# Does charge conjugation symmetry sit in the Lorentz group?

We know the Lorentz group is $$O(3,1)$$ in 4 dimensional spacetime.

We know that there are 4 disconnected components in Lorentz group $$O(3,1)$$, and https://math.stackexchange.com/q/2204349/

$$\pi_0(\mathrm{O}(1,3)) \cong \mathbb{Z}_2\times\mathbb{Z}_2.$$

My question is that in QFT we have a discrete charge conjugation symmetry $$C$$, parity $$P$$ and time reversal $$T$$.

We know the $$P$$ and $$T$$ flips the 4 disconnected components of $$O(3,1)$$ into each other.

How about the charge conjugation $$C$$, does it sit in the Lorentz group? Or does it only act on the matter field? How do we understand $$C$$ in the Lorentz group $$O(3,1)$$?

• Charge lives in spacetime but is not part of it. – G. Smith Sep 23 '19 at 3:57