# So just HOW . does magnetism work?

1. Jun 29, 2014

I'm totally confused. Everywhere I look, most explanations of how magnetism works doesn't include anything about the virtual photon interaction or whatnot.

I'm just interested in the most basic explanation of how magnetism works, and I read in a graphic physics "textbook" ish thing that they interact through virtual photons...

Is there.. a complete explanation to this?

2. Jun 29, 2014

### ZapperZ

Staff Emeritus
Clarification sought. Since you didn't ask for the quantum field theory explanation of electric field interaction, does that mean that you have understood this QFT description, and that you only need the explanation for JUST magnetism?

Zz.

3. Jun 29, 2014

### Staff: Mentor

You question really is how does electricity work because magnetism is actually a relativistic effect.

Its simply this. Consider a current carrying wire. If a charge moves relative to it because of length contraction its charge density will be greater so it will experience a greater force. That force is magnetism.

Here is the technical detail spelt out:

Now where does the the electrical force come from - the deep modern reason is requiring local gauge invariance:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

QM has global gauge invariance. However if we insist on it being local as well EM is what inevitably results.

If you can get a hold of the following you can see the details:
http://scitation.aip.org/content/aapt/journal/ajp/46/4/10.1119/1.11327

Thanks
Bill

Last edited: Jun 30, 2014
4. Jun 30, 2014

### ChrisVer

The question is very general...
what do you mean by how magnetism works?
In the quantum level, magnetism is part of Electromagnetic interactions. So electrism and magnetism are considered as one (under their unification of special relativity in the single antisymmetric tensor $F_{\mu \nu}$ ). This thing works for the photons because it describes E/M and thus you get virtual photons propagating the interaction.

5. Jul 2, 2014

Oh, um no, I guess QFT works.

What I mean is that I'm looking for a non- highschool physics explanation, but more in terms of subatomic particles, I guess QFT describes that.
I have no idea what QFT is, I have long forgotten. Sorry I'm not sure what you mean by JUST magnetism, but I basically am asking exactly how magnetism fundamentally works, and if it has particle constituents that mediate its interaction.

Edit: yes, the person above describes my question sort of well, but is there any more detailed explanation of this virtual photon interaction?

6. Jul 2, 2014

### ChrisVer

yes, in any QED or QFT textbook...
Now if you expect to read of magnetism, I hardly believe you'll find anything, because as I said you work with electromagnetic interactions, so you hardly do distinction between magnetic and electric ones...
So in case you want to understand how these two get unified, I'd recommend a special relativity textbook....if you are not so into mathematics and calculations, Griffiths's Introduction to Electrodynamics is a nice "spot" to READ everything ....

7. Jul 2, 2014

### Staff: Mentor

There is no such thing as JUST magnetism - there is only electromagnetism.

The answer to how EM works, or why EM, is at the deep level, gauge invariance. This was one of the dramatic discoveries of the great physicist Julian Schwinger - he showed in Quantum Field Theory (QFT) gauge invariance is the rock bottom essence of the QFT of electromagnetism called QED - Quantum Electrodynamics.

But that is likely cobblygook unless you have the proper background.

Sorry - the jig is up here. You are asking a very deep and fundamental question. And the answer is deep and fundamental, requiring our most powerful mathematical machinery and theories.

The following may help:
http://www.pnas.org/content/93/25/14256.full

Thanks
Bill

Last edited: Jul 3, 2014
8. Jul 2, 2014

### Staff: Mentor

Bill is right - there is no answer that is both simple and complete.

So I am most emphatically not disagreeing with what he says when I say that you might be able to get a reasonable heuristic explanation of how magnetism works from http://physics.weber.edu/schroeder/mrr/MRRtalk.html and other sites that you'll find by googling for "Purcell magnetism relativity".

You'll also find that as Bill says, "There is no such thing as JUST magnetism - there is only electromagnetism". Purcell's derivation makes it abundantly clear that the electrical and magnetic forces cannot be separated.

9. Jul 3, 2014

### nikkkom

Reminds me of one guy asking me "how computers work?".
I tried describing it several times, but got interrupted by "no, just tell me, how computers work?".
Finally I realized he thought there exists a simple explanation, as if computers were internal combustion engines.

