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  Does charge conjugation symmetry sit in the Lorentz group?

+ 2 like - 0 dislike

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 $O(3,1)$?

This post imported from StackExchange Physics at 2020-11-26 15:40 (UTC), posted by SE-user annie marie heart
asked Sep 23, 2019 in Theoretical Physics by annie marie heart (1,205 points) [ no revision ]
Charge lives in spacetime but is not part of it.

This post imported from StackExchange Physics at 2020-11-26 15:40 (UTC), posted by SE-user G. Smith

1 Answer

+ 4 like - 0 dislike

In QED, for example, charge conjugation commutes with the Lorentz group. It's an "internal" symmetry, not part of Lorentz (or Poincaré) symmetry.

However, a different kind of connection exists between charge conjugation and the Lorentz group, via the CPT theorem. The CPT theorem says that every relativistic QFT (satisfying certain axioms) has a symmetry that, among other things, reflects a timelike direction and an odd number of spatial directions. We call it the "CPT" theorem, but that's a little misleading, because it should be regarded as more fundamental (that is, more broadly applicable) than C, P, or T individually. In other words, we really ought to define charge conjugation in terms of CPT, not the other way around.

Here's the point: The general proof of the CPT theorem makes use of the complex Lorentz group (through the fact that correlation functions can be analytically continued to complex values of their spacetime arguments), and in this sense there is a kind of connection between C and Lorentz symmetry. This is reviewed in

and in the classic text by Streater and Wightman, PCT, Spin and Statistics, and All That.

This post imported from StackExchange Physics at 2020-11-26 15:40 (UTC), posted by SE-user Chiral Anomaly
answered Sep 24, 2019 by Chiral Anomaly (70 points) [ no revision ]
thanks very much vote up

This post imported from StackExchange Physics at 2020-11-26 15:40 (UTC), posted by SE-user annie marie heart

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