The h Bar - 2018-07-21

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2:31 AM

I found Wolfgang Pauli and 2 men

cds.cern.ch/record/1221536

danielunderwood

3:23 AM

Defining the Lagrangian as the integral of the density doesn't place any restriction on the Lagrangian itself does it? Well I mean not as long as you don't want some insane function for your Lagrangian

4:03 AM

arxiv.org/abs/1608.03026

It's all glyphs

@Secret lol is this real?

in Mathematics, 59 mins ago, by EnjoysMath

0

Q: Would this suffice in a visual type theory to define a list?

See the image. I got that from: wikipedia article. I want to make a visual type theory so that we aren't stuck comprehending pure text for eternity. Also, is the abstract type List a product of some sort? The diagram doesn't indicate this since it's based on a product diagram of $E \times L$...

math-software visualization type-theory data-structure

Visual type theory is not only real, it is not new

oh

hmm interesting

4:52 AM

@Chair would you mind engaging in a convo about the future of quantum information and quantum computing?

user351417

@Cows I wouldn't mind at all... They're both pretty interesting topics. But I know very little about them, so you'd be doing most of the talking.

Well the talk would not really be technical, unless perhaps in passing in some cases

I am curious as to whether quantum computation and programming will be an in demand occupation in the next 5 years

I am also curious as to whether the future of HEPTh would be in quantum information

I have noticed that a lot of companies are either publicly or secretly investing in quantum computing research and engineering,

and it turns out that I am trying to make sure I not just have fun studying but have a legitimately lucrative option should I choose to advance studies in the future

. . . . some technical thoughts

It seems to me

that the basic idea is that it one abstracts away the physical elements and thinks in terms of signals and computation, you can build an analog language for describing nature, and simulating things

I have some basic knowledge of trivial QC algos

but don't think I understand how to derive one, I don't think there is a set strategy for doing this

@Chair what are your thoughts?

just to be clear, I just try to learn a lot of these things in my free time , and claim no expertise

typically as per my earlier mention . . . . one finds some $U_f$ for a one shot eval

sorry if my lang is a bit abstrus (it's my personal lang ) for this

hmm well I guess I've spoken nonsense let me just retire to my room lol

5:51 AM

QC is done in a Hilbert space like in quantum mechanics

6:21 AM

Hmm yes

14 Year Old programmer Tanmay Bakshi talks about Quantum computing, Machine learning and Power AI

This kid makes makes me wonder my place on this planet lol

Articulate, smart and set for life

His shirt says code, although he talked qm too. I feel like he is sending a message :P

This kid is so good I am not sure how he become this good this young

Can somebody say how to write triangular delta in mathjax

I think he works at ibm

He is an IBM Q programmer

I have an IBM Q account but the last time I wrote a program on it was several years back . . maybe over 3. . Lost track

I can't seem to understand what people mean when they say a cc can't sim a qc. I think what they mean is that a cc can't sim a qsystem

@user187604 \nabla = $\nabla$ or \Delta = $\Delta$

there's also \triangle: $\triangle$

That kid is increadible. Wow!!! He talks about the high level details naturally and he does have the code up. lol he is running p100 or something in parl to a jupyt session.

Tf to keras then run in ☁️.

6:37 AM

@danielunderwood Well, strictly speaking it imposes the requirement that the integral should exist.

His epochs < 1s .

@JohnRennie is your programming job done ?

@user187604 still working I'm afraid

@JohnRennie you will do it quickly. I'm sure man. Anyway bye have a nice day.

Wow

How to pump gas without arms

Life

how to wipe without arms

People wanted to know how she wipes herself

Quadriplegic Morning Routine and WORKOUT

7:23 AM

user image

8:02 AM

@vzn another wave of Lisi-style stuff quantamagazine.org/… & motls.blogspot.com/2018/07/… & twitter.com/preskill/status/1020433911355125760

Perfect excuse to re-post motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html

'The author is not constrained by any old "conventions" and simply adds Grassmann fields together with ordinary numbers i.e. bosons with fermions, one-forms with spinors and scalars, neglecting any traces of dimensional analysis, too. He is just so skillful that he can add up not only apples and oranges but also fields of all kinds you could ever think of.'

