topsquark said:
For lack of a better term I'm referring to it as "Algebraic Physics." I am looking for more courses using Algebra, say, starting with Classical Mechanics. Does anyone know of any textbooks using this approach?
Do you really mean algebraic? Or do you mean some combination of (abstract) algebra, differential geometry, and (functional) analysis?
I am going to interpret your request very loosely.
Mathematical books on classical mechanics include:
"Mathematical Methods of Classical Mechanics" by Arnold;
"Classical Mathematical Physics" by Thirring;
"Mechanics" by Scheck.
For non-relativistic quantum mechanics, there is the outstanding "Quantum Field Theory for Mathematicians" by Hall, which assumes that the reader has undergrad background in measure theory and Hilbert spaces. It teaches grad-level functional analysis alongside QM. Did I mention that this book is outstanding?
Mathematical books on quantum field theory:
"What Is a Quantum Field Theory? A First Introduction for Mathematicians" by Talagrand;
"Quantum Field Theory: A Tourist Guide for Mathematicians" by Folland.
The books by Talagrand and Folland use mathematical rigour where possible, and where physicists' quantum field theory calculations have yet to be made mathematically rigourous (by anyone), they state the mathematical difficulties, and then formally push through the physicists' calculations.
Folland is more condensed than Talgrand, and covers more advanced physics. Folland treats QED, while Talagrand treats models that are not complicated by spin and gauge invariance. One of Talagrand's goals, inspired by Hall's QM book, was to give a treatment of QFT that is more introductory, readable, and detailed than Folland.
A book that applies applies algebraic structures and differential geometry (fibre bundles) to the standard model and non-supersymmetric grand unified theories (GUTs) of particle physics is "Mathematical Gauge Theory With Applications to the Standard Model of Theoretical Physics" by Mark Hamilton.