Mercio_and_Rudi - 2018-09-26

One of the guys who probably understood it it best said he found it remarkable that there is this correspondence with the z-component of the angular momentum and mostly that it is conserved under descend of symmetry. While the latter is thank god not that hard, I did'nt confess that we still are working on the former.

I mean on a proper proof

but as far as I see no one of those people would really care, but I do now, since I like to fully understand that.

And some said it fully possible that there is still more to discover.

No one does such kind of considerations nowadays anymore apparently :-)

But maybe its not really more than O(3) + highest order element is along z

mercio

maybe it's not really more than the representation is reducible

but since there are only finitely many irreducible cases there are probably many equivalent complicated way of stating the distinction

Rudi_Birnbaum

What I mean is that its not only O(3) but a physically meaningful O(3).

and QM can be built of position and mometum

algebraically

but position and momentum relate automatically to symmetry of the physical "objects"

When a molecule has a certain symmetry and I say I want to measure the angular momentum of the electrons in that, I will have to decide which component I want to measure.

And that decision can only be made relative to the orientation of the molecule.

mercio

can't you just rotate your measuring apparatus

:*)

Rudi_Birnbaum

7:42 PM

but now the momentum and the angular momentum "live" in the space of the O(3) reps already

I think thats the point somehow

:-)

mercio

they are operators that transform a certain way under change of coordinates

Rudi_Birnbaum

exactly

mercio

I am not done reading my quantum field theory book

:(

I didn't read today

Rudi_Birnbaum

is it also "field"

that qiute tough then

and far above me ... :(

mercio

it is spooky and magical

Rudi_Birnbaum

7:44 PM

I just know some basic concepts but never did anything with that

mercio

I have seen multiple times people saying how the calculations had no firm mathematical background so it got me curious

Rudi_Birnbaum

"renormalization"

mercio

that kind of thing

Rudi_Birnbaum

you get diverging "integrals" = inner products

mercio

reminds me of that place where he integrated exp(i(p.x))

over all of space

with some 3 vector p

Rudi_Birnbaum

7:45 PM

and to get rid of that you divide by another one

mercio

and then he innocently said "we add an 1/N normalisation constant in front of the whole thing"

Rudi_Birnbaum

sounds like theoretical physics

they do that all the time ...

mercio

and I am still not sure if the author has acknowledged that the integral is far from converging at all

I need to reread a few parts :*)

Rudi_Birnbaum

maybe not, they are happy if they get some "formula" out of it

that works

like Euler

and Euler was rocket

mercio

haha

Rudi_Birnbaum

7:47 PM

but it was the same non sense in principle

he just could get around the troubles due to his extreme intuition

it took a while until Analysis was invented that put that stuff on firm grounds

I think Physicists wait for the same with their QFT

mercio

maybe

if they have some calculations that give the right result, it is good

Rudi_Birnbaum

yes

mercio

then you just have to remove all of the calculations that give the wrong results

and show tha tthe good result always exist

Rudi_Birnbaum

lol

perturbation theory is used often in QFT without any care about convergence, so it is use in chemistry as well

and in deed often it oscillates wildly with the order and eventually diverges. But it still may be the PT2 is better than PT1 than its used

and PT3 is anyway too expensive ...

anyway after I return on Saturday I'll have a fresh look at the MS

mercio

PT3 is looking at third order terms ?

Rudi_Birnbaum

7:54 PM

third order perturbation theory

mercio

right

Rudi_Birnbaum

it contains terms like <a|O|i><i|O|b><b|O|j>/(ea-eb)(ei-ej)

or something

those are coded in the Feynman diagrams

mercio

3-cycles !

Rudi_Birnbaum

@mercio could be!

Oh yeah I forgot, the "guest star" of the meeting was talking about general magnetic interactions with non small magnetic fields

as they occure on neutron stars

you cannot treat these perturbatively but you just solve the full Hamilton equation

however one of his bottom lines was

"paramagnetism is a magnetic Jahn-Teller effect"

Which suggests another intimate connection between JT and paramagntism

it led us both to speculate if there is maybe some more general underlying theory unfying our concepts ...

but thats very hard to bring under one concept at least at the moment I have no idea about that

you see we have a "structural" Jahn Teller effect

giving rise to a certain magnetic response.

And he has a Jahn-Teller effect in the magnetism itself ..

that is induced by magnetic interactions between states

which is pretty wired ...

but maybe

maybe

I got an idea!

mercio

o..o'

Rudi_Birnbaum

8:08 PM

You see the magnetic coordinates are the "spin coordinates"

while we were treating $Q$ so far as our position coordinates

but of course, when going fully relativistic ...

then you need both of them!!

you should study your QM

especially the Dirac equation

I don't know if I ever told you

but the reason I came about this project was I was searching for SOC:=Spin-Orbit coupling effects

SOC is purley relativistic

in a way the "only" really qulatitatively new thing

when going relativistic in chemistry

in vague way

and I was searching for extreme SOC effects in closed shell systems

so I needed to understand particluar non-relativistic things first

namely the JT systems

the ones where I expected the largest SOC effects

that turned out to be true (as some test calculations show)

SO now I have almost understood the non-relativistic case

(this is our current work here)

and then my next step would be the relativistic case

until 5 minutes ago I had no idea what to expect here

but I guess now I have

its what Trygve saw

(the big guy Trygve Helgaker)

@mercio ?

mercio

it's getting late

Rudi_Birnbaum

sure! I also need to sleep, lets continue later!

mercio

neevr heard of that name

Rudi_Birnbaum

theoretical chemist, one the best world wide though I guess hes not that popular in the US but in Europe he is

among Chemists

Transcript for

$Mathematics$ Mercio_and_Rudi