The h Bar

naturallyInconsistent

00:36

@SillyGoose bohr.physics.berkeley.edu/classes/221/1112/notes/pertth.pdf

Relativisticcucumber

00:53

@Obliv @SillyGoose

Relativisticcucumber

01:03

@ACuriousMind @Minsky i think the true intersection is in complex systems applications to biology. this area does try to take the reductionist behavior of small entities and tries to create models that describe complex systems/emergent behavior, and it has worked in a number of cases. some examples are bird flocking patterns and tissue and organ formation. [...]

[...] people are trying to do this with neuronal firing patterns as well to understand cognitive functions and consciousness because the focus on neuron-to-neuron firing has not lead to insights into these higher functions. this is an area of research in the intersection of computational neuroscience and complex systems.

Silly Goose

01:14

@Obliv for math it seems (in my smol experience) that graduate level texts do not increase in difficulty but in pre-requisite knoweldge. E.g. Hall's text on Lie theory is not hard per se, but it requires being intimate with linear algebra and comfortable with topology, real analysis, and so on. but of course lacking a genuine pre-req makes things impossible to get through. I think perhaps physics is the same, except graduate physics also correlates with introducing new mathematics

i like some graduate textbooks because they are more thorough and sometimes are more logical (through use of less elementary concepts)

@naturallyInconsistent this looks nice i shall take a look!

naturallyInconsistent

01:37

Where pre-requisite knowledge involves not just a lot of theorems, but also a lot of terminology. I still cannot tell you what every one of homomorphism, holomorphism, homeomorphism, isomorphism, monomorphism, epimorphism, endomorphism, automorphism and $deities know what other morphisms exist

Relativisticcucumber

04:48

@SillyGoose x2

@Relativisticcucumber my top choice phd program that i just applied to is about this so if i get ultra lucky i can talk more about this in the future eeeeeeeeeeeeeeek

scream pray to lord of geese

naturallyInconsistent

scream scream

Ryder Rude

06:13

@Minsky Rovelli's interpretation of quantum mechanics does have a bit of panpsychism (He himself said this)

@Minsky u can find it here philarchive.org/rec/ROVRAP-2

as for whether physics theories can detect subjective experience in matter : no. not even in principle

this is different from other phenomena like biology and chemistry and computation, which can be reduced to physics, at least in principle

even in a relative state interpretation, the state vector relative to some other piece of matter is completely inaccessible to you

so u can even adopt a Solipsist viewpoint where everything except ur own brain evolves according to the Schrodinger eqn, and it wud make no difference to the predictions of this interpretation

nickbros123

06:51

@naturallyInconsistent you missed metamorphism, an (approximately) linear mapping from space of rocks to space of rocks where rock structure is preserved

naturallyInconsistent

@nickbros123 halp

nickbros123

I wonder what the basis vectors for the space of rocks are

Something to think about

Silly Goose

07:14

does anyone know of a good resource to understand how to draw basic feynman diagrams? e.g. turning a decay or reaction of up to two composite particles into a diagram

Slereah

It is in most QFT books

I learned in Peskin

Silly Goose

07:31

lol

okay i shall take al ookzie

Slereah

08:13

Currently trying to prove that if a property is true for all n inferior to N, it will also be true for all n inferior to a value smaller than N

Been a while since I have done propositional calculus

Mr. Feynman

08:29

@Slereah isn't that trivial?

Slereah

Can you prove it

Mr. Feynman

Maybe I'm misunderstanding the proposition

Slereah

It is thus

It should be $P_b \wedge b' \leq b \to P_{b'}$ I guess

Switched notation mid-proof

Ahah!

⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → ((𝜒 → 𝜃) → (𝜓 → 𝜃)))

That's the thing

Or at least it will be if I figure out how to add the universal quantifications

ekardnam_

08:50

@Slereah this is the level at which the mathematical formality gets annoying rather than useful for me

since we were speaking about it yesterday

but i acknowledge it can be fun

somebody is confident with Nekrasov's paper on instanton counting?

