# Electric and Magnetic Transforms

*Most recent answer: 11/24/2010*

- apurva (age 17)

rajkot

Consider a sheet of equal densities of charged particles of opposite signs flowing opposite directions. In our frame, E=0 everywhere. There's a B throughout the region, at right-angles to the flow direction, switching sign as you cross the sheet. Now, without worrying about the details of the E, B transform, you can see that in any other reference frame the term that comes from E in the starting frame is zero. So unless the coefficient in the transform of the term that comes from the initial B were zero, there would still be some B in the new frame.

Yes, you can sort of picture the field from the electron spin as coming from a little circulating current. However, if you calculate the circulating current from the spin angular momentum, you underestimate the magnetic field by a factor of just over 2. Calculation of that factor (the gyromagnetic ratio) is one of the great triumphs of relativistic quantum mechanics, since it's accurate to about 1 part in 100,000,000,000.

Mike W.

*(published on 11/24/2010)*

## Follow-Up #1: magnetic monopole math

- Devon (age 24)

USA

(My colleague Mike Stone points out re that paper "The author is begging the question by assuming that a global expression for the potential is required. There is no such global object, but quantum mechanics does not require a global potential --- only a potential for which the ``Dirac string" is invisible. This last condition requires the quantization of charge."In other words, we don't really need a well-defined vector potential as a function of position. All that's really require is that those properties of the vector potential which determine the physical behavior of particle be well-defined. That's a weaker condition, and one that can be met even for a monopole.)

At any rate, even if one were to accept the argument in the paper, I believe there's an entirely separate line of argument for the quantization condition, making no use of vector potentials. A combined electric field E and magnetic field B give a local momentum density (Poynting vector) proportional to ExB, with the proportionality constant depending on choice of units. Looking at the field from nearby electric and magnetic monopoles, integrating over space gives zero net momentum by symmetry. However, it gives a net angular momentum along the line between the two monopoles, with magnitude proportional to the product of the electric and magnetic monopole moments. If one uses the quantization of angular momentum, that gives a minimum (non-zero) possible value of the magnetic monopole moment, inversely proportional to the electric monopole moment. I haven't done the integral yet, but presume that gives the Dirac quantization purely via a field argument.

Mike W.

*(published on 11/30/2010)*

## Follow-Up #2: monopole follow ups

- Devon (age 24)

Lansing

Here's some semi-answers:

Yes, there are ways to search for monopoles. For example, a monopole passing through a magnetic detector loop would create a distinctive signal. Despite one suspected event years ago, it really looks like none have been seen.

As for why there should be monopoles, the obvious reason is that it would make Maxwell's equations for electromagnetism more symmetrical and hence more beautiful. That sounds stupid, but such aesthetic reasoning tends to work in basic physics. There are more serious arguments, over my head, concerning the basic symmetry groups of the grand unified theories which encompass both the electroweak force and the (strong) chromodynamics force. These arguments are described in a wikipedia article, whose accuracy I cannot personally evaluate.

It is widely suspected that the reason monopoles are so scarce is that their number was fixed prior to cosmic inflation, which then diluted the concentration enormously. In recent years, however, we've seen more and more clever ways to infer things about the early constituents of the universe, so don't rule out the possibility that some indirect test might be found.

Mike W.

*(published on 01/06/2011)*

## Follow-Up #3: First thoughts on magnetic monopole Hamiltonian

- Devon (age 24)

Lansing

I'm not smart enough (and maybe never was)to give a confident answer on this. However, I do notice one thing fishy in the argument in the paper you cite. The Hamiltonian used is strictly classical, not even close to being relativistically invariant. So it's no surprise that it doesn't work out to be consistent with the magnetic monopole. I'd like to see if the monopole were consistent with Dirac's relativistic equation for the electron. Since it was in the context of that relativistic treatment that Dirac first proposed the monopole, I suspect it is consistent. I'll try to find someone to give a better answer.

Mike W.

