Why magnetic and electric field

In summary, the most primitive mathematical reason that electromagnetic field includes magnetic and electric field, not only one field, is rooted in the representation theory of the Poincare group. This leads to the conclusion that massless particles, such as photons, have only two polarization states, corresponding to the electric and magnetic components of the electromagnetic field. This is due to the symmetry structure of space-time and the requirement of discrete intrinsic spin/polarization degrees of freedom. If only the electric field had existed without a magnetic field, the special relativity vector form would have been different. This correlation between the three-dimensional nature of photons and the three-dimensional space is still a subject of ongoing research.
  • #1
exponent137
558
33
Do anyone know, what is the most primitive mathematical reason that electromagnetic field includes magnetic and electric field, not only one field? One reason is hidden in F tensor:
http://en.wikipedia.org/wiki/Electromagnetic_tensor
but, what is the most primitive mathematical reason ...?
 
Physics news on Phys.org
  • #2
The question is what you mean by "most primitive". From the modern point of view, the forms of possible physical laws are dictated by symmetries. Starting from space-time symmetry, which is underlying all physics, you can analyze large classes of possible physical laws. Starting from the special-relativistic space-time, Minkowski space, the appropriate symmetry group is the proper orthochronous Poincare group. One particularly elegant way to realize this symmetry in terms of dynamical laws is local field theory, and the symmetry structure of space-time leads to a huge class of possible dynamical laws, among which is that of a massless vector field, which leads (under the additional assumption that there's only a discrete intrinsic spin/polarization degree of freedom) inevitably to the notion of gauge invariance, and gauge invariance then more or less determines the math, which describes the electromagnetic field. There's no electric and magnetic field, but one electromagnetic field, which has, seen from an inertial frame of reference, electric and magnetic field components.
 
  • Like
Likes Orodruin, Symmetry777, bhobba and 1 other person
  • #3
So, according to your correction, more precise question is:
"what is the most primitive mathematical reason that electromagnetic field has two components, not only one?"

(under the additional assumption that there's only a discrete intrinsic spin/polarization degree of freedom)
Do you think spin of photon, 1\hbar, or any quantum spin?

inevitably to the notion of gauge invariance, and gauge invariance then more or less determines the math, which describes the electromagnetic field.
Can this link give new information: http://en.wikipedia.org/wiki/Introduction_to_gauge_theory
Can you tell more?
 
  • #5
The reason that there are only two polarization states not three as you'd expect from a vector field is that the photon is massless.

It goes back to the representation theory of the Poincare group. Looking for irreducible unitary representations you come to the conclusion, that good states are the momentum eigenmodes. In order to be irreducible, the Lorentz transformations must bring you from any "standard momentum" to any other. This implies that ##p \cdot p=m^2## with ##m^2## a given constant for all states represented by the irrep. Physically relevant are the cases ##m^2>0## (leading to the many-body description of massive particles, when the field is quantized) and ##m^2=0## (massless particles).

For massless particles you can choose the standard momentum as ##p=\Lambda(1,0,0,1)## (with ##\Lambda \neq 0## arbitrary). The irrep. then is further characterized completely by the representation of the subgroup which leaves this standard momentum invariant (the socalled "little group"). The group leaving the light-like standard momentum invariant is not compact since this group is equivalent to ISO(2), i.e., the group built by translations and rotations of the Euclidean plane. Now, to have only discrete intrinsic ("polarization") degrees of freedom, you must represent the translations in this plane trivially and only the rotations non-trivially (if you represent them also trivially, you get massless spin-0 particles, which is mathematically fine but not observed in nature). This leads to the conclusion that the only remaining degree of freedom is the rotation around the ##z##-axis, also leaving the light-like standard vector unchanged. This is a U(1), and it's representation is given by multiplication with a phase factor ##\exp(-\mathrm{i} \lambda \varphi)##. In principle, ##\lambda \in \mathbb{R}##, can take any value, but you want to lift the representation of the little group to one of the entire Poincare group, and there you have the usual rotations in space, and thus you must have ##\lambda \in \{0,\pm 1/2,\pm 1,\ldots \}##. So for massless particles you have only two polarization-degrees of freedom, which are represented by the rotations around the direction of the momentum of the massless particle, whose generator is the projection of the particle's angular momentum to the direction of its spin, the socalled helicity. For a massless spin-1 particle you the two helicities, ##\lambda=\pm 1##. In classical electromagnetism this describes left- and right-circular polarized electromagnetic waves.
 
  • Like
Likes exponent137 and bhobba
  • #6
Until someone comes up with a metric of primitiveness, I predict this thread will go around and around in circles.

It also comes at this from the wrong direction. Electric and magnetic phenomena are not a predictions of our mathematical models of electromagnetism. These models were created to mathematically capture the observed phenomena.
 
