Am I missing something but the whole point of gauge theory (connections on a principal bundle) is that this is true, right? U(1) gauge theory gets you electromagnetism as a purely geometric result already?
Yes, but the "geometry" in question is not the geometry of spacetime, it's the geometry of spacetime plus an abstract space that's sort of "attached" to spacetime. (In the original Kaluza-Klein viewpoint, it was viewed as an extra 5th spacetime dimension, basically a circle at every point of spacetime.)
What this paper appears to be doing (although I can't make complete sense of it) is to somehow derive Maxwell's Equations (or more precisely a nonlinear generalization of them--which seems to me to mean that they aren't actually deriving electromagnetism, but let that go) as a property of the geometry of spacetime alone, without any abstract spaces or extra dimensions or anything of that sort.
Why would nonlinear generalizations be an issue? Wouldn't adding some constraints get us Maxwell's Equations? It seems significant that this can be done at all even if its not complete but maybe I'm missing something. It reminds me of Einstein's original geometrization, possibly even a breakthrough if it turns out to have uses in further development of theory.
You're right that he is just rederiving electromagnetism through local U(1) gauge symmetry. He define his metric as g_{\mu\nu}=A_\mu A\nu, which is a gauge dependent metric that gives you Maxwell's equation in the covariant formulation when you identify the gauge field A_\mu with the vector potential. Sprinkling geometric algebra in gives a feel of novelty but these results is at least one hundred years old.
>The Dirac equation can be therefore interpreted as a purely geometric equation, where the mc2 term directly relates to spacetime metric. There is no need to involve any hypothetical Higgs field to explain the particle mass term.
What happens to the Higgs field excitation and the Higgs boson, given the experiments confirming their existence? If this paper explains phenomena more effectively, does it require us to reinterpret these findings?
Any good theory probably needs to explain or reinterpret past phenomenon that has already been proven by experiments under the previous paradigm.
Part of what made Einstein's theories so good is that they reinterpreted past theories while providing new explanations that fixed major unknowns in the science of the time. Both for GR and QFT.
Purely geometric, except I suppose you still need Coulomb's law and relativity. Both of which can be easily put in a geometric framework.
The rest is just how magnetism emerges from this, and Einstein already figured it out. This guy explains it pretty well in layman's terms: https://www.youtube.com/watch?v=sDlZ-aY9GN4
Related question:
What resources are there that might teach one about Maxwell‘s equations and the electromagnetic field tensor arisig from relativity? The magnetic field is a description of the electric field with relativistic effects. Is there a way of describing electromagnetism without the magnetic field?
"As the electrodynamic force, i.e. the Lorentz force can be related directly to the metrical structure of spacetime, it directly leads to the explanation of the Zitterbewegung phenomenon and quantum mechanical waves as well."
Cool because traditional QM wave function waves are not electromagnetic waves even though they seem to be the same thing in a double slit experiment.
Forgive my ignorance but isn't this proven to be a dead end? There is this Kaluza Klein theory that proposes EM as the fifth dimension that has been ruled out, and Einstein spent large part of his later years trying to integrate EM into the GR geometric framework, with no success, mainly because he didn't know about strong and weak nuclear force as the other two fundamental force besides EM and gravity.
For people wondering what "geometric" means here, they say: "the electromagnetic field should be derived purely and solely from the properties of the metric tensor".
I'm not sure if that's exactly it.
Question: Is there any relationship between this and Axiomatic Thermodynamics? I recall that also uses differential geometry.
AFAICT the idea is that there are no "fields" or "forces" acting "in space", but the space itself bends just so that the normal mechanical motion through it looks the way the electromagnetic phenomena look.
But it can't be quite "the same deal", because gravity obeys the equivalence principle, and electromagnetism does not. (Nor do the other known fundamental interactions.) The paper does not appear to address this at all.
The classic GR line is "the stress-energy tensor tells spacetime (i.e. the metric tensor) how to bend and spacetime tells the stress-energy tensor how to move".
Okay, so this is another attempt to unify quantum field theory and gravity. By using gravity to get quantum fields, rather than by trying to quantize gravity.
If the paper is attempting to express electromagnetism in terms of the metric tensor, then it is putting it into a form that makes it potentially compatible with gravity, which is also a metric tensor. Quantum theories use a completely different type of math, and trying to express gravity in that way (quantizing gravity) results in a bunch of broken equations. If both systems can be described using differential geometry, that is a step in the direction of unifying the theories, even if it's not a hole-in-one.
