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Electric monopoles do exist, but why cannot magnetic monopoles exist?

The question is closed because I need to clarify it, but I don't know how I could ask it another way. However, I've recieved many answers that were appropiate and added something to my knowledge, so I consider it answered.

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    $\begingroup$ The reason there are no monopoles is that there are no monopoles. That's an experimental fact. You can't really "explain" that, because you can rather straightforwardly introduce monopoles into the theoretical description. $\endgroup$ – ACuriousMind Jun 7 '16 at 14:44
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    $\begingroup$ We can't even tell that they don't exist. All we can tell is that they are pretty rare, if they exist. $\endgroup$ – CuriousOne Jun 7 '16 at 14:49
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    $\begingroup$ @ACuriousMind isn't there some way to show that if magnetic monopoles do exist, then it would explain charge quantization? Also, I believe I read somewhere that we can introduce magnetic monopoles into E&M theory, but if we assume that the ratio of magnetic to electric charge is fixed everywhere then the dynamical equations wind up being the same as we have now with the total absence of monopoles. In other words, I think saying "no monopoles" is only true in a particular gauge. $\endgroup$ – DanielSank Jun 7 '16 at 15:20
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    $\begingroup$ Related: physics.stackexchange.com/q/1402/2451 , physics.stackexchange.com/q/4784/2451 and links therein. $\endgroup$ – Qmechanic Jun 7 '16 at 16:39
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    $\begingroup$ @nocomprende especially when explaining magnetism as merely a side-effect of electric under special relativity $\endgroup$ – Hagen von Eitzen Jun 8 '16 at 6:16
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There is no theoretical reason why magnetic monopoles cannot exist and indeed there are good reasons for supposing that they should exist. It's just that we have never observed one. In the past there have been various experiments to detect magnetic monopoles, though I think everyone has given up on the idea by now.

If you're asking why we can't get monopoles out of a magnet that's because the magnetic field of a magnet is built up from the individual magnetic fields of the unpaired electrons in the magnet, and those electrons have a dipole field. There isn't any way to combine the dipole fields of the electrons to create a monopole, though it's possible to make things that look locally approximately like monopoles.

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    $\begingroup$ There are also good reasons for supposing they shouldn't exist. The relativistically correct formulation of EM becomes weird to say the least if you allow them. And in fact the "symmetry" between E and B everyone harps on about is an artifact of 3 spatial dimensions. $\endgroup$ – user10851 Jun 7 '16 at 20:43
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    $\begingroup$ Relevant subject regarding things that look locally approximately like monopoles: spin ice. $\endgroup$ – leftaroundabout Jun 7 '16 at 20:56
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    $\begingroup$ @ChrisWhite When they proposed the idea of four dimensions, people argued that it was weird. Some even argued that it was impossible to be conceived but even the most basic notions in mathematics are already weird, take as an example, the definition of a point due to Euclid: "A point is that which has no part" but most people work perfectly fine with this notion and they don't even think that this is weird. Everything is weird, but as we repeat it a thousands of times, they became familiar. $\endgroup$ – Billy Rubina Jun 7 '16 at 23:36
  • $\begingroup$ @ChrisWhite Magnetic monopoles can be introduced into the relativistic EM as topological effects. Dirac, Quantized singularities in the electromagneticf field rspa.royalsocietypublishing.org/content/133/821/60 $\endgroup$ – Robin Ekman Jun 8 '16 at 13:30
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Within the framework of standard model (SM) magnetic monopoles are non-existent. It is quite subtle as to why this is not the case. To begin, look at the Dirac's famous charge quantization condition. It was first pointed out by Dirac that on the quantum level the existence of the monopole will lead to the \begin{equation} qg = 2\pi n \qquad\qquad \text{where n is an integer.} \end{equation} Where $q$ ang $g$ are the electric and magnetic charge (monopole) respectively. Now just consider the electric charge quantization within the framework of SM, which is not possible to achieve. Thats because in SM electric charges are the eigenvalues of $U(1)_{\text{em}}$ generator Q. Point is eigenvalues of $U(1)_{\text{em}}$ generator are continuous (moving on a circle) where electric charge of leptons and quraks are quantized. This simply implies that one can't get any satisfactory explanation of electric charge quantization from SM. Consider this to be true, existence of monopole (following Dirac's quantization condition) is SM also not possible.


