Are Maxwell's Equations The Most Pivotal Postulate of Classical Physics?

[maverick_starstrider]

Every textbook I read seems to follow the same logic/derivation of physics:

-Gauss' Law is observed experimentally, shows us there's this thing E
-Biot-Savart's Law is observed experimentally, shows us there's this thing B
-Ampere's Law (after fixed by Maxwell) observed experimentally, along with Faraday's law define the interrelationship between B and E

-Maxwell's equations condense these experimentally observed behaviors of these B and E things.
-Maxwell's equations are not Galilean invariant but Lorentz invariant.
-By assuming Lorentz Invariance, Linear, Temporal and Angular Homogeneity and Extremal Action we derive Relativist Mechanics which in the small energy limit gives us Classical Mechanics.

My question is this: Is there ANY WAY to divorce Maxwell's equation from those 4 experimentally observed facts (i.e. there exists something called E and it behaves this way). Is there any appeal to symmetry or some such that allows us to say "If we assume our universe has a symmetry of the form blah and blah we see that there MUST be some quantity E attached to each point in space and it MUST have these properties)?

Is there any more abstract way to motivate and derive Maxwells equations like we do for Classical Mechanics (like in the first chapter of Landau's mechanics) or MUST they be taken as 100% the result of experiment?

This seems to me to be the ultimate linchpin in the derivation of classical mechanics through the eyes of Euler-Lagrange and Extremal Action. We can motivate and derive everything else from some assumptions about the homogeneity of space and such but then we ALWAYS just tack on Maxwell's Equations and the existence of E and B as a matter of experimental fact.

In parallel in Quantum Mechanics we do the same things with Spin (we simply tack it on because experiment say it is there), however, when we generalize to quantum field theory, spin actually becomes a PREDICTION and not an experimental artifact. Can the same be done for Maxwell's equations?

 

[maverick_starstrider]

Every textbook I read seems to follow the same logic/derivation of physics:

-Gauss' Law is observed experimentally, shows us there's this thing E
-Biot-Savart's Law is observed experimentally, shows us there's this thing B
-Ampere's Law (after fixed by Maxwell) observed experimentally, along with Faraday's law define the interrelationship between B and E

-Maxwell's equations condense these experimentally observed behaviors of these B and E things.
-Maxwell's equations are not Galilean invariant but Lorentz invariant.
-By assuming Lorentz Invariance, Linear, Temporal and Angular Homogeneity and Extremal Action we derive Relativist Mechanics which in the small energy limit gives us Classical Mechanics.

My question is this: Is there ANY WAY to divorce Maxwell's equation from those 4 experimentally observed facts (i.e. there exists something called E and it behaves this way). Is there any appeal to symmetry or some such that allows us to say "If we assume our universe has a symmetry of the form blah and blah we see that there MUST be some quantity E attached to each point in space and it MUST have these properties)?

Is there any more abstract way to motivate and derive Maxwells equations like we do for Classical Mechanics (like in the first chapter of Landau's mechanics) or MUST they be taken as 100% the result of experiment?

This seems to me to be the ultimate linchpin in the derivation of classical mechanics through the eyes of Euler-Lagrange and Extremal Action. We can motivate and derive everything else from some assumptions about the homogeneity of space and such but then we ALWAYS just tack on Maxwell's Equations and the existence of E and B as a matter of experimental fact.

In parallel in Quantum Mechanics we do the same things with Spin (we simply tack it on because experiment say it is there), however, when we generalize to quantum field theory, spin actually becomes a PREDICTION and not an experimental artifact. Can the same be done for Maxwell's equations?

I would settle for a derivation from Coulomb's Law + The Requirement of Lorentz Invariance. Can you derive Maxwell's Equations from those alone?

 

[dextercioby]

Special relativity was build to offer a dynamical foundation to the theory of the electromagnetic field in vacuum. The electromagnetic field is fundamental at classical level, Maxwell's equations are postulated and interpreted in a specially relativistic mathematical setting offered by a flat 4D manifold without boundary, called Minkowski space and denoted by $\mathbb{M}_{4}$.

In other words, the e-m is postulated, if you wish to have an axiomatical description of classical dynamics.

 

[cmos]

Is there any more abstract way to motivate and derive Maxwells equations like we do for Classical Mechanics (like in the first chapter of Landau's mechanics) or MUST they be taken as 100% the result of experiment?

I believe what you are looking for is exactly what is done in Landau & Lifgarbagez, Vol. 2. After motivating a plausible form for the lagrangian of the EM field, they go on to DERIVE the Maxwell equations.

 

[homology]

and to boot, in Zee's Quantum Field Theory in a Nutshell he derives such things as opposite charge repel and the inverse square law for the Coulomb force, using the same Lagrangian mentioned above, its pretty awesome.

There's an EM book by Brau which is at a grad level but take a modern approach. It starts with SR and (as mentioned above) proposes a Lagrangian and derives Maxwell from it. Its pretty decent, a nice selection of exercises, though the book does have typos (but there's an errata somewhere).

