Deriving The Klein Gordon Equation (Relativistic Quantum)

[@michaeljburt]

I've met many physicists in my life. Many with PhD's or working on them, both theoretical and experimental particle physics. I can say very confidently that you are the clearest physics communicator I have ever seen. I hope you're pursuing a professorship. It really takes a brilliant mind to both be able to execute the theoretical mathematics and physics, and discuss it while doing it.

[@AndrewDotsonvideos]

Thanks so much!

[@KRATOS-dq6sd]

@@AndrewDotsonvideos did you really just reply an year later

[@jakwhyld4406]

@@KRATOS-dq6sd did you really just reply an eight months later

[@mevnesldau8408]

@@jakwhyld4406 did you really just replay a month later?

[@Epyriel]

@@mevnesldau8408 Did you really just reply a month later?

[@saarausmaan]

Now I found one of the clearest concepts about the Klein Gordon wave equation . Thank you so much

[@namanvats7644]

I spent all day trying studying form books , wikipedia ,detailed lectures but this video help me get rid of my every single doubt about the topic in just 14 mins.

[@mrnarason]

Do one for the dirac equation

[@mohannadislaieh3009]

I agree

[@kanhaiyalal-kv2vg]

Please make video on dirac equation

[@D-baller]

I concur

[@boriscrisp518]

yeah.. get on with the dirac eqn, you beardy hipster

[@Mr-xf9me]

Done

[@RVN_19]

i watched this video when i was just about to start uni. and i didnt understand anything. now that i am taking QFT, holy moly you are amazing at explaining things

[@alapandas6398]

Everybody is saying about the box notation. But I saw in Feynman Lectures box square was used.

[@MyTBrain]

“Klein - Gordon in under 15 minutes? “ **cracks knuckles** “Watch me...” Stellar Job Andrew!

[@theflaggeddragon9472]

You're going to want to start at the top left when presenting, split the board in two and DON'T ERASE. Don't worry I did this all the time when lecturing and have only recently corrected my habits but it really helps students/viewers on the other end. Great video ;)

[@AndrewDotsonvideos]

I'd like to start at the top left, but my light has a terrible glare around there! That's why I have a black outline in the corner in all of my math videos. So I know where not to write.

[@tracyh5751]

It's not too important for videos as we can rewind when needed. As long as you are careful in the classroom you should be okay. :)

[@garybarbourii8274]

I will not be satisfied until I understand every bit of this, and more.

[@JoeHynes284]

i am on a two and a half year journey so far, i am 44...

[@esorminihaz8269]

This equation was derived by Isaac Kleiner and Gordon Freeman. Using this equation instead of the Dirac equation caused the resonance cascade

[@renatolucas3120]

I'm in a IC(scientific initiation) of symmetry break and this video clear a lot of things that I wasn't getin. Waiting your next video !!

[@theicemancometh5492]

I love how you write your 'a' the same was as the d in a partial derivative

[@onderozenc4470]

At the end, this relativistic wave eq. comes down to expressing the probability of a particle including the photons making tunneling through a potential field acting on them to express the observability of the black holes' thermal radiation within the large wave length micro wave spectrum.

[@WaveSound123]

I've never seen the d'Alambertian writen with a square, only the box XD

[@AndrewDotsonvideos]

Hmm. Griffiths uses the squared version. I guess as long as you define what you're saying it means.

[@angelmendez-rivera351]

Liviu Stoica Box, square... eh, really minor difference

[@cyrilvidal5395]

David Tong Quantum Field Theory lecture notes www.damtp.cam.ac.uk/user/tong/qft/one.pdf => box only but both usages are allowed

[@lPlanetarizado]

first time I saw d'Alambertian I thought it was an error in the file lol

[@WaveSound123]

@@lPlanetarizado Good one😂😂😂

[@angelmendez-rivera351]

I think another, perhaps better way, to explain the significance of contraviant and covariant indeces when working with vectors and 4-vectors is to talk about the conventions of matrix calculus and to consider ordinary vectors as single-column matrices. Their components are expressed with a covariant index. Then, an expression with a contraviant index is simply a single-row matrix, or a "row-vector," also known as a covector in some contexts.

[@ashmindersingh4347]

Finally someone explained d'alembertian in form of 4 vectors

[@organicolympic1729]

Highly impressed in your presentation😊🙂

[@kamehamehaDdragon]

If you don't know quantum physics like me, this looks super crazy. If you do know physics, I bet it still looks super crazy xD. Good video.

[@thephysicistcuber175]

I've never seen the D'Alembert operator as box^2. It's always been just box :p EDIT: I've scrolled through the comments xD

[@TheBaluchiterium]

It's usually given as \Box which, however, is strange. It is more logical to make the transition from "nabla" to "quabla" (quadri-nabla) and to denote the latter as \Box which for contravariant components is equal to \partial^{\mu}. Then a Minkowski-space scalar product can be written as \Box^2 = \partial^{\mu} \partial_{\mu} which is just the d'Alembertian. I think this is the reason why some authors disagree with the more widespread convention of d'Alembertian = \Box. And I do, too.

[@petriklelel]

@@TheBaluchiterium Thank you for the explanation, as I was only familiar with the \Box notation for the d'Alembertian. Also, I really like how you wrote it with latex code :)

[@xnick_uy]

For the sake of keeping a more consistent notation, I would have written the abbreviated 2x2 metric with an overbar on top of the zeroes (since they stand for (0,0,0) ) and use -Id for what appears as a -1 with the overbar on top (since it actually stands for the minus 3x3 identity matrix).

[@CarlosKoji87]

What's interesting about the Klein-Gordon Equation is that it gives the wrong fine structure for the hydrogen atom and also gives negative probabilities. Not only that, it allows for negative energy states! I think a video on deriving the Dirac Equation would be really cool. It would be a great transition as it remedies some of the issues in the Klein-Gordon Equation.

[@teodoranirmala3163]

Carlos C Imho, Klein-Gordon equation actually only work for bosons. Electrons in hidrogen atoms are fermions, so these can be handled by Dirac Equation. May this helps.

[@angelmendez-rivera351]

Carlos C Yes, it gives the wrong results for those particles because the equation is not applicable for those particles. The Klein-Gorden equation is a boson equation, since, by having a scalar wavefunction, it assumes integer spin.

[@KhoaNguyen-os5zv]

I wonder why Caltech didn’t give u at least an email. You’re so mathematically gifted. I’m an error asian.

[@robertmorrison1657]

People there are extremely intelligent. Like sheldon cooper type people. Fine okay, maybe just leonard intelligent. However, I would suspect that to get into that university for physics, you would need an iq of like 160 at least. This guy might be very smart, but he is not a genius. He still probably has an iq from 140 to 150 though.

[@heavennoes]

@@robertmorrison1657 I don't think Caltech judges people by IQ.

[@robertmorrison1657]

@@heavennoes Not exactly. They don't have you take an iq test to see if you will do well in university. What they will do is see your grades, scores on SAT's, and other things that show how smart you are, plus maybe some extracurriculars just to see how driven and committed. So they kind of see what your iq is like, whether it is high enough or not. So they sort of do judge by iq.

[@Onegod40-v4h]

That's fantastic demonstration.Really appreciate it

[@danielbachour9987]

Great man! Enjoyed the video! i'm going to catching up with all your videos! You know, to refresh things up! Keep it going!!

[@showmethe-o4b]

Thank you very much! I was looking forward to finding the meaning of /d myu d myu/, I found it here!. I'm studying physics as a hobby. You're videos help me study physics alone because it's not easy to find lectures that teach from the scratch. I appreciate it so much! ^^*

[@AstroPC96]

This guy should have been my QM professor!!

[@Prime-o8f]

Hey Andrew, what whiteboard are you using and where can I get it from? I'm a Physics/Maths student in the UK and I've been wanting to get a good whiteboard for a while now, but when I googled it I found there's a whole quality scale based on things like erase-ability, whether it's magnetic or not etc... Only thing I care about is good erase-ability. Which yours seems to have. Thanks in advance! Great content btw, really like this sort of higher level stuff.

[@ゾカリクゾ]

ThePrimebrook same problem here, ive used 200+ pieces of paper to do problems and that is wasteful IMO. I really need a whiteboard, but I don't know which

[@kanhaiyalal-kv2vg]

Please make a video on dirac equation thanks

[@thephysicist5519]

You are genius I liked u plz continue up and I will share ur vedios

[@Alolyn]

The d’alembertian square symbol makes me imagine that the guy who introduced the equation typed it up on an iphone with an emoji for an operator, and traveled back in time to give it to physicists in the past but the emoji turned into that box with a question mark on it and they just rolled with it Or they used a japanese ロ “ro” (which makes it seem related to rho) cus they got tired of using the greek alphabet

[@AndrewDotsonvideos]

I feel completely the same

[@natepolidoro4565]

what an intelligent sasquatch!

[@darioolmos7480]

Does a tensorial field have gauge symmetries if it follows this equation?

[@AndrewDotsonvideos]

Global but not local. You always need to replace derivatives with gauge covariant derivatives if you require local gauge invariance. But yes it does still have a global phase invariance and a corresponding conserved charge

[@darioolmos7480]

@@AndrewDotsonvideos thanks for your answer!! And such a nice video 💯

[@lemniscatusofficial]

3:50 why did you use a scalar field?

[@agnaldojunior5538]

Very clean, thanks.

[@souvik9305]

I saw in a landau book that four vector transformtion are all positive sign,so why is there a minus sign in your video(-Sinhx)....plz can you clear thr doubt....what am i missing

[@jazzani95]

Thx! Very clear. Can you motivate the Sine-Gordon Equation? It has solion solutions. Understanding solitons is important when comprehending the solutions for the Kosterlitz-Thouless transition.

[@arkkaushik2642]

Hey if row vector is covariant terms shouldn't the last term should also be row vector?

[@fermiLiquidDrinker]

What if I just _don't like_ natural units?

[@as.n4244]

As far as I know, N=4 supersymmetry has spin zero s-quark, s-lepton, and LSND-neutrino :)

[@as.n4244]

And for non-SM, there’s the axion pseudoscalar for charge parity violation, all I can think of off the top of my head

[@invisibules]

I think the sign of the potential energy term is wrong in that schrodinger equation at the beginning ... isn't it?

[@AndrewDotsonvideos]

Yup you’re right, sorry about that

[@Leonlion0305]

I only have crippling intro quantum knowledge but I finished both the Schrodinger Equation @ relativistic speed and this one. I understand ~50% of these but I understand it. Pls be my physics prof lol

[@TheApostleofRock]

bold of you to call me a smart person

[@theultimatereductionist7592]

Since the Klein Gordon & Dirac equations combine relativity & quantum mechanics, then why do physicists say we are still struggling to combine the two?

[@angelmendez-rivera351]

The Ultimate Reductionist You are wrong in several aspects. Firstly, the Klein-Gordon equation and the Dirac equation are incorporations of *special* relativity into quantum physics. There is no *general* relativity involved. Secondly, quantum field theories DO *mostly* combine general relativity with quantum physics. In fact, the only real conflict between both theories is that the Einstein field equations are classical field equations, not quantum field equations, but unlike all other field equations that we have worked with, Einstein's field equations in particular cannot be quantized by any methods currently known. The problem lies with quantizing the metric tensor, which yields singularities upon singularities. These are non-regularizable and non-renormalizable.

[@theultimatereductionist7592]

@@angelmendez-rivera351 "Firstly, the Klein-Gordon equation and the Dirac equation are incorporations of special relativity into quantum physics. " Ah. Oops. Rookie mistake.

[@drbeanut]

@@angelmendez-rivera351 why need to quantize spacetime curvature? what if just let it operate on wave-function continuously?

[@angelmendez-rivera351]

@@drbeanut You cannot have wavefunctions without quantization.

[@drbeanut]

@@angelmendez-rivera351 How so? A wave-function merely assigns a continuous probability amplitude distribution to some arbitrary field.

[@soerser2428]

Why is there no minus before the V in the Schrödingerequation?

[@laneellisor7113]

Love the video! Do you think you could do a video on college labs and any tips or advice for them?

[@abacademy418]

Thanks

[@ゾカリクゾ]

One little question Andrew, or anyone else (who knows)... At 3:50 (dx/dt)^2 = d^2/dt^2 ??? I don't see why... For example: y = x (dy/dx)^2 = 1 d^2y/dx^2 = 0

[@x17exeventien79]

Its not an algebraic product, it's an operator manipulation. The linearity of derivatives allows you to get away with notation tricks like that for some things even though its not strictly correct.

[@ゾカリクゾ]

x17 Exeventien yes, i saw this in papa flammy videos, it is just using the operator twice

[@TomDixon89]

(dy/dx)^2 and d^2y/dx^2 are not the same. What he did was the squaring before applying it to wavefunction. E=d/dt i.e E as an operator which you apply to a function. an operator applied to an operator is an operator. i.e (d/dt)(d/dt)y = d/dt(dy/dt) = d^2y/dt^2 BUT (dy/dt)(dy/dt) is a function on a function ie d/dt = operator dy/dt = function. So grad^2[y]+m(y)+E^2[y] represents d2y/dx2 + my + d2t/dE2 square brackets indicate operator acting on y and regular brackets are functional, or even numeric. The important part is that the operator (the derivative operator in this case is squared not the function). The way I remember it is: if Im a function and you give me a number I can tell you the resulting number. If Im an operator and you give me a function I can give you the resulting function.

[@angelmendez-rivera351]

Francisco Russo He never wrote (dx/dt)^2. He wrote d^2(φ)/dx^2. They are different.

[@chymoney1]

Can you explain what the metric means conceptually??

[@AndrewDotsonvideos]

It tells you how not-flat the spacetime you're working in is. If it is flat, you get a diagonal of 1's (with some minus signs) which basically just tells you a^2+b^2 = c^2 still holds.

[@akokJ619]

The metric tensor is it always necessary to use the + - - - convention?