apples +oranges = orapples

Orangeapple unification of the theory of fruits

Anonymous

8:19 AM

-5

Q: Slove it with your Mind - #Paul Ardaji

Given a triangle ABC. BL is the bisector of angle ABC, H is the orthocenter and P is the mid-point of AC. PH intersects BL at Q. If ∠ABC=β, find the ratio PQ:HQ.If QR⊥BC and QS⊥AB, prove that the orthocenter lies on RS. https://i.stack.imgur.com/0tMxm.png By Paul Ardaji

mp.mathematical-physics

Anonymous

Lol, seeing something like this on MO after a long time

@ACuriousMind Felsager's book, eq. 11.34, defines a symmetry of an action in terms of push-fowards, would be surprising if what we went through yesterday wasn't perfectly right at this stage

It's amazing how easy it is to just use things like pushforwards without even realizing it, 'just plug in the transformation'

8:50 AM

"$E_8$ cannot be a grand unified group because it only has real representations which is not good enough to create chiral fermions", this is basically also a comment in GSW's appendix, seems like Lisi had no idea or ignored it or thought his magic bsrt-extended connection made the problem go away or something

6 hours later…

danielunderwood

2:31 PM

@ACuriousMind that's going to generally be the case in physics though, right? The only thing I can think of is trying to calculate the action through an infinite barrier of some sort.

@danielunderwood In field theory, there are frequently integrals over all of spacetime and no one is explicitly saying the fields should have compact support or be integrable...

danielunderwood

2:45 PM

Ahh right I forgot about the integral over all space part. I was thinking just in a finite interval. Is this generally assumed to be the case or is there a term that I should look for when searching for references?

I'm not sure there are references. Physicists ignore the issue and mathematicians usually restrict to stuff like functions of compact support. It's a technicality without any meaningful implications that I know of.

danielunderwood

2:59 PM

Aw man. I was hoping for some weird exception. Although I don't even know if you could solve the variational problem in that case if you have space derivatives. Certainly beyond my current knowledge if you can.

3:43 PM

@bolbteppa lol, my dad brought up the octonion article at dinner (he likes to look around for such stuff)

and my reaction was just bleeeh

@bolbteppa lol thx for tip she fits right in with beach bum Lisi + his surfing maybe they should consider dating each other! that is if she turns me down! :P ps you missed some big news about a experimental subquantum anomaly/ finding, any comment?

> After breaks from school spent ski-bumming, bartending abroad and intensely training as a mixed martial artist, Furey later met the division algebras again in an advanced geometry course and learned just how peculiar they become in four strokes.

in theory salon, 1 min ago, by vzn

in The h Bar, Jul 17 at 23:14, by vzn

Improved Noninterferometric Test of Collapse Models Using Ultracold Cantilevers-- experimental anomaly detected/ measured wrt standard QM favors Adler CSL emergent QM model https://www.reddit.com/r/quantum/comments/8zpzqa/improved_noninterferometric_tes‌t_of_collapse/

just for reference, that paper was published last September

Feynman's Lost Lecture (ft. 3Blue1Brown)

@Semiclassical yep, seems to be a very big deal on paper (how many massive theories pursued by (tens of) thousands have no experimental evidence?) but uncovered by any popsci writers so far. maybe octonions + orange apple unification are sexier :P

I have a hard time getting excited about octonions. Not sure why

wouldn't orange-apple unification just be "fruit" ? :P

oh dangit I didn't look at the starboard

4:00 PM

@Semiclassical yep you defn have a natural knack for physics... so go bananas with it! :P

danielunderwood

@Semiclassical they always make me think of some bizarre combination of octopus and onion

Though that's probably because I just hear about them and don't know any of the mathematical details

1 hour later…

5:22 PM

@danielunderwood presumably a generalization of quaternions which are basically just 4-vectors with vector algebra rules. maxwell used quaternions. now wondering, do they have some application in GR with x,y,z,t (minskowski) coordinates?

