Would magnetic charge be Lorentz invariant?

[Bob Eisenberg]

Would magnetic charge be Lorentz invariant (the way electric charge is) if magnetic charge existed?

 

[PeterDonis]

This question doesn't have a well-defined answer, because magnetic charge doesn't exist, so there's no way to do an experiment to test whether it's Lorentz invariant, and no way to know what theory we should use to compute the answer. There are certainly Lorentz invariant extensions of standard electromagnetism that include magnetic charges, which are Lorentz invariant in those theories; but there are also possible theories of magnetic charges in which they are not Lorentz invariant. Since magnetic charges don't actually exist, how are we supposed to tell which theory we should use to answer your question?

 

[Bob Eisenberg]

Dear Mr. or Dr. or Prof. Donis, or Peter, if first names are OK,

Thanks for a perfect answer.
I am grateful you took the time.

Bob Eisenberg
Chairman emeritus and Professor
Dept of Molecular Biophysics
Rush University Medical Center
Chicago

 

[stevendaryl]

This question doesn't have a well-defined answer, because magnetic charge doesn't exist, so there's no way to do an experiment to test whether it's Lorentz invariant, and no way to know what theory we should use to compute the answer. There are certainly Lorentz invariant extensions of standard electromagnetism that include magnetic charges, which are Lorentz invariant in those theories; but there are also possible theories of magnetic charges in which they are not Lorentz invariant. Since magnetic charges don't actually exist, how are we supposed to tell which theory we should use to answer your question?

Well, physicists have looked into the consequences of the existence of magnetic monopoles, and there are GUTs that predict their existence. So there are probably tentative theoretical answers to his question.

For example, Dirac studied the question of the affect of a magnetic monopole on charged, quantum-mechanical particles. I don't know if there is a version of QED that allows for magnetic monopoles, or not.

 

[PeterDonis]

physicists have looked into the consequences of the existence of magnetic monopoles, and there are GUTs that predict their existence

Yes, but all of these theories assume Lorentz invariance. So they don't, of themselves, address the question the OP was asking. The only way to decide that question would be if we actually discovered magnetic monopoles and could test whether their magnetic charge was Lorentz invariant.

 

[PeterDonis]

Mr. or Dr. or Prof. Donis, or Peter, if first names are OK

First names are fine. I'm not a professor and I don't have a doctorate, and "Mr." is way too formal for PF.

 

[samalkhaiat]

Yes, but all of these theories assume Lorentz invariance. So they don't, of themselves, address the question the OP was asking. The only way to decide that question would be if we actually discovered magnetic monopoles and could test whether their magnetic charge was Lorentz invariant.

Do you need to discover the colour charges of QCD in order to see they are Lorentz scalars?
In a Lorentz invariant theory, the charge (electric, magnetic or others) is given by an integral of the form $$Q_{\alpha} (t) = \int d^3 x \ J_{\alpha}^{0} ( t , x ) .$$ So, it is not at all obvious that $Q(t)$ is a Lorentz-invariant scalar, i.e., Lorentz-invariance alone is not enough to prove the scalar nature of $Q_{\alpha} (t)$. However, if the theory has another (internal) symmetry then the current $J^{\mu}_{\alpha}(x)$ will be conserved Lorentz vector, i.e., $\partial_{\mu} J^{\mu}_{\alpha}=0$. This conservation law (together with the boundary condition $|\vec{x}|^{2}x^{\mu}J^{\nu}_{\alpha}(x) \to 0$ as $|\vec{x}| \to \infty$) allow one to prove that the charge commutes with the Lorentz generators, $[M^{\mu \nu}, Q_{\alpha}] = 0$. This in turn means that $Q$ is Lorentz-invariant scalar.

Sam

 

[PeterDonis]

Do you need to discover the colour charges of QCD in order to see they are Lorentz scalars?

No, but you need to know that the theory that describes them as Lorentz scalars has been experimentally confirmed, to whatever level of confirmation you think is sufficient. AFAIK no theory that includes magnetic monopoles has any experimental confirmation; all of them are speculative extensions to the Standard Model in a regime that we have not been able to explore experimentally.

Lorentz-invariance alone is not enough to prove the scalar nature of $Q_{\alpha} (t)$.

Theoretically, it might not be, yes. Experimentally, you would just run the necessary tests to see if it behaved like one.

 

[samalkhaiat]

I don't need to know anything about any experiments. I am a theoretical physicist sitting on my desk with only pen and paper and thinking about the world. In formulating a theory, I don't care about what experiment may or may not say about the theory. Plus are you certain that there exists no magnetic charges in this universe? The OP equestion is a legitimate physics equestion.

 

[bcrowell]

The whole motivation for positing magnetic monopoles is to make E&M look more symmetric. That alone is enough to make me think that the answer to the OP's question is yes, simply because magnetic charge should behave the same way as electric charge.

