https://golem.ph.utexas.edu/category/2009/03/unitary_representations_of_the.htmlUnitary Representations of the Poincaré Group
Posted by John Baez
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John Huerta’s paper on grand unified theories made some people eager to discuss aspects of group representation theory that this paper deliberately avoided.
For example, in relativistic quantum mechanics, we classify particles as unitary irreducible representations (or ‘irreps’) of a group like this:
G×P
Here G
is a compact Lie group depending on the theory of physics we happen to be studying, called the ‘internal symmetry group’. For example, the Standard Model has G= U(1) × SU(2) ×
SU(3).
P
, on the other hand, is the Poincaré group! This is, roughly speaking, the group of symmetries of Minkowski spacetime. So, this is the same regardless of our theory, unless we posit extra dimensions or something funky like that.
A unitary irrep of G×P
is always built by tensoring a unitary irrep of G with one of P. So, the project of classifying particles splits into two parts: one depending on G, one depending on P
.
In this thread let’s talk about the second part! If we do, we’ll learn why physicists classify particles according to their mass and spin (or more precisely, helicity). So, when we hear them mutter something about a ‘massless left-handed spin-1/2 particle’, we’ll know which representation of the Poincaré group they’re talking about.
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Just to kick off the discussion, let me say exactly what group I’m calling the Poincaré group! It’s really
P=SL(2,C)?R4
If you’re feeling pedantic you might call this ‘the universal cover of the connected component of the isometry group of Minkowski spacetime’ — but let’s just call it the Poincaré group, since this is the group that matters in the classification of particles!