Slereah

One era of my life I just attempted to go through all the proofs of the Principia Mathematica

I only managed about 6 chapters tho

⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Good news, this statement is indeed true :p

ekardnam_

⊢ this symbol is a NOT right?

Slereah

That's a turnstile

It means "It is true" roughly

ekardnam_

oh i see

oh can you explain the difference between the implication $\implies$ and the implication $\rightarrow$?

i never really understood it

Slereah

$\to$ is a logical implication, $a \to b$ means that $b$ is true if $a$ is true

$\Rightarrow$ is for rules of inferences

It means that a theorem implies another theorem

ekardnam_

09:04

@Slereah doesn't this also mean that if theorem A is true then theorem B is true?

I guess the difference maybe is more technical

Slereah

It is yeah

It means that if theorem A is part of the model, then theorem B is part of the model

They are sort of similar but $\Rightarrow$ is kind of in the "metalanguage"

ekardnam_

I think intuitively I grasping a bit of the difference

Slereah

It is that Thing where in logic, we have different levels of statements

To avoid the old paradox of "This statement is false"

You have a theory for your statements, then you have a theory for the theory, then etc etc

You can't talk about the truth of a theorem within the language of the theory itself

ekardnam_

@Slereah it is somewhat similar to how classes in set theory are sets without actually being sets so you can speak of the class of all sets

Slereah

Basically yes

ekardnam_

09:12

i see i see

thanks

Slereah

As we've learned in the early 20th century it is a bad idea to have a theory that can do it all

Otherwise you get weird shit like

Curry's paradox

Curry's paradox is a paradox in which an arbitrary claim F is proved from the mere existence of a sentence C that says of itself "If C, then F", requiring only a few apparently innocuous logical deduction rules. Since F is arbitrary, any logic having these rules allows one to prove everything. The paradox may be expressed in natural language and in various logics, including certain forms of set theory, lambda calculus, and combinatory logic. The paradox is named after the logician Haskell Curry. It has also been called Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem...

ekardnam_

@Slereah I see Haskell Curry. I used to program in Haskell a bit

Slereah

"This sentence is false" has really been a neverending headache in logic for 2600 years

Slereah

10:04

Trying to figure out the sense the non-relativistic deviation of light makes, but it is tricky

It makes some sense to have the deviation of massless particles in the Galilean case, but the EM field doesn't change

But also, the EM field doesn't exist as such in the Galilean limit, since it is at least a SR phenomenon

So it is kind of some approximation of 3 theories at once

You take your optical approximation of the EM field, you do the non-relativistic limit of this optical approximation

ACuriousMind

10:25

@ekardnam_ Haskell is fun - it made me realize a lot of things I'd taken for granted about programming were, in fact, not unalterable facts but simply design choices of the more common programming languages

Slereah

Are you a big fan of monads

ekardnam_

@ACuriousMind I was making a simple parser which parser a simple language and compiles it to Lua and it was fun

anyhow @ACuriousMind

since i read your answers on instantons before, are you acquaintaned with Nekrasov's paper on instanton counting?

Slereah

I still haven't implemented my nightmare language that I designed

esolangs.org/wiki/%CE%9B%CE%B9%CE%BC%CF%80

ACuriousMind

@ekardnam_ I saw you asking about that but I didn't say anything because I'm not

Slereah

wiki.haskell.org/Category_theory/Monads

You probably won't find a lot of category theory on other language wikis

ekardnam_

10:30

and also another question: the sum over principal bundle topologies, parametrized by the instanton number $k \propto \int F \wedge F$ is somewhat analog to the sum over worldsheet topologies in string theory parametrized by the genus which is related to $\int \star R$?

