What Exactly Is The Physical Content Of Maxwell's Equations?

[bhobba]

It doesn't contain anything new, and nothing I haven't seen before, but as people who have been around a while know, I have a bit of a fascination with 'justifications' for Maxwell's equations.

I recently came acrossss the following:
https://www.vttoth.com/CMS/physics-notes/123-the-electromagnetic-field-tensor

We know the theorems by Wigner: the no-interaction theorem and the fact that all fields must be tensors (the Wigner-Eckart Theorem) strongly suggests the examination of a 4-vector field. But the above link would seem to then imply Maxwell's equations as a consequence.

Ultimately, of course, the validity of Maxwell's equations is an experimental matter, but it's hard not to get the feeling that something deep is going on here. Try as I might, though, I have no idea what it is.

Any ideas?

Thanks
Bill

 

[PeterDonis]

I recently came acrossss the following

Have you read Chapters 3 and 4 of Misner, Thorne, & Wheeler? That's where I first encountered the material that's briefly summarized in the post you reference. MTW was published in 1973.

the above link would seem to then imply Maxwell's equations as a consequence.

$F = dA$ implying $dF = 0$ is indeed a geometric property that will be possessed by any 2-form that is the exterior derivative of a 1-form. (Indeed, $d^2 = 0$ is a general identity for differential forms.)

However, that's only two of Maxwell's Equations. The other two, which Toth dismisses as "merely the defining equations of the charge density and current", are in fact the connection between the EM field and its source, which is not a "mere" definition; it's where the actual physics is. (This is a good illustration of how a "part-time physicist" can miss the boat.)

In differential form language, this can be written as $d (*F) = 4 \pi *J$, where the $*$ is the Hodge dual operator and $J$ is the charge-current 4-vector (which because of the way the Hodge dual works in 4D spacetime, makes $* J$ a 3-form, as expected for the exterior derivative of the 2-form $* F$--MTW chapter 4 goes into more detail about all this). Toth's observation that you get automatic conservation of the source (the continuity equation) from this is just another application of $d^2 = 0$--apply $d$ to both sides of $d (* F) = 4 \pi * J$ and you get $d * J = 0$, i.e., charge-current conservation. But to call all this a "mere" definition of charge and current misses the point. Automatic conservation of the source is part of the physics; it's what tells you that you've got the right connection between the field and its source.

it's hard not to get the feeling that something deep is going on here.

There's certainly important physics involved--and similar principles are used in GR to tell us that the Einstein Field Equation is the right one for the connection between spacetime curvature and its source, stress-snergy. (MTW goes into detail about this as well.) If that's "deep", then yes, there is indeed something deep going on.

 

[Sagittarius A-Star]

I recently came acrossss the following:

What also may help for understanding the EM field is to use Geometric Algebra, applied to 4D-spacetime.

The wedge product $\wedge$ (=outer product) of two vectors is a bivector.

The components of a 4-vector are projections on the four axes of the coordinate system (x, y, z, t).
The components of a bivector in 4D-spacetime are projections on the six (oriented) planes of the coordinate system (yz, zx, xy, xt, yt, zt).

The EM field bivector is $F = \nabla \wedge A$

Maxwell's equations:
$\nabla \cdot F = J$ (vector)
$\nabla \wedge F= 0$ (trivector, no magnetic monopoles)

See also

 

[FactChecker]

IMHO, as a casual amateur, geometric algebra (and geometric calculus) goes a long way to demystifying the subject. Unfortunately, it involves a long, tedious path of bookkeeping. It starts with dirt-simple observations/rules of how products of entities of different dimensions give results in other dimensions. From those rules, a lot of bookkeeping gives very satisfying results. The books by Alan Macdonald ("Linear and Geometric Algebra" and "Vector and Geometric Calculus") give a thorough explanation.
The end result, for me, was that there is no mystery in Maxwell's equations, just a lot of bookkeeping from basic geometry.

 

[weirdoguy]

What also may help for understanding the EM field is to use Geometric Algebra, applied to 4D-spacetime.

May or may not. Differential geometry is my specialisation within mathematical physics, and when I tried to study geometric algebra it only gave me headache and different forms of depression. Differential geometry, done properly (not like most physicstis do) is (for me) the way to go. Long live jet bundles!

 

[pines-demon]

We know the theorems by Wigner: the no-interaction theorem

Is it Wigner's?

 

[bhobba]

Is it Wigner's?

