Maxwell's Equations: Can They Be Summarized?

[Shaun Culver]

Can Maxwell's equations (the usual 4 equations) be summarized in the form of one equation?

 

[f-h]

*d*F(A) = J

see eg Gauge Fields Knots and Gravity by John Baez

 

[Shaun Culver]

Thank you.

 

[f-h]

Is not that the relativistic maxwell equation? Hey, there should be a mock up t-shirt for that

As opposed to what? And yes, yes there should be :)

 

[patfla]

Actually kurt you may have, in effect, answered a recent question in my mind.

Was it (is it) the case that Maxwell's original equations were not spacetime (or frame-of-reference) invariant?

That is, two different frames of reference with the same observable and Maxwell's original equations would yield different results.

And while I don't recognize *d*F(A) = J, it's my understanding that maybe the original purpose of the Lorentz transform was to fix this problem?

 

[JustinLevy]

Was it (is it) the case that Maxwell's original equations were not spacetime (or frame-of-reference) invariant?

That is, two different frames of reference with the same observable and Maxwell's original equations would yield different results.

And while I don't recognize *d*F(A) = J, it's my understanding that maybe the original purpose of the Lorentz transform was to fix this problem?

I'll try to answer this. Part of the problem in giving the full answer is giving proper historical context... I will do my best in a summary.

First, note that classical electromagnetism is contained in Maxwell's equations AND the Lorentz force law. Amazingly these equations were already Lorentz invariant, and historically led to learning about Lorentz symmetry, first through Lorentz's aether theory (which for classical electrodynamics agreed with relativity) finalized sometime in 1904, later postulated as a symmetry of all physics (Einstein 1905), and finally thought of in terms of a spacetime symmetry (Minkowski 1907).

The point is, our understanding evolved over time. In particular, even our understanding of coordinate systems... something Galileo, Newton, and Maxwell pretty much took for granted. The idea however of Galileo's relativity, and the idea of inertial frames at that time, fit well with people's intuition, and probably even "seemed obviously true" to them. So when Maxwell's equations predicted light to travel at the speed c, a very reasonable question at the time was: According to whom?

People considered - maybe it was just traveling at c relative to the source, or maybe according to some "medium" of light. What turned out amazing to them, and Lorentz slowly figured out over a couple decades, is that if you assume there is some "special medium" and only in the rest frame of this medium is it allowable to apply Maxwell's equations as written... that it turns out there is a whole class of coordinate systems, very similar to what they then thought of as "inertial frames", in which Maxwell's equations and the Lorentz force law ended up having the same form. Lorentz, even after Einstein's relativity papers, considered this more of a mathematical variable substitution trick to help solve electrodynamics in an experiment moving in the "aether".

So, us looking back, the line is clear to us (the "essence" of the physics hasn't changed), but to Maxwell seeing how we apply his equations in different inertial frames... we probably wouldn't agree on much of it. In a sense, we'd have to teach him how to apply his own equations.

So I would answer: Yes, the equations were relativistic (in the sense that they were lorentz invariant). BUT they were not originally written in a coordinate free notation, AND it wasn't clear how to apply them in "moving" frames -- even to scientists at the time.

As science is something of a continually evolving intellectual debate consuming many individuals... a history of science is a history of ideas, and does not have a simple path/time line as at any point the ideas are as varied as the individuals. Therefore understanding the history of science is not easy, and I do not claim to be an expert, so in the end you will have to make your own conclusions of what individuals believed back then. But that is my opinion of it.

 

[patfla]

Hi Justin,

That was useful. So the answer (while more subtle than how I asked the question) is: yes. Yes that is: if one is trying the apply Maxwell's original equations from different coordinate systems. I.e. you would get different results (on the same observable). I would assume that at that time, the 'fundamental' inertial frame was space (or the aether) while the moving frame-of-reference (with a different coordinate system) was the Earth.

