  I describe the logical basis of the method that I developed in 1962 and 1963
to define a quantum operator corresponding to the observable particle number of
a quantized free scalar field in a spatially-flat isotropically expanding
(and/or contracting) universe. This work also showed for the first time that
particles were created from the vacuum by the curved space-time of an expanding
spatially-flat FLRW universe. The same process is responsible for creating the
nearly scale-invariant spectrum of quantized perturbations of the inflaton
scalar field during the inflationary stage of the expansion of the universe. I
explain how the method that I used to obtain the observable particle number
operator involved adiabatic invariance of the particle number (hence, the name
adiabatic regularization) and the quantum theory of measurement of particle
number in an expanding universe. I also show how I was led in a surprising way,
to the discovery in 1964 that there would be no particle creation by these
spatially-flat FLRW universes for free fields of any integer or half-integer
spin satisfying field equations that are invariant under conformal
transformations of the metric. The methods I used to define adiabatic
regularization for particle number, were based on generally-covariant concepts
like adiabatic invariance and measurement that were fundamental and determined
results that were unique to each given adiabatic order.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Theory