# Representation Theory of $SL(2, \mathbb{C})$

I'm a PhD. in mathematics (working mainly in complex algebraic geometry), but I'm looking for a "convincing" answer concerning the various applications of representation theory of the group $SL(2, \mathbb{C})$ in physics. I know some things about it, like for instance we are primarily interested in the study of $SU(2, \mathbb{C})$, and its associated Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ (as a real algebra), which complexification produces the complex Lie Algebra $\mathfrak{sl}_2(\mathbb{C})$ and through category theory we can reduce the problem of representations (irreducible ones) of the aforementioned group to the rather easier study of the repersentations of its corresponding Lie Algebra $\mathfrak{sl}_2(\mathbb{C})$ . I know that the motivation of that rather interesting geometrical/algebraic object (Lie Group in other words) and its study, comes directly from physics but I didn't manage to find out a convincing answer on internet. I'm mentioning the word "convincing" because my knowledge in physics is quite limited (maybe some basics as an amateur). Also I know that has something to do with the spin of some classes of particles but this quite obscure too.. So whoever wants to answer that question must assume that they reply to an undergraduate student in physics who understands mathematics (into a certain extend always, that's the problem with both mathematics and physics unfortunately :( ).

EDIT: This is my first time asking something here, so if I have done a kind of mistake please do let me know! (although I think works as math stack exchange)

• Related/possible duplicates: physics.stackexchange.com/q/108212/50583, physics.stackexchange.com/q/28505/50583. For why we care about the Lorentz/Poincaré group to begin with, consult any text on relativity. – ACuriousMind Mar 31 '17 at 13:57
• The primary reason the group $\mathrm{SL}(2,\mathbb{C})$ is important in physics is because, up to a $\mathbb{Z}_2$ quotient, it is isomorphic to the (proper orthochronous) Lorentz group, which is the group of symmetries of flat spacetime. – gj255 Mar 31 '17 at 13:59
• @ACuriousMind thank you for your reply, you're right this seems to be quite related, so I'll check it. It doens't show up if you ask that question on google or on pshysics stack exchange to be honest. – mayer_vietoris Mar 31 '17 at 13:59
• You might, or might not, appreciate this WP article. – Cosmas Zachos Mar 31 '17 at 16:18
• Have a look at Chapter 1 "Relativistic Transformation Laws" of the book "PCT, Spin and Statistics, and All That" by Streater and Wightman. It has a nice discussion of $SL_2(\mathbb{C})$ in relation to the representations of the Lorentz/Poincare group needed for quantum field theory. – Abdelmalek Abdesselam Apr 10 '17 at 13:42