Poincare Group

This page shows something that an author used and it was very helpful. It turns out there is a lot more to the Poincare Group than the generalization of the idea shown below:

(x^1)' = L^1_1 v^1 + L^1_2 v^2 + L^1_3 v^3 + L^1_4 v^4 + a^1

(x^2)' = L^2_1 v^1 + L^2_2 v^2 + L^2_3 v^3 + L^2_4 v^4 + a^2

(x^3)' = L^3_1 v^1 + L^3_2 v^2 + L^3_3 v^3 + L^3_4 v^4 + a^3

(x^4)' = L^4_1 v^1 + L^4_2 v^2 + L^4_3 v^3 + L^4_4 v^4 + a^4

Hopefully, from prior studies you recognize this as being similar to a 4×4 matrix operating on an old 1×4 vector to make a new 1×4 vector, and after that the result moves by constants in all four dimensions.

Recall how we have the following from Algebra:

  • y = mx
  • y = mx + b
  • y=mx+b is analogous to the Poincare Group
  • y=mx is analogous to the Lorentz Group

Appendix P

More content will be added to be “fair” to the idea.