Did contradictions actually arise (I assume they must have)? Something in the way that the perihelion of Mercury tipped people off to flaws in Newtonian dynamics that were then remedied (much later of course) by Einstein's relativity (special, not general I believe). I see that Maxwell first submitted his work to the Royal Society in 1866 whereas maybe the first formulation of the Lorentz transformation was by Voigt in 1887. Plenty of time for scientists to work with Maxwell's equations and discover ways in which they broke down.

And understanding things historically has often worked well for me. E.g. (in math) how the first the quadratic, then the cubic and quartic were all solved. Leaving the problem of the quintic. Which led to Abel and Galois - the latter producing group theory. Have always meant to figure out how Abel's proof (the unsolvability of the quintic) compares with Galois'. Fundamentally I think what they both observed and then structured is that there are patterns in the solutions to polynomials.

I'm a programmer (aka computer scientist) and work a great deal with data (as opposed to 'just' programming). It was interesting to go back and see just why Codd needed, in 1970, to write his paper proposing a relational structure for data.

pat

 

[JustinLevy]

Yes that is: if one is trying the apply Maxwell's original equations from different coordinate systems. I.e. you would get different results (on the same observable). I would assume that at that time, the 'fundamental' inertial frame was space (or the aether) while the moving frame-of-reference (with a different coordinate system) was the Earth.

Hmm... just to be clear, Maxwell's equations today are the same as Maxwell's "original" equations (although we have found more convenient ways of writing them, for example the equations usually seen in undergrad textbooks are how Heavyside rewrote them with vector notation).

So if we apply Maxwell's original equations in two inertial coordinate systems we do get the same predictions of experiments. Back then however, they though that inertial coordinate systems were related by Galilean transformations, and yes those would have given conflicting results.

So Maxwell's equations didn't change, but our understanding of how to apply them changed as our understanding of coordinate systems changed.

Did contradictions actually arise (I assume they must have)?

Oh yes, there was all kinds of confusion. Check out the Michelson-Morley experiment, Trouton-Noble experiment, Trouton-Rankine experiment, just to name a few. What is interesting to point out is that the last experiment listed there was after Einstein's special relativity paper. It took quite awhile for people to adjust to these new ideas.

If you go back and read Lorentz's papers, you'll notice something else as well. In addition to changing the concept of how inertial frames were related (Galilean vs. Lorentz transformations), special relativity also showed that Newton's laws themselves needed changing. So even after people stubled upon the Lorentz transformations, there was still a lot to figure out even when applying the equations just in one frame. For instance, momentum is not proportial to v as Newton suggested, therefore forces (F=dp/dt) seemed to behave weird, especially for electrons as they were so light, they can easily obtain relativistic velocities (such as in your TV set if you have a CRT - the TV's with a "tube"). Lorentz made strange modifications like giving the electrons different masses depending on the direction of the force, etc.

This must have all seemed very strange to them at the time.
Because Maxwell's equations and the Lorentz force law happenned to be Lorentz invariant as they were found, they were one of the few things that did not need to be changed through all this.

Hope that helps clarify somewhat.

 

[f-h]

"Maxwell's equations today are the same as Maxwell's "original" equations"

That's always been my understanding as well with the caveat that we today often just deal with the conceptually and mathematically much simpler case of EM in a vacuum whereas Maxwells equations were written in a medium. And EM in a medium does have a preferred frame: the rest frame of the medium.

Can anyone familiar with the original papers confirm this?

 

[patfla]

Hmm... just to be clear, Maxwell's equations today are the same as Maxwell's "original" equations (although we have found more convenient ways of writing them, for example the equations usually seen in undergrad textbooks are how Heavyside rewrote them with vector notation).

