  Canonical transformations are ubiquitous in Hamiltonian mechanics, since they
not only describe the fundamental invariance of the theory under phase-space
reparameterisations, but also generate the dynamics of the system. In the first
part of this work we study the symplectic structure associated with linear
canonical transformations. After reviewing salient mathematical properties of
the symplectic group in a pedagogical way, we introduce the squeezing
formalism, and show how any linear dynamics can be cast in terms of an
invariant representation. In the second part, we apply these results to the
case of cosmological perturbations, and focus on scalar field fluctuations
during inflation. We show that different canonical variables select out
different vacuum states, and that this leaves an ambiguity in observational
predictions if initial conditions are set at a finite time in the past. We also
discuss how the effectiveness of the quantum-to-classical transition of
cosmological perturbations depends on the set of canonical variables used to
describe them.