10. Jul 3, 2014

### Staff: Mentor

Highly visual people, even mathematically advanced ones, often have difficulty with non visualizeable explanations. As great as Tesla was, and he most certainly was mathematically quite sophisticated calling Maxwell's treatises poetry, he never understood relativity because he couldn't visualize it. Its a theory about space-time geometry so can't be understood in such terms. But Tesla's mind simply didn't work that way. He had to have a mechanical picture.

To the OP. Its the same here. These days we do know the deep why of EM, but its not of the visualisable type. You simply have to accept it unless you want a detour into advanced math.

Thanks
Bill

Last edited: Jul 3, 2014
11. Jul 4, 2014

Thank you for emphasizing my ignorance of this, It's been much appreciated.

Thanks then, I think I understand what you're talking about.

That's very unfortunate that there's no simple answer, my conviction was that magnetism worked specifically through virtual photon interactions as read from a textbook.

12. Jul 4, 2014

### ChrisVer

magnetism is a result of the virtual photons interactions because electromagnetism is a result of it.
Now why we have electromagnetism, again I am going to address you to a special relativity book, or Griffiths introduction to electrodynamics - because there he is trying to show mainly with words how magnetism is a result of the electrism depending on the reference frame of the observer. The mathematical result of his reasoning is the ability/existence of a 2rank antisymmetric tensor which contains the Electric and Magnetic field.
Another way to see how those things are connected, is via the Maxwell's equations- there you can see how the electric field can really generate magnetic field and vice versa. Of course everything becomes much simpler and clear in special relativity.
In fact there's a joking comment from most people who teach the Maxwell equations, that what Special Relativity intended to do, was to make the Maxwell Equations frame invariant (they are not Gallilean invariant), as can be easily checked. I hope this thread helps somewhat:
The main problem with Gallilean transformations is that they don't change the time....

Nevertheless... Once you are able to work in special relativity, you are able as a said to define the antisymmetric 2rank tensor (field strength tensor field):
$F_{\mu \nu} = - F_{\nu \mu}$

$F= dA$
or in components:
$F_{\mu \nu} = \partial_{[\mu} A_{\nu]} \equiv \partial_{\mu} A_{\nu} -\partial_{\nu} A_{\mu}$

Where $A^{\mu}= (\Phi, \vec{A})$
is the electromagnetic 4 potential. The zero-th component is just the electric potential field, where the spatial components is the magnetic potential vector field.

There are many ways to define the above $F_{\mu \nu}$. One way, in particular in particle physics, is via the Invariance of the lagrangian under local U(1) transformations, which accept such terms. This is a nice way because it's also able to be extended for other groups (non-abelian.... U(1) is an abelian group) and thus even give you the corresponding field strength tensors for SU(3) (QCD), or SU(2) (somewhat the Weak Interactions- of course not exactly since you need to learn Higgs Mechanism for it). Nevertheless, I shall continue with the particle way of seeing things. The Lagrangian for a fermion for example, is just the Dirac Lagrangian:
$L= i \bar{\Psi} \gamma^\mu \partial_{\mu} \Psi$
If you try to use a local U(1) transformation, to the Dirac spinors, which is equivelant to making the change:
$\Psi \rightarrow e^{ig a(x)} \Psi$

(a global U(1) is in case $a$ doesn't depend on spacetime points $x^{\mu}$...In that case your lagrangian is invariant as it is).

You can see that the above Lagrangian is not invariant... it will change. In order for it not to change you have to allow a minimal coupling- that means to change the way you do partial derivatives... The same thing you do with General Relativity, where instead of the normal derivatives you have to allow a minimal coupling to the gravitational field through the Christoffel symbols/connections.
It happens to be a correct way to allow:
$\partial_{\mu} \rightarrow \partial_{\mu} + ig A_{\mu} \equiv D_{\mu}$
After replacing the partial derivatives with $D_{\mu}$ the Lagrangian is able to become invariant, if you allow $A_{\mu}$ to accept a U(1) transformation:
$A_{\mu} \rightarrow A_{\mu}'= A_{\mu} + G \partial_{\mu} a(x)$
$G$ some factor...