ACuriousMind

@Slereah not really, I think e.g. Rust's various "map" functions on its generic types are much easier to grasp than Haskell's monads while largely achieving the same goals

ekardnam_

@ACuriousMind oh okay thank you still

didnt notice you were here earlier

Slereah

@ACuriousMind But then it won't have the same ideological purity

Ryder Rude

chatgpt can write programs these days

but theyre not always correct

ACuriousMind

@Slereah good thing I'm not a functional purist, then!

I find the powerful type system much more interesting than functional purity

Slereah

10:36

I'm not even sure what monads to rly

Seems like output is still a side effect?

ACuriousMind

I/o is a confusing example because you have to learn both "monads" and how functionally pure languages try to hide i/o

the simplest monad is Maybe (or Option(al) in other languages)

Slereah

Apparently you have to consider the "category" of the states of the computer

ACuriousMind

to be frank I think the categorial description helps no one (in particular because Haskell doesn't actually enforce the identities that an actual categorial monad has to obey)

Slereah

"To the programmer, handling a RealWorld directly is too dangerous—in particular, if a programmer gets their hands on a value of type RealWorld, they might try to copy it, which is basically impossible"

D:

@ACuriousMind Presumably it is for people like these

ekardnam_

if you have never watched it you should check this video out youtube.com/watch?v=ADqLBc1vFwI&t=1s speaking of functional programming

ACuriousMind

10:40

The thing is just: If you have a Maybe T, and a function $f : T\to S$, then you'd like to promote that to a function $f' : \mathrm{Maybe} T \to \mathrm{Maybe} S$ simply by saying that f'(Nothing) = Nothing

that's all the bind and unit for the Maybe monad do

and everything else is just more complicated instances of this, where the additional structure in Monad(T) compared to T is more complicated than just having the option to be Nothing

Ryder Rude

how can u know so many things : physics, biology, programming, math, history, philosophy

Slereah

I get bored :p

Ryder Rude

i cant even know even one of these properly

do u also know finance

Slereah

If I did I wouldn't have gone into physics

Ryder Rude

lol

ekardnam_

11:24

omg the website is asking me to review a question

Ryder Rude

it's not mandatory as far as i recall

Ryder Rude

11:41

can there be an in between philosophy to moral relativism and objectivism

sorry the term is moral realism. it asserts that mind independent moral truths are in the universe

Slereah

12:55

Trying to figure out why contact geometry is so called, but the original paper is Zur Theorie partieller Differentialgleichungen

Das ist nicht gut

From context I think it is because it relates to the notion of contact equivalence for maps, but any map is 0 under the contact form

Is there something specific that happens if you have two maps $f,g$ in contact at $p$ with the contact form $\theta$

ACuriousMind

@Slereah I'm pretty sure it's because of the hyperplane interpretation of the contact elements

if you have $\mathbb{R}^n \times \mathbb{R}$, then the objects in the kernel of the contact form on $T^\ast \mathbb{R}^n\times \mathbb{R}\cong \mathbb{R}^{2n+1}$ are hyperplanes that "make contact" at one point of the $\mathbb{R}^n\times \mathbb{R}$

Slereah

I see

ACuriousMind

I guess the name "tangent" was already taken so someone chose "contact" :P

Ryder Rude

13:29

in rep theory of lorentz grp, we want reps of $[J_i ^+, J_j^+]=\epsilon ^{ijk} J^+_k$, identically for $J^- _i$ and $[J^+_i, J^-_j]=0$

one way to do it is to set $J_i ^+= J_i ^{(p)} \otimes I$ and $J_i ^-= I \otimes J_i ^{(q)}$

another way to do it is to set $J_i ^+= J_i ^{(q)}\oplus O$ and $J_i ^-= O \oplus J_i {(p)}$

why dont books discuss the second possibility

$O$ is the null matrix

by $A\oplus O$ i mean making a block diagonal matrix with left block A and right block O

this second way takes a p dimension rep and a q dimensional rep of rotation group, and gives a p+q dimensional rep of lorentz group