*(published on 01/30/2011)*

## Follow-Up #4: more monopole issues

- Devon (age 24)

Lansing

You don't really need to be able to construct a unique well-defined vector potential. In a particular choice of gauge, it's only the field (essentially the curl of the potential) and certain properties of the vector potential which show up in the behavior of particles. In the famous Aharonov-Bohm effect, in which interference effects are seen in particles which don't directly go through a field region, but do see a vector potential, the interference is periodic in the potential. Basically, if the potential is ill-defined in a way that only allows integer multiples of 2π arbitrariness in phases, that doesn't matter.

You're wondering whether those Dirac strings in the vector potential have an energy that's proportional to length, like the links between a quark and an anti-quark, or like a magnetic vortex in a superconductor. If so, monopoles wouldn't really exist as fully separate particles. The Dirac strings are a sort of mathematical fiction, so monopoles really would be the sorts of things which could bounce around inside a box. Quarks are definitely not magnetic monopoles themselves.

Mike W.

*(published on 02/18/2011)*

## Follow-Up #5: conservation of magnetic charge

- Devon (age 24)

Lansing

The same would hold for a magnetic monopole. The difference would be that monopoles are so extremely rare that a monopole of one sign would have an extremely low chance of encountering an opposite monopole, so annihilation events would almost never occur under current conditions.

You may also be asking why this magnetic charge is conserved. I think there are supposed to be good reasons for the different charge conservations, involving connections to underlying gauge symmetries in the field theories. Noether's Theorem says that for each continuous symmetry there's a corresponding conserved quantity. So yes, it is connected to the way the forces are unified.

Mike W.

*(published on 02/22/2011)*

## Follow-Up #6: positrons, monopoles, and all that

- Devon (age 24)

Lansing

It's true that monopoles change Maxwell's equations, adding a divergence to B and a current term to the curl of E. Since they already have a divergence to E and a current term in the curl of B, this makes them more symmetrical. It's not like tearing up a simple theory and replacing it with some hodgepodge.

Mike W.

*(published on 02/25/2011)*

## Follow-Up #7: relativistic limits on monopoles?

- Devon (age 24)

Lansing

"The talk page discussion is reminiscent of Brillouin's criticism of relativity: Given a solution to Einstein's equations, just change the co-ordinates and you get another, so the theory has no concrete predictions. This is wrong of course, The physical quantity is the interval int_geodesic sqrt{ds^2} and this is coordinate independent. Similarly with A_mu. The physical quantities (forces, phase-shifts) extracted from it must be scalars, but A_mu itself is gauge variant and not usually globally defined. I wonder if it is worth contributing to the talk page. Are you registered? I've often edited Wiki pages, but I've never participated in the discussions.

A criticism about the way string theory books and papers describe strings is more valid. Stringers used to say (many still do) that string physics is not coordinate independent unless d=26 (or d=10). This is because they tacitly tie the UV cutoff on the x^mu(sigma, tau) word-sheet fields to the sigma, tau coordinate grid. So changing coordinates does more than passively change the description. I was confused about this for ages, until Marc Goulian explained it to me. The real physics requiring d=26 etc is that for other dimensions, the string has longitudinal (internal) degrees of freedom, and these are not desired. " (Mike Stone)

I understand the first part of what he wrote as a sensible response to the first paragraph in the discussion you sent. I will ask Mike again about the second paragraph, which sounds to an ignorant outsider like a potentially more serious challenge, although he did not find it so on first reading.

Mike W.

*(published on 03/08/2011)*

## Follow-Up #8: more on monopoles

- Devon (age 24)

Lansing

and Chen Ning Yang, Phys. Rev. D
14,
437–445
(1976)

Dirac's monopole without strings: Classical Lagrangian theory

The non-quantum-mechanical interaction of a Dirac magnetic monopole and a point charge through the electromagnetic field is studied. A classical action integral which is multiple-valued is found. Stability of this action integral against variations of the world lines of the point charge and the monopole, and against variations of the electromagnetic potentials, yields the correct Lorentz equations of motion of the particles and the Maxwell equations for the field. No strings are introduced in the formalism.

Mike W.

*(published on 03/27/2011)*