@ACuriousMind In fact, I don't think that equation can even be true. For instance, if $f(x)=|x|$ (Euclidean norm), then $f^{-1}(1)$ is compact (it's a sphere) and since $L$ is linear it is continuous, so $L(f^{-1}(1))=\mathbb R^{m-n}$ is definitely wrong

@ACuriousMind Some linear algebra musings: Let $V,W$ be vector spaces and $L:V\to W$ be linear. Conjecture: If $M\subset V$ (set theoretic subset, not linear subspace) and $L(M)=W$, then $M$ is a vector space.

Let $v,w\in V$ such that $L(v)=x,L(w)=y$. Then $\exists c\in V$ such that $L(c)=ax+by=L(av+bw)\implies c\cong av+bw$ $\mod\mathrm{ker}\,L$.

So, $M/\mathrm{ker}\,L$ is a vector space?

> Barack Hussein Obama II (reportedly born in Honolulu, Hawaii on August 4, 1961)

Conservapedia is AWESOME

Yeah, that's definitely not true that $M$ is a vector space.

But $M$ with the kernel modded out is definitely one.

@ACuriousMind math.stackexchange.com/questions/1815263/…

> According to a NBC News poll, from August of 2014, Obama's approval rating was a dismal 40%[14], showing that six in ten Americans disapproved of his liberal policies. Given that liberals frequently engage in deceit, the actual approval ratings could be much lower.

Haha

@0celo7 lol

@user507974 it's not wrong

this is an awesome site

liberals are so much more annoying to engage in a debate with than non-religious conservatives

religious is like half-half

@user507974 why would engaging with a rational human being be annoying

@0celo7 which one is the rational?

YES conservapedia.com/Causes_of_Homosexuality

I'm probably in violation of some hate speech rule at my school right now

you know, im not gonna lie. I still dont know which is politically correct to say, that homosexuality is nature or nurture or both. Apparently even trying to answer that question is bigotted

i wouldnt be surprised if saying "i dont know" would be considered hate speech

How about "I don't care, I'm here for the lulz"

your bigotted for not caring for the plights of the homosexual you evil funny person

cuz now humor is problematic

and (i say with respect for the respectful parts of your institution) fuck your school if they cant take a joke

lol that site is not a joke

the dude who runs it is 100% cereal

i just dont have any respect for universities impinging on free speech

@0celo7 maybe, im about 50-50 on that

but in all seriousness, trying to engage in an intellectual debate with the average liberal is like watching a person without teeth try to eat popcorn

conservapedia.com/Feminism

they literally cannot digest facts and change their opinion on something

> often condemn the God-given order of gender roles, as laid out in the Holy Bible

God dammit

Why are so many conservatives religious

@0celo7 traditionalism

conservapedia.com/Quantum_mechanics

still though that part is upplayed in the media

> Quantum mechanics is predicted in several additional respects by the Biblical scientific foreknowledge.

Haha

what uni do you go to (or at least how conservative/liberal is it)

cf. e.g. my profile

im guessing its pretty conservative?

no

once you get urban university and southern its like flipping a coin

much more conservative probably than a california university

we have an "office of diversity" that paints rainbows and shit on our sidewalks

and they tried to ban gender-specific pronouns but the governor told them to fuck off

(governor is technically the chair of the school board)

its pretty bad in CA

thank god im not at Berkeley though, wouldnt be able to stomach that probably

I can't stomach knowing OF Berkeley

haa

but yea, when you are on student apartments board you here so much of this diversity stuff it makes you sick of it

and I promise at least everyday i get one stupid email from some organization for LGBTQWTFOMGBBQ1337 whatever else

BBQ?

Wait, don't tell me

We'll probably get banned

I had the misfortune of hosting one of these diversity workshops in our complex so I peeked in on it

what the hell is that

I know our RAs have to do that, one of my friends is an RA

Like 5 seconds after they finished defining what a micro-aggression is (which took WAY TOO LONG), and how it is a bad thing to generalize people and their experiences and say that just because someones x gender or race they will do something, they go on to say in the next slide how men all men harass women by talking to them because they want to bang them of something like that

I like harassing my gf and she likes being harassed

We're both terrible people

like im not kidding, i wrote this down somewhere it was so hypocritical and ironically done in the presentation