 

[dextercioby]

Though putting the Lagrangian on first place may seem elegant, I still prefer the old traditional approach employed by some books, namely postulating the 4 equations in vacuum then, from the proof of incompatibility with Galilei's relativity principle (Newton's first law), constructing a new dynamics with new geometry and observables and expressing the so-called non-covariant formalism of Maxwell into the covariant one of Einstein and Minkowski.

Or could just invent the Lagrangian density from thin air and use to obtain everything.

 

[dextercioby]

I believe what you are looking for is exactly what is done in Landau & Lifgarbagez, Vol. 2. After motivating a plausible form for the lagrangian of the EM field, they go on to DERIVE the Maxwell equations.

I don't have their book right now, but can you summarize their <motivation> and final form of the Lagrangian ?

 

[jtbell]

Is there any appeal to symmetry or some such that allows us to say "If we assume our universe has a symmetry of the form blah and blah [...]"

If you're willing to start with quantum field theory, you can start by assuming local U(1) gauge invariance of the fields and invoke Noether's theorem. You get a conserved quantity which turns out to be electric charge, and a gauge field which turns out to be the electromagnetic 4-vector potential, from which you can derive the E and B fields.

 

[Matterwave]

Landau and Lifgarbagez gives:

$$S=\int_a^b\{-mcds-\frac{e}{c}A_i dx^i-\frac{1}{16\pi}\int_V F^{ik}F_{ik}dVdt\}$$

The form of the first piece of the Lagrangian is gotten by assuming merely the principle of relativity and the invariance of ds, the third term of the Lagrangian can be gotten by supposing that the fields must obey the principle of superposition (and therefore, the Lagrangian must be quadratic in the field terms, this principle itself is based on experiment) and the principle of relativity (and therefore, the variation of the action must use a scalar).

The second term; however, requires experimental verification. The exact form of the second term cannot be gotten from first principle considerations alone (even Landau admits this). We know that the second term is correct because it gives us the Lorentz force law which is experimentally verified.

It should be noted that the constants (pi, and c) are in this Lagrangian to make the units work out to Gaussian units.

I myself see no reason to elevate action principles to be somehow "fundamental"; however. In the end, the action principle itself must be experimentally verified.

 

[dextercioby]

And what's the connection between F and A ? How is this justified ? How is the use of A justified ?

 

[Matterwave]

$$F_{ik}=\frac{\partial A_k}{\partial x^i}-\frac{\partial A_i}{\partial x^k}$$

Landau put "A" there to be just some 4-vector which characterizes the field. He then shows the equations of motion that are derived from such a Lagrangian, and makes a definition of the E and B fields (the equations come to be the Lorentz force law except instead of E and B, you have A's in there, and then you make the usual definitions so as to coincide with the rest of the physics community).

The F's arise if you try to cast the Lorentz force law in 4-D notation. I don't want to give the full derivation here.

 

[dextercioby]

So essentially there's no advantage of postulating the Lagrangian density, because at every step you must justify yourself that: <if we don't do this, we don't end up with what's known since 1895>.

I dislike this, really.

 

[Matterwave]

Well, the Lagrangian method MUST give equivalent solutions as the regular method, so in some sense, there has to be some justifications in what to put in the Lagrangian. For relativistic mechanics, the Lagrangian is no longer necessarily just T-V (as can be shown that the first term in the Lagrangian is NOT the kinetic energy)

The advantage of using the Lagrangian method, is for "the attainment of maximum generality, unity, and simplicity of presentation" as Landau states.

 

[dextercioby]

I don't and didn't contest the elegance and utility of the Lagrangian approach, but rather the presentation: starting with it requires from my perspective an unpleasant justification of each result. Normally, the Lagrangian should be derived already knowing who E,B, A, phi are and how the field equations look like in the noncovariant formalism, then define A and F in the covariant formalism. Finally, post the Lagrangian of the field and deduct its simplest coupling to matter.

 

[Matterwave]

The justification for a term in the Lagrangian is that it produces the correct equations of motion, I do see how this can seem "unpleasant", but I don't think it's any worse than just supposing the equations of motion (Lorentz force law) and the field equations (Maxwell's equations) are just axiomatically true based on experimental fact.

If you know of a better "derivation" I would surely like to see it!

 

[RedX]

Landau and Lifgarbagez gives:

$$S=\int_a^b\{-mcds-\frac{e}{c}A_i dx^i-\frac{1}{16\pi}\int_V F^{ik}F_{ik}dVdt\}$$

The second term; however, requires experimental verification. The exact form of the second term cannot be gotten from first principle considerations alone (even Landau admits this). We know that the second term is correct because it gives us the Lorentz force law which is experimentally verified.

I don't see why the second term requires experimental verification. Isn't that just the coupling of a 4-current (matter) to a field? Plugged into Lagrange's equation, that would just give you a source term.