Yes of course. There are, as we've seen, various formulations (various ways of expressing the same mathemtical relations). I believe Maxwell's original '4' are these (at the top):

http://simple.wikipedia.org/wiki/Maxwell's_equations

f-h: the *d*F(A) = J you referred to earlier is (I believe) the first formula we see under "Covariant Forumulation" at the same:

http://simple.wikipedia.org/wiki/Maxwell's_equations

This section, though, says that there is another equation that accompanies *d*F(A) = J.

$$0 = \partial_c F_{ab} + \partial_b F_{ac} + \partial_a F_{bc}$$

How then do we 'fit this together' with *d*F(A) = J?

pat

 

[patfla]

Oh also f-h Gauge Fields Knots and Gravity by John Baez is, unfortunately, a book. Unfortunate in the sense that I can't immediately lay my hands on it.

But JB does seem to explain the terms and operators (in *d*F(A) = J) here:

https://www.physicsforums.com/showthread.php?t=141493

It would seem, for example that d is what Baez calls the exterior derivative. And * would be the Hodge star operator. I'll have to look up what the 'star operator' is, but I have a guess that it perhaps related to the wedge product.

I've never had any formal physics past high school, but in college had a whole lot of things including math and computer science (oh and Japanese also). So it's not as if I'm completely devoid of instincts with which to guide myself.

pat

 

[masudr]

patfla

In this formulation, F(A) is a one-form.

 

[f-h]

Actually F(A) is a two-form (so d*F is a 3-form and *d*F is a 1-form to match the current 1-form). The other equation is dF = 0 that's why I wrote F(A) it means F is the curvature of a connection A and therefore dF = 0 holds automatically. (The equation you write is always true if you write the Field equations in terms of the vector potential).

I guess one could also write F(A) explicitly: *d*(dA + [A,A]) = J

 

[patfla]

OK thanks f-h. Actually the JB link doesn't mention your full expression *d*F(A) = J but rather *d*F = J in combination with dF = 0.

You said that current (J) is a 1-form. So I would assume the whole of the expression *d*F(A) reduces to a 1-form?

 

[f-h]

exactly. roughly: the hodge star turns an n-form into a 4-n form. So we start with a 1-form A take the exterior derivative: dA (2-form) -> *dA (2-form) -> d*dA (3-form) -> *d*dA (1-form)

J is a 1-form, locally a linear map from the tangent space to the reals. So given any spacetime direction given by a vector v, J(v) gives you the magnitude of the current in the direction v.

the hodge star is defined such that w wedge *v = <w,v> vol where <w,v> is a sort of inner product turning the n-forms into a function and vol is the volume form of the metric. Since the volume form is an m-form in m-dimensional spacetimes in you can see that the hodge star needs to turn the n-form v into an (m-n)-form *v such that w wedge *v can be the volume m-form.

 

[jambaugh]

Along the line of thought of the OP, any series of equations can with suitable convention and definition be written as "one equation" e.g.:

(a,b,c,d,...) = (w,x,y,z,...)

But of course the devil is in the details... one can indeed "state" Maxwell's equations by stating the starting point from which they can be deduced, e.g. given SR, Maxwell's equations are equivalent to: "There exists a spin-1 massless U(1) gauge field".

 

[masudr]

Actually F(A) is a two-form...

$"^@ oops. Sorry for the error. f-h is right, of course.

 

[patfla]

roughly: the hodge star turns an n-form into a 4-n form. So we start with a 1-form A take the exterior derivative: dA (2-form) -> *dA (2-form) -> d*dA (3-form) -> *d*dA (1-form)

Thanx f-h - that's perfect. Although '4-n': is this just for Minkowski spacetime (that is: 3D+1)? From here:

http://en.wikipedia.org/wiki/Hodge_dual

I got the impression that the hodge star (aka dual) was, for any n dimensionial space, where you were 'projecting' (transforming, whatever) from a k subspace into the 'opposite' or whatever n-k subspace. This seemed possibly intuitive to me when one considers that the wedge product generalizes upon the cross product. And one way in which I think of the cross product is a two vectors in a 2-plane (in 3-space) where, when you take the cross product another, 3rd, vector, suddenly pops out into the 3rd dimension (unless the first two vector lie exactly atop one another - but then I suppose one could look at the resulting 3rd vector as being of length 0). That is, the a generalization of the cross product (the hodge star operator) will take existing vectors filling some subspace and pop them out into the remaining n-k subspace for a space of dimension n.