ACuriousMind

@RyderRude Because the second option is trivially reducible just to the separate representations on the left and right summand

you're not constructing any interesting new representations with this

Ryder Rude

oh

so it is accounted for

Cosinux

Hi.
When I studied optics, diffraction was explained using the Huygens-Fresnel principle. Our equations were based on integrals over the diffraction surface, where each point was considered to act as a source of spherical waves.
In my mind I made a connection between this model and the 3rd and 4th Maxwell's equations. As it seemed to be commonly stated, a time-varying E field would create a curl in the B field and vice-versa, which would lead to a self-propagating wave. The way I understood it was that that process created new wave sources everywhere along the wave, so it was natural to ass…

Ryder Rude

@ACuriousMind what do u mean explicitly

oh so the direct sum rep decomposes into two reps

Slereah

$A \oplus 0 \cong A$

ACuriousMind

13:40

@Cosinux Jefimenko's equations don't really describe what we want to describe in optics, namely the propagation of EM waves without any particular interest in their source

Ryder Rude

yeah

i feel even while finding reps of multi particle heisenberg algebra, we have these two options

Slereah

Not gonna decribe the motion of charges in the sun to find out what happens in your magnifier

ACuriousMind

there are no free incoming EM waves that can diffract in Jefimenko's formalism because you need to describe the source current in full detail to get the waves from them

Ryder Rude

[X_1, P_1]=i, [X2,P_2]=i and $[X_1, X_2]=0$

and $[P_1,P_2]=0$

we choose the rep $X_1=x\otimes I$

but theres this other option too which would give a direct sum rep of this algebra

but i guess this option is ruled out by experiments?

the direct sum rep describes two separate wavefunctions

ACuriousMind

@RyderRude what are you talking about? You're just again constructing a reducible representation where each half of the algebra acts trivially in one of the two subrepresentations

Ryder Rude

13:43

yes

ACuriousMind

but what we want are irreducible representations

Ryder Rude

oh but the tensor product rep is reducible too?

we decomposed it yesterday

ACuriousMind

why do you think so?

@RyderRude no, you have to pay attention to the specific setting

Ryder Rude

yeah. it was very different. sorry

ACuriousMind

in the setting where you have one group $G$, then for any two irreps $V_1,V_2$, $V_1\otimes V_2$ is in general a reducible representation of $G$

but when you have two groups $G,H$ with irreps $V_G,V_H$, then $V_G\otimes V_H$ is an irrep of $G\times H$

Ryder Rude

13:46

oh

ACuriousMind

both your cases here are the second case where $G=H$ (=SU(2) or Heisenberg groups)

Ryder Rude

yeah so my second way produces reducible reps

we hav to do tensor product

@ACuriousMind this also means that the half-integer pairs classify irreducible reps

for (n,m) pair, the rotation sub-group still decomposes, right? becuz of CV coefficients

but the whole group doesnt decompose

or maybe not. i will need to think about this

ok so it doesnt decompose when one of n or m is 0

ACuriousMind

see this answer of mine

Cosinux

@ACuriousMind I see. But provided that one fully describes the source, can the equations also reproduce diffraction? If so, how? Is my intuition that the movement of charges inside the wall would create waves that would interfere with the original radiation to produce the expected diffraction pattern correct?

Ryder Rude

yeah. this is what i meant. the rotation sub-group is block diagonalsable

thankss

ACuriousMind

14:03

@Cosinux note that Jefimenko's equations assume that you already know the full 4-current at arbitrary times. But in the case of diffraction, there's some kind of interaction going on between the EM waves and the material that reflects/diffracts them and all of this happens on a microscopic scale where the model of macroscopic currents is questionable to begin with

Note also that Jefimenko's equations are very straightforwardly not made to describe EM waves, since while plane EM waves are valid solutions to Maxwell's equations in vacuum, the unique result of Jefimenko's equations for $j = 0$ is $E=B=0$

EM waves and diffraction happen exactly in the idealized corner of electromagnetism for which these equations are not a good model

Cosinux

@ACuriousMind How about using the Heaviside-Feynman formula to model the effect of each particle on every other particle (or the E and B field contribution at that particle's position). Would that also be a misuse of the formula?

