# Lorentz group generators and its dimensionality

I am unsure about the generators of the Lorentz group and its dimensionality.

I believe any Lorentz transformation can be written as the product of a proper, orthochronous Lorentz transformation with an element of $$\{1,P,T,PT\}$$ where $$P$$ is space inversion and $$T$$ is time reversal. The proper, orthochronous Lorentz group has 6 generators, so with $$\{1,P,T,PT\}$$ it seems like one needs 7 pieces of information to describe a general Lorentz transformation. Why, then, does the group have only six dimensions, i.e. only the number of generators counts towards it?

Also, what is the significance of the fact that the other three components cannot be made from these generators directly (because they are not connected to the identity), but rather need an additional $$P$$, $$T$$, or $$PT$$? Does this mean that they do not have the same Lie algebra?

• No, the proper orthochronous Lorentz group and the full Lorentz group have the same Lie algebra Dec 14, 2020 at 13:49
• There are no generators for discrete groups. The definition is made only for Lie groups, so manifolds locally homemorphic to $\mathbb R ^n$. Dec 14, 2020 at 13:51
• @DanielC although (perhaps confusingly) discrete groups do have a different concept called a generator. In that sense, the group $1,P,T,PT$ is isomorphic to $\mathbf{Z}\times\mathbf{Z}$ and has two "generators". I don't think there's any obvious connection and you'd never want to talk about both concepts at once afaik. Dec 14, 2020 at 17:33

The Lorentz group has four disconnected parts. Each of those parts had six dimensions. The reason you need P and T in addition to the generators to get to every part of the Lorentz group is that you cannot move along a continuous path between any of the four disconnected parts. If you will pardon some sloppy physicist math, the generators represent "infinitesimal" group elements near 1. You can create a continuous path away from 1 by stringing together a bunch of them together. But you cannot jump across to another piece of the group.

The fact that there are multiple pieces does not mean that there is an extra dimension in any normal sense. I can't move along the P direction. I'm either in P or I'm not.