16

Q: Quaternions and 4-vectors

I recently realised that quaternions could be used to write intervals or norms of vectors in special relativity: $$(t,ix,jy,kz)^2 = t^2 + (ix)^2 + (jy)^2 + (kz)^2 = t^2 - x^2 - y^2 - z^2$$ Is it useful? Is it used? Does it bring anything? Or is it just funny?

special-relativity vectors metric-tensor complex-numbers

Quaternions in University-Level Physics Considering Special Relativity/ Horn arxiv.org/abs/physics/0308017

in Mathematics, 24 mins ago, by Ted Shifrin

Well, conceptually, our example we've been talking about is very different. You can't add something to "fill in a hole," but the phenomenon is analogous — one "line" is the upper bounds for the other, and, vice versa, the other is the lower bounds for the second.

currently trying to build a partially ordered set as a container so that the hole is so big it can fit in an inaccessible without overflowing

Infinities are super fun once you know how to work with them

@Secret btw wondering how you ran across the amazing Vinante paper if you have any story behind it, it escaped my own radar for ~¾yr journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.110401

well simple, because NewScientist mentioned about it in its last week's issue

@Secret oh very cool what was the title of the article? so maybe popsci did discover it... bought an issue of that recently & installed the app on my ipad...

The whisper of reality

5:34 PM

lol wow awesome so new scientist is all over it already, feature article... Is this our first clue to a world beyond quantum theory? Our best theory of physical reality is exquisite – but inexplicable. A low, unexplained experimental noise could herald a revolution in the making newscientist.com/article/…

5:47 PM

How well known is $\pi^2 = g$ among physicists?

Well known, given that both are 10 :P

3

@Abcd that is wrong. Only #JEEThings

Mathematicians will kill you for that

in Mathematics, 29 secs ago, by Ted Shifrin

annoyed

Ha!

in Mathematics, 3 mins ago, by Ted Shifrin

It's not even close as an approximation. This is garbage.

5:52 PM

Also, anyone willing to give recommendations for a minor degree to pursue?

in Mathematics, 1 min ago, by Secret

at least I will roast you if you put an imprecise answer lol

proving that I am part mathematician :D

If I were active in Math.SE, I would surely downvote an answer that uses approximations for something that can be rather easily proved.

@Sid AI

Heard from my cousins

That AI is the thing right now

Hmm. Okay. That sounds Computer Science.

Jul 2 at 16:14, by John Rennie

Though I suppose I would recommend New Scientist as a source of entertainment, though not as a way to learn about physics. It is irredeemably superficial.

@Sid Computer Science or Engineering, Mathematics, Physics (STEM) etc

danielunderwood

6:04 PM

CS will certainly get you a job, though I've seen multiple people come out of physics with more capability than some CS students to do things in the real world. Those people generally did things like simulations in their research and had an interest in programming anyway though.

yes ML + datamining + datascience + AI will be solid for decades

danielunderwood

Applied math/statistics would also be up there due to the rise of data analysis people are doing. I don't know how that will be in 5-10 years though.

CS and Math are top on my priority order, yes.

danielunderwood

Some places will let you do a concentration in computational mathematics I believe

And my numerical analysis class was one of my favorites in college

Anonymous

@Sid lesbian dancy theory is an interesting subject these days...

danielunderwood

6:09 PM

Oh you said minor...a concentration probably wouldn't be a thing then

@Blue um.. don't think we have that here. :P

Just in case anyone was wondering what's lesbian dancy theory

urbandictionary.com/define.php?term=Lesbian%20dance%20theory

Anonymous

@Sid <jokes on not mentioning available minors when asking for minor recommendations> ;)

@Blue lol just lesbians are interesting (just ask howard stern) :P

@AvnishKabaj don't quote urban dictionary here!

6:11 PM

@Sid why?

It's the everlasting source of truth

@avnish if you are interested, type Sid on the Urban dictionary homepage. You will see lots of ... interesting results

@Sid Every single name has a definition

For some reason

danielunderwood

FWIW I have a math minor and focused on programming in my research. The programming has made it kind of trivial to get a job and I don't think the math has been much of an influence since someone with a physics degree already has the math skills necessary for most things.

@AvnishKabaj I will neither confirm nor deny any of the definitions you see if you search for "Sid" in Urban Dictionary.