See:
https://journals.aps.org/pr/abstract/10.1103/PhysRev.182.1397

It's been on my to-do list for ages to get down to my old uni library to see the proof. Getting old and disabled these days - do not even have a car anymore.

Thanks
Bill

 

[pines-demon]

See:
https://journals.aps.org/pr/abstract/10.1103/PhysRev.182.1397

It's been on my to-do list for ages to get down to my old uni library to see the proof. Getting old and disabled these days - do not even have a car anymore.

Thanks
Bill

Oh I see, I thought it was Currie-Jordan-Sudarshan's. Wigner's cites them, they write

are all these no interaction theorems different?

 

[Sagittarius A-Star]

What also may help for understanding the EM field is to use Geometric Algebra, applied to 4D-spacetime.

Addition: Geometric Algebra can also be applied to curved spacetime. I did not yet have a closer look to this.
Source:
https://davidhestenes.net/geocalc/pdf/Curv_cal.pdf

 

[FactChecker]

May or may not. Differential geometry is my specialisation within mathematical physics, and when I tried to study geometric algebra it only gave me headache and different forms of depression. Differential geometry, done properly (not like most physicstis do) is (for me) the way to go. Long live jet bundles!

Can you recommend a reference for this approach? I have limited knowledge of differential geometry and (at my age) consider myself only a casual amateur.

 

[robphy]

Addition: Geometric Algebra can also be applied to curved spacetime. I did not yet have a closer look to this.
Source:
https://davidhestenes.net/geocalc/pdf/Curv_cal.pdf

From an old thread of yours on Geometric Algebra

Undergrad Thread 'SR: GA Multivector vs. Tensor notation, Maxwell's equations'

Jan 20, 2025

Geometric algebra (a subset of Clifford algebra) can be consistently used for all branches of physics, that are based on Euclidean space+time or Minkowski spacetime. It uses the following concepts for Minkowski spacetime:

• Geometric product (Dot product + Wedge product)
• Multivectors ( = scalar + vector (with 4 components) + bivector (with 6 components) + trivector (with 4 components) + pseudo-scalar)

Geometric product definition (edit: not valid for all kinds of multivectors):
$$ ab = a \cdot b + a\wedge b$$Dot product (commutative): ##a \cdot b = \sum_{r} \sum_{s}a_\hat...

• Sagittarius A-Star
• Geometric algebra Maxwells equations Tensor notation
• Replies: 7
• Forum: Special and General Relativity

I said...

I think part of the issue is a balance between
how much one has to learn in terms of mathematical preparation
in order to solve certain kinds of problems.

In addition, some notations are too-compact.
I like and am comfortable with the abstact-index-notation
because I can see what tensor-types I am working with.
(Needless to say, some instead want "indices that one sums over" or maybe just a list of components.)

Finally, I have to communicate my results in a language and notation that my reader can understand.

Geometric algebra is interesting... but I'm not there yet.
(One stumbling block for me are various conventions I've seen in different references.
I'll sort that out when I can... but that's not a high priority right now.)

There's needs to be a table of sign conventions [and other conventions (like notation)] to help translate between the works of GA different authors (similar to the inside cover of Misner-Thorne Wheeler (MTW) Gravitation).

Here's a start...
https://geometrica.vialattea.net/en/conventions/
...but, in my opinion, needs more if one wants more people to use (i.e., think and speak) it.

 

[bhobba]

are all these no interaction theorems different?

I only know of Wigner's version from a comment in Ohanian's Gravitation and Space-Time.

It was actually my first serious book on GR after I graduated with my math degree. It's different in approach from other books like Wald (my favourite) and MTW (it's the standard).

Thanks
Bill

 

[bhobba]

Can you recommend a reference for this approach? I have limited knowledge of differential geometry and (at my age) consider myself only a casual amateur.

A really great book that integrates it with multivariable calculus and linear algebra is:
https://matrixeditions.com/5thUnifiedApproach.html

Can be done after Calc BC (or equivalent, e.g., specialist math in Australia).

Here in Australia, my old HS (not when I went there, but now) allows you to do Calc BC in grade 10 and 11 and university math year 12 - it could be done then. Excellent preparation for college. My favourite math book.