  • Like
Likes Symmetry777
  • #7
True, but the OP is about the "most primitive mathematical reason", and that's representation theory of the Poincare group. Of course, all natural science is empirical. Sometimes mathematical consistency allows to make nice predictions (like the introduction of Maxwell's "displacement current" into the electromagnetic system of equations, making them to what's know known as "Maxwell equations", which lead to the prediction of electromagnetic waves and the identification of light with such), but the general scheme of the physical laws are indeed all due to observations.
 
  • Like
Likes bhobba and exponent137
  • #8
Vanadium 50 said:
Until someone comes up with a metric of primitiveness, I predict this thread will go around and around in circles.

It also comes at this from the wrong direction. Electric and magnetic phenomena are not a predictions of our mathematical models of electromagnetism. These models were created to mathematically capture the observed phenomena.

Maybe the answer with Poincare Group is this, what I searched. I need to read more. But as "primitive" I thought, for instance, that McIrvin explained why the same charges reppel and opposite charges attract. http://math.ucr.edu/home/baez/physics/Quantum/virtual_particles.html
This is more primitive explanation. Weizsacker explained that Pauli matrices are three independent and this is (maybe) cause for three space dimensions.

The next question can be: if only electric field had existed without magnetic one, what would have been special relativity vector form instead of F tensor? If this form is clumsy, this is also reason for two components, E and B.

The basic motivation for the question was: "photon is 3 dimensional (B, E, direction of speed), space is three dimensional. Maybe this is correlated and connected?

In the case I will not find direct answers on basic motivation, I will anyway better understand QFT.
 
Last edited:
  • #9
exponent137 said:
The next question can be: if only electric field had existed without magnetic one,

That's not possible.

The existence of magnetic fields is a relativistic effect from Lorentz contraction.


Here is a link that gives the full mathematical detail of how Maxwell's equations follow from electric fields:
http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm

EM is as it is because of the existence of electric fields and space-time geometry.

Although understanding all these different views of EM is worthwhile I am with Vanhees - that is the most fundamental reason, also unfortunately the most advanced.

Thanks
B ill
 
  • Like
Likes Symmetry777 and exponent137
  • #10
exponent137 said:
So, according to your correction, more precise question is:
"what is the most primitive mathematical reason that electromagnetic field has two components, not only one?"
There is only one thing: the electromagnetic field.
The definitions of electric and magnetic fields have historic reasons and they can be convenient in many setups, but there is nothing fundamental about this convention.
 
  • Like
Likes exponent137
  • #11
bhobba said:
That's not possible.

The existence of magnetic fields is a relativistic effect from Lorentz contraction.


Here is a link that gives the full mathematical detail of how Maxwell's equations follow from electric fields:
http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm

EM is as it is because of the existence of electric fields and space-time geometry.

Although understanding all these different views of EM is worthwhile I am with Vanhees - that is the most fundamental reason, also unfortunately the most advanced.

Thanks
B ill

Yes, very simple answers. The question was a lapse. But, anyway, I still hope to better understand the most fundamental reason.

Thus a photon has three dimensional structure, B, E and direction of waves.
Gravitometric force also exists at gravitational wavings: http://en.wikipedia.org/wiki/Gravitoelectromagnetism
How it is here with B_g direction?
 
  • #12
mfb said:
There is only one thing: the electromagnetic field.
The definitions of electric and magnetic fields have historic reasons and they can be convenient in many setups, but there is nothing fundamental about this convention.
Because of two components, the photon has 3 dimensional structure, B, E and direction of waves. Thus two components are not only virtual? Or am I wrong?
 
  • #13
exponent137 said:
hus a photon has three dimensional structure, B, E and direction of waves.

What a photon is, is only elucidated in Quantum Field Theory which requires, unfortunately, many years of study. Anything you read outside a QFT textbook is likely wrong - certainly the above is wrong.

If you would like to undertake that journey the following is accessible after a first course in QM:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

By first course I mean something like Susskind that explains the full theory - not a lay version:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

It will however take time and require close attention.

And once that's done you can start to understand why U(1) symmetry is the key idea.

Thanks
Bill
 
  • #14
exponent137 said:
Because of two components
Or 3, or 4, or whatever number you like if you make arbitrary "components" of the electromagnetic field.
the photon has 3 dimensional structure, B, E and direction of waves. Thus two components are not only virtual? Or am I wrong?
I don't think that question makes sense.
 
  • #15
bhobba said:
What a photon is, is only elucidated in Quantum Field Theory which requires, unfortunately, many years of study. Anything you read outside a QFT textbook is likely wrong - certainly the above is wrong.

If you would like to undertake that journey the following is accessible after a first course in QM:
https://www.amazon.com/dp/019969933X/?tag=pfamazon01-20

By first course I mean something like Susskind that explains the full theory - not a lay version:
https://www.amazon.com/dp/0465036678/?tag=pfamazon01-20

It will however take time and require close attention.

And once that's done you can start to understand why U(1) symmetry is the key idea.