@Blue lol. I am a bit confused what Preference order I will apply for. I am not sure what the industry wants right now(or will want in a few years)

Why worry about industry

Startup

And create the industry

Anonymous

6:18 PM

@Sid "I am not sure what the industry wants right now" --- as far as industry is concerned, programming skills will always be very highly valued as others mentioned, at least for the next half-century. But then again, unless you mention what options are available to you, it's difficult to recommend anything :P

Anonymous

If algorithms classes are available take as many as possible, because that's one area where EE majors lag behind CS majors

Anonymous

@AvnishKabaj "Startup" would still be a subset of "industry" ....all the skills needed in industry are also needed for startups, if not more

You see, there are only 40 students allowed to pursue a CS Minor. So, I am not even sure I will get one. Might get waitlisted.

Anonymous

So, what are the other options?

I was looking to go for either Math or Electronics. Not sure how different Electrical and Electronics courses will be, though.

6:24 PM

@Blue plis let me say stuff I don't understand so that I can feel self important and wise

Anonymous

@Sid If it were me, I'd blindly take up math (probably with a focus on statistics) :)

Anonymous

@AvnishKabaj lol

@AvnishKabaj lol

Anonymous

You're already an EEE major. I don't see any use of taking up more electronics courses :P

Anonymous

Unless you're obsessed with electronics, of course

6:27 PM

@Blue Yeah. That's what I am thinking. Going for Math over EC now. Well, at least that's the preference I have given

@vzn there is a guy you might love trying to find a quaternion theory of everything

Worst case scenario, I will get an Electronics Minor. Not at all bad.

Anonymous

@Sid If you take up a minor with a focus on statistics, it would help you a lot if want to later transition in ML or AI stuff too

Anonymous

@Sid Yup!

The Stand-Up Physicist: Why Quantum Mechanics is Weird

GEM Blogs: GEM Unification Rescinded, Official

Anonymous

6:31 PM

@Sid Is this minor thingy new? I didn't know NITs offer minors too

This stuff is hilarious, as you'd expect from a stand-up physicist

Using Quaternion for Lagrangians in EM and GEM, Cartesian

@Blue yup. It was introduced this year. Along with a revised syllabus

Anonymous

Nice! :D

All this stuff is of course terrifying at how easy it is to delude oneself and a nice sharp reminder

Anonymous

We too are having a syllabus revamp this year, I think. Will be introduced from next semester onwards

Anonymous

6:34 PM

Some elective entrepreneurships courses and stuff

Anonymous

Not sure I'd want to take them though

Open Electives?

What are entrepreneurship courses?

@Sid sell me a pen

Wolf of the wall street ?

Anonymous

@Sid What do you mean by "Open". I guess they'll be like normal graded courses only

Anonymous

What the syllabus will be, I don't know

6:37 PM

On a serious note

How much programming does an EE student end up learning

Wrt a cse kid?

Anonymous

@Sid Something like the MBA courses

@Blue Open as in, it's open to everyone.

Anonymous

@AvnishKabaj CS students get more practice since they have many more programming assignments. But then again, a lot of people from ECE and EE also take up software jobs immediately after graduation (those are mostly self-preparation cases)

@AvnishKabaj obviously lesser. But if you are interested, you can always learn stuff up yourself

Anonymous

A good CS course will teach you the fundamentals behind programming rather than teaching programming in any specific language(s), though

Anonymous

6:41 PM

@Sid I don't know yet

Thanks

@bolbteppa some of the "delusions" are about highly established theories aka strings etc

Jul 2 at 17:45, by bolbteppa

@vzn this fluid = everything stuff is not even science let alone new science

↑ lol betcha Adler + Vinante theory has some fluid paradigm angle... whos gonna be the 1st to find it? apparently not you :D :P

3 hours later…

9:41 PM

@EmilioPisanty Ideally yes, questions like this would go to Signal Processing. However, folks on Signal Processing will likely be confused by the solid state physics context, e.g. they won't know what "structure factor" means. That being the case, I think probably we're best off leaving the question here. In cases where the question author can de-imbed the essence of the question from the physics context, going to other more directed specialist SE sites makes a lot of sense, IMHO. — DanielSank 23 mins ago