She enables my harassment

I decided to call him out on that one then after he says, you know, i didnt think of it like that maybe youre right but lets move on, some putz in the audience says that by definition it cant be applied to straight males because they come from a position of privilege

and they move on

correct

I was once told blacks cannot be racist to whites because the people they are allegedly being racist against are in a position of power/privilege

@0celo7 its just sad to see how much they've perverted the language

ive always said the english language is used to conceal meaning (ty George Carlin for this great line)

I don't get Carlin

is he a liberal?

he might just be a massive misanthrope

or he was just here for the ride

but probably the misanthrope was part of it

i mean he said he loves individual people

he just understood how sickeningly destructive group think is

i mean for as much as liberals like to harp on being secularist feminism has kind of become another religion

and racial equality movements

and just like religions you cant hope to accomplish the goals but disagree on opinions, anything is excommunication worthy an offense

@0celo7 he was probably just an anti-neocon if you think about him politically

yeah he was isolationist

which ironically is functionally a libertarian, though he would not get along with a lot of them

hmm?

@MAFIA36790 do $t\mapsto -t$ and see what happens

(I think)

(not sure exactly what the context is)

uhhhh

@MAFIA36790 well how is he defining 4-vector?

ah, ok, and how do these coordinates transform

and that is?

so how does $(v_t,v_x)$ transform under that transformation

@MAFIA36790 i know

@MAFIA36790 wait, where are you reading this?

where exactly

I found it

@MAFIA36790 ah, this is what he means

(I think, I don't know any relativity)

What he means is that

$$(\frac{dx}{dt},\frac{dt}{dt})$$

it does NOT become

$$(\frac{dx'}{dt'},\frac{dt'}{dt'})$$

I think, lol

This is not very clear

I think so...

Let's see if I have any relativity books on my shelf

just one

@MAFIA36790 Mr. @JohnRennie will know for sure

I don't know any physics I'm afraid

@0celo7 Just maths and engineering? :-)

@JohnRennie Not even that

I can pick my nose successfully

Well it's 05:45 here, and I'm not even awake enough to scratch my bum, let alone pick my nose.

I'd probably end up scratching my nose and picking my bum

depending on how clean it is, that might be an enjoyable experience

Ugh, I feel sick now

@MAFIA36790: what were you asking?

I suspect he just means that the $t$ coordinate is being treated differently to the other three.

In the four velocity all four coordinates are differentiated wrt $\tau$ so this is symmetric.

@JohnRennie Poor you

@JohnRennie I like how in the mathematical literature $t$ is the proper time

@0celo7 that seems an odd convention. What do they use to denote coordinate time?

@JohnRennie they don't?

coordinate free

Ah, yes. One day I must get comfortable with the coordinate free approach.

@JohnRennie I can help you

@JohnRennie and if I needed to use coordinates I would use $s$ for proper time

I suppose having both proper time and proper distance is a historical artefact, though to be fair both have their uses.

proper distance?

that's a physicist thing

never shows up in math

no respectable mathperson writes $\mathrm ds^2$

@JohnRennie I might use $l$ too

Is it really necessary to $\mathrm d$ everywhere? @MAFIA36790 even goes back and edits it into my old posts :-) Does anyone really care?

@JohnRennie no

when I need to do a lot of them I won't bother

see above

@MAFIA36790 I don't care either way. I'm just surprised people care enough to type the extra eight characters :-)

using $d$ as a variable is still poor form

when I see $d$ I think of a topological metric

@MAFIA36790 o/

yo

Which volume of the Feynman lectures are you reading?

I see.

@JohnRennie when are you going to read a "mathematically sophisticated" GR book and what will it be

@0celo7 the trouble is that to the extent I'm a physicist at all I'm a phenomenologist not a theoretician.

I'm interested in what GR can tell me about the world, but less interest in the details of the theory itself.

I went through a brief phase of being really interested in maths for its own sake at about your age, but these days I see it as just a tool

What was your specialty in Physical Chemistry?

milk @Sᴋᴜʟʟᴘᴇᴛʀᴏʟ

@JohnRennie what % milk do you drink

Computational chemistry. I did my final year project on distributed multipole analysis for the decsription of scattering.

@0celo7 full fat.

@JohnRennie wow

Interesting.

If you look at the calorific content then the difference between full fat and skimmed is small in absolute terms unless you drink gallons of the stuff.

oh really

And skimmed milk leaves your coffee a chaming babyshit brown colour.