Thanks
Bill
I try still to find clearer connection between U(1), gauge theory and electromagnetism.
U(1) is pretty simple, it is based on basic circle. For instance:
http://uw.physics.wisc.edu/~himpsel/449group.pdf
A, (and Q) are probably included in wave functions?
And transformation is gauge transformation?
What gauge transformations
##A_\mu=A_\mu'+q^{-1} \partial_\mu \varphi## etc. have to do with the unit circle? Or how to include it in system of explanation with U(1)?
 
  • #16
exponent137 said:
have to do with the unit circle? Or how to include it in system of explanation with U(1)?

Have a look at the first link I gave:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

Its got to do with invariance to complex phase ie invariant to exp^iy - which is basically a circle - recall the phaser representation of complex numbers - as y varies it traces out a circle.

To go any further than this you need to start into much more advanced math - areas I am not even conversant with. Can we know your level of math training eg have you done multi-variable calculus to the level to understand the first link? That's so I can recommend some texts.

Thanks
Bill
 
Last edited:
  • #17
bhobba said:
Have a look at the first link I gave:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

Its got to do with invariance to complex phase ie invariant to exp^iy - which is basically a circle - recall the phaser representation of complex numbers - as y varies it traces out a circle.

To go any further than this you need to start into much more advanced math - areas I am not even conversant with. Can we know your level of math training eg have you done multi-variable calculus to the level to understand the first link? That's so I can recommend some texts.

Thanks
Bill
Yes I understand your link, as I wrote in PM. But, more I need, why electromagnetism is U(1), except, that wave function describe unit circle.
 
  • #18
exponent137 said:
Yes I understand your link, as I wrote in PM. But, more I need, why electromagnetism is U(1), except, that wave function describe unit circle.

As explained in the link its the requirement from local phase symmetry ie if you require the equation to be locally invariant to exp^iy then you need to have a term e/cA to cancel the terms that appear because of that requirement. exp^iy when graphed in the complex plane is a circle. Thus we require the equation to be invariant to the symmetry of a circle which by definition is U(1) symmetry.

I can't explain it better than that. If you still don't get it I will need to leave it to others to explain.

Thanks
Bill
 
  • #19
bhobba said:
As explained in the link its the requirement from local phase symmetry ie if you require the equation to be locally invariant to exp^iy then you need to have a term e/cA to cancel the terms that appear because of that requirement. exp^iy when graphed in the complex plane is a circle. Thus we require the equation to be invariant to the symmetry of a circle which by definition is U(1) symmetry.

I can't explain it better than that. If you still don't get it I will need to leave it to others to explain.

Thanks
Bill
The answers above are now clear.
I want to understand on the next level.

Why names bundles and fibres, where it is similarity with prolonged structures? What is analogy with curvature and with »connections«? Which components of curvature are B and E? Can anyone give any link which visulizes these things? Links like those http://www.math.toronto.edu/selick/mat1345/notes.pdf are not visulized enough.
 
  • #20
I think that this is sufficiently different from the OP to warrant a separate thread. Since the questions in the OP are now answered, let's close this thread.

exponent137, please try to formulate your question about bundles and fibres as clearly and as simply as possible, and avoid a "shotgun" series of questions. You also may be better off simply re-reading the material from this thread for a while until it has had a chance to really sink in.
 

1. Why do magnetic and electric fields exist?

Magnetic and electric fields exist because of the fundamental forces of nature. These forces are responsible for the interactions between charged particles and can be explained by the laws of electromagnetism.

2. How are magnetic and electric fields related?

Magnetic and electric fields are closely related. A changing electric field will produce a magnetic field, and a changing magnetic field will produce an electric field. This relationship is known as electromagnetic induction and is the basis for many technologies, such as generators and transformers.

3. What are the applications of magnetic and electric fields?

Magnetic and electric fields have numerous applications in our daily lives. They are used in technology, such as motors, generators, and transformers. They also play a crucial role in medical imaging, such as MRIs, and in particle accelerators used in scientific research.

4. How do magnetic and electric fields affect matter?

Magnetic and electric fields can affect matter in various ways. They can cause particles to move, align, or interact with each other. In certain materials, they can also induce a current, which leads to the production of light or heat.

5. Are magnetic and electric fields harmful?

Magnetic and electric fields are all around us, and at low levels, they are not harmful. However, at high levels, they can cause damage to living organisms. This is why there are safety regulations in place to limit exposure to strong magnetic and electric fields, such as those found in power lines or medical equipment.

Similar threads

Replies
9
Views
1K
Replies
3
Views
773
  • Electromagnetism
Replies
5
Views
221
Replies
41
Views
3K
  • Electromagnetism
Replies
8
Views
753
  • Electromagnetism
Replies
7
Views
890
  • Electromagnetism
Replies
8
Views
763
  • Electromagnetism
Replies
5
Views
1K
Replies
27
Views
983
Back
Top