MAGNETIC MONOPOLES AND MATHEMATICAL REALITY

In 1931, the great physicist Paul Dirac published an article that would have a huge impact on the scientific community. On the one hand, his work postulated the existence of a new and strange physical entity that no one had ever detected; on the other, this new physical entity possessed extraordinary qualities, including the existence of a kind of mathematical "artifice" undetectable by any possible physical experiment. But then, why did physicists pay so much attention to such a seemingly speculative work?

The answer is fascinating: because if this new physical entity existed, the quantization of the charge of all particles in the Universe would be explained. In Dirac's own words: "The existence of even one of these entities in the Universe would explain why all existing electric charges are a multiple of e." But there's more: Maxwell's equations present an "uncomfortable" asymmetry: they are not invariant under the interchange of electric and magnetic fields. However, if these new entities existed, Maxwell's equations would become beautiful symmetric equations, and the electric and magnetic fields would be on an equal footing. In addition to these arguments, there are physical theories such as GUT theories and cosmic inflation that predict the existence of these entities. Is all this "overwhelming" theoretical evidence sufficient to believe in the existence of these new entities? The reader can judge for themselves. In this article, we will explain all these arguments step by step with simple mathematics so that the reader can find the answer to this question on their own.

Magnetic monopoles and Dirac strings

Electric fields (EF) and magnetic fields (MF), despite being associated with the same fundamental physical process, have some differences. One of the most important is that EFs emanating from a charged particle travel freely in all directions, while CMs always travel from an initial charge (one pole) to a final charge (the opposite pole). This is why it is said that CMs have sources and sinks, while EFs do not. If we wanted to "unify" EFs and MFs, we should first analyze what would happen if EFs behaved like EFs, that is, if there were isolated magnetic poles without sources or sinks. What would their magnetic field and potential be like? To try to answer this last question, let's consider a magnetic charge at the end of a very long solenoid.

The magnetic charge is located at the point z=0 and the solenoid is situated along the negative part of the z-axis. Let S be the solenoid cross-section, and "landa" the current per unit length. The solenoid's magnetic field can be considered the sum of a large number of small magnetic dipoles dm. The magnetic dipole between a length z and a length z+dz will be:

A magnetic dipole located at the origin produces the potential:

The total potential vector will be:

The quantity (Lambda*S/c) would represent a hypothetical fundamental magnetic charge g that generates the potential. The above potential would be created by a new physical entity, the new entity we mentioned at the beginning: a magnetic monopole.

If we look at the previous expression, we encounter a problem: for an angle of 0 or PI, the denominator is 0, and therefore the potential diverges and tends to infinity. This is what is called a singularity. This angle corresponds to the -z axis of the infinite solenoid; this line is very special and is called a Dirac string . In his original work of 1931, Dirac called them nodal lines and discovered that these lines actually constituted singularity points of the electromagnetic field since they were present in all charged particles. The existence of Dirac strings poses a major problem not only for the possible existence of magnetic monopoles but for the entire formulation of electromagnetic theory. How could this problem be solved?

Magnetic monopoles and the quantization of all charges in the Universe

Next, we can ask ourselves the following question: How can we physically detect Dirac strings? Since the electromagnetic field outside the solenoid is zero, we don't expect to find any physical experiment that would allow us to detect the presence of the solenoid. However, there is a very special experiment based on the so-called Aharow-Bohm effect that allows us to detect phase shifts in a particle's wave function. As we know, the complex wave function of a particle can be broken down into two parts: the magnitude and the phase. Changes in the phase of an individual particle's wave function cannot be detected since the probability of finding a particle at a specific position is given by the square of the wave function's magnitude. However, we can measure the phase difference between two particles if we make them interfere. In the Aharow-Bohm experiment, two beams of particles pass around a very long solenoid. Despite the fact that the electromagnetic field outside the solenoid is zero, the particles in both beams acquire a different phase shift when passing near the solenoid:

Therefore, if we interfere with both beams at the end of the process, we will obtain a measurement of the acquired phase difference and will therefore be able to detect a Dirac string. The phase shift of a particle passing around the solenoid will be:

Let's look at the previous expression. If the electromagnetic flux of the solenoid phiB had a value that was a multiple of e/hc, then the phase shift would be zero, and therefore the Dirac string would be undetectable, and the singularity would have no physical effect!

Dirac strings are another example of the intricate relationship between mathematics and real physical quantities; these entities seem to have a "mathematical" rather than a "physical" existence since they are not physically detectable.

In effect, we have that the flow that passes through the solenoid is:

If we require that this flow is also a multiple of e/hc (phi0) then we have:

Therefore, if g = Nhc/2e, then the Dirac string is undetectable, and we therefore solve the problem of the singularity of the magnetic potential. But we also achieve something even more important: the above expression implies that e = Nhc/2g

and thus we explain why all charges in the Universe appear in multiples of a fixed amount.

In his original work of 1931, Dirac deduced from a set of heuristic but rigorous steps that the end of all nodal lines (the monopole) must be the same for all wave functions. This insight led Dirac to write his famous phrase, "The existence of a single magnetic monopole in the Universe would explain the quantization of all electric charges in the Universe."

The Coulomb attraction-repulsion force between two magnetic monopoles is approximately 1173 times stronger than the force produced between two electrons or an electron and a proton. Dirac claimed that this could explain why no one has ever seen a magnetic monopole.

The symmetry of Maxwell's equations

Maxwell's 4 equations can be written as follows:

If magnetic monopoles exist and therefore the fundamental magnetic charge unit g then the equations become:

The equations are now symmetrical under the interchange of E and B!

This is undoubtedly another powerful argument supporting the existence of these new entities. The reader can judge for themselves whether this (overwhelming) theoretical evidence has convinced them that magnetic monopoles must exist.

More recently, physicist Joseph Polchinski stated that the existence of magnetic monopoles constitutes " one of the safest bets that one can make about physics not yet seen."

Do you agree with this opinion?

Sources: Quantized Singularities in the Electromagnetic Field Magnetic Monopoles Quantization and Quasiparticles Dirac quantisation condition