Slereah

@Cosinux ambitious

ACuriousMind

@Cosinux mostly it would be impossible :P

you will not understand diffraction quantitatively by trying to model every charge inside the material on which the light diffracts

Slereah

You just need to know the position and momentum of every particle in the universe

That is how God does it

Mr. Feynman

@Slereah that guy has been doing it like forever, it's unfair

Slereah

14:29

"Groups feel sad unless they are acting as symmetries of something."

:(

Cosinux

I suppose what I'm interested in is whether it can be reasonably expected that solving the equations for the entire system would yield diffraction.
Is diffraction a well modeled optical phenomenon that has however never been derived from Maxwell's equations? Do we know how to model diffraction around arbitrary geometry?

ACuriousMind

@Cosinux Of course we can derive diffraction from Maxwell's equations. See e.g. this question and its answers

Mr. Feynman

I just finished doing five A3 sheets of computations

I feel kinda empty now

what's the meaning of life

ekardnam_

@Mr.Feynman about what?

Mr. Feynman

Nothing special actually, just the Riemann-tensor and the EFE for a general spherically symmetric metric

But I couldn't find that elsewhere written in terms of Misner-Sharp mass, except Poisson with a "you may see that"

I've been procrastinating this for too much time indeed :P

ekardnam_

14:45

@Mr.Feynman oh yeah it is there. I trusted the book the computation seem to boring

maybe it could be fun doing it with mathematica though

Mr. Feynman

@ekardnam_ I'm still so noob with it that I would spend less doing it by hand

Well, I did that actually

Is the GR exam in Bologna an oral exam?

ekardnam_

@Mr.Feynman im still very noob to, maybe it will take longer but you learn useful stuff along the way

@Mr.Feynman yes, most exams here are actually

Mr. Feynman

What would you do if the professor asked you to write the Kerr metric?

More specifically, what katana would you choose

ekardnam_

only QFT 1, QFT 2 and Theory of the standard model where written I think

Mr. Feynman

Where I am anything except a QM exam is oral at the master's

ekardnam_

14:48

@Mr.Feynman I'd ask in which coordinates they want it (?)

@Mr.Feynman QFT 2 was written for me

my exercise was to compute the g-2 correction due to a yukawa coupled scalar in QED

I filled 4 A4 pages of just calculations

Mr. Feynman

@ekardnam_ I refuse to remember it in any coordinate system :P

ekardnam_

I did the exercise at home before and I could never finish it correcly, but on the day of the exam I got it right

Mr. Feynman

@ekardnam_ is that easier than finding the $g-2$ correction to electron magnetic moment in spinor QED?

ekardnam_

@Mr.Feynman it is spinor QED

there is just an additional scalar coupled via a Yukawa interaction to the electron

Mr. Feynman

God, I remember that without Yukawa it took like 20+ pages of calculations

ekardnam_

14:51

I honestly do not remember much but I'd say it is not very different, you just have a scalar running in the loop rather than a photon

Mr. Feynman

Oh ok, so you didn't have to consider the photon too

ekardnam_

no no just the contribution from the scalar

basically the vertex correction due to the scalar

Mr. Feynman

I think that makes it easier than working with photons, which cause wild infrared divergences and the loops are hellish

(I would fail that test right now, that's for sure)

ekardnam_

Anyhow I honestly would not remember the Kerr metric right now, but I did know it in Boyer-Lindquist and the coordinates Kerr found it in (I dont remember the name) at the time

Mr. Feynman

Kerr-Schild

I refuse to know both, it's a waste of time imho :P

ekardnam_

14:56

@Mr.Feynman knowing the expression itself is a waste of time in many cases

Mr. Feynman

It's more about balance

For Schwarzschild it's so easy that it's worth

ekardnam_

Schwarzschild, AdS, dS, Reissner Nordstrom are kinda easy to remember at least in some coordinates

not that i remember them though

Mr. Feynman

Yes, those I remember now that you mention it

naturallyInconsistent

15:15

One day I will terrorise yet another class of students with the 2 hour lecture just deriving one term in the Schwarzchild solution.

