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Q: Which guidelines does this question violate?

This question is one close vote away from closing, all four marking "Community-specific reason" - which is a new substitute for "Off-topic". The question clearly is on-topic. What guideline, then, is being violated? My guess for the 'real' reason is the opening statement, which some users found ...

Nobody recognizeable

The nuclear radius is $(r0)*a^(1/3) $ now if a=1 we are essentially talking about hydrogen nucleus . Then$r_0$ should be 0.8775fm but my book writes $r_0$=1.2 fm. Why?

bolbteppa

00:34

The eigenfunctions for a particle in a box are not eigenfunctions of the Hamiltonian of the harmonic oscillator

Charlie

00:58

@bolbteppa thank you for your help

skullpatrol

06:23

Time is running out for super symmetry and string theory

4

John Rennie

@skullpatrol it feels as though this debate is getting a bit tired now. People keep saying the same things, most of which are just their opinion and none of which can be proved.

skullpatrol

We should go into neuroscience :-)

ACuriousMind

11:50

Argh, this meeting has been postponed by half an hour every half hour for the last two hours and now it's on Monday

It's fine, I didn't want to use the last two hours productively anyway, I guess :P

Astyx

12:57

Hi, I am trying to compute the feynman rules for Gross Neveu as in this question : physics.stackexchange.com/questions/544358/…

I was hinted to compute $\langle\psi_\alpha(x)\bar\psi_\beta(y)\psi_\gamma(x')\bar\psi_\delta(y')\rangle$, which I proved to be $S_F(x-y)_{\alpha\beta}S_F(x'-y')_{\gamma\delta}$ minus the same term with indices permutated

I do not know how I could continue from here

Here $S_F$ is the Feynman propagator for fermions

bolbteppa

13:44

That post is not really asking about the Feynman rules, compare e.g. to QED the Feynman rules in momentum space are just things like the propagator and what you attach at a vertex

Astyx

I'm having trouble understanding how to compute what to attach at a vertex

bolbteppa

@ACuriousMind this paper (sec. 2) seems to say that the Henneaux view of primary first class constraints generating gauge transformations is wrong (even explicitly references the book)

In qed in momentum space it's like $-i e \gamma_{\mu}$ at a vertex, where the interaction term in the Lagrangian is $-e A_{\mu} j^{\mu} = -e A_{\mu} \overline{\Psi} \gamma^{\mu} \Psi$, so in GN it's probably $\pm ig^2$ right

Skimming the GN paper it seems this is more or less how they do it

Astyx

Could you please link the paper you're talking about ?

bolbteppa

13:59

Gross–Neveu model

The Gross–Neveu model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu as a toy model for quantum chromodynamics, the theory of strong interactions. It consists of N Dirac fermions, ψ1, ..., ψN. The Lagrangian density is L = ψ ¯ a...

First reference

Astyx

Thank you

ACuriousMind

@bolbteppa A cursory reading does not convince me because the author demonstrates the "inconcistency" without considering the full extended action. The gauge transformation generated by the constraint acts not only on the field variables but also on the variables of the extended phase space. When you look to the reduced phase space you need to be careful. In particular eq. (1) is wrong, cf. eqs. (19.13) in H/T's book. Using (19.13), the variation of $F$ correctly works out to zero.

The author is essentially trying to use (19.11) for only $\epsilon^1$ on the reduced phase space where there are no longer two different $\epsilon^i$.

Slereah

@ACuriousMind Why is a presheaf a functor on the opposite category?

rather than the category itself

ACuriousMind

This of course is inconsistent - in the passage to the reduced phase space, $\epsilon^1 = \epsilon^2$ needs to be imposed, as H/T correctly write.

@Slereah The arrows are the wrong direction

Slereah

Wrong direction for what?

ACuriousMind

14:08

For the categorical presheaf to coincide with the topological notion of presheaf when considered on the category of open subsets of a space with inclusions as morphisms.

Sheaves (at least for me) are much easier to understand if you first understand how they work in their topological incarnation

That is, forget all about categories and consider (pre-)sheaves as assignments of certain data (e.g. rings of functions, or vector spaces) to open subsets of a space

Then, since you already are familiar with bundles, understand that bundles and sheaves are the same idea of local data gluing together in some way - every bundle defines a sheaf by assigning to each open subset the space of bundle sections over it

Conversely, the gluing data that differentiates a pre-sheaf from a sheaf is exactly the gluing condition that glues together a global bundle from its local trivializations

Slereah

What field are topological presheaves discussed in?

