@Slereah No - the two indices come because your $\mathrm{SU}(3)$-spinor ends up as a section of a bundle associated to $\mathrm{SU}(3)\times\mathrm{Spin}(3,1)$. Writing association as $G\rtimes V$ (to not confuse association with the direct product), the strongly interacting spinor is a section of an $(\mathrm{SU}(3)\times\mathrm{Spin}(3,1))\rtimes (\mathbb{C}^3 \otimes \mathbb{C}^2)$-associated bundle.
(To be clear, I'd write the Weyl-spinor bundle as $\mathrm{Spin}(3,1)\rtimes_{(1/2,0)} \mathbb{C}^2$ if I wanted to be really formal, and I'm not sure why you just used $\times$)
I did my own notation wrong :P. What I really wanted to say is this: If you have a $G$-principal bundle $P$, then you denote the associated bundle to the representation space $V$ as $P\rtimes_G V$ (whether you use $\rtimes$ or $\times$ is not essential, really)
But it's neither a product nor a semi-direct product
Whenever you have two groups $G,H$ acting independently on the same object in representations $V_G, V_H$, you can express this equivalently as $G\times H$ acting on $V_G\otimes V_H$.
So in bundle language, Coleman-Mandula is applied to products of $\mathrm{Spin}(3,1)$ with 'internal' Lie groups i.e. compact ones like $\mathrm{SU}(3)$?
Coleman-Mandula has nothing to do with bundles. It's just saying the following: If the physical theory has the Poincaré group as a spacetime symmetry, then it must be a direct factor in the total symmetry group, i.e. the full symmetry must be $\mathrm{Spin}(1,3)\times\text{stuff}$
Note: My statements above for Coleman-Mandula should really say "Poincaré group" instead of $\mathrm{Spin}(1,3)$
(Since the Poincare group is a semi-direct product of $\mathrm{Spin}(1,3)$ and $\mathbb{R}^{1,3}$, which would contradict the statements as I made them)
In quantum mechanics we know that the canonical position $\hat x$ and momentum operator $\hat p$ satisfying
\begin{align}
[\hat x,\hat p] = i \quad (\hbar = 1)
\end{align}
have continuous spectrum.
We also know what the spectrum of the operator
\begin{align}
H = \frac{1}{2}\left(\hat p^2 + \ha...
Funnity (and perhaps frustratingly), the one-line computation ignoring all sorts of pathologies gives exactly what Valter also gets with a much more complicated analysis (except that all of us seem to have forgotten that $p = -\mathrm{i}\partial_x$, not $p = \partial_x$ :P)
I have no physics background, which is the genesis of my question.
In pop-science, it is frequently mentioned that Newton's apple didn't fall toward his head, but rather that his head came up and smacked the apple. Or, put another way, if you jump out of a window, you don't crash into the Earth...
Everytime I hear the word "orthodoxy" applied to mainstream physics, I can't help but think its original meaning fits perfectly: Belief in that which is correct.
Does anyone else have the problem of thinking, constantly, about space, and how black holes and neutron stars look very close up etc? I must see them one day. These thoughts are in my mind about 87% of the day and night.
I am a bit skeptical of hyper dense cosmic objects, though. We've all heard how a thimble of neutron star matter would weigh....what is it? It changes with every article.......a billion tonnes, or something. I mean, look: Imagine even trying to squeeze a t-shirt into the size of an atom.
And the Big Bang theory. Trillions of galaxies at least, and the countless stars and planets within, all packed into a speck the size of an atom. It's almost as mad as saying God made everything.
(Rather, the equivalent material of said galaxies etc)
But what really takes the bourbon biscuit is saying space itself also came from the primordial atom. Infinite empty space clearly has always existed.
Little Note: That oddly placed t-shirt comment was clumsy.
I genuinely believe future generations will laugh at the fact people actually believed the Big Bang theory. Not that we humans will survive for too long anyway.
@danielunderwood Indeed. A small BH will spaghettify you long before your close to the EH, a SMBH is pretty boring if it's quiescent and way too deadly to be anywhere nearby if its active.
@danielunderwood Indeed. A small BH will spaghettify you long before you're close to the EH, a SMBH is pretty boring if it's quiescent and way too deadly to be anywhere nearby if its active.
A fresh neutron star is also pretty scary, especially if it's a magnetar, but I guess an old cold one wouldn't be too bad, if you don't get close enough for the tidal effects to hit you.
In quantum mechanics we know that the canonical position $\hat x$ and momentum operator $\hat p$ satisfying
\begin{align}
[\hat x,\hat p] = i \quad (\hbar = 1)
\end{align}
have continuous spectrum.
We also know what the spectrum of the operator
\begin{align}
H = \frac{1}{2}\left(\hat p^2 + \ha...
I have no physics background, which is the genesis of my question.
In pop-science, it is frequently mentioned that Newton's apple didn't fall toward his head, but rather that his head came up and smacked the apple. Or, put another way, if you jump out of a window, you don't crash into the Earth...
