For those enjoying the history of early quantum mechanics, I've been following Dr. Jorge S. Diaz on YouTube[1].
He has a great video series going on the people, experiments and discoveries that lead to quantum mechanics.
The videos are very accessible, but he does go into some details like various key derivations and such. Well worth watching for casual physics fans IMHO.
Diaz is awesome. For me, he manages to strike just the right balance between making the topic interesting in a historical/human-interest sense and including enough technical detail to cultivate actual insight. Both his Franck-Hertz and Stern-Gerlach videos are the best I've run across on YouTube (and I see the latter has a part 2 now, so that'll be next on my watchlist.)
I also just love his energy. He's unapologetically excited about the stuff that excites him, and even if I don't share all his interest, it makes his videos an absolute joy to watch.
>"Months later, Schrödinger — inspired by Louis de Broglie’s idea that matter behaves like a wave — proposed an entirely different, but mathematically equivalent, formulation of particle behavior based on the better-known mathematics of waves."
[...]
More surprising results unfurled when Dirac extended his equation to describe an electron interacting with an electromagnetic field. Experimentalists had confirmed that the electron’s intrinsic angular momentum, or spin, was equal to 1/2, but theoreticians couldn’t figure out how to properly incorporate it into their theories. With his new equation, Dirac had found, almost as an afterthought, that the spin emerged naturally.
[...]
The Dirac equation was simple and elegant, yet dense with implications. Perhaps its most profound feature was that,
instead of producing two components for negative and positive spin states, it produced four: a negative and positive spin state for each of two particles with positive and negative energy states."
Dirac was not working in vaccum . Klein-Jordan equation was the simplest and the most obvious extension of Schrodinger equation in relativistic manner.
So historically, Dirac was focused on correcting the Klein-Gordon equation, which had issues with negative probabilities and describing electron behavior. His goal was to find a relativistic equation that resolved these problems while maintaining consistency with his own matrix mechanics formulation of quantum mechanics.
By extending his matrix mechanics formalism, Dirac derived an equation that not only addressed the issues with the Klein-Gordon equation but also predicted the existence of antimatter. I would argue that Dirac's approach was consistent with his established framework, and while he found renormalization mathematically unsatisfactory, it does not diminish the validity of his method in deriving the Dirac equation. I doubt he focused on any elegant solutions, he was actually quite happy working with matrix mechanics framework.
Bohr was a big shot, Nobel prized establishment authority. In Weimberg QFT book he recalls a fragment of Dirac's memoirs:
"I remember once when I was in Copenhagen, that Bohr asked me what I was working on and I told him I was trying to get a satisfactory relativistic theory of the electron, and Bohr said 'But Klein and Gordon have already done that!' That answer first rather disturbed me. Bohr seemed quite satisfied by Klein's solution, but I was not because of the negative probabilities that it led to. I just kept on with it, worrying about getting a theory which would have only positive probabilities."
Thank you for posing the quaternion formulation. It inspired me to search for a geometric algebra version of the same equation and was happy to find that it also exists[1].
I very like this playlist https://www.youtube.com/watch?v=xLfFWXUNJ_I It's good introduction in quantum mechanics with minimum posible math. It's on Russian but as I see It has English subtitles.
I know nothing of physics, but it seems to me that rotation fingerprints are everywhere in physics. Is this just me or is there something more tangible in this remark?
Rotations and spin are deeply tied into the geometrical nature of a space. It's not just you. It's core to understanding the nature of matter itself.
Cartan had only just invented spinors as an object in themselves (ignoring clifford) so a lot of the physics stuff was done in parallel or even without the knowledge the mathematicians had.
It's not just you. Dirac fields are constantly rotating. In fact the solutions are called spinors. (e.g. things that spin). There are a lot of rotations at the quantum level. It's also why complex numbers show up a lot in q.m.
I've been trying to get an intuitive understanding of why multiplying by e^ix leads to a rotation in the complex plane, without going into Taylor series (too algebraic, not enough geometric). I tried to find a way to calculate the value of e in a rotational setting, maybe there is a way to reinterpret compound interests as compound rotation. Any insight ?
Of course. The solutions of the Dirac equations live in space and space has rotation symmetry. These solution have to transform in some way under rotations.
There's a lot of rotations and stuff in QM, yes. That you should do it with quaternions is mostly an internet thing. I have a PhD in physics, and I never encountered quaternions in any course I took, and from seeing curricula in other universities I haven't seen it there either.
