How is group theory related to particle physics? I've always heard of such statements as $SU$ group describing our world even before I seriously learned physics.   When learning about spin, someone spoke of $SU(2)$ but did not seriously explain. When learning isospin and hypercharge for particles made of $u, d, s,$ someone spoke of $SU(3)$ but did not give detailed explanation. When talking about the discovery of J/ψ, teacher (a experimentalist) say, well now with the charm quark, we need $SU(4)$.   I am crazy, $SU$ is just special unitary matrix group, what are they talking about when relating these matrix with spin or particles? I hope someone could give me a detailed explanation about how these group is related to real world substances or recommend a book.
What is the significance of Lie groups $SO(3)$ and $SU(2)$ to particle physics? I was hoping someone could give an overview as to how the Lie groups $SO(3)$ and $SU(2)$ and their representations can be applied to describe particle physics?  The application of Lie groups and their representations is an enormous field, with vast implications for physics with respect to such things as unification, but I what specifically made these groups of physical importance and why there study is useful.  I have just started studying these two groups in particular, but from a mathematical perspective, I'd very much appreciate understanding some sort of physical motivation.
Is the universe isomorphic to a universal turing machine? I often think about problems that require an understanding of the very essence of computation and its inherent limitations. So, my questions are as followed:   Is the universe isomorphic to a universal turing machine? Is the universe isomorphic to a universal quantum computer? What evidence do we have within our universe to support an answer to such a question? What corollaries follow from a universe that is or is not isomorphic to a universal turing machine (or universal quantum turing machine)? Is this question really answerable, and what is the current literature on this subject?   I'm mostly interested in the implications and corollaries that follow from a universe that is or is not isomorphic to a turing machine (or universal quantum computer) and the insights that it would provide us with, about the nature of computation.