It always feels good when a student or two realises that they had witnessed such a calculation.

Ryder Rude

r u a lecturer @naturallyInconsistent

naturallyInconsistent

No

Ryder Rude

oh. maybe u give lectures as a side thing

naturallyInconsistent

No.

I was giving that set of classes as a lecture module.

It was just my job.

Mr. Feynman

15:34

@naturallyInconsistent I'm waiting for the day I'll tell my students stuff like "you may easily convince yourself that spacetime is 7.4 dimensional"

Having spent all my life proving that

naturallyInconsistent

lol

grrrr

im almost done with a rather beautiful and in-depth derivation of Thomas precession and I have some dayum sign errors

Ryder Rude

16:04

is the demand for "irreducible representations" a motivation for entanglement or the other way around?

irreducible reps of multi particle algebra makes entanglement possible

ACuriousMind

that's a very strange question :P

Ryder Rude

i mean is there any reason we shud prefer irreducible reps of the multi particle algebra, or is it purely for physical reasons?

real world likes irreducible reps

ACuriousMind

still a strange question; what do you think in general the significance of irreducible representations is?

Ryder Rude

other reps are built from irreducible reps?

ACuriousMind

exactly

Ryder Rude

16:09

so these are like the atoms of reps

ACuriousMind

so if we had a reducible representation of an algebra of observables, that would mean there's a smaller irrep inside it that represents a completely independent system, since the action of no observable can take states out of it

Ryder Rude

oh

so like two independent wavefunctions

ACuriousMind

this has nothing to do with entanglement or systems with subsystems, it's just generally not very clear why you'd look at a reducible rep and call it a single system, since there's an independent system inside of it

Ryder Rude

but i feel the hamiltonian on the space can still be irreducible even if the heisenberg group is reducible

so this means two independent wavefunctions can interact?

ACuriousMind

there are cases where you need a reducible representation of the algebra of observable to correctly model the theory; the physics name for the irreducible components of such a reducible representation is a superselection sector

Ryder Rude

16:13

ooh but the hamiltonian is made up of the individual observables

ACuriousMind

@RyderRude "the Hamiltonian can be irreducible" doesn't mean anything

Ryder Rude

i mean the time evolution group can be irreducible

ACuriousMind

that also doesn't mean anything

it's the representation that is irreducible, not the group or elements of it

Ryder Rude

yeah i mean the rep of the time evolution on the space of two separate wavefunctions can be irreducible

so it's like two separate wavefunctions but interacting

ACuriousMind

if what you want to say is that we can decompose the space of states into irreps of the time evolution group generated by the Hamiltonian, then of course we can: Those are just the one-dimensional eigenspaces of the Hamiltonian

Ryder Rude

16:15

oh so it's always reducible.. sorry

yeah. this makes sense

@ACuriousMind oh

ive heard of superselection sectors but dont know much. i jsut thought they cant interfere becuz the phase difference is physically meaningless

but nice to know this ties to irreducible stuff

ACuriousMind

sure, but the only way in which it can be physically meaningless is if no observable can detect it, which is only the case when all the observables are block-diagonal in the superselection sectors

Ryder Rude

really fascinating

so these act like independent physical systems

Slereah

17:55

Question is

If the ancient greeks used sand to draw geometric figures

how did they trace lines?

I can see how you'd draw a circle in the sand, but a line seems trickier

Can't put your ruler directly on the sand

ACuriousMind

why not

just push the ruler into the sand, there's your line

Slereah

You could use a stretched string, but that would require three hands

@ACuriousMind Maybe I guess?