@DanielSank given @dmckee's first comment, I have zero inclination to re-engage with that OP

I still think that question should be closed. As you said, it is about how to match the indices in the DFT. It might have a physics context but it doesn't have any physics content as such. You could cut out all of the references to structure factors and whatnot and keep the core question intact, from what I can see.

but

I have voted to close

it's in the hands of the queue

if it gets closed and OP doesn't fix it, then w/e

if it gets closed and OP fixes it, then w/e

if OP fixes it before it gets closed, then w/e

if it doesn't get closed, then w/e

9:55 PM

hmm

within QM, I know of two systems where the propagator can be written in closed form: the case of a free particle, and the case of a harmonic oscillator

are there some other cases that I should know about?

Hydrogen atom

Not pretty at all

preferably 1D

I would be surprised if anybody has ever even written down the eigenfunctions of the Harmonic oscillator starting from the path integral even

It's in Feynman's QM book, very ugly

@bolbteppa That's an exercise in Shankar, actually

specifically, he has them use the fact that $K(x,t;x',t')=\sum_n\psi_n(x)\psi_n^*(x')e^{-i E_n(t-t')/\hbar}$

then set $x=x'$ in the closed-form for the exact propagator (obtained from the path integral) and obtain the first few eigenvalues/eigenfunctions by looking at the time-dependence

@Semiclassical think Feynman goes a good bit further iirc getting the general formula

Found another source kind of doing Feynman's thing, skimmed Shankar hoping to find it ages ago

10:04 PM

yeah. Shankar specifically doesn't compute the prefactor of K(x,t;x',t')

Gave up on it

at least not in that case

and just says "look at Feynman-Hibbs"

Path integrals are useful in qft, not really in qm

(heck, he basically just says "use this prefactor" in the problem I'm referencing)

Yeah

I'm mostly just curious

Apparently Feynman never got the hydrogen atom path integral

10:05 PM

The fun thing about the propagator of the harmonic oscillator is that it's explicitly periodic

and that it behaves badly at $\omega t=0,\pi,...$

I have a 30 page write-up of compton scattering amplitude for qed starting from path integrals, that might be easier than the qm harmonic oscillator path integral :\

lol

sounds right

I'm guessing that "compute $\sum_n \psi(x)\psi^*(x') e^{-i E_n(t-t')/\hbar}$" isn't the kind of derivation you have in mind

(Not that I know how easy that calculation is in the first place)

I say that, to be clear, because that calculation is oooold: en.wikipedia.org/wiki/Mehler_kernel

huh, the remarks on the probability version are neat

It may all be equivalent, maybe Shankar does the same thing as Feynman, not sure

The calculation I'm tempted to do for the hell of it is to rederive the HO propagator via Bohmian trajectories :P

well, for the evaluation of the prefactor of the HO propagator, he reduces it to a path integral and then states "The evaluation of this integral is discussed by the book of Feynman and Hibbs referred to at the end of this section."

so his treatment is definitely not as complete as F-H

10:23 PM

I see, Feynman gets the $\psi_n$ wave functions anyway

one thing I did manage to derive (from the HO propagator, so not exactly from scratch) is the Bohmian trajectories for the case where $\psi(x,0)=\delta(x-x_0)$

i.e. the particle is initially localized at x=x0 in a harmonic oscillator potential

Does spin mean anything to Bohm

depends what you mean

at one level, spin just means what it usually does

replace scalar wavefunctions with spinor wavefunctions

on the other hand, spin is supposed to be derived from the Dirac equation, and therefore relativistic in origin

in which case I'm a lot less sure what spin means for Bohm

Spin is not relativistic.

Yeah that's a historical misconception

In one sense spin should be fine in Bohm, 3-D wave functions encode representations of the rotation group so spin should just fall out in one sense

But there's a big problem with spin also apparently to Bohm, don't understand it

10:38 PM

the main weirdness of spin in the Bohmian story is that it's necessarily contextual

this preprint does a nice job laying it out, I think: arxiv.org/pdf/1305.1280.pdf

Hmm... Stern-Gerlach involves magnetism, i.e. quantum field theory, it's kind of crazy even in normal QM books people give this example, you just need the Schrodinger equation for something like the hydrogen atom problem to set up spin

@ACuriousMind what if I stand on my office chair, spin myself to almost-relativistic speeds, and then pull in my arms?

