Geometric/group-theoretic interpretation of charge conjugation symmetry

Question

In quantum field theory, time reversal $(T)$ symmetry and parity $(P)$ symmetry have a nice group-theoretic interpretation: they correspond to the elements of the discrete group $O(3, 1) / SO^+ (3, 1) = \{1, P, T, PT \}$. However, I haven't seen such an interpretation for charge conjugation $(C)$ symmetry. Does there exist a nice geometric/group-theoretic interpretation of $C$-symmetry?

Answer 1

There are different operations that are called "charge conjugation". They implement distinct, albeit often conflated, transformations on various fields. These are:

1. Charge conjugation is what maps particles to anti-particles. This is a bad definition. It doesn't work in theories that have no notion of "particle", e.g., anything that is strongly coupled, or conformal theories, etc. Most QFTs are of this type. Don't use this definition.$^1$

2. In a gauge theory, "charge conjugation" is by definition an outer automorphism of the gauge group, if it exists. In $SU(N)$ or $U(1)$ gauge theories, this corresponds to complex conjugation. In e.g. $SO(N)$ theories, it corresponds to "target space parity".

3. In a general QFT, "charge conjugation" is whatever is left after you apply a PT transformation. I will explain what this means below. For the time being, let me note that this definition turns out to agree with the previous one, in most QFTs we care about, but only a posteriori. Even in theories without a gauge group (e.g., Yukawa), if we pretend that fields are charged under some gauge group, there is a basis where the gauge outer automorphism happens to take the same form as the PT-based operation. This is why we give these two operations the same name. They are conceptually different operations though.

So, what is the PT definition of C? In a Lorentz invariant theory, we can always Wick rotate to euclidean signature, where the external symmetry goes from $SO(1,d-1)$ to $SO(d)$. In this signature, consider the operation of reflecting along a given axis, call this $R$. This need not be a symmetry: $R$ is not part of $SO(d)$ (it is part of $O(d)$, which sometimes is a symmetry, sometimes is not).

Consider, instead, reflections along two orthogonal axes, $R_1$ and $R_2$. Again, separately these need not be symmetries, but when applied together, this is the same as a 180-degree rotation around the plane that these two axes generate. In other words, $R_1R_2\in SO(d)$ which, by assumption, is a symmetry of the theory.

How does $R_1R_2$ act on fields? It is a Lorentz transformation, so fields transform according to $\phi\to D\phi$, where $D=D(R_1R_2)$ is the action of the rotation $R_1R_2$ on $\phi$ -- the specific form of this matrix depends on the spin of $\phi$. We define $C:=D^{-1}$.

In conclusion, we learn that if we reflect along two distinct axes, and then act on fields with a certain matrix $C$, this is equivalent to the "do nothing" operation.

Let us now Wick rotate back to lorentzian signature. We get the most mileage if we let one of the two axes above be the euclidean time direction. Then, the corresponding reflection becomes "time reversal", and the other reflection stays a spatial reflection. The matrix $C$ will potentially pick up a phase after Wick rotation. In any case, we learn that in a relativistic theory, if we apply time-reversal $T$, plus a spatial reflection $R$, plus a certain operation $C$ that leaves spacetime points invariant but acts on spin indices, then the whole thing is equivalent to doing nothing at all. This is the CPT theorem. (We call it $P$ for historical reasons, but the operation that goes into the theorem is actually a single reflection instead of a complete reflection along all axes; the two are equivalent in even spacetime dimensions but not in odd).

So, according to definition 3, charge conjugation is whatever you need to do to your fields to undo the action of parity and time-reversal. In a relativistic theory, this "whatever" is guaranteed to be a 180-degree rotation, so it is just a reshuffling of the spin components of your fields.

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Footnote 1: there is a version of this definition that is ok, and is not uncommon, especially in cond-mat. If you have a system with symmetry, say, $U(1)\rtimes \mathbb Z_2$, with non-trivial action of $\mathbb Z_2$ on $U(1)$, it is common to call this $\mathbb Z_2$ "charge conjugation". This may or may not agree with the "global" charge conjugation which should flip the signs of all charges. For example, a certain system may have $U(1)\rtimes \mathbb Z_2$ symmetry plus various accidental extra $U(1)$ symmetries. If we only care about the first $U(1)$, and $\mathbb Z_2$ flips its sign, then we would still call it charge-conjugation even if it does not flip the sign of the other $U(1)$'s.