When we particles particles, the action is obtained by integrating the Lagrangian over time:

In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.

Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.

E.g. for a 2-dimensional field $f(x,y,t)$:

Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with $x\in {\mathbb{R}}^2$:

And in the context of special relativity, people condense that even further by adding $t$ to the spacetime Four-vector as well, so you don't even need to write that separate pesky $t$.

The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.

Ciro Santilli's energy throughout the day varies as follows:

Ciro has low tolerance to sleep deprivation which makes him very irritable, and low ability to sleep if there is any light. It must have to do with those damned ganglion cell photoreceptors. On the other hand, Ciro Santilli's wife can sleep without any problems with some morning light! It is definitely genetic. Ciro conjectures that people from very Northern parts of the world must have a gene that allows them to sleep even if there is some light, while more equatorial people don't. Maybe: pubmed.ncbi.nlm.nih.gov/33049062/

Ciro has mild olfactory synesthesia for star anise (八角, bajiao), which is widely used in Chinese cuisine and makes Ciro think uncontrollably of the color blue. Ciro does not have any other known synesthesias. He is also prone to nerd sniping form time to time.

Ciro is a reptilian-like being with cold hands and feet and low blood pressure. For this reason he believes that he will die of cancer or some respiratory problem. If the Chinese government doesn't get him first that is. This also partly explains why Ciro is not a big fan of swimming.

Besides Chinese food, Ciro really likes eating fruits and roasted nuts, maybe partly because he was born in Brazil, and partly because of monkey nature, see his Chinese name. At home he is known as "水果大王" (the big king of the fruits). Ciro is also a sucker for yoghurt (natural without added sugars and full fat, fat-tree yoghurt is terrible, often eaten with fruits). Ciro's "favorite drink" could be tonic water with freshly squeezed lemon. Tied with fresh fruit juices. Chocolate-wise, although not a huge fanatic, a Lindt dark chocolate with whole hazelnut pieces bar will do the job.

Ciro does not like receiving or giving gifts on expected social situations like birthdays or Christmas. Ciro believes that every day is equally precious, and can be a day to give, be it through awesome open source software contributions, or if you find something that your friend will like

Ciro has some respiratory allergies. When he was around 5, he had relatively serious asthma crisis which scared his parents to death. Throughout his life, he appears to be allergic at an intermediate level to: mold or dust mites (or whatever it is that old books/pillows have), cats (itching on touch), hay fever (in May in the UK, likely grass pollen). But even outside of hayfever season, Ciro's nose is constantly either running in the cold, or often partially blocked while sleeping throughout the year. Ciro believes however that this also gives him higher resistance to viral infections, since it has been many many years since he had a cold/flu, and when everyone in the office is going down with it, he's just fine. Ciro wonders if his active immune system will actually kill off cancers early, which he ranks as his most likely causes of death, along with respiratory and gastro-intestinal problems. Ciro has low blood pressure and cannot get fat, so cardio vascular problems seem much less likely.

Ciro is generally democrat due to his high compassion level. He believes that politics is highly genetically determined, and that just like you enter a room full of people and immediately like some and dislike others, the same goes for politics. People just vote for whoever they want to see more of because their way of speaking makes them feel good. There is not rationality involved in it at all.

Ciro self diagnoses a slight graphomania in the early 2020's. This is largely what led him to create OurBigBook.com, and contribute to Stack Overflow. Literature Nobel Prize laureate Naguib Mahfouz also suffers from the condition however, so maybe good can also come out of it:

When Ciro was quite young, maybe around 7-10, when he got very angry or sad for some stupid reason (bullying perhaps? Ciro forgot), he would have a psychosomatic manifestation: his spine would become visibly curved sideways (scoliosis). While writing this paragraph, Ciro Googled it, and found e.g. medium.com/@michaelrosen_94192/the-root-cause-of-scoliosis-5c461002b634 that describes:so it is a somewhat well known thing! Incredible. Can you imagine the level of the passions that lead to such physical deformations? But of course, it was all for nothing.

Count: 0.

Not a fun of giving up control for such a low-maintenance cost venture... but keeping a list just in case...

Fridge: Lamona LAM6800 fridge freezer.

Bedroon door glass pannel dimensions: 36cm x 59.6cm

Middle separator length: 2.5cm.

So total glass area: 36 * 2 + 2.5 x 59.6 * 2 + 2.5 = 74.5 x 121.7

Bin: TODO size.

Washing machine: Lamona, someting along LAM8603: www.manualslib.com/manual/1049371/Lamona-Lam8603.html

~1TB.

Internal hard drive likely removed from some old computer I lost track of, kept in a crappy case, incredible stuff.

Ubuntu 20.04 gnome-disks benchmark, NTFS partition: 40MB/s.

Bought 2021-11, grey, size EUR 44.5, 80$ (later found cheaper online): www.sportsdirect.com/nike-run-swift-2-running-shoes-mens-121052

The China front was of course the Russia front this time: stackoverflow.com/users/895245 (web.archive.org/web/20220322230513/https://stackoverflow.com/users/895245/ciro-santilli-Путлер-Капут-六四事)

Examples:

Basically the operators are just analogous to the classical ones e.g. the classical:becomes:

Besides the angular momentum in each direction, we also have the total angular momentum:

Then you have to understand what each one of those does to the each atomic orbital:

There is an uncertainty principle between the x, y and z angular momentums, we can only measure one of them with certainty at a time. Video 1. "Quantum Mechanics 7a - Angular Momentum I by ViaScience (2013)" justifies this intuitively by mentioning that this is analogous to precession: if you try to measure electrons e.g. with the Zeeman effect the precess on the other directions which you end up modifing.

TODO experiment. Likely Zeeman effect.

Second quantization also appears to be useful not only for relativistic quantum mechanics, but also for condensed matter physics. The reason is that the basis idea is to use the number occupation basis. This basis is:

Bibliography: