A subset of Spacetime diagram.
The key insights that it gives are:
- future and past are well defined: every reference frame sees your future in your future cone, and your past in your past coneOtherwise causality could be violated, and then things would go really bad, you could tell your past self to tell your past self to tell your past self to do something.You can only affect the outcome of events in your future cone, and you can only be affected by events in your past cone. You can't travel fast enough to affect.Two spacetime events with such fixed causality are called timelike-separated events.
- every other event (to right and left, known as spacelike-separated events) can be measured to happen before or after your current spacetime event by different observers.But that does not violate causality, because you just can't reach those spacetime points anyways to affect them.
Animation showing how space-separated events can be observed to happen in different orders by observers in different frames of reference
. Source. 
Amazon's informtion about their own intances is so bad and non-public that this was created: instances.vantage.sh/
Check which you you have:Tested on Ubuntu 23.10 I see:which means I have GNOME Display Manager.
systemctl status display-manager.service● gdm.service - GNOME Display Manager
Loaded: loaded (/lib/systemd/system/gdm.service; static)
Active: active (running) since Sun 2023-12-24 10:34:50 GMT; 23min ago
Process: 1827 ExecStartPre=/usr/share/gdm/generate-config (code=exited, status=0/SUCCESS)
Main PID: 1850 (gdm3)
Tasks: 4 (limit: 71817)
Memory: 6.8M
CPU: 119ms
CGroup: /system.slice/gdm.service
└─1850 /usr/sbin/gdm3And if you really can't make money from a subject, there is only one other thing people crave: beauty.
You have to give the beauty motivations upfront, before boring people to death with endless prerequisites, otherwise no one will ever want to learn it.
Free borrow on the Internet Archive: archive.org/details/inwardboundofmat0000pais/page/88/mode/2up
The book unfortunately does not cover the history of quantum mechanics very, the author specifically says that this will not be covered, the focus is more on particles/forces. But there are still some mentions.
Leads to the Dirac equation.
The dual space of a vector space , sometimes denoted , is the vector space of all linear forms over with the obvious addition and scalar multiplication operations defined.
Since a linear form is completely determined by how it acts on a basis, and since for each basis element it is specified by a scalar, at least in finite dimension, the dimension of the dual space is the same as the , and so they are isomorphic because all vector spaces of the same dimension on a given field are isomorphic, and so the dual is quite a boring concept in the context of finite dimension.
Infinite dimension seems more interesting however, see: en.wikipedia.org/w/index.php?title=Dual_space&oldid=1046421278#Infinite-dimensional_case
One place where duals are different from the non-duals however is when dealing with tensors, because they transform differently than vectors from the base space .
We can reach it by taking the rotations in three directions, e.g. a rotation around the z axis:then we derive and evaluate at 0: therefore represents the infinitesimal rotation.
Note that the exponential map reverses this and gives a finite rotation around the Z axis back from the infinitesimal generator :
Repeating the same process for the other directions gives:We have now found 3 linearly independent elements of the Lie algebra, and since has dimension 3, we are done.
Man-made virus!
TODO: if we had cheap de novo DNA synthesis, how hard would it be to bootstrap a virus culture from that? github.com/cirosantilli/cirosantilli.github.io/issues/60
Is it easy to transfect a cell with the synthesized DNA, and get it to generate full infectious viral particles?
If so, then de novo DNA synthesis would be very similar to 3D printed guns: en.wikipedia.org/wiki/3D_printed_firearms.
It might already be possible to order dissimulated sequences online:
The projects you do must always aim to achieving some novel result.
You don't have to necessarily reach it. But you must aim for it.
Novel result can be taken broadly.
E.g., a new tutorial that explains something in a way never done before is novel.
But there must be something to your project that has never been done before.
You can start by reproducing other's work.
Leads to the Proca equation.
Let's do a sanity check.
Searching for "H" for hydrogen leads to: physics.nist.gov/cgi-bin/ASD/lines1.pl?spectra=H&limits_type=0&low_w=&upp_w=&unit=1&submit=Retrieve+Data&de=0&format=0&line_out=0&en_unit=0&output=0&bibrefs=1&page_size=15&show_obs_wl=1&show_calc_wl=1&unc_out=1&order_out=0&max_low_enrg=&show_av=2&max_upp_enrg=&tsb_value=0&min_str=&A_out=0&intens_out=on&max_str=&allowed_out=1&forbid_out=1&min_accur=&min_intens=&conf_out=on&term_out=on&enrg_out=on&J_out=on
From there we can see for example the 1 to 2 lines:
- 1s to 2p: 121.5673644608 nm
- 1s to 2: 121.56701 nm TODO what does that mean?
- 1s to 2s: 121.5673123130200 TODO what does that mean?
We see that the table is sorted from lower from level first and then by upper level second.
So it is good to see that we are in the same 121nm ballpark as mentioned at hydrogen spectral line.
TODO why I can't see 2s to 2p transitions there to get the fine structure?
Split in energy levels due to interaction between electron up or down spin and the electron orbitals.
Numerically explained by the Dirac equation when solving it for the hydrogen atom, and it is one of the main triumphs of the theory.
Unlike the simple case of a matrix, in infinite dimensional vector spaces, the spectrum may be continuous.
The quintessential example of that is the spectrum of the position operator in quantum mechanics, in which any real number is a possible eigenvalue, since the particle may be found in any position. The associated eigenvectors are the corresponding Dirac delta functions.
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