Magnetic Monopoles: Exploring Existence & Impact

[wam_mi]

Can anyone explain to me why it would not be possible to write magnetic field (B) in terms of a vector potential (A) if magnetic monopoles exist please?

Why aren't there any magnetic monopoles? Why can't we find them? And Why do some physicists support the idea of the existence of magnetic monopoles?

Cheers guys!

 

[xepma]

The first question comes from the fact that a magnetic monopole acts as a source of magnetic flux. This means that in the presence of a monopole we have to change one of Maxwell's equations:

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

becomes

$$\nabla\cdot\mathbf{B} = \rho$$, where $$\rho$$ represents the magnetic monopole, similar to a charge density.

But when we write $$\mathbf{B}$$ in terms of a vector potential we have the identity:

$$\mathbf{B} = \nabla\times\mathbf{A}$$
$$\Longrightarrow \nabla\cdot\mathbf{B} = \nabla \cdot \nabla\times\mathbf{A} = 0$$

That last statement is an identity in vector calculus. We might think that we can simply forget about the vector potential, but we cant. In quantum mechanics we really need the vector potential to handle the effect of an electromagnetic field on charged particles.

But as it turns out, this problem can be resolved. We can still work with the identity $$\mathbf{B} = \nabla\times\mathbf{A}$$, provided we only allow it to be valid locally, instead of globally. This is where things get quite interesting, but also a more complex (it leads for instance to a quantization argument of Dirac, which states that magnetic monopoles can only exist if both the magnetic and electric charges are quantized).

What makes magnetic monopoles so interesting is the fact that instanton solutions of Yang-Mills theory exists which indeed are predicted to be monopoles. Better yet, any Yang-Mills theory based on some compact Lie group (like SU(3)xSU(2)xU(1)) in combination with a Higgs field (i.e. the Standard Model) predicts these monopoles. These are called 't Hooft-Polyakov monopoles.

Now there is some statement, but I'm not really sure of it, that when some symmetry breaking through the Higgs effect in YM-theory occurs you always end up with a bunch of magnetic monopoles. (but correct me if I'm wrong). Since this is pretty much how the Standard model works one might wonder where these monopoles are...

 

[wam_mi]

The first question comes from the fact that a magnetic monopole acts as a source of magnetic flux. This means that in the presence of a monopole we have to change one of Maxwell's equations:

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

becomes

$$\nabla\cdot\mathbf{B} = \rho$$, where $$\rho$$ represents the magnetic monopole, similar to a charge density.

But when we write $$\mathbf{B}$$ in terms of a vector potential we have the identity:

$$\mathbf{B} = \nabla\times\mathbf{A}$$
$$\Longrightarrow \nabla\cdot\mathbf{B} = \nabla \cdot \nabla\times\mathbf{A} = 0$$

That last statement is an identity in vector calculus. We might think that we can simply forget about the vector potential, but we cant. In quantum mechanics we really need the vector potential to handle the effect of an electromagnetic field on charged particles.

But as it turns out, this problem can be resolved. We can still work with the identity $$\mathbf{B} = \nabla\times\mathbf{A}$$, provided we only allow it to be valid locally, instead of globally. This is where things get quite interesting, but also a more complex (it leads for instance to a quantization argument of Dirac, which states that magnetic monopoles can only exist if both the magnetic and electric charges are quantized).

What makes magnetic monopoles so interesting is the fact that instanton solutions of Yang-Mills theory exists which indeed are predicted to be monopoles. Better yet, any Yang-Mills theory based on some compact Lie group (like SU(3)xSU(2)xU(1)) in combination with a Higgs field (i.e. the Standard Model) predicts these monopoles. These are called 't Hooft-Polyakov monopoles.

Now there is some statement, but I'm not really sure of it, that when some symmetry breaking through the Higgs effect in YM-theory occurs you always end up with a bunch of magnetic monopoles. (but correct me if I'm wrong). Since this is pretty much how the Standard model works one might wonder where these monopoles are...

First of all, thank you for your reply!

That all sounds very interesting. But if magnetic monopoles are predicted in the present theories, what makes it so difficult for the detection of a single monopole?
Would the LHC in Cern be able to tell us perhaps magnetic monopoles have existed with the condition similar to the early age of the univerese?

Sorry I don't know much about the subject, but I am eager to find out more because I think monopoles are one of the most fundamental objects we've ever encountered.

Thanks a lot