I don't know what Coleman Mandula theorem is, however if I were forced to say something about it, I will say it is a statement that suggests that internal and spatial symmetries have no unique combined structure. As it were, supersymmetry seems to pop translated fermions from bosons and what not. The truth is I have read a little about this and am looking for someone to give me a simplified mathematical and wordy excursion to the proper statement of the theorem and its (simplified) mathematical formulation. Usually, there are simpler versions or analogs, for most hard things to grasp. Is there one for this?


Basically, the Coleman-Mandula theorem says that given some reasonable assumptions (which do not always hold), the most general Lie algebra $\mathfrak g$ of symmetries of S-matrix is a direct product of Poincare algebra and an algebra of internal symmetries (perhaps they also show that the latter is reductive, I am not quite sure). The assumptions are stated in the abstract of their paper

  1. $\mathfrak g$ contains a subalgebra isomorphic to Poincare algebra,
  2. there is only a finite number of one-particle states below any given mass,
  3. an assumption on analyticity of S-matrix,
  4. an assumption on non-triviality of S-matrix (needed in part to exclude highly symmetric free theories),
  5. a technical assumption about the kernels of symmetry generators.

Also, they give a precise definition of what they mean by a symmetry -- for them it is a unitary operator which

  1. maps one particle states to one particle states,
  2. transforms multi-particle states as if they were tensor products,
  3. commutes with S-matrix.

I am not very familiar with the proof, but it can be of course found in the original paper. The essence of the theorem is that (given the assumptions) one cannot mix space-time symmetries with internal symmetries in a non-trivial way. E.g. one cannot have non-trivial commutators $[P_\mu, B]\neq 0$ for $P$ the momentum and $B$ an internal symmetry. It also says that one cannot have higher-spin conserved charges (i.e. a "momentum" with two indices $P_{\mu\nu}$). The latter actually follows from rather simple kinematical considerations, which show that scattering has to be trivial if such charges exist.

An extension of Coleman-Mandula theorem is Haag–Lopuszanski–Sohnius theorem, which relaxes the assumption that $\mathfrak g$ is a Lie algebra, and allows it to be a super-Lie algebra (i.e. to have anti-commutators). In this case they find the same result with Poincare algebra replaced by super-Poincare algebra. The latter contains Poincare algebra, supersymmetry generators (fermionic supercharges), and $R$-symmetry generators. $R$-symmetry is a kind of "internal" symmetry which mixes with ferminonic space-time symmetries -- the supercharges are charged under this symmetry. Still, Poincare generators do not mix with it, and the bosonic part of $\mathfrak g$ obeys Coleman-Mandula theorem.

There are some modifications for gapless theories, which give rise to (super-)conformal algebras, but for me the meaning of it is not clear, since conformal field theories do not have a S-matrix in canonical sense. Super-conformal cases can be more rigorously dealt with by looking for super-Lie algebras which have a conformal subalgebra. This leads to a classification of superconformal algebras in any spacetime dimension (one of the results is that there are no superconformal algebras in higher than 6 spacetime dimensions).

I want to add that the original Haag–Lopuszanski–Sohnius paper is very easy to read and I totally recommend it.

  • $\begingroup$ Does super Lie = graded Lie algebra in your language? $\endgroup$ – innisfree Mar 9 '16 at 9:15
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    $\begingroup$ @innisfree, yes, I believe that is the standard terminology in physics literature $\endgroup$ – Peter Kravchuk Mar 9 '16 at 9:17
  • $\begingroup$ No, there is a difference between Z/2-graded Lie algebras and super Lie algebras: the former is still a Lie algebra, only enjoying the additional poperty that the Lie bracket respects a given Z/2-grading on the underlying vector space. But a super Lie algebra is in general not an ordinary Lie algebra: it also has an underlying Z/2-graded vector space, but the key difference is that the bracket of two odd-graded elements in this space is not skew-symmetric, but symmetric. $\endgroup$ – Urs Schreiber Aug 24 '16 at 18:24

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