When I studied representation theory for the first time it was only focused on finite groups. It was the second half of a one semester course in group theory, and the book employed was "Representation Theory: A First Course" by Harris and Fulton.
Currently I'm studying Quantum Field Theory, and regarding spinors and quantum fields with spin, I'm interested in the representation theory of the Lorentz group.
I know the description of the fields with various spins is tightly connected to the irreducible representations of the Lorentz group.
Unfortunately, though, most traditional textbooks in QFT take a stand to not do a rigorous treatment of the matter, nor present a mathematicaly pleasant notation or definitions. Most of them speak quite loosely of the subject, that it doesn't even look connected to the representation theory I saw in the course I took.
Considering this, what I'm looking for is a book or lecture notes which presents a mathematicaly rigorous approach to the representation theory of the Lorentz group and also establish in a mathematicaly correct way the connection to quantum fields with spin and the representations of the Lorentz group. Is there such a resource out there?
A classical and fairly complete (even if it is considered somewhat outdated by someone) reference is [1]: instead of describing its content, let me report the first sentences of the "Phisics Today" review of the book by J. E. Mansfield
Perhaps, even if it is a quite old reference, it is a nice place to start, even considering that a quite inexpensive Dover paperback reprint has been recently published.
Addendum. In the preface of of [1], the Authors recommend the monograph [2] (see here for a review published in the Journal of the London Mathematical Society) "... to the reader wishing to study the representation of the Lorenz group more thoroughly and at a greater length", in their own exact words ([1] p. xviii).
References
[1] I. M. Gel’fand, R. A. Minlos and Z. Ya. Shapiro, Representations of the rotation and Lorentz groups and their applications. Translated from the Russian by G. Cummins and T. Boddington (English), Oxford-London-New York-Paris: Pergamon Press. xviii, 366 pp. (1963), MR0114876, Zbl 0108.22005.
[2] M. A. Najmark, Linear representations of the Lorentz group. Translated by A.Swinfen and O.J.Marstrand. (English) International Series of Monographs in Pure and Applied Mathematics. 63. Oxford etc.: Pergamon Press. XIV, 447 p. (1964), MR0170977, Zbl 0116.32904.