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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]

#131

Post by PaulHoule »

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…

The fact that you can write it in one equation shows that the theory is very simple because it is an expression of symmetry. E and B are not these two different things related by an inscrutable cross product but just two aspects of the same thing.

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

#132

Post by hnfong »

mjburgess wrote: post #77
discoinverno wrote: post #74

Earlier quoted context omitted.

It is true that there is no experimental evidence, but I think there are some convincing arguments that something must happen at the Planck scale (for very short distances) in a full quantum-gravity theory. Here are some quotes from "Covariant Loop Quantum Gravity", Rovelli and Vidotto (slightly redacted). I suggest the whole chapter 1, in particular 1.2 to get an idea of why fundamentally spacetime may be discrete.…

You're talking about minimum lengths, not discrete spacetime. It may be the case that there's a minimum length beyond which "no meaningful laws of physics apply", but it really says nothing about whether real numbers are indispensable in the formulation of physics, or about whether spacetime is continuous. There being a minimum length doesnt mean that everything is a discrete multiple of this length, or that space is…

I agree with all your claims here in the literal sense, but suppose there's a minimum length, then it would seem to be at least theoretically possible to use discrete mathematics to formulate an approximation to the "real number formulation of physics"?

The fact that we don't have already a full system using discrete maths doesn't mean it is impossible, because our current system is based on a long tradition of belief in real numbers, and assuming physical space is continuous.

I'd argue (admittedly unhelpfully) that unless we have actually tried to formulate physics using discrete mathematics and found a barrier that we prove unequivocally that it is impossible to overcome, we can't claim that physics must be formulated using real numbers/continuous math. There's a difference between "we don't know how to do this" vs "we know we can't do this".

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

#133

Post by wch4999 »

Mikhail_K wrote: post #87

OK, I do not understand prof.Dyson's argument at all. "This does not mean that an electric field-strength can be measured with the square-root of a calorimeter. It means that an electric field-strength is an abstract quantity, incommensurable with any quantities that we can measure directly." Electric field-strength is measurable no less directly than energy, it is a force experienced by a unit charge placed within t…

You need to read the paragraph prior to the quote. He is talking about which one of field and mechanical stress is more "fundamental" or less "direct". If one measure the force exerted by electric field using a unit charge, one measures the field by measuring the mechanical stress first.

Of course, the context matters. Often if one compares potential and field, field would be the one directly measured. It is just semantics really.

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

#134

Post by lupire »

mjburgess wrote: post #69
omnicognate wrote: post #66

Earlier quoted context omitted.

> Cognition is discrete There's little evidence even of this, except in the trivial sense that language (minus prosody) is composed of discrete units.

It is discrete insofar as we're talking about sequences of thoughts, ie., reasoning. What offends the minds of some people is the world might not be like their mind at all. They want always to analogise everything to Reason. Everything should be countable, everything should be knowable, etc.

And everything should be... "quantizable"?

https://www.energy.gov/science/doe-explainsquantum-mechanics

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

#135

Post by hnfong »

omnicognate wrote: post #58
yard2010 wrote: post #55

Earlier quoted context omitted.

But isn't the whole point of QM is that this assumption doesn't hold in some scale? I mean it's literally in the name. Care to explain? :)

No, position and time are typically continuous variables in quantum mechanics. You can have formulations in which they are discrete but they are not required and are relatively exotic. QM certainly doesn't say they must be discrete.

I feel that a lot of people here are confusing the math and "reality".

You're definitely correct about the math, i.e. the systems that we humans have invented to model reality. But I guess most of us don't really care about what mathematical model scientists like to use (especially not whether they're "exotic" or not), but rather what reality could be like.

And the quantum properties of QM do seem to suggest that there's some sort of fundamental discreteness in reality. And it seems to run contrary to the resolute claims that reality must be continuous as if it were a proven fact. What I understand is that the math most commonly used by scientists is definitely continuous, but whatever we can measure seems to have some kind of planck limitation.

So are we talking about empirical science or science-flavored theology here? Have we actually found empirical evidence or proven the continuousness of space/time?

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

#136

Post by wch4999 »

ThePhysicist wrote: post #124
toth wrote: post #104

Earlier quoted context omitted.

>You can make the electric field disappear by choosing the right gauge. Same goes for the magnetic field (can't make both disappear together though). What? No you can't. The fields are invariant under gauge transformations.

You're right, sorry I was thinking of a Lorentz transformation that would make either the magnetic or electric field disappear under certain conditions.

"transform into each other" would be more appropriate. The gauge choice you mentioned is not totally wrong. The gauge freedom can be used to set the electric field to zero, but only once at a single point.

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

#137

Post by lupire »

psychoslave wrote: post #73
lll-o-lll wrote: post #56

Earlier quoted context omitted.

> the world is continuous Is it though? Does it matter one way or the other? Do we think reality is the math in some way, or is the math a really darn good model of the reality?

Given that no one, or at least no human, can experiment what reality in its whole, and as far as we want to honestly recognize the effective scope of our knowledge, probably we will never know in absolute terms. What matter is a subjective topic. What we all have in common is logistics constraints. So if some people set as a goal something that requires to settle if reality is more easily handled when modeled in cont…

Continuity just hides the ball.

You say you can't comprehend how something can move from 1 to 2 discretely. But the paradoxical notion of infinite continuous change has been known since antiquity. It's faith either way.

Discrete doesn't mean state changes are wholly globally arbitrary. Imagine a graph with nodes and edges, a state machine as computer sciences call it. I think it's easy to agree that the universe could be parsed by a regex ;-) Heck, imagine an integer on the number line that can go up or down.

Worlfram has written a ton about this. Despite all his issues, his math is solid. (Which is not to say his physics is true.)

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

#138

Post by lupire »

omnicognate wrote: post #71
mjburgess wrote: post #69

Earlier quoted context omitted.

It is discrete insofar as we're talking about sequences of thoughts, ie., reasoning. What offends the minds of some people is the world might not be like their mind at all. They want always to analogise everything to Reason. Everything should be countable, everything should be knowable, etc.

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.

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

#139

Post by greysphere »

chr1 wrote: post #94
greysphere wrote: post #67

Earlier quoted context omitted.

Imagine there was a grid for space. For simplicity consider a regular grid of size 1unit in one direction and 1unit in a perpendicular direction. If such a grid existed, using one unit of ?something? would move you 1 unit along the axes of the grid, but you'd need 2 units of ?something? to move root2 units 45deg to the grid. Any discrete grid of any shape or size or pattern would have something like this, some sort o…

You don't have to imagine an ordered grid. If grid unit is small enough (say plank length 1,6 10^-35) and the grid is chaotic, for the distances of ~ 10^-16 that we can measure, everything will look the same in all directions. This happens the same way in which steel demonstrates isotropic behavior although its microscopic structure is anisotropic. So there is no easy way to prove or disprove continuity of space.

A chaotic grid would be macroscopically observable because random + random != 2 random, it's equal to 'bell curve'. Everything would be smeared as a function of distance, which we don't see.

This characteristic is observable for metals as well. Steel becomes less flexible as it's worked because it's grains become smaller and more chaotic - A microscopic property with a macroscopic effect.

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

#140

Post by lupire »

mjburgess wrote: post #84
discoinverno wrote: post #82

Earlier quoted context omitted.

I don't understand your point, I never said that "everything is a discrete multiple of this length, or that space is broken into units of it, or that objects have to be aligned on grid boundaries defined by it", I just wanted to mention that "continuity of spacetime is a convenient approximation" may be a correct sentence in the context of quantum gravity. Also, for what is worth, in QM the space of wavefunctions can…

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.

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