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Ryder Rude

08:29

@MoreAnonymous i think the mass independence idea only holds for expectation values. if we initialise two different wavefunctions with same initial $\langle X \rangle, \langle V\rangle$, then they have the same $\langle X(t) \rangle$ regardless of their mass. this is just Ehrenfest's theorem

idk how else to define a notion of "same initial velocity" without using expectation values

Ryder Rude

08:48

in classical mech too, two different masses behave the same under gravity only if u initialise them with the same x and v. if u initialise them with the same x and p, then they behave differently

Ryder Rude

09:09

i think the core of equivalence principle is whether or not the laws in zero gravity frames are the same as the laws in free fall frames. Mass dependence/independence stuff is not that important

Tobias Fünke

morning

qwerty

09:38

gday

Signor Feynman

09:49

Why, hello

qwerty

hello hello :)

Ryder Rude

3 hours later…

Ryder Rude

13:08

pseudoscience believers r always trying to convince u that their ideas r scientific

especially astrology believers. and they just think u r not open minded

ACuriousMind

13:22

@User198 I'm not sure what makes a canonical transformation "cool" :P Locally, it's effectively just a coordinate change, and if you're lucky you can find coordinates in which the equations of motion are particularly simple (cf. Action-angle coordinates), this is not different from any other formalism of mechanics.

Ryder Rude

13:35

@ACuriousMind i have never encountered these co ordinates

ACuriousMind

I'm not sure what you expect me to do with that information :P

Ryder Rude

this looks really advanced. looking forward to learning it

it uses group theory stuff physics.stackexchange.com/a/440133/156987

@User198 u can also just do algebra

see this comment chat.stackexchange.com/transcript/message/67118141#67118141

it uses algebra to solve for $x(t)$ of the free particle

it is really strange that one can solve these particular classes of differential equations using just algebra

i was once looking for a generalisation of this idea. this idea seems to work when the diff equations can be expressed using commutators, which is a very specific class of differential equations

ACuriousMind

13:56

It's just Lie theory.

Ryder Rude

yes. but how to get a general diff equations solving techniques out of this

it is a really unique technique

Signor Feynman

There is one book that Qmechanic always mentions about DE and Lie groups

Ryder Rude

usually, one can solve a coupled system of ODEs when the coefficients are constant

like matrix exponentiation

but I think this idea allows u to solve more complicated differential equations

Signor Feynman

Applications of Lie Groups to Differential Equations - Olver

Ryder Rude

but u need to be able to look at a differential equations and find. Lie -algebra in there

@SignorFeynman oh

i was once trying to get a general technique out of this. but I couldn't get very far. it seemed to apply to a very specific class of diff eqns

@SignorFeynman thanks

the idea I had was that, if i have a diff eqns dx/dt = f(x,p) and dp/dt=g(x,p), and if I am able to map x and p to operators on a vector space, and if I am able to re-write the equation as dx/dt=[O, X] for some O, then I can use this technique

qwerty

14:09

@SignorFeynman that book seems to fill a unique niche. I started on it last year but haven't made much progress

Signor Feynman

I haven't read it so far, I only know of it from Qmechanic's answers :P

And I'm a bibliography maniac

qwerty

iirc there's some stuff on Noether I couldn't find elsewhere

Signor Feynman

That's the context of those answers

Probably it was related to Noether's second theorem

qwerty

when I read the intro to the Olver book it was so surprising and seemingly profound that I texted two maths postdocs asking if they knew this stuff and if so why I had never been taught it before :P and at least one answered "cos I didn't know it either"

Signor Feynman

On average, math people are not very accustomed with Lie theory, unless they are geometers

qwerty

14:14

that seems surprising to me

Signor Feynman

I would dare to say that the average physicist has a superior running knowledge about Lie groups :P

I may be mistaken, though. That's just the sample of mathematicians that I know

qwerty

yeah it's so central in physics

Signor Feynman

I'm bewildered. I've just found one book about applications of Lie groups to DIFFERENCE equations

I suspect they mean something else by "difference equation"

qwerty

weird indeed

Signor Feynman

Oh, no. They really mean difference equations

Applications of Lie Groups to Difference Equations - Vladimir Dorodnitsyn

I'll never open that again :P

qwerty

14:27

I thought lie groups would imply CTS stuff and difference eqns discrete stuff...

Ryder Rude

i think the crucial idea here is carrying over the Lie bracket structure from the vector fields to the functions

like, carrying over the commutator of vector fields to the Poisson bracket of functions

and once we have a Lie algebra structure on the functions, we can use this technique

so, in what ways can one inherit a Lie algebra on the functions using the commutator on vector fields. the Poisson bracket is one such way

like, the Poisson bracket satisfies $[X_f, X_g]=X_{\{f,g\}}$

@qwerty same

@SignorFeynman xD. it sounds intriguing tho

fqq

15:22

@SignorFeynman as in "know how to run away from it"

Signor Feynman

16:04

I was thinking about the running coupling, but I guess that'll do :P

3 hours later…

imbAF

19:13

gauge field of SU(3) does such a field exist? Isn't a gauge field a QFT that is invariant under gauge transformation. And isn't SU(3) a transformation that changes coordinates, which is the opposite of a gauge transformation ?