If magnetic monopoles do not exist the answer is no, we cannot save a magnetic monopole inside a hermetic box. If they do exist and the hermetic box walls are infinitely far from the monopole the answer is Yes we can save it inside the box. Clearly, if magnetic monopole will be one day produced in labs, the function of a hermetic box will be solved by the walls of the experimental apparatus whose nature is not given in the question. Today we know that "Observation of Dirac monopoles in a synthetic magnetic field" Nature 505, 657–660 (30 January 2014) has been accomplished in a confined spinor Bose-Einstein condensate. Although this result helps to achieve an answer to the question, it should be noted that the experiment refers to synthetic magnetic monopoles.

Dear Vincenzo,

Obviously the Dirac magnetic monopole in a synthetic magnetic field always could be within the hermetic box, but what could happen with the magnetic Dirac monopole particle defined with real magnetic and electric fields?

Dear Daniel, I start to believe that you are skeptic about the possibility that Dirac monopoles can be isolated. The Dirac magnetic monopole observed in the synthetic magnetic field behaves locally as a magnetic monopole. Let's go to "real monopoles". A Dirac string stretches between two magnetic monopoles or from a magnetic monopole to infinity. If the walls of the box are hermetic even to a Dirac string, my thinking (on a Sunday) suggests me that the string will end on an other monopole. The result is that single magnetic monopoles cannot exist isolated in a hermetic box. However, the question is deep and deserves to be elaborated more seriously possibly with some mathematical proof, that unfortunately at the moment is lacking on my side. What is your idea on the question?

Dear Vincenzo,

I am not only skeptic of isolating magnetic monopoles, I am sure if we go to the original idea of Dirac. What I do not know is how the Dirac string could behave with the walls of the box and this could be the more difficult part of my question.

Dirac magnetic monopoles are topological particles which cannot be defined only locally. But nowdays we have more forms of understanding the magnetic monopole as the one that you refers in a Bose-Einstein condensate with "electric" and "magnetic" designed fields, which cannot follow globally the Maxwell's equations . Nice that if we introduce a designed monopole in a hermetic box, then the equation of div B is not zero at diference of what it ought to be.

The string introduced by Dirac is only a mathematical artifice. Dirac wanted to derive the magnetic field B(x,y,z,t) (of the monopole) from a vector potential A(x,y,z,t). He had then to introduce this string, near which A(x,y,z,t) becomes infinite. The string path (from infinity to the monopole) can be changed without changing the magnetic field B. So there is a kind of arbitrariness in the string path, analogous to the arbitrariness of in the choice of A(x,y,z,t), which results from gauge transformation.

Thus, at first sight, the string seems unobservable, therefore unphysical.

But this is not the end of the story. The string can be revealed by the Aharonov-Bohm effect, unless the product {magnetic charge g of the pole} times {the elementary electric charge e} is a multiple of the Planck constant, that is

e . g = n .h (n=integer). This is the Dirac condition. If it is fulfilled, the string has absolutely no physical existence.

Indeed there is a way to reformulate the theory of the Dirac monopole without introducing the Dirac string. It is less simple mathematically : one has to introduce different vector functions A(x,y,z,t) in different domains of the space-time, with some overlap between the domains.

Monopoles also arise as topological defects of some gauge theories ('t Hooft, Poliakov). No Dirac string is used, nevertheless the Dirac condition is fulfilled.

What about Aharonov-Bhom effect with quarks (which have fractional charges ?). I considered this effect in Nucl. Phys. B85 (1975) 442 and B129 (1977) 415, and wrote that it could be the "raison d'être" of the strings which bind quarks together. However I worked without the light of QCD theory. Nevertheless the idea was not irrelevant. As far as I understand, there is a connection between fractional (chromoelectric) charges and QCD strings.

The question is that div B=o must be followed locally as Maxwell equations say. Therefore if you apply the integral Gauss theorem this means that the magnetic flux have to be zero too and therefore you cannot define any magnetic pole or monopole in electrodynamics.

The idea is to avoid to have a closed surface which could surround the magnetic source and this can be done in many different forms as you said. One of them (first historically) is the Dirac string which is infinite string of singularities for the magnetic field or what is the same, it put a veto to the existance of any charged particle in contact with it. This is a way to say that you cannot have a hermetic box which completely surrounds it, isn't?

Obviously the topology can be introduced in many different forms as Wu-Yang or others for avoiding the model of the string but in any case you cannot surround the magnetic source.

All the theories use the the non trivial homotopy pi(U(1)) of the electromagnetic unitary group for measuring this non trivial topological solution. And in QCD the unitary groups are SU(2) or SU(3) or even higher but which contains as abelian solution the U(1) and therefore the magnetic monopole.

There is no compelling theoretical reason to impose div B = 0. The only serious reason is the lack of observation magnetic monopoles so far. But that may change one day... Of course, assuming div B = 0 one can derive B from a vector potential A, derive the Maxwell equations from a local Lagrangian. This makes quantization of QED much more easier.