Thanks
Bill

 

[robphy]

In my websearch for conventions in Geometric Algebra,
I stumbled upon two vocal critics of GA (listed below).
Since I am just learning about GA,
I can't vouch for the correctness of anything these folks say.
... but it might be worth a look

• The Case Against Geometric Algebra
  https://alexkritchevsky.com/2024/02/28/geometric-algebra.html ,
  which links to the next document
• Poor Foundations in Geometric Algebra
  https://terathon.com/blog/poor-foundations-ga.html

On a positive note, this might be of interest:

• https://projectivegeometricalgebra.org/ (by the author of the previous site)
• https://bivector.net/ ( https://bivector.net/merch.html )

 

[FactChecker]

In my websearch for conventions in Geometric Algebra,
I stumbled upon two vocal critics of GA (listed below).
Since I am just learning about GA,
I can't vouch for the correctness of anything these folks say.
... but it might be worth a look

• The Case Against Geometric Algebra
  https://alexkritchevsky.com/2024/02/28/geometric-algebra.html ,
  which links to the next document
• Poor Foundations in Geometric Algebra
  https://terathon.com/blog/poor-foundations-ga.html

I don't think their criticisms address what appealed to me about geometric algebra. I liked that it started with a basic geometrical definition of oriented areas, proceeded to examine their properties in different number of dimensions, and set up the mathematical framework to work with them. There was no mystery in it at all.
After that, the bookkeeping rules and fundamental results were examined and led to standard results when applied to physics problems.
I can't say that I would want to apply GA in a serious way. The manipulations are tedious, but I didn't plan to do them by hand. If I had to use GA, there are computer tools to do the algebra.
What I liked best was the unified way to arrive at many results which, if done the traditional ways, each required an eerily similar, but not exactly identical, approach. Each of the traditional diverse approaches were named after a famous mathematician/physicist of old. I found the single, unified approach of GA to be aesthetically pleasing. I think it might also appeal to @bhobba for the same reasons.
That is my "two cents" on the subject. Since I am old, retired, and never used GA, I will leave this discussion to others.

 

[Hornbein]

A nice thing about geometric algebra is that there is no such thing as "the right hand rule." The sign comes about naturally.

An n-vector is an n-dimensional linear subspace with a scalar attached. The scalar can be used for whatever you like. If you don't want the scalar you can set it to 1. A bivector is a 2D linear subspace with a scalar attached, trivector is 3D, etc.

It was very useful for what I was doing. I don't know that it would be worthwhile to rewrite all of physics in GA. Maybe you would find something useful by doing so, but maybe not. You don't know until you try, eh?

 

[A.T.]

A nice thing about geometric algebra is that there is no such thing as "the right hand rule." The sign comes about naturally.

Yes, it avoids introducing things like pseudo-vectors into the analysis, which are asymmetrical between two physically symmetrical situations.

See also this previous thread:

Undergrad Post in thread 'How deep does it need to go for a physics theory to be consistent?'

Mar 15, 2025

I would rather prefer answer on the metter...

You asked what the point of pseudovectors is. I gave you a practical example from physics, where it matters if something is a vector or pseudovector.

You can find more examples here:
https://en.wikipedia.org/wiki/Pseudovector

Here, when you mirror $\vec{I}$, then $\vec{B}$ is not just mirrored, but mirrored and negated. So $\vec{I}$ and $\vec{B}$ behave differently under a mirror transformation.

This anti-symmetric behavior might seem somewhat weird. But it becomes clearer, when you consider that in physics...

• A.T.

• 

 

[Sagittarius A-Star]

I have a bit of a fascination with 'justifications' for Maxwell's equations.

This can be found in chapter 38 of the book "Introduction to Special Relativity, 2nd Edition" by Wolfgang Rindler:
https://www.amazon.com/-/he/Introdu...-Publications/dp/0198539525?tag=pfamazon01-20

A shortened version of chapter 38 can be found under this link.

 

[robphy]

...I have a bit of a fascination with 'justifications' for Maxwell's equations.

