Conjugate of Lie Algebra
Conjugate of Lie Algebra A little article noting the similarities between momentum conservation (continuous) and charge conservation (discrete). We all know that the conjugate coordinate to position is momentum . Thus a term such as ∫O(-k)O(-q)O(k+q)dk 4 dq 4 = ∫O'(x)O'(x)O'(x)dx 4 But a lie group given by it's root vectors (charges) also has a similar form which looks a lot like momentum: O {-A} O {-B} O {A+B} + (A n )O {A} O {-A} O n So we should be able to find conjugate variables to the root vectors. So we can write the above as: O' C O' C O' C The O' C being linear combinations of the Os. In the case of E 8 , the {A} are 8 dimensional unit vectors. One difficulty are the neutral particles, O n , which seemingly have no equivalent in the momentum picture. Such a term would look like: ∫O(k)O(-k)kμAμ(0)dk 4 = ∫O(x)Bμ∂μO(x)dx 4 which is not Lorenz invariant.... [ unless O(x) were spinors and Bμ were Dirac matrices?? B