ACuriousMind

That kind of presheaves are often not very useful (precisely because they don't define a bundle, hence define no topological space), I don't think anyone discusses them standalone before jumping to sheaves

It's kinda like how much more people discuss groups than semigroups :P

In fact the only context off the top of my head in which I've ever met examples of presheaves is in constructions where you define some operation on sheaves and the naive outcome is only a presheaf, but then you sheafify and that's it.

bolbteppa

This course on QG references the paper and "ABC before D: Issues with Dirac's view on first class constraints and gauge transformations" so maybe there is some technicality

I'd say it's fine to ignore to a first approximation anyway

@Slereah holy shi... I think I'm convinced Category theory is useful after reading section 1 of this, you're implicitly using category theory everytime you think about QM

A symmetric monoidal category is an obvious construction, what's happening...

ACuriousMind

14:24

By that notion, you're "implicitly doing category theory" every time you do math (not saying that viewpoint is wrong, just saying it's not special to physics or QM at all :P)

Slereah

That's the npov

but then you could say the same of any foundation of mathematics

you're implicitely using a Turing machine!

ACuriousMind

for instance, classical physics is also happening with a monoidal operation combining state spaces, just the monoidal operation is the direct product and not the tensor product

bolbteppa

My hope was that category theory would turn subjects like algebra, topology etc... into one subject where one category proof covers some theorem in each subject (e.g. Cayley's theorem is just Yoneda's lemma) but I'm not smart enough to see how to do that yet or if it's even possible, but I don't see why it shouldn't be and why it wouldn't let you find new theorems in new subjects by just looking for analogues

ACuriousMind

I'd say it more makes it possible to view the subjects through a unified lens, e.g. many "existence and uniqueness" proofs are essentially proofs that the category under consideration "has" certain universal objects

bolbteppa

Category theory could be the 'string theory' of mathematics :p

Slereah

14:31

I'm afraid you'll have to read some nlab for that!

also set theory already does that :p

or any other foundational system

ACuriousMind

I mean, I'm definitely a fan of e.g. describing constructions in term of universal properties. They capture the essence of what a "direct product" or "direct coproduct/sum" really is much better than specific definitions interior to a specific category, imo.

But that doesn't mean I find everything clearer or more obvious in categorical language. For instance, as I said, I find sheaves much easier to understand in their specific incarnation on topological spaces than in the abstract setting

Slereah

Well

I guess I'd better read more algebraic topology!

@ACuriousMind I do wish the category people would more often attempt to do actual diagrams of categories

ACuriousMind

Well, everything except the most boring categories are infinite, unfortunately :P

I think the "nicest" at least mildly interesting ones are rigid ones like vector spaces where you have exactly one isomorphy class for each cardinality and everything is a sum of the one-dimensional vector space

But something like the category of groups I would have no idea how to even begin to draw

Also, if you've ever seen how algebraic geometers "draw" schemes, you really don't want them to draw categories :P

I've tried to read explanations of why the generic point is a fuzzy ball at least a dozen times and it never really clicked

bolbteppa

14:48

Not sure if this makes sense: so if the states A, B, C, ... of an electron and an associated collection of measurements $f, g, h, ...$ can be described as a category, then (trying to motivate a functor) a change of coordinate system can be described by a functor since it not only changes what the states A, .. look like but also what the f, ... operations look like. Then we can say any coordinate transformation is 'isomorphic' to any other using a 'natural transformation'

ACuriousMind

so intuitive

@bolbteppa usually people think of the state spaces of systems as the objects of the category, i.e. the category is that of (finite, if we're doing quantum information) Hilbert spaces

Then depending on what morphisms you allow (e.g. unitary, positive, POVM-type, etc.) you get different ideas of what a transformation/measurement might be

bolbteppa

Hmm, re-reading the start of that pdf they are a bit fuzzy, I mean A and B don't seem to be different states, it seems like they could be saying A is the classical data of an electron before the measurement, then B is the classical data after the measurement, then C would be classical data after another measurement, and one could frame A, B, C... as the Hilbert space + different classical data to encode both the hilbert space but still make them look different...

qph.fs.quoracdn.net/main-qimg-631360a81abdc596f2905aabc4c46b82

Madness

Slereah

@ACuriousMind You can attempt to sketch an infinite category!

ACuriousMind

15:05

Sure, but as I said for anything less rigid than vector spaces I'd struggle

Slereah

Hm

What are the objects for $\mathrm{Set}$

sets?

