Post #4446771

𝐇𝐀𝐑𝐃𝐂𝐎𝐑𝐄 𝐏𝐇𝐘𝐒𝐈𝐂𝐒 𝐏𝐎𝐒𝐓 It's cool how the SU(5) grand unified theory uses a Higgs to break down the symmetry to the Standard Model gauge group. The Standard Model gauge group is really S(U(3) x U(2)): that is, the subgroup of SU(5) that preserves a 3d subspace of ℂ⁵, or equivalently the 2d subspace orthogonal to that. So the desired symmetry breaking amounts to a choice of 2d subspace in ℂ⁵: that is, a point in a certain Grassmannian called Gr(2,5). So we're looking for a field, a 'Higgs field' H taking values in some vector space V, and a potential on V whose space of minima is this Grassmannian! The usual choice is to take V to be the Lie algebra 𝔰𝔲(5). Multiplying by i, we can think of elements as 5×5 traceless self-adjoint matrices. These matrices are characterized (up to conjugation) by 5 real eigenvalues that sum to zero. So we want a polynomial on V whose minima occur when two of these eigenvalues are +1 and three are -1, or the other way around. This polynomial should be invariant under conjugation so it should be a linear combination of tr(H²), tr(H³), tr(H⁴), etc. In fact it's enough to take V(H) = a tr(H²) + b tr(H²)² + c tr(H⁴) Here's how to pick the right a,b,c. tr(H²) is the sum of squares of eigenvalues of H. If we imagine that's fixed, tr(H⁴) will measure how "spread out" the eigenvalues are - so it will be smallest when they're split into two blocks of opposite sign with roughly equal size... that is, two eigenvalues of one sign and three of the other! It turns out V(H) = - tr(H²) + tr(H²)² + tr(H⁴) works, as do lots of other choices. The minima of this in the space of 5×5 traceless self-adjoint matrices should form the Grassmannian Gr(2,5).