Timeline for Does the $U(1)$ charge of a scalar particle flip under charge conjugation?
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11 events
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| May 17, 2020 at 20:14 | comment | added | knzhou | @Hermitian_hermit To get the original symmetry back, I could allow the Earth to be translated too, so $\phi(\mathbf{x}) = gz \to g(z - \delta z)$. But then have I not "illegally" changed a "fixed" "background" field? Again, you can see how we're going in circles here. If you don't let yourself change some things, you have less symmetries. If you do change them, you have more symmetries. What is the "correct" choice? Whatever is most convenient for you at the moment. | |
| May 17, 2020 at 20:12 | comment | added | knzhou | @Hermitian_hermit Here's another example from freshman physics. Newtonian gravity has translational symmetry. But now suppose I treat the gravitational potential of the Earth as a fixed background. Then near the surface of the Earth, $\phi(\mathbf{x}) = g z$, which is no longer symmetric under translations along $z$! What is going on? By insisting I keep the Earth's field fixed (i.e. translating everything except for the Earth) I threw out the original symmetry. | |
| May 17, 2020 at 20:11 | comment | added | knzhou | @Hermitian_hermit If you use a fixed background field and insist that it doesn't change under symmetry operations, you've broken the original symmetry, so it doesn't apply anymore. | |
| May 17, 2020 at 20:07 | comment | added | Hermitian_hermit | Suppose I downgraded $A_\mu$ to a background field. The Lagrangian is $\mathcal{L} = (D_\mu \phi)^* D^\mu \phi $. This has global $U(1)$ symmetry with the same current as above. However, as the vector potential is fixed, the Lagrangian does not have the symmetry $(\phi, A_\mu) \rightarrow (\phi^*, -A_\mu)$, but instead $(\phi,e) \rightarrow (\phi^*, -e)$ and this does indeed flip the current. So if we are working with a dynamical $A_\mu$ charge conjugation must flip $A_\mu$ too as the particles are the source, while for a background $A_\mu$ charge conjugation must flip $e$? | |
| May 17, 2020 at 18:45 | comment | added | knzhou | @Hermitian_hermit Indeed, suppose you had a theory with a discrete symmetry where $A_\mu$ didn't flip, but did involve complex conjugations on the complex variables. We wouldn't say, "aha, we've proven that charge conjugation doesn't truly flip $A_\mu$". We would say "huh, there's a funny discrete symmetry in this theory. It looks similar to what we would call charge conjugation in other contexts, but it doesn't flip $A_\mu$." | |
| May 17, 2020 at 18:43 | comment | added | knzhou | @Hermitian_hermit But actually that's kind of a slippery question. It's better to say that there exist certain symmetries of the Lagrangian, regardless of what we call them. Then we decide to call the one that involves a lot of complex conjugations, but doesn't involve changing, e.g. momenta and spins, "charge conjugation". For more on this see here. | |
| May 17, 2020 at 18:41 | comment | added | knzhou | @Hermitian_hermit So I guess the real issue is, what is charge conjugation? What is the true, original, primal, metaphysically valid, proper, rigorous, legitimate definition of charge conjugation, right? | |
| May 17, 2020 at 10:38 | comment | added | Hermitian_hermit | @Prof.Legolasov (Thank you, likewise!) I am assuming an interacting theory here because I am assuming that both the EM field and scalar field are dynamical as they both contain their own dynamical terms in the Lagrangian. | |
| May 17, 2020 at 10:26 | comment | added | Prof. Legolasov | @Hermitian_hermit (loving your username!) which theory are you considering here? Is it the theory of a complex scalar field in an external electromagnetic field, or is it the interacting theory of electromagnetism + the complex scalar? | |
| May 17, 2020 at 9:44 | comment | added | Hermitian_hermit | This argument seems circular to me. Suppose I didn’t know that complex conjugation was charge conjugation, then what would be telling me that $A_\mu$ flips sign for this transformation? As the current $j^\mu$ is the source of the EM field by the equations of motion, I would expect the following logic. Complex conjugation => the sign of $j^\mu$ flips => the sign of $A_\mu$ flips. The essence of my question is I am trying to show that complex conjugation IS charge conjugation by showing the current flips sign, but flipping $A_\mu$ by hand is assuming the current flips sign which is circular? | |
| May 17, 2020 at 5:03 | history | answered | knzhou | CC BY-SA 4.0 |