I guess it kind of is no different in a sense

I think the issue with spin is more conceptual, seems like you can plug it into the formalism because it's just a Schrodinger equation

hmm: quantumchymist.blogspot.com/2014/04/…

the weird thing is that I remember seeing what they call the Levy-Leblond equation in my advanced QM class, but they didn't name it as such

(Another interesting thing is they are taking a relativistic model here and getting spin even though Bohm has huge problems with relativity, but I think we can ignore it because ultimately spin comes from the Schrodinger equation here)

10:53 PM

or maybe it was in Shankar

(i'd feel conflicted about looking up a pdf of Shankar except that I do own a copy, it's just not at home)

My sense is neither author appreciates how spin arises from group theory

Took me ages to make sense of it, still making sense of it I guess

Seems like if you take the Schrodinger equation, you're going to end up with spin anyway, so I think the problem is about extra degrees of freedom in a problem that shouldn't have them or something idk

I'm not following. In what sense does spin come from the Schrodinger equation? (I'd understand if you replaced Schrodinger with Dirac)

If you take something like the Hydrogen atom problem, the wave functions have to be invariant under representations of rotations, because it's a (central) $1/r$ potential, well, the rotation group is not simply connected, therefore representations of the rotation group will be multi-valued, in this case double-valued, which is why spin arises with spin up and spin down

hmm

Same logic for the Dirac equation, wave functions are invariant under representations of the Lorentz group, which splits into two parts, each of which has this problem

11:05 PM

I'd object along the lines of "consider a pion with spin-0" except that pions are neutral

and more generally there's no known elementary spin-0 charged particles

The difference is this happens for the free Dirac equation, but only an interacting Hydrogen atom problem, so it's easy to compute the angular momentum operator for the Dirac equation and see it is not conserved without adding the extra term that is the spin angular momentum operator

Well, a cop out answer is if you try to derive the spin angular momentum operator for Klein-Gordon it's zero

It might seem like KG should have spin the way I said it, hmm

well, what I more mean is: at first glance, it seems like there should be nothing terribly odd about having a hydrogen atom but with the electron replaced by a spin-0 particle

in which case a 'derivation' of spin by analysis of the hydrogen atom problem would make no sense

but if you can't have a spin-0 version of the hydrogen atom, then that objection can't be sustained

Basically, I was trying to think globally, there is a local way of ending up with spin that's perfectly fine also

Haven't fully thought about why the relativistic KG Hydrogen atom problem doesn't end up with spin, hmm

the blog post I linked earlier seemed to link spin with 'wave equation has first-order spatial derivatives'

danielunderwood

I hate to interrupt your conversation, but is there any difference between a function $f: \mathbb{R} \to \mathbb{R}$ and a quantity $q \in \mathbb{R}$ labeled by an index $\lambda \in \mathbb{R}$ like $q_\lambda$. I think I confused myself over something quite silly.

11:15 PM

$q_{\lambda} = q(\lambda)$, with $q : \mathbb{R} \to \mathbb{R}$

typically you'd use the subscript when it's an index i.e. $\lambda\in \mathbb{N}$

otherwise you'd be able to write stuff like $q_{0.5841}$ which just looks goofy

(huzzah for conventions)

Yeah, but sometimes people do still do it, and especially when labelling sets

danielunderwood

Well the context was going from $q_i, i \in \mathbb{N}$ to $q_\lambda, \lambda \in \mathbb{R}$ to get the field EL equations and I had no idea the step between that and $\phi(x)$...turns out there isn't one. I still don't know why the label should be the position in that case, but I'll get there

yeah

@danielunderwood that's sorta icky

danielunderwood

I'm going to start labeling all my functions like that $x_t$ :D

And in my case, I chose the label really poorly and ended up with something on the blackboard that looked like

$$
\int \left( \frac{\partial L}{\partial q^\mu} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}^\mu} \right) d\mu
$$