In fact, isn't that term responsible for Huygen's principle, something desirable for waves? That's what Green's functions are - Huygen wavelets created at sources - and don't Green's functions appear in all differential equations (not necessarily just wave equations) so long as they're linear?

 

[Matterwave]

The coupling from the second term to the last term gets you the field equations, and the coupling from the second term to the first term gets you the Lorentz force law. Experimental verification is needed, if only to specify that you can parameterize the equations for your particle with a single parameter (charge).

I'm not entirely sure I get your argument though, how would you arrive at that exact form for the second term from first principles?

 

[RedX]

The coupling from the second term to the last term gets you the field equations, and the coupling from the second term to the first term gets you the Lorentz force law. Experimental verification is needed, if only to specify that you can parameterize the equations for your particle with a single parameter (charge).

I'm not entirely sure I get your argument though, how would you arrive at that exact form for the second term from first principles?

You're probably right. When just varying the fields, a coupling JA leads to the differential equation:

$$DA=J$$

where D is a linear differential operator, and DA=0 comes from setting the first and second Lagrangian term equal to zero.

It is natural to think of the source J as some type of 4-current coming from the charge.

For example suppose the electron is at x in the 1st integral. Then source can be: $$e \delta(x-x')$$. Connect it to the scalar potential:

$$\int \phi(x',t') e \delta(x-x') dV'dt'= \int e\phi(x,t') dt'$$

Now you have to make that term into a 4-vector, so you can just assume:

$$e A^i dx_i$$ is the right term.

 

[cmos]

Regarding peoples' dissatisfaction with a postulated field Lagrangian and it's necessity to be experimentally verified (e.g. Landau & Lifgarbagez, Vol. 2):

This seems to be the same exact thing that must be done in classical mechanics (e.g. Landau & Lifgarbagez, Vol. 1). There is motivation for the free-particle Lagrangian to be proportional to v², but the same motivation seems to also apply to v⁴, v⁶, and so forth. Ultimately, it's experimental verification that leads to v². Just as experimental verification leads to the Lagrangian for the EM field.

Please let me know if I've misinterpreted the physics, but it seems to me that what makes physics physics, is it's experimental verification.

 

[homology]

I figured that the desire to derive the Maxwell equations from a Lagrangian density was to show the consistency of a method. That method seems to be to try to form the simplest density with certain symmetries.

We could start this way: Suppose we have a 4-vector field $$A_{\alpha}$$ and we want to construct some equations for it. We want the action to be Lorentz invariant. We'll limit ourselves to terms no higher than second order in the field and its derivatives (so our resulting equations are linear). Also let's include a source term for the field, call it $$J_{\alpha}$$. So here's a possible density:

$$L=c_1F_{\alpha\beta}F^{\alpha\beta}+c_2A_{\alpha}A^{\alpha}+c_3A_{\alpha}J^{\alpha}$$

where F=dA and so has first derivative info. And the products of the two tensors gives us second order derivatives in the density and so our variation will give equations linear in the field. Why do we want this? Well let's start with that and see what happens.

The neat thing I just read about a few weeks ago was that the coefficient c_2 only vanishes if we have gauge invariance and that its proportional to the mass of a photon. I didn't know that photon mass, gauge invariance and charge conservation were all related.

I only wrote the above because I think its a reasonable 'derivation' in that its investigating the consequences of our assumptions. We certainly didn't need it to come across EM, however there are certainly fields in nature that we have a hard time experiencing and so we have to 'guess' and then design experiments to see how our guesses fare. Putting classical physics in this framework seems to serve the purpose of a unification methods. EM is a field theory and so can be constructed from reasonable assumptions placed upon a density.

In any event, its pretty awesome.

 

[RedX]

$$L=c_1F_{\alpha\beta}F^{\alpha\beta}+c_2A_{\alpha}A^{\alpha}+c_3A_{\alpha}J^{\alpha}$$

where F=dA and so has first derivative info.

How did you know to use F as the differential of a 1-form? $$F_{\alpha\beta}F^{\alpha\beta}$$ has two distinct terms, one involving the divergence of the (vector) field (after integration by parts), and one involving mixed partials of the (vector) field, and they are both given equal weight. For example, why not add to your Lagrangian: $$7*A_\nu \partial^2 A^\nu$$?

The neat thing I just read about a few weeks ago was that the coefficient c_2 only vanishes if we have gauge invariance and that its proportional to the mass of a photon. I didn't know that photon mass, gauge invariance and charge conservation were all related.

I don't think the photon was known in Maxwell's time. Also gauge invariance is arbitrary. Sure, once you have Maxwell's equation you can show gauge symmetry. But can one really guess gauge symmetry and then derive Maxwell equations, given that gauge symmetry doesn't correspond to a symmetry that can be observed?

Also I think a term like $$\epsilon_{ijkl}F^{ij}F^{kl}$$ is also Lorentz invariant, since $$\epsilon_{ijkl}$$ is Lorentz invariant.