1-form, 2-form, red-form, blue-form

Sorry, myow memory is failing meow again, but I'd like some confirmation: If in QED we integrate out the photon modes, the resulting theory of merely electrons will acquire the ugliness of being non-local and everywhere failing to have a sensible CCR, but are there other problems? Will it fail to be unitary? Will it have infinitely many interaction terms? Could we extract some useful information out, e.g. mass gap and other renormalised values?

oh my god this asshole outbid me at the last 10 seconds

meowth?

hi sorry i have no clue what the answer is

hope you are well nI

good nightly

@Allie eBay :-)

@naturallyInconsistent should be non-renormalizable - already a 4-fermion vertex is non-renormalizable in 4d, so this only makes sense as an effective theory

I think there was a section on Schwartz about the Euler-Heisenberg Lagrangian. I don't enjoy that book but it may be helpful

morning

hi

Hello people

So at maximum one could give arbitrary values to 3 of the 6 components of E and H, while two of them are fixed by the invariance relations

Note, that E and H here are Electric and Magnetic fields respectively

@ACuriousMind ah, that's even more annoying, thanks

if anyone knows a mathematical quantum field theorist, please share this post physics.stackexchange.com/q/837045/156987

and give me their feedback

it is not rigorous yet but it is traces of a result

@Arjun in what context are these statements made? Obviously you can't give E or H "arbitrary values" for a fixed physical situation...

@SillyGoose any reason to assume it's not just the normal meaning, cf. en.wikipedia.org/wiki/Forward_scatter?

A nice little Lenz's law demo:

I just found out about Poisson algebra! That you can do whole classical mechanics, and more, without using any analysis or have any knowledge about physics. Just algebraic manipulations following the axioms. How fascinating is that ? :)

What is your opinion, people who are further up the road than me in their physics journey, about that formulation of mech.? How fascinating is that formulation from 1 to 10 (10 is really fascinating).

I think it is really cool.

11 for me

@User198 I mean, it's not really pure algebra - if you want to know the evolution of $f$ under the transformation generated by $g$, you still have to solve the differential equation $\partial_\epsilon f = \{f,g\}$. It's a synthesis of algebra and analysis, i.e. the theory of Lie algebras and groups.

@Arjun I don't see that in L&L page 68, are you sure that's the page?

You can do it without derivatives as long as you know the algebraic rules for every $\aleph_0$ elements of the ring of smooth functions

@ACuriousMind But if I have some simple systems and I input $q$ and $p$ for my $f$ than the LHS is just the velocity and $ma$. All the work is algebraic my manipulating the brackets.

Is this used for solving with computers? You just change the input Hamiltonian and the comp just does algebraic manipulations. Seems low complexity. Right?

@User198 I'm not sure what you mean - if you choose $f = q$ and $g = p$ then it's easy, sure, but all you're showing then is that momentum generates translation in position and vice versa

In general you actually need to solve the first-order coupled differential equations $\dot{q} = \{H,q\}$ and $\dot{p} = \{H,p\}$ where $H$ is an arbitrary function of $p$ and $q$. This is not a purely algebraic exercise

The Hamiltonian formulation is certainly "more algebraic" than the presentations you've likely seen before this, but it doesn't remove differential equations from the picture entirely

@User198 The N E-L equations are second-order. These are equivalent to 2N first-order equations.

Ah

Right

Thank you xD

sure, but in general you might not

The $\dot p$ is the one that described the dynamics of the system. At least that is as I observed.

@ACuriousMind Ok ok Thanks

guys!

I was accepted to NYU!!!!

congrats!

Congratulations

Am I the only one who is disturbed that words can kinda be approximate by vectors in LLMs?

I mean words aren't mathematical objects

Makes me wonder maybe we doing the same thing in physics :p

everything is a vector space if you're brave enough

What???

or, more properly, many things are locally $\mathbb{R}^n$

@Allie niiiice, congratz :) I am happy for you. Good luck with your further studies!

@ACuriousMind The words of a boogieman

@ACuriousMind I love the phrases with ... if you're brave enough hehe

really what ML algorithms use is just that you want to group the space of "words" (it's not really words) you want a notion of closeness. "Closeness" between points of a set is what a topology models, and $\mathbb{R}^n$ is the easiest topological space you could try (in particular because matrix operations can be implemented very efficiently)

turns out if you put $n$ high enough this is actually useful :P

(for some values of "useful")

I read in high dimensional vectorland king minus man + woman approximately equals queen

the idea of mapping the space of possible outputs as a vector space is not the breakthrough that enabled the craze about LLMs, this has been more or less the model machine learning operated on years before the boom

and it goes back more or less to the beginnings of neuronal networks decades ago

What is the breakthrough that enabled LLMs?

@MoreAnonymous "+" and "-" are essentially just statistical correlations here. All you're saying is that if you ask "what's the most probable world to occur in a context where words like 'king' would occur, but also a context where no words relating to 'man' but words relating to 'woman' occur?" the answer is "a word like queen". It's very interesting that you can train this into an ML model, but it doesn't really reveal anything about language we didn't already know.

@MoreAnonymous A much more efficient architecture for these neural networks, transformers - the 'T' in GPT stands for 'Transformer'.

Does the high dimensionality of LLMs tell us anything about how a species uses language to describe reality?

I don't know what that means

we don't know any other species that use "language" except humans

Fair..

If only we could train LLMs on alien text

Have you read Godel escher bach by any chance @ACuriousMind ?

nope

I think you would love it

nope

@ACuriousMind why do u think so?

@MoreAnonymous All I've heard about it indicates that at best it's perhaps one of the better examples of pop-science, but I have no interest in pop-science.