If our society were sane, rational, advanced, the headlines would be all about scientific and technological progress. The fusion power breakthrough of 2022 by Lawrence Livermore National Lab would still dominate the news. Large corporations would compete to create the first Star Trek replicator (at least for organic matter, food, etc) by advancements in nanofabrication. Politicians would debate R&D topics and strategy, figuring out which path leads to greater broad-sector economic progress.
One can dream. :) Instead, we have a society almost entirely dependent on many kinds of technology, and yet very few understand any of it, nor care to. Wonder how long this trend can persist until some sort of phase transition appears on the horizon.
We aren't spherical philosophers in a vacuum. We are emotional animals trying our best. This fact requires constant consideration and management lest it all come crumbling down.
The LLNL fusion result was not a breakthrough. The fusion output was about 1% of the energy input. The exaggerated press release was just a PR ploy to get support for continued DoE funding, which was expiring at the end of 2022.
And while we talk about fusion, even when the energy output surpass the energy input and the reaction is stable enough, how to you harness the energy? Because the reaction happen within a vacuum, the only way is to capture expelled neutrons and make electricity from it somehow.
> The fusion power breakthrough of 2022 by Lawrence Livermore National Lab would still dominate the news.
If our society were sane, rational, advanced, it would recognize that that "breakthrough" was a minor, arbitrary improvement in reaction efficiency, that realistically brings us no closer to commercially viable fusion power, and doesn't prove anything about the possibility of that.
That reaction still consumed something like 100 times the power that it produced, and the "power" that it produced was just heat energy, which would still entail losses when converted into usable form.
On top of that, the nature of the Livermore reaction is not one that's even intended or suitable for commercial power production.
At this point, we simply don't even know whether controlled, commercially viable fusion will ever be able to produce more power than it consumes. There's no guarantee that it will.
While this might all seem like an irrelevant aside to the point being made above, it's relevant because it shows how pervasive misinformation is, even when coming from supposedly scientific sources.
The announcement was correct and precise. I am not sure what misinformation you are describing here.
Regarding your 100 more energy claim. It overlooks key facts about the NIF breakthrough. The fusion reaction itself achieved net energy gain, producing 3.15 MJ compared to 2.05 MJ of input laser energy - far from consuming "100 times the power it produced." While the total facility power usage was indeed higher due to laser inefficiencies, this misses the crucial scientific achievement. This was basically humanity's first controlled fusion reaction producing more energy than was directly input to the fuel. Dismissing this as a "minor, arbitrary improvement" understates its significance. This wasn't just about efficiency metrics - it demonstrated fusion ignition was possible, a fundamental physics milestone that had eluded scientists for decades. Though challenges remain for commercial fusion power, the breakthrough proved a critical theoretical concept that many thought impossible. Many critics before that were referring to this point as the reason why it isn't worth it to keep researching. And they were proved wrong.
Trying to redefine the announcement and experiment result to mean something else so that you can attack is a dishonest behavior.
NIF is first and foremost a thermonuclear weapons research facility. The "breakthrough" you're talking about doesn't bring us an inch closer to fusion power.
I think that a 'breakthrough' would be to realize that harnessing fusion outside of a solar environment is a hopeless dream that, carefully fed, is very good at provoking research grants.
Some many years ago some people collected some negative traits to describe the foibles of people. Unfortunately, these negatives seem to dominate much of the news:
Pride, Greed, Lust, Anger, Gluttony, Envy, Sloth.
If we could somehow dim the influence of these human traits, we might get closer to the world you described
For those enjoying the history of early quantum mechanics, I've been following Dr. Jorge S. Diaz on YouTube[1].
He has a great video series going on the people, experiments and discoveries that lead to quantum mechanics.
The videos are very accessible, but he does go into some details like various key derivations and such. Well worth watching for casual physics fans IMHO.
[1]: https://www.youtube.com/@jkzero/
Diaz is awesome. For me, he manages to strike just the right balance between making the topic interesting in a historical/human-interest sense and including enough technical detail to cultivate actual insight. Both his Franck-Hertz and Stern-Gerlach videos are the best I've run across on YouTube (and I see the latter has a part 2 now, so that'll be next on my watchlist.)
Hugely underrated YouTuber.
I also just love his energy. He's unapologetically excited about the stuff that excites him, and even if I don't share all his interest, it makes his videos an absolute joy to watch.
Thank you. Just in time for the weekend!
>"Months later, Schrödinger — inspired by Louis de Broglie’s idea that matter behaves like a wave — proposed an entirely different, but mathematically equivalent, formulation of particle behavior based on the better-known mathematics of waves."
[...]
