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Why is Maxwell's theory so hard to understand? (2007) [pdf]

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Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#151

Post by adrian_b »

toth wrote: post #105
adrian_b wrote: post #99

Earlier quoted context omitted.

The originally published equations were "20 or so" because one equation was written for each scalar component. Rewriting the equations in vector form reduces the number to the modern number. Moreover, the original equations are the complete system. The variant with 4 equations is the simplified variant for vacuum, which is mostly useless, except for the purpose of studying the propagation of electromagnetic radiation…

> The originally published equations were "20 or so" because one equation was written for each scalar component. > Rewriting the equations in vector form reduces the number to the modern number. And if you use the differential form or 4d tensor notation they get reduced to 1 equation. Of course, for a lot of practical problems this is not very useful and it's better to work with the 3d vector form. > The variant with…

Even the vacuum version is incomplete without adding an equation for force or energy, because no meaning can be assigned to the electromagnetic field or potential otherwise than by its relationship with the force or energy.

Even today, there exists no consensus about which is the correct expression for the electromagnetic force. Most people are happy to use approximate expressions that are known to be valid only in restricted circumstances (like when the forces are caused by interactions with closed currents, or the forces are between stationary charges).

Moreover, when the vacuum equations are written in the simplified form present in most manuals, it is impossible to deduce how they should be applied to systems in motion, without adding extra assumptions, which usually are not listed together with the simple form of the equations (e.g. the curl and the divergence are written as depending on a system of coordinates, so it is not obvious how these coordinates can be defined, i.e. to which bodies they are attached).

While the vacuum equations are fundamental, they may be used as such only in few applications like quantum mechanics, where much more is needed beyond them.

In all practical applications of the Maxwell equations you must use the approximation of continuous media that can be characterized by averaged physical quantities that describe the free and bound carriers of electric charge. The useful form of the Maxwell equations is that complete with electric polarization, magnetization, electric current of the free carriers and electric charge of the free carriers. It is trivial to set all those quantities to zero, to retrieve the vacuum form of the equations.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#152

Post by lupire »

namaria wrote: post #59
xhstephen wrote: post #53

i have tried to transform the pdf into a presentation with AI, may help you read faster

I don't know what's more concerning, the fact that you find a six page paper written by one of the greatest communicators in science hard to digest or that you think an automatically chopped up version with colorful shapes is equivalent to the original.

Reformatting makes a document more digestible, especially when not reading on a printed sheet of paper, which the OP is exclusively designed for.

What's concerning (aside from your callous disregard for people who have small screens) is that the PP created a new document, and didn't show it, suggested that we might find it more digestible.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#153

Post by eigenket »

saalweachter wrote: post #142
eigenket wrote: post #103

Earlier quoted context omitted.

Quantum mechanics does not mean everything is quantized. It got its name because the first predictions of quantum mechanics were quantized energy levels in some example systems, but that does not even mean that all energies are quantized in quantum mechanics. There are many systems you can study where energies are continuous, and many examples where other quantities are continuous in quantum mechanics.

Wasn't it more the observation the theory was designed to explain than the first prediction?

I think its a linguistic difference only. At least where I studied it was quite normal to call phenomena you can derive from a physical theory "predictions" even if they have been observed before. I agree the photoelectric effect strongly suggested some quantization before quantum mechanics was formalised.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#154

Post by simiones »

lupire wrote: post #140
mjburgess wrote: post #84

Earlier quoted context omitted.

minimum lengths arent relevant to whether things are continuous or not. these arent related.

That's literally the definition of continuity. You have an object at position p, and the behaviors of the system are discretely different between P and P + h, without an intermediary at P+h/2.

Discreteness would mean that there exists some base distance p such that the distance between any two objects is Np, with N being a natural number (and any surface is some Mp^2 and any volume is Qp^3 and so on). Continuity is simply the opposite of that. It could be that objects can be at arbitrary real-valued distance d from each other, but that d > p is a precondition for any other law of physics.

By contrast, discretness has various unintuitive mathematical properties that mean it's not easy to fit into some other theories (particularly those relying on differential equations).

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#155

Post by lupire »

> Instead of thinking of mechanical objects as primary and electromagnetic stresses as secondary consequences, you must think of the electromagnetic field as primary and mechanical forces as secondary.

Feynman explained this nicely. He said essentially, you ask me to explain what is electmagnetism. Is it like two hands pushing on other? Well, if it is, then what is "pushing"? Pushing is just the result of electomagnetism in your hands! It is impossible explain electromagnetism to you in terms of anything simpler that you already understand. It is a fundamental force.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#156

Post by teleforce »

MichaelZuo wrote: post #11
bsder wrote: post #3

Maxwell's theory is not hard to understand--once you have the proper tools. The problem is that because of trying to cram a degree into 4 years, you wind up having a class on electromagnetics without any understanding of vector fields . Electrical engineering is particularly bad about this. You never get exposed to the Hamiltonian formulations of classical mechanics, and you never get exposed to vector analysis. Cons…

I've always thought the Heaviside notation is a bit bizarre... is there any advantage to them at all?