ACuriousMind

yes

Slereah

In which case can you do, say, a sequence

Starting from the empty set

and using the various set building items

ACuriousMind

Okay, that's right. Let's say it like this - it's easy for something as structureless as sets or as rigid as vector spaces. It's really hard for the interesting middle ground like groups, modules, manifolds, etc

bolbteppa

So it seems like the reason GR is not a 'gauge theory' in the same way EM is a gauge theory, from a Lagrangian perspective, is that the EM Lagrangian has variables whose 'accelerations' are not determined, whereas for the GR Lagrangian the 'accelerations' are all determined, not sure if that's true haha

Slereah

15:24

Is it?

Isn't GR famously underdetermined

dumb as a rock

good evening everyone

https://imgur.com/a/BMbu2EH

Can anyone explain how does author makes the conclusion ?
"There is little chance of finding two parallel spin (triplet state) electrons together and the electrons act as if they repel each other"
9th line
in the image

Slereah

I do know that GR has that weird thing where it is all constraints

bolbteppa

The text describes that the wave function becomes zero when they are parallel so the probability of finding them parallel is 'practically zero'

ACuriousMind

@Slereah You mean because it has zero "real" Hamiltonian? Any system can be made (space)time-reparametrization invariant, i.e. made to consist only of constraints in the Hamiltonian, see e.g. physics.stackexchange.com/a/194925/50583

Eliminating the constraints gives you back a system with non-zero Hamiltonian. The Hamiltonian - zero or not - is not an intrinsic property of a physical system, but a matter of the choice of variables

bolbteppa

15:47

Seems like GR does actually have 'gauge' variables as described above (not fully sure yet), but:

"Relativistic theories of gravitation - such as Einstein's theory of general relativity - may also be considered as gauge theories. The bundle $P$ consists in this case of orthonormal linear frames (tetrads) of the space-time manifold $M$ and $G$ is the Lorentz group. Alternatively, one can take $P$ to be the bundle of orthonormal affine frames, in which case $G$ is the inhomogeneous Lorentz group.

There are, however, important differences between Einstein's theory and gauge theories such as electrodynamics or the Yang-Mills theory. First of all, the bundle of frames is soldered to the base $M$ whereas in other gauge theories the bundle is rather loosely connected to $M$...

The soldering results in the appearance, in theories of gravitation, of torsion, in addition to curvature, which occurs in any gauge theory. (Torsion is zero in Riemannian geometry, but being zero is different from not existing at all.) Moreover, the form of Einstein's equations of gravitation is different from the "generic" form of the field equations assumed in gauge theories.

The latter are derived from Lagrangian's quadratic in curvature, whereas the former are based on a linear Lagrangian. The possibility of constructing such a linear Lagrangian is also related to the existence of the soldering form on $P$."

Abhas Kumar Sinha

@bolbteppa sir, you've seen this? writings.stephenwolfram.com/2020/04/…

^a different intrepretation...

bolbteppa

Yes I made comments on it yesterday and the day before in here

Abhas Kumar Sinha

okay...

@bolbteppa Sir, Is there any way to write electromagnetic tensor without $F_{\mu \nu}$ thing?

bolbteppa

Not sure what you mean, but it's easy it's just $F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$ and you can easily derive it 'from thin air' by looking for the equations of motion of a particle in an electromagnetic field if you want a natural example of where it could come from

Abhas Kumar Sinha

@bolbteppa I totally didn't get that... :-O Where that $\partial_{\mu}A_{\nu} - \partial_{\nu} A_{\mu}$ thing came?

bolbteppa

15:57

What is $F_{\mu \nu}$?

Abhas Kumar Sinha

Electromagnetic tensor

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely. == Definition == The electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A,...

^this one I mean

ACuriousMind

@bolbteppa If you search for "solder form" or "soldered" in the chat transcript you'll find me or Slereah saying pretty much this on various occasions :P

bolbteppa

"A vielbein or solder form on a manifold $X$ is a linear identification of a tangent bundle with a vector bundle with explicit (pseudo-)orthogonal structure."

Vivek

@dumbasarock when the the spin of the two electrons are parallel, the space part must be antisymmetric so the complete wave function of the electrons that is fermions remains antisymmetric.

Slereah

it means it's a real vector and not a fake vector :p

bolbteppa

16:01

So using the word vielbein instead of solder form, the GR field is vielbeined to the base, while the others aren't, eh..

Vivek

now the author assume that the two two electrons are almost near to each other such that their space coordinates are equal because of which the space wave function that was antisymmetric becomes zero due to the negative sign

if you calculate the probability density quantum mechanically, that would be almost zero, meaning the two electrons with parallel spin cant be near to each other because of which the probability of finding such a case becomes almost zero and the case here specific is is the case of parallel spin

dumb as a rock

@bolbteppa but even if the wavefunction are nearly equal the triplets wavefunction are still additive so they can't be zero as they are symmetric.
it's singlet which becomes zero.
but text says finding triplet close to each other is not probable

Vivek

@dumbasarock the space probability, is zero,...