The most irritating kind of junior devs to work with are the ones who refactor code into abstraction oblivion that nobody can decipher in the name of code deduplication or some other contrived metric.
That phenotype is well-represented in mathematical physics.
> Mathematicians and computer programmers use abstraction to opposite ends
I claim to be qualified in both disciplines. With this background, I disagree.
If you are very certain what you want to model, abstractions are often very useful to shed light on "what really happens in the system" (both in mathematics and computer science, but also in physics).
The problem with applying abstractions in computer programs (in this way) lies somewhere else: in business, users/customers are often very "volatile" what they want from the computer program, instead of pondering deeply about this question (even though this would be a very good idea). Thus (certain kinds of) abstractions in computer code make it much harder to adjust the program if new, very different requirements come up.
In math (i am not a mathematician), abstractions are a base to build on. You define some concept (e.g. a group, a set, whatever) then you prove things about it, building ever more complexity around your abstraction.
This works great because in math your abstractions don't change. You are never going to redefine what a group is. If you need something different,maybe you define some related concept, a ring, a semigroup, or whatever, but you never change your original abstraction. It is the base you build on.
As a result you can pack a lot of complexity. E.g. if something is a group, what are all the logical consequences of that? Probably so many you can't list them all, and that's ok. The whole point of math is to pick out some pattern and figure out what that entails.
In contrast in computer programming, the goal of abstraction is largely isolation. You want to be able to change something in the abstraction, and it not affect the system very much. The things the abstraction entails should be as limited as reasonably possible as everything it entails is a potential depedency that will get messed up if anything changes. Ideally you should be able to understand what the abstraction does by only looking at the abstraction's code and not the whole system.
Just think about the traditional SOLID principle in OOP design. Its all about ensuring abstractions are as isolated as possible from each other.
To summarize, i think in math abstractions are the base of the complexity pyramid. All the complexity is built on top of them. In computers its the opposite. They should be the tip of the complexity pyramid.
P.S. my controversial opinion is that this is the flaw in a lot of reasoning haskell fans use.
I disagree with the idea that computer science has an inverted use of abstractions. Unfortunate naming aside, computer science is basically mathematics applied to computation and data (still theoretical!) and software engineering (a good name, if only more people followed it) is applied computer science. Abstractions (models) must be the basis of the codebase. The JVM is an abstraction. Assembly is an abstraction. Threads are an abstraction. And so on. Of course, software engineering adds the complication of changing specifications and hence changing abstractions. Don't confuse poor abstractions for a reason to not have abstractions. Indeed, we have abstractions everywhere.
I have a feeling that our arguments are not that different (though not identical), but just phrased in very different words:
> This works great because in math your abstractions don't change.
This is just a different formulation about the "volatility" of a lot of requirements of software by the users/customers.
> In contrast in computer programming, the goal of abstraction is largely isolation. You want to be able to change something in the abstraction, and it not affect the system very much.
Here my opinion differs: isolation is at best just one aspect of abstraction (and I would even claim that these concepts are even only tangentially related). I claim that the better isolation is rather a (very useful) side effect of some abstractions that are very commonly used in software development. But on the other hand, I don't think that it is really hard to come up with abstractions for software development that would be very useful, but don't lead to better isolation.
The central purpose of abstraction in computer programs is to make is easier to reason about the the code, and being able to avoid having to write "related" code multiple times. Similar to mathematics: you want to prove a general theorem (e.g. about groups) instead of having to prove one theorem about S_n, one theorem about Z_n etc.
You actually partly write about the aspect of reasoning about the code by yourself:
> Ideally you should be able to understand what the abstraction does by only looking at the abstraction's code and not the whole system.
In this sense using more abstractions is a particular optimization for the goals:
- you want to make it easier to reason about the code abstractly
- you want to avoid having to duplicate code (i.e. save money since less lines have to be written)
But this is not a panacea:
- If the abstraction turns out to be bad, you either have to re-engineer a lot, or you will have a maintenance nightmare (my "volatility of customer requirements" argument). Indeed, I claim that the question of "do we really use the best possible abstractions in our code for the problem that we want to solve" is nearly always neglected in software projects, because the answer is nearly always very inconvenient, necessitating lots of re-engineering of the code.
- low-level optimizations become harder, so making the code really fast gets much more complicated
- since abstractions are more "abstract", (depending on the abstraction) you might need "smarter" programmers (who can be more expensive). For an example consider some complicated metaprogramming libraries of Boost (C++): in the hands of really good programmers such abstractions can become "magic", but worse programmers will likely be overwhelmed by them.