the hodge star is defined such that w wedge *v = <w,v> vol where <w,v> is a sort of inner product turning the n-forms into a function and vol is the volume form of the metric. Since the volume form is an m-form in m-dimensional spacetimes in you can see that the hodge star needs to turn the n-form v into an (m-n)-form *v such that w wedge *v can be the volume m-form.

This I'll have to think about a little more. Homework.

jambaugh
When I first came across (or rather understood) SU(3)XSU(2)XU(1) as the standard model I, of course, thought that was just wonderful. With U(1) being electromagnetism. JB though gives a refinement here:

http://math.ucr.edu/home/baez/week253.html

G = (SU(3) × SU(2) × U(1))/(Z/6)

That is, we mod out by Z/6. What exactly that means I understand at a first level (Z6 is common to all of SU(3), SU(2) and U(1)), but I'm still studying it to understand further.

pat

 

[agkyriak]

Can Maxwell's equations (the usual 4 equations) be summarized in the form of one equation?

The possibility of the formal representations of the linear Maxwell equations in the form
of the Schreudinger and the Dirac electron equations was mentioned in several articles and books [1,2,3,4,5].
For instance, let us consider the plane electromagnetic (EM) wave moving, for example, on
$y$- axis.
It is not difficult to show that in the case when we choose the $\Phi$-matrix in the following form:
$\label{eq1}<br /> \Phi =\left( {{\begin{array}{*{20}c}<br /> {{\rm E}_x } \hfill \\<br /> {{\rm E}_z } \hfill \\<br /> {i{\rm H}_x } \hfill \\<br /> {i{\rm H}_z } \hfill \\<br /> \end{array} }} \right),<br /> \quad<br /> \Phi ^+=\left( {{\begin{array}{*{20}c}<br /> {{\rm E}_x } \hfill & {{\rm E}_z } \hfill & {-i{\rm H}_x } \hfill & {-i{\rm <br /> H}_z } \hfill \\<br /> \end{array} }} \right),$
we can obtain the electromagnetic equations in the operator forms.
Consider e.g. the wave equation.
The EM wave equation has the following view [6]:
$\label{eq3}<br /> \left( {\frac{\partial ^2}{\partial t^2}-c^2\vec {\nabla }^2} \right) \vec <br /> {\Phi }(y)=0,$
where $\vec {\Phi }(y)$ is any of the above electromagnetic wave fields (\ref{eq2}).
In other words this equation represents four equations: one for each wave
function of the electromagnetic field.

We can also write this equation in the following operator form:
$\label{eq4}<br /> \left( {\hat {\varepsilon }^2-c^2\hat {\vec {p}}^2} \right)\Phi (y)=0,$
where $\hat {\varepsilon }=i\hbar \frac{\partial }{\partial t}, \quad \hat {\vec <br /> {p}}=-i\hbar \vec {\nabla }$ are the operators of the energy and momentum
correspondingly and $\Phi$ is some matrix, which consists four components
of $\vec {\Phi }(y)$.
Taking into account that $\left( {\hat {\alpha }_o \hat {\varepsilon }} <br /> \right)^2=\hat {\varepsilon }^2, \quad \left( {\hat {\vec {\alpha }}\hat {\vec <br /> {p}}} \right)^2=\hat {\vec {p}}^2$, where [7,8]
$\hat {\alpha }_0 ;<br /> \quad<br /> \hat {\vec {\alpha }};<br /> \quad<br /> \hat {\beta }\equiv \hat {\alpha }_4$ are Dirac's matrices and $\hat {\sigma <br /> }_0$,$\hat {\vec {\sigma }}$ are Pauli matrices, the equation (\ref{eq4}) can also
be represented in the matrix form of the Klein-Gordon-like equation without
mass:
$\label{eq5}<br /> \left[ {\left( {\hat {\alpha }_o \hat {\varepsilon }} \right)^2-c^2\left( <br /> {\hat {\vec {\alpha }}\hat {\vec {p}}} \right)^2} \right]^\Phi =0$