Heaviside and his proponents avoided quaternion like a plague, and like a classic false messiah he somehow convinced people not to use it. If we want to easily and completely model the EM waves its entirety including polarization we need to embrace quaternion, there is no two ways about it. The intuitive understanding of EM can be only developed by using quaternion and me personally waiting for someone to write a Pozar's book version in quaternion approach.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#157

Post by simiones »

lupire wrote: post #138
omnicognate wrote: post #71

Earlier quoted context omitted.

That's the "trivial sense" I'm talking about. If we restrict "cognition" to the stuff we know is discrete then trivially it's discrete. But cognition is a hell of a lot more than that.

I don't see how what we know is discrete. A word doesn't even have a discrete meaning, except locally in relation to other words. Saying A = B + C looks discrete, just by hiding any potential non-discreteness inside B and C.

Discrete here means "non-continuous" - i.e. there is no smooth transition function between ideas/thoughts/rationalizations.

For example, a formal proof is a discrete process: it follows step-wise rules that you can assign natural numbers to (this is the first step, this is the second step, this is the third step). A non-discrete process, a continuous one, would have a smooth transition between these steps, which is hard to even imagine.

While I am not convinced it is correct to say that "human reasoning is discrete", human language is definitely discrete. Words don't blend smoothly into each other. If you don't believe me, try to define a function f:[0,1] -> Words, such that f(0) = "red" and f(1) = "blue" and tell me what is f(sqrt(2)/2), or what is df/dx.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#158

Post by namaria »

lupire wrote: post #152
namaria wrote: post #59

Earlier quoted context omitted.

I don't know what's more concerning, the fact that you find a six page paper written by one of the greatest communicators in science hard to digest or that you think an automatically chopped up version with colorful shapes is equivalent to the original.

Reformatting makes a document more digestible, especially when not reading on a printed sheet of paper , which the OP is exclusively designed for. What's concerning (aside from your callous disregard for people who have small screens) is that the PP created a new document, and didn't show it, suggested that we might find it more digestible.

I see hundreds of students consuming PDF files on smartphones everyday for their classes. I read this paper on my phone just now. I am callously disregarding people who cannot bring themselves to read six pages and have to make it small and cute first.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#159

Post by simiones »

TheOtherHobbes wrote: post #127
mjburgess wrote: post #52

Earlier quoted context omitted.

> continuity of spacetime is a convenient approximation I disagree, and there's no evidence for this. This is computer science leaking out; physics has no formulation of spacetime in discrete terms, and indeed, all of physics presumes continuity. In QM, the space of wavefns is infinite-dim continuous, and if wasnt, QM wouldnt be linear. Cognition is discrete, but the world is continuous.

This seems very unlikely. If you're from Copenhagen every measurement is a lossy discontinuity that resets the wavefunction. This is not an abstraction, it's directly observable. As for discrete formulations: https://en.wikipedia.org/wiki/Causal_dynamical_triangulation

Interpretations of QM and measurement have very little to do with whether space-time is discrete or continuous. The simple fact is that no common QM formulation uses discrete mathematics for space-time, and it's unclear if any that does would even work.

Also, your link is not a formulation of QM, it is a different theory which makes different predictions (it is a quantum gravity theory). And, per the sounds of the Wikipedia article at least, it is not actually proven equivalent to QM in the regimes where it needs to be ("There is evidence [1] that, at large scales, CDT approximates the familiar 4-dimensional spacetime", or in other words, it is not fully worked out if this is the case).

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#160

Post by guardingit »

nyssos wrote: post #145
guardingit wrote: post #123

Earlier quoted context omitted.

Thanks for the pointer on Geometric Algebra. This looks to be a promising path to understanding relativity/QM/EM, and goes some way to explaining my unease with cross products and imaginary numbers. Disclaimer: maths degree, so my unease was not a plain lack of understanding.

> maths degree Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM. The hype around geometric algebras (Clifford algebras over R) just comes from the fact that it's not the plug'n'chug explicit numbers and coordinates approach, which is all most people ever see. They do not do a good job of tracking the physical structure of electromagnetism, and…

Thanks for your comment.

> They do not do a good job of tracking the physical structure of electromagnetism

What do you mean by tracking the physical structure?

By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?).

> and in fact end up baking in a lot of assumptions about the setting that fail to generalize.

Can you explain a bit more what you mean here please?

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