- fighting about the "right" abstraction can become very political (for low-level code there is often less of such a fight, because here "what is more performant is typically right").
---
Concerning
> To summarize, i think in math abstractions are the base of the complexity pyramid. All the complexity is built on top of them.
This is actually not a bad idea to organize code (under my stated specific edge conditions! When these specific edge conditions are violated, my judgment might change). :-)
---
> P.S. my controversial opinion is that this is the flaw in a lot of reasoning haskell fans use.
I am not a particular fan of Haskell, but I think the Haskell fans' flaw lies in a very different point: they emphasize very particular aspects of computer programming, and, admittedly, often come up with clever solutions for these.
The problem is: in my opinion there exist aspects of software development that are in my opinion far more important, but don't fit into the kind of structures that Haskell fans appreciate. The difficulty is thus in my experience convincing Haskell fans that such aspects actually matter a lot instead of being unimportant side aspects of software development.
I think sometimes you have to build the abstraction hell to completion and live with it for a while to truly realize it is in fact inferior. And even then, in science sometimes it never dies fully but lives on in some niche where it has desirable qualities.
Reminds me of Feynman Checkerboard:
https://en.wikipedia.org/wiki/Feynman_checkerboard
and the work of David Hestenes:
Zitterbewegung in Quantum Mechanics
https://davidhestenes.net/geocalc/pdf/ZBWinQM15**.pdf
Zitterbewegung structure in electrons and photons
https://arxiv.org/abs/1910.11085
Zitterbewegung Modeling
https://davidhestenes.net/geocalc/pdf/ZBW_mod.pdf
Am I missing something but the whole point of gauge theory (connections on a principal bundle) is that this is true, right? U(1) gauge theory gets you electromagnetism as a purely geometric result already?
Yes, but the "geometry" in question is not the geometry of spacetime, it's the geometry of spacetime plus an abstract space that's sort of "attached" to spacetime. (In the original Kaluza-Klein viewpoint, it was viewed as an extra 5th spacetime dimension, basically a circle at every point of spacetime.)
What this paper appears to be doing (although I can't make complete sense of it) is to somehow derive Maxwell's Equations (or more precisely a nonlinear generalization of them--which seems to me to mean that they aren't actually deriving electromagnetism, but let that go) as a property of the geometry of spacetime alone, without any abstract spaces or extra dimensions or anything of that sort.
Why would nonlinear generalizations be an issue? Wouldn't adding some constraints get us Maxwell's Equations? It seems significant that this can be done at all even if its not complete but maybe I'm missing something. It reminds me of Einstein's original geometrization, possibly even a breakthrough if it turns out to have uses in further development of theory.
You're right that he is just rederiving electromagnetism through local U(1) gauge symmetry. He define his metric as g_{\mu\nu}=A_\mu A\nu, which is a gauge dependent metric that gives you Maxwell's equation in the covariant formulation when you identify the gauge field A_\mu with the vector potential. Sprinkling geometric algebra in gives a feel of novelty but these results is at least one hundred years old.
*typo
>The Dirac equation can be therefore interpreted as a purely geometric equation, where the mc2 term directly relates to spacetime metric. There is no need to involve any hypothetical Higgs field to explain the particle mass term.
What happens to the Higgs field excitation and the Higgs boson, given the experiments confirming their existence? If this paper explains phenomena more effectively, does it require us to reinterpret these findings?
Any good theory probably needs to explain or reinterpret past phenomenon that has already been proven by experiments under the previous paradigm.
Part of what made Einstein's theories so good is that they reinterpreted past theories while providing new explanations that fixed major unknowns in the science of the time. Both for GR and QFT.
Purely geometric, except I suppose you still need Coulomb's law and relativity. Both of which can be easily put in a geometric framework.
The rest is just how magnetism emerges from this, and Einstein already figured it out. This guy explains it pretty well in layman's terms: https://www.youtube.com/watch?v=sDlZ-aY9GN4
That's a great explanation. Thanks!
Related question: What resources are there that might teach one about Maxwell‘s equations and the electromagnetic field tensor arisig from relativity? The magnetic field is a description of the electric field with relativistic effects. Is there a way of describing electromagnetism without the magnetic field?
I'm pretty sure this is what you want:
"Collective Electrodynamics: Quantum Foundations of Electromagnetism" https://www.amazon.com/Collective-Electrodynamics-Quantum-Fo...
Thanks!
Atom by Asimov?