Now for the Dirac equations:

$\label{eq6}<br /> {\begin{array}{*{20}c}<br /> {\left[ {\left( {\hat {\alpha }_o \hat {\varepsilon }+c\hat {\vec {\alpha <br /> }}\hat {\vec {p}}} \right)+\hat {\beta }m_e c^2} \right]\psi =0,} \hfill \\<br /> {\psi ^+\left[ {\left( {\hat {\alpha }_o \hat {\varepsilon }-c\hat {\vec <br /> {\alpha }}\hat {\vec {p}}} \right)-\hat {\beta }m_e c^2} \right]=0,} \hfill <br /> \\<br /> \end{array} }$

Using (1) we obtain electromagnetic form of the equations (6):
$\label{eq31}<br /> {\begin{array}{*{20}c}<br /> {\left\{ {\begin{array}{l}<br /> \frac{1}{c}\frac{\partial E_x }{\partial t}-\frac{\partial H_z }{\partial <br /> y}=-ij_x^e \\ <br /> \frac{1}{c}\frac{\partial H_z }{\partial t}-\frac{\partial E_x }{\partial <br /> y}=ij_z^m \\ <br /> \frac{1}{c}\frac{\partial E_z }{\partial t}+\frac{\partial H_x }{\partial <br /> y}=-ij_z^e \\ <br /> \frac{1}{c}\frac{\partial H_x }{\partial t}+\frac{\partial E_z }{\partial <br /> y}=ij_x^m \\ <br /> \end{array}} \right.,} \hfill & {\left\{ {\begin{array}{l}<br /> \frac{1}{c}\frac{\partial E_x }{\partial t}+\frac{\partial H_z }{\partial <br /> y}=-ij_x^e \\ <br /> \frac{1}{c}\frac{\partial H_z }{\partial t}+\frac{\partial E_x }{\partial <br /> y}=ij_z^m \\ <br /> \frac{1}{c}\frac{\partial E_z }{\partial t}-\frac{\partial H_x }{\partial <br /> y}=-ij_z^e \\ <br /> \frac{1}{c}\frac{\partial H_x }{\partial t}-\frac{\partial E_z }{\partial <br /> y}=ij_x^m \\ <br /> \end{array}} \right.,} \hfill \\<br /> \end{array} }$
where j are the imaginary currents,

If we will use instead of Dirac matrics the 3x3 photon matrices we can obtain the full six Maxwell
equations
References
[1]. W.J. Archibald. Canadian Journal of Physics, 33, 565, (1955).
[2]. A.I. Akhiezer and W.B. Berestetskii. Quantum electrodynamics. Moscow, Interscience publ., New York, 1965.
[3]. T. Koga. International Journal of Theoretical Physics, 13, No 6, p.p. 377-385 (1975).
[4]. A.A. Campolattoro. International Journal of Theoretical Physics, 19, No 2, p.p. 99-126, (1980).
[5]. W.A., Jr Rodrigues. E-print arXiv: math-ph/0212034 v1 (2002).
[6]. J.D. Jackson. Classical Electrodynamics, 3rd ed., 808 pp. Wiley, New York, 1999.
[7]. L.T. Schiff. Quantum Mechanics, (2nd edition), McGraw-Hill Book Company, Jnc, New York, 1955.
[8]. H.A. Bethe. Intermediate Quantum Mechanics, W. A. Benjamin, Inc., New York -Amsterdam, 1964.