More surprising results unfurled when Dirac extended his equation to describe an electron interacting with an electromagnetic field. Experimentalists had confirmed that the electron’s intrinsic angular momentum, or spin, was equal to 1/2, but theoreticians couldn’t figure out how to properly incorporate it into their theories. With his new equation, Dirac had found, almost as an afterthought, that the spin emerged naturally.
[...]
The Dirac equation was simple and elegant, yet dense with implications. Perhaps its most profound feature was that,
instead of producing two components for negative and positive spin states, it produced four: a negative and positive spin state for each of two particles with positive and negative energy states."
Related:
https://en.wikipedia.org/wiki/Chirality
https://en.wikipedia.org/wiki/Angular_momentum
https://en.wikipedia.org/wiki/Noether%27s_theorem
> Yet the technique employed to make the theory useful — renormalization — repulsed Dirac because he found it mathematically ugly.
Perhaps if he had used quaternion the solution will not be mathematically ugly or can even be beautiful [1].
[1] A quaternion formulation of the Dirac equation:
https://mauritssilvis.nl/research/publications/silvis-rug10....
Dirac was not working in vaccum . Klein-Jordan equation was the simplest and the most obvious extension of Schrodinger equation in relativistic manner.
So historically, Dirac was focused on correcting the Klein-Gordon equation, which had issues with negative probabilities and describing electron behavior. His goal was to find a relativistic equation that resolved these problems while maintaining consistency with his own matrix mechanics formulation of quantum mechanics.
By extending his matrix mechanics formalism, Dirac derived an equation that not only addressed the issues with the Klein-Gordon equation but also predicted the existence of antimatter. I would argue that Dirac's approach was consistent with his established framework, and while he found renormalization mathematically unsatisfactory, it does not diminish the validity of his method in deriving the Dirac equation. I doubt he focused on any elegant solutions, he was actually quite happy working with matrix mechanics framework.
Bohr was a big shot, Nobel prized establishment authority. In Weimberg QFT book he recalls a fragment of Dirac's memoirs:
"I remember once when I was in Copenhagen, that Bohr asked me what I was working on and I told him I was trying to get a satisfactory relativistic theory of the electron, and Bohr said 'But Klein and Gordon have already done that!' That answer first rather disturbed me. Bohr seemed quite satisfied by Klein's solution, but I was not because of the negative probabilities that it led to. I just kept on with it, worrying about getting a theory which would have only positive probabilities."
5 replies →
That reformulation doesn't let you avoid renormalization, does it?
No, it doesn't.
Thank you for posing the quaternion formulation. It inspired me to search for a geometric algebra version of the same equation and was happy to find that it also exists[1].
1. https://fondationlouisdebroglie.org/AFLB-342/aflb342m679.pdf
I very like this playlist https://www.youtube.com/watch?v=xLfFWXUNJ_I It's good introduction in quantum mechanics with minimum posible math. It's on Russian but as I see It has English subtitles.
> Quaternions
I know nothing of physics, but it seems to me that rotation fingerprints are everywhere in physics. Is this just me or is there something more tangible in this remark?
Rotations and spin are deeply tied into the geometrical nature of a space. It's not just you. It's core to understanding the nature of matter itself.
Cartan had only just invented spinors as an object in themselves (ignoring clifford) so a lot of the physics stuff was done in parallel or even without the knowledge the mathematicians had.
It's not just you. Dirac fields are constantly rotating. In fact the solutions are called spinors. (e.g. things that spin). There are a lot of rotations at the quantum level. It's also why complex numbers show up a lot in q.m.
I've been trying to get an intuitive understanding of why multiplying by e^ix leads to a rotation in the complex plane, without going into Taylor series (too algebraic, not enough geometric). I tried to find a way to calculate the value of e in a rotational setting, maybe there is a way to reinterpret compound interests as compound rotation. Any insight ?
8 replies →
Of course. The solutions of the Dirac equations live in space and space has rotation symmetry. These solution have to transform in some way under rotations.
There's a lot of rotations and stuff in QM, yes. That you should do it with quaternions is mostly an internet thing. I have a PhD in physics, and I never encountered quaternions in any course I took, and from seeing curricula in other universities I haven't seen it there either.
Looking for possible citations for
>... renormalization — repulsed Dirac because he found it mathematically ugly.
I found
https://physics.stackexchange.com/questions/375209/dirac-onc...
