The h Bar

7:34 AM

In an unusual reversal of events, I first found this paper - I don't remember how I came to that - that seemed to make strange claims (and is also weirdly disparaging of the people allegedly making mistakes), and then found an 8 year old question here that asked about the same confusions I had so I had somewhere to write down my thoughts

I am unable to arrive at the equation at the bottom starting from 5.7. Starting from eq 5.7, I expanded the mu index which would give four terms. In total, there would be five terms inside: four gamma terms and m. Assuming we're working with four components, I can see four differential equations.

The last equation looks like the four individual differential equations have been summed up to zero. But isn't this a weaker condition than requiring each differential equation to equate to zero?

@Yashas Even if it's a weaker condition, it's still true, isn't it?

@ACuriousMind Yeah but it might admit more solutions than what the original Dirac equation would have?

So? I'm not sure why the author writes that equation down, but I don't see a claim there that it's equivalent to (5.7), so what's the problem?

7:44 AM

@ACuriousMind Oh, I am not sure why I thought it's supposed to be equivalent but now I don't see why that summed form was even mentioned.

I think it's probable the author just make a mistake :P

8:07 AM

@ACuriousMind maybe you learned about it here :p

@bolbteppa Huh, I had completely forgotten that, but no, I didn't find this in the depths of my download folder, I stumbled across it a few weeks ago

I vaguely remember a discussion about this point in the book the other answer cites and that book is decades old so at the very least it's a known question mark

but good to see that me from one year ago agrees with me today :P

@Yashas the thing at the bottom is just (5.7) with the matrix indices written out?

@bolbteppa $\psi_k$ makes it look like they are summing over all components?

each component of $\psi$ should have it's own differential equation, right?

oops, I think I understood

8:17 AM

$\psi$ is a column vector, with components $\psi_1,...,\psi_4$, so the equation is just usual matrix multiplication equation $M_{jk} \psi_k = 0$ for the matrix $M_{jk} = (i \gamma^{\mu} \partial_{\mu} - m)_{jk}$

@bolbteppa Yup, got it.

So I need to write the last equation four times by varying j from 1 to 4 to get all four equations.

Yes, of you can write the whole out as a big $4 \times 4$ matrix, where the entries of the matrix $\gamma^{\mu}$ are in general just $\gamma^{\mu}_{ik}$ so it's ugly, but if you have a standard explicit representation for the $\gamma^{\mu}$'s it simplifies in appearance

@MoreAnonymous there's so much to say about what's going on in that video before even clicking on it

Q: Is it okay to edit your answer to make it better after seeing some point that is addressed in another answer or is this bad etiquette/stealing?

9:16 AM

ncatlab.org/nlab/show/asymptotic+safety

Talking about some issues with that 'Higg's mass' paper

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Let's say i have written an answer to a question. I feel like i have done a good job of answering it and covered all the points that needed to be covered. Now, if i see some another answer that i feel brings up some point, that i realise i should have covered in my answer as well. Is it okay to e...

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