I am in need of milk

I will investigate this

@JohnRennie coffee?

@MAFIA36790 You're a man. Without sugar, too?

Coffee is an essential part of my wake-up strategy

@JohnRennie Damn, you sound like an old man

@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ less interesting than you might think. It was mostly calculating the electron density and fitting it with a multipoles not centred at the centre of the molecule. It turned out not to be very useful. At least I don't know of anyone who used it in real calculations.

@MAFIA36790 correct on all three counts :-)

But you forgot to add devastatingly witty and handsome :-)

This chat is a brilliant way to avoid doing any real work :-)

But then, what does it mean for something to have below zero kelvin that is not negative kelvin (i.e. not the type where $\frac{\partial E}{\partial S} < 0$)...?

@0celo7 Constructive does not mean finitist

and even finitists admit an infinite number of naturals/primes

\o

@JohnRennie the saddest thing was in the comments for the Einstein ring was in the comments, someone was like "Astrology is great. Keep it coming"

Just start throwing scorpions at people everytime they mix up the two

watch the problem disappear in a big %$*#ing hurry

:-/

o/ @Secret

Obivously that's a last resort...

Most people (even cranks I have met so far) are reasonable enough to not go into mad anger without losing focus on the problem

@Secret worst case scenario you get to watch a fight break out

Well... then I will quickly leave the place (or if it is really serious, start rallying the bystanders when in a safe situation and call the police)

@user507974 you couldn't make it up :-)

In other news, I also read NewScientist, but I also read Sciencedaily and Scientific American

these 3 sources are how I maintain my broadness in the science despite going deeper like typical researchers do

One thing to be aware about Sciencedaily, though, is that for some physics articles, they like to make eye catching but inaccurate titles. Other times, they like to recycle some previously reported research

A guide to Sciencedaily is that for each week it often has some focus on some genre. E.g. in one of the weeks you will see most articles talking about e.g. skymions, other days they wil be talking about Zika virus

@Secret i really dont like newscientist

at least a few sections of it really show off their donors interests

sciencedaily.com/releases/2016/06/160602083602.htm

@Secret all your boson are belong to us?

lol

25 million people

Hi everyone! Why the photon field does not couple to the axial current in the massless fermion limit? Just consider the dirac lagrangian with m=0. This is chiral symmetric $U(1)_L \times U(1)_R$. Then, we have two currents. The sum of the currents $J_L + J_R = J_V$ is the vector current that is coupled to the photon when we gauge the theory. But, we have another current that is the axial one $J_A = J_L - J_R$. Why this current does not couple in the gauged theory?

you forgot your $

A massless fermion field is basically two fields

In effect, it does couple to the "axial current"

how do you show this?

But of course, the Maxwell equation is still gonna be $\approx j_R + j_L$

If you want a gauge theory that couples to the axial current, you need a different gauge

It seems strange because QED conserve parity and an axial current transforms with a minus sign

$\psi(x) \rightarrow e^{i\gamma^5 \alpha(x)} \psi(x)$

That's the axial gauge

try to be more specific

what you wrote is an axial transformation

the axial gauge should be a condition on the photon field

that is $A_\mu^3=0$

Well that's a gauge transformation

You won't couple to the EM field with the regular $U(1)$ gauge

@0celo7 Why not? It is nowhere claimed that $T_x M$ is a vector subspace.

Hey @ACuriousMind

Heyhey

Tell me

Do you know what mathematically the difference is between a commuting spinor field and an anticommuting one

@0celo7 I think you're implicitly supposed to choose $L$ surjective.

Is a commuting spinor field still a Clifford/Grassman algebra

@Slereah Sure, it's the same classical field

Whaaaat

The commuting/anti-commuting property is something you impose on its quantization

I thought the Grassman algebra was specifically to make it anticommuting in the quantum limit

I mean I'm talking about path integral quantization here

There is no CCR defined

Oh...let me perhaps think a bit more, then

I have this suspiscion that if we have a commuting spinor field, then maybe it's not a part of a Grassman algebra, and maybe not a Clifford one???

But that's all vague hunches

Wait, when even is a "commuting spinor field"?

Yeah that's part of what I wonder

You can't have a spinor field that commutes and have a well-defined quantum theory, that's the whole reason people turned to the anti-commutator, no?