Topological considerations lead to the general result that stable monopole solutions occur for any gauge theories in which a simple gauge group G is broken down to a smaller group $H = h \times U(1)$ containing an explicit $H = h \times U(1)$ factor. For a review of the topological arguments see (Coleman 1975, 1981). Clearly this is compatible with the fact that expectation of charge quantization and existence of monopole are related and that charge quantization follows from the spontaneous symmetry breaking of a simple gauge group. In the grand unified theories where the symmetry is broken from some large simple group, e.g. $SU(5)$, to $SU(3)_{\text{c}} \times SU(I){\text{em}}$, there are also monopole solutions of the 't Hooft-Polyakov type. The monopole mass is determined by the mass scale for the symmetry breaking $M_{X}$ (mass of the color changing gauge bosons). In the $SU(5)$ GUT, $M_{X}\equiv 10^{15}$GeV. This means that this type of monopole is out of reach for its production by accelerators (LHC energy scale is 14 TeV). So even if any monopole do exist, its out of our reach. Keep in the mind that $SU(5)$ has already ruled out!

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The historical basis of this belief is embodied in Gauss' Law:

$$ \nabla \cdot \mathbf{B} = 0 $$

This form is widely accepted for classical electromagnetism (as opposed to the form modified to allow for magnetic monopoles). It implies that the net magnetic flux over any surface is zero. A magnetic monopole would cause the magnetic flux of a surface to be non-zero, and so it would violate this law.

It correctly predicts the results of experiments where classical physics can be applied. As with all physics, this formula is subject to new theories and experiments, but it is correct enough to become widely accepted as part of classical physics. This is one of Maxwell's equations, though he used a different form.

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    $\begingroup$ That law is the mathematical version of the statement 'monopoles don't exist', and saying that the law correctly predicts experiments is the same as saying 'we have not observed monopoles', both of which are implicit in the OP. The question at hand is why they don't exist. $\endgroup$ – Emilio Pisanty Jun 8 '16 at 8:34
  • $\begingroup$ @EmilioPisanty I think that the first comment to the Question is the best answer: they don't exist = they are not observed. There is no way to explain why something doesn't exist. We have theories that explain the characteristics of things that we observe (which is very different from them existing) but existence in itself can never be given a reason. Physics is not a Religion, it is not Teleological. Someone did address the question of why cutting a magnet in two creates new poles. That was good. $\endgroup$ – user95006 Jun 8 '16 at 13:41
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    $\begingroup$ @nocomprende you can answer "why something doesn't exist" with "because it would contradict deeply established beliefs in the current theory." It's not always necessary to qualify "explanation" "with respect to some theory." $\endgroup$ – djechlin Jun 8 '16 at 13:49
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    $\begingroup$ @EmilioPisanty But the formula has applications beyond monopoles not existing. It predicts certain behaviors from a variety of configurations, which are observed. A physical theory is never based solely on things we don't observe. That would be absurd. You can't experimentally verify a negative. But this theory, which is verified by things we do observe, predicts that monopoles do not exist. As I explicitly state, it is subject to new theories and understanding, but this at minimum has historical significance in explaining where the belief comes from. $\endgroup$ – jpmc26 Jun 8 '16 at 16:59
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    $\begingroup$ @EmilioPisanty I saw no need to duplicate content from other answers that I am unable to present as well as they have. I am merely contributing a historical basis for the belief (which modern theories suggest may have been mistaken). This answer was never intended to be definitive, and not every answer has to be. It only needs to contribute relevant knowledge. (As the help page says, "Any answer that gets the asker going in the right direction is helpful...") $\endgroup$ – jpmc26 Jun 8 '16 at 17:13
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I think that John Rennie's is a good answer, specially due to the link given where it is shown the reasons why physicists expect the magnetic monopoles to exist. However I would like to stress that this is not simply the case of "explaining the non existence of something that we never saw". It is rather how to explain the lack of observation of something that it is expected to exist.

The strongest reason they are expected to exist is the fact they are predicted in Grand Unified Theories (GUT's), such as $su(5)\rightarrow su(3)\oplus su(2)\oplus u(1)$ or $so(10)\rightarrow so(6)\oplus so(4)\rightarrow su(3)\oplus su(2)\oplus u(1)$. They naturally appear as localized and stable solution to the equations of motions for the theory.

So a natural question to do is: Why do not we observe these theoretically predicted monopoles? Once they are predicted by a consistent theory we no longer can answer "they don't exist because they are not observed". We do need to explain why we do not see them even though the theory says they exist.

A simple but not interesting answer would be that the Universe did not show a spontaneous symmetry breaking patterns which allow for stable monopoles (such as the above ones).

A much more interesting scenario is the following one, called monopole problem. The monopoles predicted by GUT's are in general heavy and abundant. Its abundance is calculated to be far greater than the observed density of the Universe. This was one of the Alan Guth's motivation for proposing an Inflationary Universe. The idea is that these far too many monopoles were diluted during an infationary epoch.

A third explanation which have been proposed is that there was an intermediate phase during which monopoles anihilate themselves by means of flux tubes, which are other topological solutions predicted by GUT's.

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