broadly useful and important that the physics community has collectively decided to give them special names. Bogoliubov transformations are one example; Fourier transforms (both continuous and discrete) are another.

Answer 2

"...it's just a simple basis transformation" — yes and no. Note that in the multi-particle case (what we are usually interested in) the Bogoliubov transformation is defined by its action on single-particle modes in a Fock state.

When we say "a transformation" we may refer to two slightly different things. For a particular basis in a Hilbert space, we can imagine a matrix acting on a basis state $|i\rangle$ as $A_{ij}|i\rangle$. In reality, however, we typically deal with operators, whose action on basis states is defined in a rather peculiar way. Once the action of an operator on states is specified, we can surely calculate its matrix $u_{ij}$, but it's just important to keep in mind that oftentimes we don't really think it terms of such matrices.

For example, if second quantization is used, the basis vectors are the so-called Fock states, the (anti-)symmetrized tensor products of single-particle basis states. Various operators acting on Fock states are then typically defined in terms of creation/annihilation operators acting on individual modes. While the action of an operator on a single mode may look simple, its action on Fock states, constituting the basis of the Hilbert space, is actually quite non-trivial.

So the beauty of the Bogoliubov transformation is in that, while being defined at the level of single-particle operators, it ends up being highly useful when studying the multi-particle system. However, the complexities described above don't bother us that much, as we do most things at the level of the Hamiltonian operator — substitute the new operators instead of the old ones, and then directly solve it in the new basis (i.e. a basis of new Fock states defined in terms of new single-particle states).