"As the electrodynamic force, i.e. the Lorentz force can be related directly to the metrical structure of spacetime, it directly leads to the explanation of the Zitterbewegung phenomenon and quantum mechanical waves as well."
Cool because traditional QM wave function waves are not electromagnetic waves even though they seem to be the same thing in a double slit experiment.
What makes them different when they perform the same way in a double slit? They act differently at different scales or something else?
I think they're referring to quantum wavesfunctions being in configuration space rather than real spacetime.
Forgive my ignorance but isn't this proven to be a dead end? There is this Kaluza Klein theory that proposes EM as the fifth dimension that has been ruled out, and Einstein spent large part of his later years trying to integrate EM into the GR geometric framework, with no success, mainly because he didn't know about strong and weak nuclear force as the other two fundamental force besides EM and gravity.
Coming up with some "good enough" theoretical approximations could be extremely useful though.
For people wondering what "geometric" means here, they say: "the electromagnetic field should be derived purely and solely from the properties of the metric tensor".
I'm not sure if that's exactly it.
Question: Is there any relationship between this and Axiomatic Thermodynamics? I recall that also uses differential geometry.
AFAICT the idea is that there are no "fields" or "forces" acting "in space", but the space itself bends just so that the normal mechanical motion through it looks the way the electromagnetic phenomena look.
That is, the same deal as with gravity in GR.
> the same deal as with gravity in GR.
But it can't be quite "the same deal", because gravity obeys the equivalence principle, and electromagnetism does not. (Nor do the other known fundamental interactions.) The paper does not appear to address this at all.
What bends the space?
The stress-energy tensor.
What is affecting the stress-energy tensor?
The classic GR line is "the stress-energy tensor tells spacetime (i.e. the metric tensor) how to bend and spacetime tells the stress-energy tensor how to move".
Okay, so this is another attempt to unify quantum field theory and gravity. By using gravity to get quantum fields, rather than by trying to quantize gravity.
I don't think so. The paper doesn't talk about gravity at all. It talks about electromagnetism.
If the paper is attempting to express electromagnetism in terms of the metric tensor, then it is putting it into a form that makes it potentially compatible with gravity, which is also a metric tensor. Quantum theories use a completely different type of math, and trying to express gravity in that way (quantizing gravity) results in a bunch of broken equations. If both systems can be described using differential geometry, that is a step in the direction of unifying the theories, even if it's not a hole-in-one.
The most irritating kind of junior devs to work with are the ones who refactor code into abstraction oblivion that nobody can decipher in the name of code deduplication or some other contrived metric.
That phenotype is well-represented in mathematical physics.
Mathematicians and computer programmers use abstraction to opposite ends
> Mathematicians and computer programmers use abstraction to opposite ends
I claim to be qualified in both disciplines. With this background, I disagree.
If you are very certain what you want to model, abstractions are often very useful to shed light on "what really happens in the system" (both in mathematics and computer science, but also in physics).
The problem with applying abstractions in computer programs (in this way) lies somewhere else: in business, users/customers are often very "volatile" what they want from the computer program, instead of pondering deeply about this question (even though this would be a very good idea). Thus (certain kinds of) abstractions in computer code make it much harder to adjust the program if new, very different requirements come up.
Perhaps i should say why i think this.
In math (i am not a mathematician), abstractions are a base to build on. You define some concept (e.g. a group, a set, whatever) then you prove things about it, building ever more complexity around your abstraction.
This works great because in math your abstractions don't change. You are never going to redefine what a group is. If you need something different,maybe you define some related concept, a ring, a semigroup, or whatever, but you never change your original abstraction. It is the base you build on.
As a result you can pack a lot of complexity. E.g. if something is a group, what are all the logical consequences of that? Probably so many you can't list them all, and that's ok. The whole point of math is to pick out some pattern and figure out what that entails.
In contrast in computer programming, the goal of abstraction is largely isolation. You want to be able to change something in the abstraction, and it not affect the system very much. The things the abstraction entails should be as limited as reasonably possible as everything it entails is a potential depedency that will get messed up if anything changes. Ideally you should be able to understand what the abstraction does by only looking at the abstraction's code and not the whole system.
Just think about the traditional SOLID principle in OOP design. Its all about ensuring abstractions are as isolated as possible from each other.
To summarize, i think in math abstractions are the base of the complexity pyramid. All the complexity is built on top of them. In computers its the opposite. They should be the tip of the complexity pyramid.
P.S. my controversial opinion is that this is the flaw in a lot of reasoning haskell fans use.