The last is also an indication that your field is somewhat a photon - it accepts the same transformations you can find for a photon in special relativity. By the above, also a local U(1) invariant term in the Lagrangian is the quantity: $F^2 = F_{\mu \nu} F^{\mu \nu}$
This gives the photon's kinetic terms..
$L_{photon kin}= - \frac{1}{4} F^2$
Why? because by making a change of the $A_{\mu}$ as above, and because of the antisymmetry you will have:
$F_{\mu \nu}' = F_{\mu \nu} + G \partial_{\mu} \partial_{\nu}a(x)-G \partial_{\nu} \partial_{\mu}a(x)= F_{\mu \nu}$

All the above might seem like magic, that we are able to reconstruct the electromagnetic tensor from imposing a local U(1) transformation. Also most of the points can be exctracted only by looking at the topological properties of U(1) gauge group (seems like geometry), something you can find in Peskin & Schroeder Chapter 15 ( particularly paragraph 1). That's why some people mentioned that Electromagnetism is in fact a result of gauge invariance.

Nevertheless... The next step for someone is to see how fields interact with the photons... In fact this is more difficult for me to explain, because the vollume of what must be said is huge and can result in writing a whole book- which I am unable to do at the moment... I already wrote too much in order to help you see how this is connected to gauge symmetry group.
In order for someone though to check how it results, is to write down the full lagrangian - allowing all the fields that you want, and allowing them to interact somehow. The charged fields will act as a source of electromagnetic field (you can check that by looking at the equations of motion for $F_{\mu \nu}$ (in fact in the easiest way when you have no interactions of photons with fields, the equations of motion for F are going to give you the Maxwell equations. Some clever mind used this fact in order to create a t-shirt with:
"And god said $F=dA$ and so there was light"
much more simple than writing the whole 4-set of Maxwell's equations, isn't it?
In $F$'s equations of motion alone, are not enough to allow both the 4 equations to be reconstructed. To get them you need either to allow the dual $F_{Dual}^{\mu \nu} = \epsilon^{\mu \nu \rho \sigma} F_{\rho \sigma}$ or $dF=0 \rightarrow \partial_{[\mu} F_{\nu \rho]}=0$ which you can see it holds because $dF= d(dA)=0$.
$\epsilon$ is the Levi Civita symbol in 4 dimensions (or totally antisymmetric element).
$\epsilon^{\mu \nu \rho \sigma}= +1,~~ (\mu \nu \rho \sigma)$ even permutations of $(0123)$ cyclic group.
$\epsilon^{\mu \nu \rho \sigma}= -1,~~ (\mu \nu \rho \sigma)$ odd permutations of $(0123)$ cyclic group
$\epsilon^{\mu \nu \rho \sigma}= 0,~~$ else (in fact if any index is the same as other)

eg
$\epsilon^{0123}=+1, ~~ \epsilon^{1032}=+1, ~~ \epsilon^{1023}=-1, ~~\epsilon^{1302}=-1, ~~ \epsilon^{0120}=0,~~ \epsilon^{1013}=0$
etc

Nevertheless, once you have the full lagrangian, you can start looking at Feynman Diagrams. The Lagrangian will indicate you what couplings you can have in a Feynman Diagram, and from that you can start looking at charged particle interactions with the Electromagnetic field - there you will need the photons as propagators (virtual/off-shell particles) and thus you see how particles interact with those virtual photons.
That's the electromagnetic interactions- in them you have hidden both the electric and magnetic interactions, but they are identical : two views of the same coin.

I hope this is enough to motivate you into getting to study things like differential geometry, group theory and quantum field theory.

Last edited: Jul 4, 2014
13. Jul 4, 2014

### Staff: Mentor

EM is the result of that. But what underlies even that is this gauge invariance I talked about.

All forces, not just EM, is mediated by virtual particles. But this deep idea of gauge invariance is what distinguishes one type of force from another.

Thanks
Bill