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)dk⁴dq⁴ = ∫O'(x)O'(x)O'(x)dx⁴

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}Oⁿ

So we should be able to find conjugate variables to the root vectors. So we can write the above as:

O'^CO'^CO'^C

The O'^C being linear combinations of the Os. In the case of E₈, the {A} are 8 dimensional unit vectors. One difficulty are the neutral particles, Oⁿ , which seemingly have no equivalent in the momentum picture. Such a term would look like:

 ∫O(k)O(-k)kμAμ(0)dk⁴ =  ∫O(x)Bμ∂μO(x)dx⁴  

which is not Lorenz invariant.... [unless O(x) were spinors and Bμ were Dirac matrices?? But then you can't have 3 spinors in an interaction. So maybe O(x) are super-particles?] But note we never have 3-scalar particle interactions anyway.

Q. What is the equivalent of a vector (or spinor) representation for root vectors.

Need somehow to connect momentum with root-vectors into one system. Because root-vectors are easy to understand as that is conservation of charge. Also, conservation of momentum is very similar. Spins are the different representations for momentum. What is the equivalent for charge? (Supersymmetry?)

Table of equivalence

Momentum                               Charge
Constants????                          Neutral particles
Spin                                         ?????
Lorenz symmetry (k.k=m2)      (A.A=4) ????
????                                         Group symmetry


Charges can come from Kaluza-Klein compactification hence the equivalences.