The exact tools people used is a little hard to find in details

Imagine that you wrote your amazing proof in the sand and then sneezed

Woody Allen - Annie Hall - Coke Scene

►

classic gag

I can see the point of using a medium like that because when I did my master thesis I would just redo the same steps over and over slightly differently for a proof

Imagine having to buy a new papyrus scroll every time

handcrafted by Egyptian artisans

ACuriousMind

for ephemeral writing the ancient world probably mostly used wax tablets

just put it near the fire to erase

Slereah

Also that yeah

but apparently the sand table was nearly synonymous with doing geometry?

ACuriousMind

It's pretty hard to draw neat lines on wax

Slereah

18:08

I can't say that I've tried

ACuriousMind

we did the writing on wax tablets in Latin class once

Slereah

what did you write

ACuriousMind

I don't remember

probably something like "My name is <name>. I'm <age> years old." in Latin

Slereah

What is your latin name, Spirito Inquisitous?

ACuriousMind

I'd have said mens inquirens, though mens curiosa would be more direct

Slereah

18:21

perhaps a question for the Latin stackexchange

Mr. Feynman

19:23

@Slereah wym just take a circle and remove a point :P

Slereah

You'd have to make a circle of infinite radius

and according to Aristotle the universe is finite

Mr. Feynman

@Slereah what's wrong with homemorphisms

Slereah

Doesn't preserve the metric

Silly Goose

22:23

bleh i am still troubled by how to deal with singularities encountered in non-relativistic, time-independent QM perturbation theory

Perhaps my error is in the argument I want to make. I am trying to approach this topic from the angle that the non-uniqueness of a solution (set of eigenstates) within a degenerate subspace leads to problems.

ACuriousMind

22:41

@SillyGoose What's your problem with the standard resolution of this?

Silly Goose

to my understanding (which may be wrong), the standard resolution is to set the action of the operator of interest $O$ on any singular points $\lvert \psi \rangle$ to $0 = O\lvert \psi \rangle$. But this seems like a solely operational and, more problematically, arbitrary procedure. My problem is in this procedure being arbitrary.

ACuriousMind

I'm not sure what you mean

I'd have said the standard resolution is to solve (for first-order degeneracies) for the eigenvectors of the perturbation $H_1$ in this subspace

Silly Goose

hm well then maybe I am mixing up problems in applying the method. To my understanding, choosing to use the eigenstates of $H_1$ as the 0th order kets for the degenerate subspace is only done in order to resolve a singularity issue.

ACuriousMind

...but you said you wanted to resolve a singularity issue

Silly Goose

oh, I am saying that even in making this choice one is only doing so to hand-wavely throw out singular points

in other words, I am actually not even sure how making this choice resolves at all the singularity issue

ACuriousMind

22:48

what about this is a "hand-wave"?

Silly Goose

it makes the LHS in equation (7) equal to $0$ in the degenerate subspace. which makes the operator one wants to invert non-invertible in the degenerate subspace, but this is the problem in the first place. and then, sakurai hand-wavely disregards all contribution to corrections coming from the degenerate subspace by redefining the operator to only act on the non-degenerate subspaces and to set all contributions from degenerate subspaces equal to $0$

sorry by "it makes.." I mean this choice of eigenstates for the degenerate subspace.

but we could have just disregarded all the singular points in the first place (if we were going to do that eventually) without imposing a constraint to enforce the choice of eigenstates for the degenerate subspace !!

hm maybe i am mixing up some issues

ACuriousMind

I think you're misunderstanding the kind of handwave that happens here - essentially, degenerate perturbation theory consists of two separate problems: Computing the contributions from the non-degenerate states and computing contributions from states degenerate with it. What you've written down there is the logic for the non-degenerate case

see e.g. this for an explanation of the degenerate perturbation terms that doesn't have to divide by zero, as you seem to be worried about

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