Well you can have a commuting spinor field in QFT

It just has a few problems

what you can have is anticommuting scalars (e.g. FP ghosts)

No lower bound of the Hamiltonian and break of causality

Also I think the path integral diverges

But I'm interested more classically here

@Slereah That's a called "not having a well-defined quantum theory" :P

I'm wondering the exact link between the spin group, the clifford algebra and the grassman algebra

Like is it mandatory that the spin group's rep is associated to a clifford algebra?

@Slereah Classically, it's just a Grassmann-even valued field transforming as a spinor, then

But if it's Grassman, isn't it anticommuting

@Slereah Grassmann-even.

Like are we not imposing anticommutation because of the quantum theory

The anticommuting things are Grassmann-odd.

Hm

@Slereah No. E.g. the Weyl spinors are not reps of the Clifford algebra

It's the Dirac spinor (and only the Dirac spinor) that is obtained as the unique irreducible representation of the Clifford algebra

Is there a direct link between the clifford algebra and the grassman algebra, too?

Isn't the clifford algebra a type of grassman algebra (or one of its subalgebra)

You can see the Clifford algebra as a deformation of the Grassmann algebra (the Grassmann algebra is a Clifford algebra for a "metric" that's just zero)

I don't think the appearance of the Clifford algebra for the Dirac spinor and the Grassmann algebra for anticommuting things in general are related, though

@FrancescoS The gauge field always couples exactly to the current that is conserved by the global version of the gauged symmetry.

Know what couples to the axial current tho?

Torsion.

The axial current is not conserved anymore as soon as you gauge the vector symmetry due to the ABJ anomaly, so if you try to gauge the axial symmetry, you get an inconsistent theory (gauged symmetries must not be anomalous)

why do spinors have so many groups and algebras involved

"Clifford (1878) introduced his ‘geometric algebras’ as a generalization of Grassmann algebras, complex numbers and quaternions"

good ol' Clifford

Was he related to Clifford the big red dog

Are there any use of spinors in physics outside of fermions, by the way

Oh wait

you can do GR with spinors

@ACuriousMind But in the massless limit, the global symmetry group is $U(1)_V \times U(1)_A$ not? So there are two independent currents which are conserved

@FrancescoS No, there aren't. Gauging the vector symmetry makes the axial symmetry anomalous.

That is, you are not allowed to make both of these symmetries gauge symmetries because gauge symmetries must not be anomalous (anomalous global symmetries are fine)

Anomalies can be cool, tho

The scale anomaly lets you violate the ANEC!

Well, but an anomalous gauge symmetry is just doomed because the unphysical d.o.f. no longer decouple

So I'm trying to show that Robert Low's spacetime does indeed feature compactly generated CTCs without any closed null geodesics

But this involved the most awful thing

BASIC GEOMETRY

It's just showing that lines will not intersect!

It is quite odd that his solution is so simple

It's your basic Minkowski surgery wormhole, but

rotated 90°

@ACuriousMind It actually is

@ACuriousMind Mike Miller helped me figure it out

It's retarded

for some reason "$f^{-1}(y)$ corresponds, under $F$, to the hyperplane $\{y\}\times\mathbb R^{m-n}$" does NOT mean $F(f^{-1}(y))=\{y\}\times\mathbb R^{m-n}$ like any reasonable person would think

rather, it means $F^{-1}(\{y\}\times\mathbb R^{m-n})=f^{-1}(y)$

@ACuriousMind Ok perfect, I understood that was the axial anomaly the problem. Thanks

@ACuriousMind so, the local axial transformation is not a symmetry of the gauged theory, right?

Because the gauge field transforms only under the vectorial gauge U(1); right?

@FrancescoS No, it is not

@ACuriousMind it is not right or it is right what I said?

eheh :)

It's correct that the local axial transformation is not a symmetry.

The global one is, but only classically - it's anomalous in the quantum theory.

ook

yes, i know

perfect. thank you

What is the axial anomaly, btw

What is not symmetric in the quantum theory by an axial gauge transformation

@Slereah In the path integral formalism, it's the fermionic measure that is not invariant.

Dang it, Grassman algebra again!

So, the global axial transformation in quantum theory is again not a symmetry?

In the canonical formalism, you just find that a certain triangle diagram that corresponds to the axial current conservation doesn't vanish

I could check I suppose

My Peskin is right next to my keyboard

But Stephani is on top

And my wrist is sprained

So it's hard to lift it

weak