(2019) for the "Cosmic Galois group" (edited) as well as Anixx's comments/Neumaier's responses
If our society were sane, rational, advanced, the headlines would be all about scientific and technological progress. The fusion power breakthrough of 2022 by Lawrence Livermore National Lab would still dominate the news. Large corporations would compete to create the first Star Trek replicator (at least for organic matter, food, etc) by advancements in nanofabrication. Politicians would debate R&D topics and strategy, figuring out which path leads to greater broad-sector economic progress.
One can dream. :) Instead, we have a society almost entirely dependent on many kinds of technology, and yet very few understand any of it, nor care to. Wonder how long this trend can persist until some sort of phase transition appears on the horizon.
We aren't spherical philosophers in a vacuum. We are emotional animals trying our best. This fact requires constant consideration and management lest it all come crumbling down.
Alright, so we're spherical cow-philosophers... (jk, I like your point!)
> trying our best
I strongly question this part. Most people just want comfort. More is never enough for them.
Spherical bearded philosophers. You forgot the bearded part.
Don’t worry, we can create a priesthood caste with secret knowledge of technology whose purpose is to guide the human race toward a brighter tomorrow!
Maybe someone could write a foundational science fiction novel about this.
Influx by Daniel Suarez comes to mind (though it has a slightly different spin).
Anathem by Neal Stephenson?
3 replies →
The LLNL fusion result was not a breakthrough. The fusion output was about 1% of the energy input. The exaggerated press release was just a PR ploy to get support for continued DoE funding, which was expiring at the end of 2022.
And while we talk about fusion, even when the energy output surpass the energy input and the reaction is stable enough, how to you harness the energy? Because the reaction happen within a vacuum, the only way is to capture expelled neutrons and make electricity from it somehow.
1 reply →
> The fusion power breakthrough of 2022 by Lawrence Livermore National Lab would still dominate the news.
If our society were sane, rational, advanced, it would recognize that that "breakthrough" was a minor, arbitrary improvement in reaction efficiency, that realistically brings us no closer to commercially viable fusion power, and doesn't prove anything about the possibility of that.
That reaction still consumed something like 100 times the power that it produced, and the "power" that it produced was just heat energy, which would still entail losses when converted into usable form.
On top of that, the nature of the Livermore reaction is not one that's even intended or suitable for commercial power production.
At this point, we simply don't even know whether controlled, commercially viable fusion will ever be able to produce more power than it consumes. There's no guarantee that it will.
If you're not aware of what I'm referring to, this article is a starting point: "Why the nuclear fusion ‘net energy gain’ is more hype than breakthrough": https://whyy.org/segments/why-the-nuclear-fusion-net-energy-...
While this might all seem like an irrelevant aside to the point being made above, it's relevant because it shows how pervasive misinformation is, even when coming from supposedly scientific sources.
The announcement was correct and precise. I am not sure what misinformation you are describing here.
Regarding your 100 more energy claim. It overlooks key facts about the NIF breakthrough. The fusion reaction itself achieved net energy gain, producing 3.15 MJ compared to 2.05 MJ of input laser energy - far from consuming "100 times the power it produced." While the total facility power usage was indeed higher due to laser inefficiencies, this misses the crucial scientific achievement. This was basically humanity's first controlled fusion reaction producing more energy than was directly input to the fuel. Dismissing this as a "minor, arbitrary improvement" understates its significance. This wasn't just about efficiency metrics - it demonstrated fusion ignition was possible, a fundamental physics milestone that had eluded scientists for decades. Though challenges remain for commercial fusion power, the breakthrough proved a critical theoretical concept that many thought impossible. Many critics before that were referring to this point as the reason why it isn't worth it to keep researching. And they were proved wrong.
Trying to redefine the announcement and experiment result to mean something else so that you can attack is a dishonest behavior.
5 replies →
NIF is first and foremost a thermonuclear weapons research facility. The "breakthrough" you're talking about doesn't bring us an inch closer to fusion power.
I think that a 'breakthrough' would be to realize that harnessing fusion outside of a solar environment is a hopeless dream that, carefully fed, is very good at provoking research grants.
Some many years ago some people collected some negative traits to describe the foibles of people. Unfortunately, these negatives seem to dominate much of the news:
Pride, Greed, Lust, Anger, Gluttony, Envy, Sloth.
If we could somehow dim the influence of these human traits, we might get closer to the world you described
> Pride, Greed, Lust, Anger, Gluttony, Envy, Sloth
The greatest popular innovation of our time appears to be to have extended the above list with Falsehood, Cruelty, and Pollution.
2 replies →
No they would not. Not all and probably most progresses are not technological. Are you living under a rock?
what are you talking about? the most important thing is to make sure senators use the correct bathroom!