Ciro Santilli @cirosantilli 37

The dual space of a vector space $V$, sometimes denoted $V^{\ast }$, is the vector space of all linear forms over $V$ 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 $V$, 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 $V$.

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:$L_z$ 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 $L_z$:

Repeating the same process for the other directions gives:We have now found 3 linearly independent elements of the Lie algebra, and since $SO(3)$ has dimension 3, we are done.

Leads to the Dirac equation.

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.

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:

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.

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.

NIST database for spectral line: physics.nist.gov/PhysRefData/ASD/lines_form.html

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:

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?

Same motivation as Galilean invariance, but relativistic version of that: we want the laws of physics to have the same form on all inertial frames, so we really want to write them in a way that is Lorentz covariant.

This is just the relativistic version of that which takes the Lorentz transformation into account instead of just the old Galilean transformation.