@dumbasarock means, the probability of finding the two electrons in such a case, with parallel spins, and that too close to each other, in unit volume of measurement

Slereah

@bolbteppa it's part of the tensor bundle, and not the tangent bundle of the principal bundle!

Vivek

look, the singlet state has a factor of root two...., i mean, that's a good probability than that of with almost zero

dumb as a rock

16:11

triplet is symmetric and parallel case.
the approximation will not result in cancelling out of the overall wave-function
thus when we will find the probability density with it we'll get a value.
singlet case is antisymmetric and antiparallel case
equating the wace-function will result in the overall wave-function in value zero.
thus probability density will result in zero too as wavefucntion itself is zero
so how come author says that
"There is little chance of finding two parallel spin (triplet state) electrons together and the electrons act as if they repel each other"

Vivek

@dumbasarock don't calculate the probability density from the complete wave function... i.e., don't consider spin part in it

dumb as a rock

finding the anti-parallel ones close should be less probable

I just want to know how being close results in triplet probability being less

ACuriousMind

@dumbasarock The text you quote explicitly says that the spatial wavefunction of triplets is anti-symmetric.

So when the two positions are close together, the spatial wavefunction - and hence the overall wavefunction - becomes close to zero.

What exactly is your problem with that statement?

dumb as a rock

but on previous page here it says the triplets are symmetric

ACuriousMind

It is probably talking about the spin wavefunction.

Vivek

16:17

@dumbasarock being close in a triplet probability is less because when a triplet is form both the spins are up that is is the spin part of the wave function is symmetric. show the electrons can have the overall wave function as antisymmetric, the space path must be antisymmetric

ACuriousMind

Even in your screenshot it says that at the top "there are three possible spin wave functions which are all symmetric"

bolbteppa

The wave function is anti-symmetric, $\frac{1}{\sqrt{2}}[\psi_a(1) \psi_b(2) - \psi_b(1)\psi_a(2)]$, if $\psi_a(1) \approx \psi_a(2)$ and $\psi_b(1) \approx \psi_b(2)$ then $\psi_a(1) \psi_b(2) \approx \psi_a(2) \psi_b(1)$ which is just a complex number whose value is the same value you get from $\psi_b(1) \psi_a(2)$ so subtracting them is basically zero which means the wave function is basically zero

ACuriousMind

You need to pay close attention to the difference between "total wavefunction", "spatial wvefunction" and "spin wavefunction"

Vivek

total = space . spin

total (anti for electrons, fermions) = spin (symm for triplet, because both up) . space (must be antisymmetric, i.e., negative sign, i.e., when electorns are close, ~ 0)

dumb as a rock

okay to make the total antisymmetric for triplet the space wave-function should be antisymmetric because the spin is symmetric already

bolbteppa

16:20

Unless I'm missing something, the point of that sentence is nothing more than explaining what happens when the electrons are close together so that the values in both parts of the anti-symmetric wave function are basically the same so you basically get zero on subtracting them

Vivek

@dumbasarock you got it

and remember, probability density, what is it?

dumb as a rock

@bolbteppa yes that is the meaning but the wavefunction used for puttin values was of singlet while the less chances were told for triplet that was bugging me

Vivek

probability density is the probability of finding the particle at a particular position or in a region of Space at a particular time

dumb as a rock

@Vivek absolute value of square of total wave-function

ACuriousMind

@dumbasarock I don't understand that sentence(?).

Vivek

16:22

so see, it has nothing to do with spin...

only space

ACuriousMind

@Vivek No, it's not. A probability density is not a probability, its integral is a probability.

Vivek

on the other hand, there's a good factor of root 2, hmm, makes a good probability

dumb as a rock

I am definitely sure I am wrong but where ?
@Vivek what you are describing is something true but different

ACuriousMind

And you can perfectly well consider a probability density on the combined classical space-spin state space $\mathbb{R}^3 \times \{-1/2,1/2\}$ so that it is a proper probability density on states and not only on the spatial part.

Vivek

@ACuriousMind yp, got confused

ACuriousMind

16:26

@dumbasarock I can't tell you where you're wrong because what you said is not a sentence I could parse.

dumb as a rock

I am explaining the doubt once again from beginning.
Take a look at this image
https://imgur.com/a/cVWitON

ACuriousMind

Those are the spin wavefunctions, yes.

Vivek

@dumbasarock what does that beta indicates? spin?

dumb as a rock

now going back to assumption, for close electron the ${\beta ^(+)}_1 = {\beta ^(-)}_1$ and ${\beta ^(+)}_2={\beta ^(2)}_2$
Those equal signs are approximately equal ( I forgot latex code for approximately equal)
so which of the state goes zero in above photo

it;s the singlets

ACuriousMind

No.

