quantum field theory - Can someone give a simple expose on Coleman Mandula theorem and what Mandelstam variables are?

Can someone give a simple expose on Coleman Mandula theorem and what Mandelstam variables are?

Coleman-Mandula is often cited as being the key theorem that leads us to consider Supersymmetry for unification. An overview discussion with sufficient detail is missing in many popular texts. So how does Coleman Mandula theorem actually meet its no-go claim?

Answer

This is a bit of a sketch;

The $S$-matrix acts on shift the state or momentum state of a particle. A state with two particle states $|p, p’\rangle$ is acted upon by the $S$ matrix through the $T$ matrix $$ S~=~1~–~i(2\pi)^4 \delta^4(p~–~p’)T $$ So that $T|p, p’\rangle\ne0$. For zero mass plane waves scatter at almost all energy. The Hilbert space is then an infinite product of n-particle subspaces $H~=~\otimes_nH^n$. As with all Hilbert spaces there exists a unitary operator $U$, often $U~=~exp(iHt)$, which transforms the states S acts upon. $U$ transforms n-particle states into n-particle states as tensor products. The unitary operator commutes with the $S$ matrix $$ SUS^{-1}~=~[1 – i(2\pi)^4 \delta^4(p~–~p’)T]U[1~+~i(2π)^4 \delta^4(p~–~p’)T^\dagger] $$$$ = U~+~i(2\pi)^4 \delta^4(p~–~p’)[TU~–~UT^\dagger] ~+~[(2\pi)^4 \delta^4(p~–~p’)]^2(TUT^\dagger). $$ By Hermitian properties and unitarity it is not difficult to show the last two terms are zero and that the S-matrix commutes with the unitary matrix. The Lorentz group then defines operator $p_\mu$ and $M_{\mu\nu}$ for momentum boosts and rotations. The $S$-matrix defines changes in momentum eigenstates, while the unitary operator is generated by a internal symmetries $A_a$, where the index a is within some internal space (the circle in the complex plane for example, and we then have with some $$ [A_a,~p_\mu]~=~[A_a,~M_{\mu\nu}]~=~0. $$ This is a sketch of the infamous “no-go” theorem of Coleman and Mundula. This is what prevents one from being able to place internal and external generators or symmetries on the same footing.

The way around this problem is supersymmetry. The generators of the supergroup, or a graded Lie algebra, have 1/2 commutator group elements $[A_a,~A_b]~=~C_{ab}^cA_c$ ($C_{ab}^c$ = structure constant of some Lie algebra), plus another set of graded operators which obey $$ \{{\bar Q}_a, Q_b\}~=~\gamma^\mu_{ab}p_\mu, $$ which if one develops the SUSY algebra you find this is a loophole which allows for the intertwining of internal symmetries and spacetime generators. One might think of the above anti-commutator as saying the momentum operator, as a boundary operator $p_\mu~= -i\hbar\partial_\mu$ which has a cohomology, where it results from the application of a Fermi-Dirac operator $Q_a$. Fermi-Dirac states are such that only one particle can occupy a state, which has the topological content of $d^2~=~0$. This cohomology is the basis for BRST quantization.