The "Dirac veto" has no importance. Classically, there is zero probability for a point-like electric charge to touch the Dirac string (I don't understand why Dirac made this veto. May be he was thinking about an extended charge).

Quantum-mechanically, there are well-defined states of a charged particle in the field of a monopole, although the charge density |Psi(x,y,z,t)|^2 is extended.

In conclusion I think that a magnetic monopole, if it exists, could be kept alone in a box (may be not on Earth, because of a too big weight).

Div B=0 is one experimental law. It isn't a theoretical assumption for finding a vector potential A. If you cut a magnet, you only can obtain two new poles in such a form that you cannot never isolate only one magnetic pole.

The magnetic monopole is something more because it solves the classical difficulties introducing quantum mechanics and topology. The most appealing result is that you could justify the quantization of the electric charge and explain how one proton is with the same electric charge than an electron changed of sign in spite of being so different particles. This explains how most of the bodies are electrically neutral although they are made of so different and huge amount of charged particles.

The veto condition for the Dirac's string is absolutely necessary because it is equivalent to introduce the non-triviality of the topology associated and therefore to overcome the difficulties related with the Gauss theorem.

Dear Daniel and Vicenzo,

I may misinterpret your messages. Could you clarify your opinions about the question " Is it theoretically allowed or forbidden to sequester one monopole (without compensating anti-monopole) in a box ?"

My opinions are

- "yes we can" ( sequester one monopole)

- The Dirac string does not exist physically. It is just a mathematical tool. Its direction and shape, from the monopole to infinity (or to an anti-monopole) can be chose arbitrarily.

- The Dirac condition e . g = n .h applies to free particle (not necessarily to quarks)

- If monopoles exist, one could re-formulate the Dirac approach by assigning Dirac strings to electric charges instead of magnetic charges (duality principle).

I think the above opinions are "orthodox" for most scientists in this field. Besides, I have more questionable opinions :

- A monopole may violate the Dirac condition with the quarks, for instance

(e/3) . g / h = 1/3, or 2/3, 4/3, 5/3, etc.

- The monopole can make a Bohm-Aharonov effect with the QCD string which binds the quarks.

@Xavier. Perhaps I do not fully understand your question: " Is it theoretically allowed or forbidden to sequester one monopole (without compensating anti-monopole) in a box ?"

1- From my point of view this question is wrong because the existence of the magnetic monopole is absolutely independent of the existence or not of "anti-monopole". The same that I can have electrons independently of the existence or not of positrons. Notice that it is even much more difficult with monopoles because the monopole mass is very huge respect to the electron one.

2- The string is not only a mathematical trick, in fact it is what defines the magnetic monopole PHYSICALLY, although there are many forms of seeing it. But what is necessary is to hava a non-trivial topological physical background. Perhaps you are thinking and seeing a "magnetic pole" obtained with the duality rotations of electrodynamics.

3- The magnetic monopole that I am thinking is always using the electromagnetic interaction associated to U(1) unitary group and the monopole defined as pi-1(S-1) solutions. Say the interaction is made through photons are not gluons. The non-abelian solutions that you suggest and see their difficulties, it is logic pure. This needs to refine the quantization conditions as t'Hooft-Polyakov has done.

4- My question is simpler: is posible to sourrands a magnetic monopole? Notice that this is related information send without depending of the velocity and therefore out of the relativity as nowadays is discussed with the quantum entanglement.

Dear Xavier,

Sorry because I see that I didn't answer you clearly to your idea of sequester a monopole isolated.

The answer is not because there must be always a hole or a singularity of the electromagnetic field.

I add some information more:

1. Maxwell's equations hasn't duality between electricity and magnetism for the sources of the fields, in fact one of his equations div B= 0 forbids it. It is true that electrodynamics could kept invariant under what are known as duality rotations symmetry. That is to say, its classical physical observables could be invariant under such fictitious rotations as what observed by Oliver Heaviside in the XIX century. This symmetry allows to make classifications of electricity and magnetism within electrodynamics as are made in:

D.Baldomir and P.Hammond, Geometry of Electromagnetic Systems, Clarendon Press, 1996. With these transformations you can get a divergence of B different of zero but you do not have magnetic monopoles at all. This is only a different form of having the electric and magnetic fields defined into electrodynamics.

2. In 1931 Dirac showed that it was possible to have Maxwell's equations besides a particle which produced the magnetic field as the electrons or other electric particle in such a form that this could explain how so different particles as the electron and the proton had exactly the same electric charge in spite to be so differente (lepton,hadron). Later more models were developped such as the one of J.Schwinger, Wu and Yang or t'Hooft and Polyakov.

3. Every model of monopole has a picture of the magnetic monopole and the simplest one is due to Dirac. The magnetic monopole has the minimum magnetic charge g= 137 e/2= 68.5 e, in Schwinger model is the double g= 137 e, etc and the mass is obtained assuming that the classical radius of the electron is the same of the one of the monopole obtaining a huge minimum mass of 2.4 GeV, But in all that I know the mass of the monopole is still an unsolved question.