Here's a different approach by Fred Hehl:

• (1999) Classical Electrodynamics: A Tutorial on its Foundations
  Friedrich W. Hehl, Yuri N. Obukhov†, Guillermo F. Rubilar
  https://arxiv.org/pdf/physics/9907046
  
  
• (2000) "A gentle introduction to the foundations of classical electrodynamics: The meaning of the excitations (D,H) and the field strengths (E, B)"
  Friedrich W. Hehl, Yuri N. Obukhov
  https://arxiv.org/abs/physics/0005084
  
  
• (2003) Foundations of Classical Electrodynamics (Progress in Mathematical Physics)
  by F. W. Hehl (Author), Yuri N. Obukhov (Author)
  https://www.amazon.com/Foundations-...-Mathematical/dp/0817642226?tag=pfamazon01-20

Also interesting:

• On the geometry of electromagnetism
  Alain Bossavit
  https://faculty.washington.edu/seattle/physics544/2011-lectures/bossavit.pdf

For me, I began a trip down this rabbit-hole on electromagnetism from books and writings by Bill Burke on relativity and "twisted differential forms" as I tried to understand the drawings of differential forms in Misner-Thorne-Wheeler's Gravitation. (As a grad student, he wrote back when I asked him about differential forms in thermodynamics.)

• https://www.ucolick.org/~burke/home.html
  especially "Twisted differential forms"
  "Drawing Vectors: The Eight Icons"
  https://www.ucolick.org/~burke/forms/dgcad.html
• (1980) https://www.amazon.com/Spacetime-Geometry-Cosmology-William-Burke/dp/0486845583?tag=pfamazon01-20
• (1985) https://www.amazon.com/Applied-Differential-Geometry-Burke/dp/0521269296?tag=pfamazon01-20
• "Manifestly parity invariant electromagnetic theory and twisted tensors" J. Math. Phys. 24, 65–69 (1983)
  https://doi.org/10.1063/1.525603
• (1993, 1994, 1995) "Div, Grad, Curl are Dead"
  https://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf
• Burke and Scott Special Relativity Primer
  https://physics.stackexchange.com/q...for-burke-and-scott-special-relativity-primer

Here's an old poster of mine on (2007) "Visualizing Tensors" from
https://www.opensourcephysics.org/CPC/
https://www.opensourcephysics.org/CPC/abstracts_contributed.html
https://www.opensourcephysics.org/CPC/posters/salgado-talk.pdf

 

[pervect]

The thing I found most interesting was how that if you postulate magnetic monopoles, the duality between the homogeneous and non-homogenous parts of Maxwell's equations when expressed in the language of differential forms becomes even clearer. Maxwell's equations, which can be written in the language of forms as dF=0 and d*F=J then become dF=J_m and d*F=J_e, where J_e is the usual charge-current density and J_m is the monopole analogue. I'm omitting some constant factors by the use of geometrized units. F is of course the Faraday tensor, d is the exterior derivatrive, and * is the Hodges dual operator.

I think this was included in MTW, but I'm not 100 percent positive that's where I heard about it.

 

[weirdoguy]

Can you recommend a reference for this approach? I have limited knowledge of differential geometry and (at my age) consider myself only a casual amateur.

I have a problem with coming up with something at hand, I learned most of these things from different papers and from people I worked with. And from lecture notes, but these are in polish. But I do have a lot of materials from that period (it's been over 10 years ago) on my backup disc, I'll check that later. Maybe I'll find something interesting.

 

[Sagittarius A-Star]

The thing I found most interesting was how that if you postulate magnetic monopoles, the duality between the homogeneous and non-homogenous parts of Maxwell's equations when expressed in the language of differential forms becomes even clearer. Maxwell's equations, which can be written in the language of forms as dF=0 and d*F=J then become dF=J_m and d*F=J_e, where J_e is the usual charge-current density and J_m is the monopole analogue. I'm omitting some constant factors by the use of geometrized units. F is of course the Faraday tensor, d is the exterior derivatrive, and * is the Hodges dual operator.

I think this was included in MTW, but I'm not 100 percent positive that's where I heard about it.

In Geometric Algebra, the Hodge * operation is replaced by an algebraic operation, a multiplication with the unit pseudo-scalar $I$ (=$\gamma_0\gamma_1\gamma_2\gamma_3$ = geometric product of all basis vectors).

The dual of the EM-field bivector $F$ is $G=FI$.
An xt-component becomes i.e. a yz-componnent, which is a swap between electric and magnetic components.

Maxwells equations with magnetic monopoles:
$\nabla \cdot F = J_{e}$ (vector)
$\nabla \wedge F= IJ_{m}$ (trivector)

The 2nd equation can be re-written as
$\nabla \cdot (FI)= J_{m}$ (vector)

Source (see equation 6.16):
https://davidhestenes.net/geocalc/pdf/SpaceTimeCalc.pdf

The EM-field bivector with an additional magnetoelectric potential is $F = \nabla \wedge A_e + I(\nabla \wedge A_m)$