The $\beta$s are not the total wavefunctions, they are the spin wavefunctions

You can't plug a position into them

dumb as a rock

16:32

now I got it

ACuriousMind

Your total wavefunction is a product $\beta \cdot \psi(x)$, where $\beta$ is one of the four expressions in that image and $\psi$ is the spatial wavefunction.

dumb as a rock

thank you very very much @Vivek @ACuriousMind

I just read the line "in order to have total wave-function antisymmetric...."

in the image I posted earlier

@Vivek Thank you very much for this @Vivek

Vivek

17:01

@dumbasarock wlcm...

bolbteppa

17:17

Imagine trying to work out the Hessian of the Einstein-Hilbert action...

Slereah

Do it in $0+1$ dimensions

bolbteppa

I really can't imagine directly working out the Hessian or examining the EOM directly for the metric variables to figure out if some of the accelerations are missing to show whether the EH action is really a constrained Hamiltonian system, but the way it seems to be done is:

"This action is invariant under general coordinate transformations (or passive spacetime diffeo-morphisms, $_P \mathrm{Diff}(M)$) and immediately, as the passive diffeomorphisms are a subset of the local Noether symmetries one can conclude, by Noether’s second theorem, that GR is a singular system and the equations of motion will have this symmetry. We therefore expect to find a constrained Hamiltonian system once we have completed the canonical analysis."

Noether's second theorem

In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least action. Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differentia...

Slereah

18:18

btw @bolbteppa

cds.caltech.edu/~marsden/bib/1992/05-GoMa1992/GoMa1992.pdf

Pretty good article

bolbteppa

The way I understand (1.1) and (1.2) of that article is that (1.1) is derived by varying the coordinates $x'^{\mu} = x^{\mu} + \varepsilon$ directly in the Lagrangian and then requiring it to be invariant, while (1.2) is derived by asking how $g_{\mu \nu}(x)$ changes under $x'^{\mu} = x^{\mu} + \varepsilon$ and then requiring the variation to be invariant, so both really come from the same underlying translational invariance idea

I can't imagine carrying around the rest of that paper in my head when the derivation using what I said above is so simple :p

Slereah

18:58

@bolbteppa The interesting part is 1.3 though!

bolbteppa

The (easy) proof of (1.2) shows why it reduces to (1.1) up to a total derivative and so takes the form (1.3)!

JingleBells

The simpler quadratic formula | Lockdown math ep. 1

►

bolbteppa

mathonline.wikidot.com/geometric-proof-of-the-quadratic-formula

JingleBells

19:13

@Slereah, how are you? I don't know what happened between us but I want to apologize and I want peace. I'm sorry for whatever I've done. This is my last attempt to make things right, if you decide to respond, it would be great, if not, I will not be trying to contact you anymore.

bolbteppa

45

Q: Bourbaki's definition of the number 1

According to a polemical article by Adrian Mathias, Robert Solovay showed that Bourbaki's definition of the number 1, written out using the formalism in the 1970 edition of Théorie des Ensembles, requires 2,409,875,496,393,137,472,149,767,527,877,436,912,979,508,338,752,092,897 $\approx$ 2.4 $\...

set-theory lo.logic ho.history-overview bourbaki

"(I conjecture that Bourbaki's proof of 1+1=2, written on paper, would not fit inside the observable Universe.)"

skullpatrol

21:09

@bolbteppa the OP has just posted an answer

There seems to be a rule of thumb in the logic community: if Solovay says it, it's correct. He seems to have garnered the reputation of being very, very careful. For example, he was the one who alerted Nash to a hole in his (Nash's) embedding theorem. — Todd Trimble ♦ 2 hours ago

bolbteppa

Interesting reading about this stuff

37

Q: Were Bourbaki committed to set-theoretical reductionism?

A set-theoretical reductionist holds that sets are the only abstract objects, and that (e.g.) numbers are identical to sets. (Which sets? A reductionist is a relativist if she is (e.g.) indifferent among von Neumann, Zermelo, etc. ordinals, an absolutist if she makes an argument for a priviledge...

mathematical-philosophy ho.history-overview set-theory bourbaki

19

Q: Is the Bourbaki treatment of Set Theory outdated?

On page 5 of the following write up, the author asks why the Bourbaki did not notice that their system of Zermelo set theory with AC was inadequate for existing mathematics. Throughout the rest of the discussion, the author asserts that the Bourbaki group never acknowledged Godel's results on inc...