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Electromagnetism and the weak force are intertwined as two parts of the electroweak force and some links still exist even at low energy levels one of the being weak hypercharge which is determined by a combination of electric charge and weak isospin in the form $$Y_w=2(Q-T_3) $$ Where $Q$ is electric charge and $T_3$ is weak isospin. Magnetic monopoles are not known to exist but I understand that predictions about their properties if they do exist have been made and that both the electromagnetic and weak force as well as their unification as the electroweak force are very well understood. If monopoles and magnetic charge do exist would they have a similar link to the weak force that electrons and electric charge do?

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The standard model predicts the existence of particles that are not represented directly by any individual quantum field in the lagrangian. As an example, it predicts the existence of protons, even though there is no proton field in the lagrangian. The standard standard model also predicts magnetic monopoles, even though there is no magnetic-monopole field in the lagrangian.

Here's some background to explain what I mean by standard standard model:

  • The gauge group of the standard model is often written $SU(3)_c\times SU(2)_L\times U(1)_Y$. Weak isospin $T_3$ is like a "charge" with respect to the $SU(2)_L$ part of the gauge group, and hypercharge $Y$ is the "charge" with respect to the $U(1)_Y$ part of the gauge group. The usual charge $Q$ is associated with the $U(1)_\text{EM}$ electromagnetic gauge group, which is a mixture of the $SU(2)_L$ and $U(1)_Y$ factors. The model looks simplest when written in terms of $SU(2)_L$ and $U(1)_Y$ instead of $U(1)_\text{EM}$, so in hindsight we usually consider $T_3$ and $Y$ to be inputs and $Q$ an output, even though $Q$ is the thing we measure more directly.

  • Actually, the gauge group $SU(3)_c\times SU(2)_L\times U(1)_Y$ is just one of a few possibilities that cannot be distinguished from each other in a small-coupling expansion (Feynman diagrams). A few of the possible variations are reviewed in arXiv:hep-ph/0609029. The possibility relevant to this question is that the $U(1)_Y$ factor (and therefore the resulting electromagnetic gauge group) could be noncompact — in other words, it could be $\mathbb{R}$ instead of $U(1)$. We're confident that it's really the compact group $U(1)$, though, because this explains why the electric charges of electrons and protons have precisely the same magnitude.$^\dagger$

By standard standard model, I mean the standard model using the compact group $U(1)_Y$ for the gauge interaction associated with hypercharge. In this version of the model, magnetic monopoles automatically exist, even though there is no magnetic-monopole field in the lagrangian. This is reviewed by Polchinski in arXiv:hep-th/0304042. It's relatively easy to see in lattice QED, and it plays a prominent role in Polyakov's classic book Gauge Fields and Strings.

Based on that, asking whether magnetic charge would contribute to weak hypercharge is like asking whether rocks would contribute to weak hypercharge. Weak hypercharge is an input to the theory. Rocks are an output: they are phenomena that the theory predicts. Similarly, magnetic monopols are an output: they are something that the theory predicts, using inputs like weak hypercharge.


Footnotes:

$^\dagger$ On page 76 in arXiv:1810.05338, Harlow and Ooguri mention that the electric charges of electrons and protons have the same magnitude to within one part in $10^{21}$, with this comment: "By far the most plausible explanation of this remarkable agreement is that the gauge group of electrodynamics is indeed $U(1)$, which presumably is why this is the terminology most people use." (In the context of the standard model, assuming the compact group $U(1)_Y$ for weak hypercharge is equivalent to assuming the compact group $U(1)$ for electrodynamics.)

By the way, the coexistence of quantum physics and gravity may give us another reason to be confident that the gauge group is compact, and therefore that magnetic monopoles exist. This connection is reviewed at length in Harlow and Ooguri's paper, which states the connection as conjecture 3 on page 1: If a quantum gravity theory at low energies includes a gauge theory with gauge group $G$, then $G$ must be compact.

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  • $\begingroup$ Does that mean that electric charge and weak isospin are "caused" by weak hypercharge rather that the other way around? Or is even attempting to apply the notion of causation to something as abstract as the laws of physics a category error? $\endgroup$ Jun 21 '21 at 22:39
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    $\begingroup$ @AdamLincolnSteele I wouldn't call it cause-and-effect, but I think of it in terms of inputs and outputs. Even though electric charge is the thing we measure more directly, the model looks simpler when it's expressed in terms of weak isospin and hypercharge. That's why I wrote the answer the way I did... but your comment made me realize that it wasn't written as clearly as I'd like, so I edited it to try to make this part a bit more clear. $\endgroup$ Jun 21 '21 at 23:24

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