11:21 PM

:S

danielunderwood

yeah turns out notation can be cruel

@bolbteppa I think that, within the pilot-wave story that I know, the question is largely one of how the probability currents are defined

once you've got that you can deduce appropriate velocities (as ratios of current/density) and take the flowlines generated by such to be the allowed trajectories

nicely, L-L's original paper on his equation includes the probability conservation equation: projecteuclid.org/download/pdf_1/euclid.cmp/1103840281

Basically it seems like the KG equation for a coulomb electric field should produce spin 1/2 wave functions, wtf

well, note what I said earlier: a particle obeying the KG equation must be spin-0, and there's no elementary spin-0 particles with electric charge

hence why it seems like that kind of scenario doesn't actually appear in nature?

i dunno, it's weird

$$\mathbf{j} = \frac{1}{2mi}[\varphi^*(\nabla \varphi)-\varphi(\nabla \varphi^*)]+\frac{1}{2m}\nabla\times(\varphi^*\boldsymbol{\sigma}\varphi)$$

ugh, everytime I try to do \varphi i inevitably do \var\phi

also, $\rho=\varphi^*\varphi$

11:38 PM

Each component of the Dirac equation satisfies the Klein-Gordon equation, even though the whole thing also satisfies the Dirac equation, I think the same principle is going on here

so the 'obvious' thing to do from the pilot-wave perspective would seem to be $\mathbf{v}^{\varphi} = \mathbf{j}/\rho$

the first term in $\mathbf{j}$ is just the usual probability current, so that's nothing new

not so sure about the second term

danielunderwood

wow it hurts more when someone else does it

@Semiclassical wait, what?

$\var$?

hmmmm.

i mean that when I try to type it

i reflexively insert that extra \

that printed as $\delta$ on latex on my machine

wtf

11:46 PM

huh

that's

huh

danielunderwood

Have you globally defined a command or something?

for me it's just that I end up typing \var\phi instead of \varphi because halfway through my brain sees phi and says "oh, you want \phi"

danielunderwood

Oh no I was saying for @EmilioPisanty having it create a delta

Yeah, so as counter-intuitive as it sounds, the Klein-Gordon equation with a central 1/r potential can be used to try describe spin 1/2 wave wave functions, just as you do with non-relativistic wave functions, it just turns out to give incorrect values

user image

man, TKAM

watch out for it

it's going to be the bomb

11:49 PM

You just do it by working on each component separately and using the representation theory of angular momentum to argue you should go beyond one component

maybe \var for variation?

danielunderwood

Just be sure to leave that bit in

@Semiclassical probably. It's almost certainly \usepackage{physics} causing it

makes sense

ah, yep, top of page 6: mirrors.ibiblio.org/CTAN/macros/latex/contrib/physics/…

danielunderwood

Is there a difference in using $S$ and $\mathcal{S}$ for action? A couple wikipedia use both for whatever reason

Woah automatic parenthesis/brackets/braces...that could have saved me so much time

11:53 PM

maybe the same as how $L$ is the Lagrangian and $\mathcal{L}$ is the Lagrangian density?

@Semiclassical that'd be stupid

action density is (surely?) pretty useless

hmm

maybe just silly aesthetics then?

danielunderwood

That's what I thought, but one of the examples is $\mathcal{S} = \int L dt$...I think it's because people with different conventions edited or something. I've never really seen $\mathcal{S}$ before though.

I don't think I've ever seen it.

danielunderwood

Or a plain $S$ wasn't fancy enough for their precious action lol

11:55 PM

Hmm, I think it may come down to what value of $L$ you choose for angular momentum, for an electron you can set it equal to $1/2$, for a scalar particle you set it equal to $0$

@danielunderwood DiFrancesco use $\mathcal{S}$

My guess would be you'd use $S$ if you were doing like $S = \int dt L = \int dt \int d^3 x \mathcal{L}$ while you'd use $\mathcal{S}$ for $\mathcal{S} = \int d^4 x \mathcal{L}$

but those are the same thing...

Yeah but the former is acting like it's treating time as different

hmm

No DiFrancesco don't even do that, they literally use $\mathcal{S}$ even in a $\int dx dt$ case, scratch that :p

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