First of all, I am neither a physicist nor a mathematician. And I am afraid that mathoverflow is not a suitable place for my question, but having asked similar questions on math SE it is obvious that this question is not appropriate for math.SE.
As far as I have searched in mathematical physics literature, historically, physicists such as Pauli and Dirac pioneered the concept of spin for a particle. Dirac developed his theory for spin-1/2 electrons by factorizing the Klein-Gordon equation to find a linear relativistic equation that is compatible with Schrodinger's wave equation but doesn't give negative probability density. He factorized the Klein-Gordon equation and found algebraic constraints that gave the Clifford algebra $Cl(1,3)$.
Now that's the algebraic part of the story. What I don't understand is how the concepts in Spin manifolds and Spin geometry were developed from the point of view of differential geometry and topology.
Apparently, Elie Cartan was one of the pioneers and he has written a book about it (I have read its first few chapters). But his language is very different from the language of differential geometry that we use today. He doesn't talk about covering spaces, vector bundles or connections the way that has become common in today's math literature. So, I find it very difficult to trace the chain of thoughts that has led physicists and mathematicians to develop spin geometry in its current language.
I'd like to know how our today's mathematical physics literature has been developed historically and what is the aim of using and further developing this sophisticated language.