I disagree with the idea that computer science has an inverted use of abstractions. Unfortunate naming aside, computer science is basically mathematics applied to computation and data (still theoretical!) and software engineering (a good name, if only more people followed it) is applied computer science. Abstractions (models) must be the basis of the codebase. The JVM is an abstraction. Assembly is an abstraction. Threads are an abstraction. And so on. Of course, software engineering adds the complication of changing specifications and hence changing abstractions. Don't confuse poor abstractions for a reason to not have abstractions. Indeed, we have abstractions everywhere.
I have a feeling that our arguments are not that different (though not identical), but just phrased in very different words:
> This works great because in math your abstractions don't change.
This is just a different formulation about the "volatility" of a lot of requirements of software by the users/customers.
> In contrast in computer programming, the goal of abstraction is largely isolation. You want to be able to change something in the abstraction, and it not affect the system very much.
Here my opinion differs: isolation is at best just one aspect of abstraction (and I would even claim that these concepts are even only tangentially related). I claim that the better isolation is rather a (very useful) side effect of some abstractions that are very commonly used in software development. But on the other hand, I don't think that it is really hard to come up with abstractions for software development that would be very useful, but don't lead to better isolation.
The central purpose of abstraction in computer programs is to make is easier to reason about the the code, and being able to avoid having to write "related" code multiple times. Similar to mathematics: you want to prove a general theorem (e.g. about groups) instead of having to prove one theorem about S_n, one theorem about Z_n etc.
You actually partly write about the aspect of reasoning about the code by yourself:
> Ideally you should be able to understand what the abstraction does by only looking at the abstraction's code and not the whole system.
In this sense using more abstractions is a particular optimization for the goals:
- you want to make it easier to reason about the code abstractly
- you want to avoid having to duplicate code (i.e. save money since less lines have to be written)
But this is not a panacea:
- If the abstraction turns out to be bad, you either have to re-engineer a lot, or you will have a maintenance nightmare (my "volatility of customer requirements" argument). Indeed, I claim that the question of "do we really use the best possible abstractions in our code for the problem that we want to solve" is nearly always neglected in software projects, because the answer is nearly always very inconvenient, necessitating lots of re-engineering of the code.
- low-level optimizations become harder, so making the code really fast gets much more complicated
- since abstractions are more "abstract", (depending on the abstraction) you might need "smarter" programmers (who can be more expensive). For an example consider some complicated metaprogramming libraries of Boost (C++): in the hands of really good programmers such abstractions can become "magic", but worse programmers will likely be overwhelmed by them.
- fighting about the "right" abstraction can become very political (for low-level code there is often less of such a fight, because here "what is more performant is typically right").
---
Concerning
> To summarize, i think in math abstractions are the base of the complexity pyramid. All the complexity is built on top of them.
This is actually not a bad idea to organize code (under my stated specific edge conditions! When these specific edge conditions are violated, my judgment might change). :-)
---
> P.S. my controversial opinion is that this is the flaw in a lot of reasoning haskell fans use.
I am not a particular fan of Haskell, but I think the Haskell fans' flaw lies in a very different point: they emphasize very particular aspects of computer programming, and, admittedly, often come up with clever solutions for these.
The problem is: in my opinion there exist aspects of software development that are in my opinion far more important, but don't fit into the kind of structures that Haskell fans appreciate. The difficulty is thus in my experience convincing Haskell fans that such aspects actually matter a lot instead of being unimportant side aspects of software development.
I think sometimes you have to build the abstraction hell to completion and live with it for a while to truly realize it is in fact inferior. And even then, in science sometimes it never dies fully but lives on in some niche where it has desirable qualities.
It's not my fault the universe is built on a hell of abstractions, I just model it.
You ignore the reality of nature at your own peril.
Besides, you can just use computers automate the wrangling of this hell. It's what they are good at, after all.
"charge density is a field, which propagates at the speed of light."
Uh ...
Couldn't get past the robot wall.
I recommend https://www.youtube.com/watch?v=Sj_GSBaUE1o instead
https://www.researchgate.net/publication/390640631_Electroma...
And then I got a bot check on researchgate, first time and I download a lot of papers from them.
> Couldn't get past the robot wall.
Nothing like that for me. I just clicked the big "article pdf" button at the bottom of the page.
Direct link to full pdf:
https://iopscience.iop.org/article/10.1088/1742-6596/2987/1/...
Links to the article and the PDF both are behind this human test. Guess today is the day I learned I'm a robot.
At this point, should failing the test be an indicator of being human, rather than success?