Here is a further argument that I think is definitive. As pointed out by Dirac, if you have one electric charge $e$ and one magnetic monopole with magnetic charge $g$, then the angular momentum of their fields is finite, proportional to the product $eg$, and independent of the distance between them. Since both angular momentum and electric charge are quantized, it follows that magnetic charge, if it exists, must also be quantized. If it's quantized, then we can't have it changing when we do a Lorentz transformation.

It's probably also possible to make purely classical arguments. The original motivation for SR was to make Maxwell's equations Lorentz invariant. If charge wasn't invariant under a boost, that would break the Lorentz invariance of Maxwell's equations. I think the same would apply to magnetic charge.

 

[PeterDonis]

In formulating a theory, I don't care about what experiment may or may not say about the theory.

No, but you do if you're going to use the theory to make claims about reality, which is what I read the OP's question as asking for. If all that the OP is asking for is what theories that include magnetic monopoles predict, that's a different question. Perhaps the OP should clarify exactly what kind of answer he's looking for.

are you certain that there exists no magnetic charges in this universe?

No. So what? We don't have any evidence of them, so anything we say about them can only be based on speculative theories.

 

[PeterDonis]

magnetic charge should behave the same way as electric charge.

I agree that this is a reasonable theoretical expectation, but reasonable theoretical expectations don't always turn out to be true when the relevant experimental regimes are actually explored.

if you have one electric charge $e$ and one magnetic monopole with magnetic charge $g$, then the angular momentum of their fields is finite, proportional to the product $eg$, and independent of the distance between them. Since both angular momentum and electric charge are quantized, it follows that magnetic charge, if it exists, must also be quantized. If it's quantized, then we can't have it changing when we do a Lorentz transformation.

This is an interesting argument which looks to me to be largely independent of any specific theory about monopoles; it just relies on quantization of angular momentum and electric charge, both of which are experimentally well confirmed. My only question would be what assumptions are required to show that the angular momentum is proportional to $eg$.

 

[samalkhaiat]

Here is a further argument that I think is definitive. As pointed out by Dirac, if you have one electric charge $e$ and one magnetic monopole with magnetic charge $g$, then the angular momentum of their fields is finite, proportional to the product $eg$, and independent of the distance between them. Since both angular momentum and electric charge are quantized, it follows that magnetic charge, if it exists, must also be quantized. If it's quantized, then we can't have it changing when we do a Lorentz transformation.

First of all, in field theory with electric and magnetic charges, the numbers $(e , g)$ are coupling strengths. They are not the Noether charges associated with the symmetry group $U_{e}(1) \times U_{g}(1)$. If we put these couplings in the corresponding $U(1)$ transformation, they simply appear as coefficients in the corresponding charge $$Q_{\alpha} = q_{\alpha} \int d^3 x \ J^{0} (x) ,$$ where $q_{1} = e, \ \ q_{2} = g$. If the OP was talking about (g), then I would not have bothered myself with his question.

It's probably also possible to make purely classical arguments. The original motivation for SR was to make Maxwell's equations Lorentz invariant. If charge wasn't invariant under a boost, that would break the Lorentz invariance of Maxwell's equations. I think the same would apply to magnetic charge.

It is not as simple as you put it. With monopoles comes naturally Dirac’s string which, in field theory, forces us to introduce an arbitrary fixed 4-vector into our Lagrangian which in turn breaks Lorentz invariance of the theory. Restoring Lorentz invariance will (depending on the type of matter fields in the theory) impose further conditions on the coupling strengths $e$ and $g$.

 

[bcrowell]

My only question would be what assumptions are required to show that the angular momentum is proportional to $eg$.

None. The Poynting vector is proportional to $eg$, and the angular momentum density is proportional to the Poynting vector.

 

[pervect]

By symmetry, I'd expect that magnetic charge density would be conserved, but would not be a Lorentz invariant scalar, which is how I interpreted the question, though I could be misunderstanding it. The mathematical statement of electric charge conservation is that $\partial p / \partial t + \nabla \cdot J=0$ where J here is the 3-vector current. https://en.wikipedia.org/w/index.php?title=Charge_conservation&oldid=659868173. This can be formulated in 4-vector terms, which is more in the spirit of relativity, but probably unnecessarily advanced for the thread. See https://en.wikipedia.org/w/index.php?title=Four-current&oldid=654524633

Anyway, I'd expect the magnetic charge would behave a lot like electric charge. I don't have a specific reference to quote on this point to check my memory, though.

 

[bcrowell]

By symmetry, I'd expect that magnetic charge density would be conserved, but would not be a Lorentz invariant scalar, which is how I interpreted the question, though I could be misunderstanding it.

It's electric charge that acts like a scalar, not electric charge density.