Is there a relationship between the negative probabilities of Klein and the negative energy of Dirac? Did his formulation just move the problem? If so, does it imply anything? Like are probability and energy related?

Euler's formula is a specific case of the exponential map from Lie theory. This means e^x can be used with all sorts of interesting x types, and it often has surprisingly intuitive behavior! When x is a real number you get continuous growth. When x is a purely imaginary number you get continuous rotation. When x is complex you get continuous growth and rotation. When x is a matrix you get a continuous linear transformation (growth, rotation, and shear). What's the similarity here? Euler's formula treats it's argument as a transformation which gets continuously applied in infinitesimal amounts. This also explains the formula for calculating the value of e:

    e = lim (1 + 1 / n) ^ n where (x -> infinity)

https://en.wikipedia.org/wiki/Matrix_exponential

https://www.youtube.com/watch?v=O85OWBJ2ayo

Complex numbers and (Pauli/Dirac) matrices not required if you use Geometric Algebra. I highly recommend the book by Doran and Lasenby [0], or you can get the details from their papers, notably [1].

[0] Geometric Algebra for Physicists, CUP, 2003

[1] https://arxiv.org/abs/quant-ph/0509178

My favorite intuitive explanation was actually written by science fiction author, Greg Egan. It takes the exact approach you're asking for, reinterpreting compound interest in a 2d rotational context on the complex plane, and doesn't use more than high school math:

https://gregegan.net/FOUNDATIONS/04/found04.html#s2

Fig. 7 is the money shot.

IANAM but I'd go with "it's implicit in how complex numbers are defined". Complex numbers are a thing made up by humans (as are negative numbers), and we got to define i as "up the y-axis". Once you do that, and note that a rotation is therefore cos angle plus i sin angle, add in that e^something is an eigenfuncion of differentiation, and you're pretty much there.

Fwiw I think it's Maclaurin series for this.

Edit: obviously should be j not i.

The first thing to understand is that multiplying a complex number by i rotates the complex number by 90 degrees counter-clockwise around the origin. For example, 1 * i = i (e.g. 1 + 0i is mapped to 0 + 1i). And i*i = -1 (e.g. 0 + 1i is mapped to (-1 + 0i) and so on. e^ix is a continuous generalisation of this discrete rotation, as I understand it.

One possibility: take the unit circle, and a vertical line tangent to the circle at (1,0). Then e^ix takes that line and wraps it around the circle. This

Foundation Series by Isaac Asimov

I’ll have to check that one out

Yes, they forgot that part, so there will be further energy losses (at least 70%).

They also have to up the repetition rate from once a week to at least 1k /s.

Note that ASML lithography machines accurately dilate and irradiate molten tin droplets with lasers at ~ 50k /s.

https://www.asml.com/en/technology/lithography-principles/li...

Nobody ever doubted that fusion ignition was physically possible. It happens in stars all the time, and people have achieved it in thermonuclear weapons.

This was the first time fusion ignition was achieved in a laboratory setting, i.e. in a controlled fashion. Is that seen as a fundamental physics milestone? To me it seems more an incremental engineering achievement.

> The announcement was correct and precise.

"The" announcement? There were several announcements, with varying degrees of scientific rigor.

Here's one typical example: https://www.llnl.gov/article/49301/shot-ages-fusion-ignition...

Quote: "...achieved fusion ignition — creating more energy from fusion reactions than the energy used to start the process."

That is not "correct and precise." In fact, without any mention of the additional context that at least 300 MJ of power was used to produce 3.15 MJ of not directly usable heat energy, it's incorrect, imprecise, and misleading at best.

It's also misleading because it doesn't tell you that NIF's definition of "ignition" is significantly different, in essential respects, from the term's use in other fusion contexts. For example, ignition at NIF doesn't mean that a self-sustaining reaction has been achieved. As such, the use of this term at all is dubious. It has no fundamental meaning here, it's just a name being used for an arbitrarily defined efficiency target.

Realistically, the term is being used to try to connect what NIF is doing, in a facility ostensibly intended for nuclear weapons research, to what fusion power research efforts are doing. It's a hype-driven word game, it's not meaningful.

Back to the quote above: it's carefully worded to sound as though it's saying something that not true. No layperson without prior knowledge of nuclear fusion issues is going to correctly understand that statement - and indeed, most of the initial press about this didn't, i.e. the journalists reporting it didn't understand what it meant, which is what the article I originally linked to was responding to.

That brings us to the main point: I didn't say anything about an announcement. I responded to someone who was talking about what our society would do if it "were sane, rational, advanced".

I'm saying that it's extremely unfortunate that our society is too scientifically illiterate to correctly report on and understand what ultimately was a somewhat routine scientific achievement, reaching a defined efficiency target that has no particular fundamental meaning in the context.

> 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."

It used at least 300 MJ of power to drive the lasers[1]. 300 / 3.15 = 95. But that factor of 95 would just be to reach a break even point with the heat energy produced, it's not directly usable energy.

For actual usable energy, according to a 2023 presentation at the LLNL High Energy Density Science Seminar[2], "For a power plant, gain would need to be increased ~1000x relative to current NIF performance."

None of the announcement about this so-called "ignition" event mentioned any of this, and nor did most (any?) of the mainstream press about it.

The reality here is that in order to maintain public interest in nuclear fusion, and keep getting funded, it has to be presented as though fusion power is just around the corner - "5 years!". What I was pointing out is that "if our society were sane, rational, advanced," we would not need to play such games. We would not need to continually mislead the public, we would not need to pretend that facilities being used to do nuclear weapon "stockpile stewardship" research have some relevance to fusion power, and so on.

I also found it ironic that the commenter who wanted a "sane, rational, advanced" society appeared themselves to be a victim of the misleading hype around the NIF event, saying that it should "still dominate the news." It simply wasn't that significant.

> This wasn't just about efficiency metrics - it demonstrated fusion ignition was possible, a fundamental physics milestone

This is incorrect, as explained above. "Ignition" here is a term defined by LLNL to apply to their particular weapons-oriented fusion facility. There's nothing "fundamental" about it. It's a defined target for experimental efficiency, that's all.

> ... that had eluded scientists for decades

And still does, at any facility that's trying to achieve nuclear power generation, and not just a weapons research facility blasting a pellet with 300 MJ from 192 lasers. The NIF result is simply not transferable to any other fusion scenario.

> Trying to redefine the announcement and experiment result to mean something else so that you can attack is a dishonest behavior.

It's not clear that you yet understand the full extent of the deception that you've been subjected to, so you're trying to shoot the messenger.

[1] https://ww2.aip.org/fyi/2022/national-ignition-facility-achi...

[2] https://heds-center.llnl.gov/sites/heds_center/files/2023-03... (bottom of 59th slide)

Falsehood Cruelty and Pollution are results of the 7 sins. Cruelty is typically caused by Anger or Envy, Pollution from Gluttony and Sloth, Falsehood from Pride and Envy etc.

I assure you falsehood, cruelty, and pollution have existed long before our time.