[@oni8337]

nope you can use - + + + instead if youd like but keep it consistent

[@akokJ619]

@@oni8337 didn't know 2 years ago. I know now XD

[@ionlyazir3662]

Nothing better than quantum mechanics at 3 a.m. btw im only 15. Thats probably the reason that i have no friends -.-

[@ionlyazir3662]

Francisco Russo thx for support :D

[@frankmoscatelli4555]

Students are so used to seeing an arrow as denoting a 3 vector that when I teach 4 vectors I use it, not a bar, to represent the 3 part of the 4 vector. Otherwise, great video.

[@bonchiengooddog3791]

What is this equation trying to achieve?

[@tony_narchuk]

Isaac Kleiner - Gordon Freeman equation

[@archi124]

why can the klein gordon equation used to describe scalar bosons if it leads to a negative probability density?

[@angelmendez-rivera351]

I do not believe it does that if you work with bosons, only with fermions. That is because the equation only works with bosons.

[@caterinacevallos9822]

I'm a bit confused when you say something is Lorentz-covariant? Does it mean that the "form" of the equation stays the same? What would it mean for a transformation to be then contra-variant? (Do you have an example of that?)

[@thephysicistcuber175]

Caterina Cevallos I'm pretty sure in this case covariant refers to the form of the equation. Indeed there are no indices in the final form of the KG equation. I don't think "contravariant equation" is a thing

[@AndrewDotsonvideos]

Sorry I'm just seeing this now. When I say Lorentz-covariant, I mean it transforms properly under a Lorentz transformation. The equation will still change, but under the prescription given by the L.T.

[@zoltankurti]

@@thephysicistcuber175 maxwells equations in vacuum are covariant, and they have unpaired indices... Covariant in this context means the equation is tensorial with respect to the Poincaré group. In other context covariant and contravariant just refers to tangent vector and cotangent vector components.

[@thephysicistcuber175]

@@zoltankurti I wasn't implying that there must not be unpaired indices, I was just saying that in this case the equation is obviously covariant because of the absence of such indices

[@zoltankurti]

@@thephysicistcuber175 many not invariant objects have no indices. Pressure, energy density, and I don't need to go on because I'm sure you can list a bunch of these too.

[@chymoney1]

For the “ wave function” in your Klein gorgon equation you said I.T was as a scalar field. A probabilistic scalar field? What is it??

[@TheDutchPhysicist]

For the schrodinger equation you have the wave function whichs depends on space and on time and moves through space and time seperately, here you get a field that propagates in space-time. So its like the wave function, except now its a field in space-time.

[@angelmendez-rivera351]

Sebastiaan van Rijk Yes, but he is asking what does the field stand for. The answer is that it is a probability amplitude.

[@havefun4493]

thank you very much!

[@mohannadislaieh3009]

U are genius, I likef that

[@thephysicist5519]

Mohannad Islaieh u are right

[@theultimatereductionist7592]

Why does nobody ever show the Klein Gordon Equation applied to 3, 4, 5 or general N particles? I want to see exact solutions for SYSTEMS of particles. NOT just 1 particle.

[@zoltankurti]

Qft is all about that. But you automatically get all integer particle numbers and even their superpositions.

[@mastershooter64]

I want that beard :(

[@soulsilencer1864]

I entered an "AP physics practice test" today and they asked me fucking relativistic quantum mechanics

[@sidddddddddddddd]

Correct me if I’m wrong. That box shouldn’t be squared. No?

[@AndrewDotsonvideos]

It's just a definition. You could define a smiley face operator to equal the right hand side so long as it's consistent with what you're using it for. But, Griffiths also uses the box squared for what its worth.

[@sidddddddddddddd]

Andrew Dotson Yeah, I get that. But it’s always convenient to have some kind of a universal notation. It’s like the metric, you can use + - or - + notation. There should be some SI equivalent of such stuff. Anyway, keep it up, Andrew.

[@ahmedkarrar25]

I still don't understand why did u write phi instead of psi? (4:11)

[@navneetmishra3208]

Phi usually represents a complex function that is not a scalar field.

[@MultiDaniella14]

Thanks. Help alot

[@MultiDaniella14]

helped*

[@Salmanul_]

Idk why but the box notation looks a bit uncomfortable, it feels like I have to write something inside it. Like fill in the blanks questions haha

[@arifsarkar7328]

Cool man!! this is 👌

[@Riteshbachhar]

why here we have scaler field instead of wavefunction?

[@angelmendez-rivera351]

Ritesh Bachhar The wavefunction is a scalar field.

[@angelmendez-rivera351]

Smit Shilpatul Yes, but the Klein Gordon equation is only valid for spin 0 particles, so your point is moot.

[@MM-ei7xv]

god! you're great thanks a lot

[@zoltankurti]

It's interesting... On what basis do you call this quantum mechanics? You don't even have one operator in association with this classical Klein Gordon field.

[@angelmendez-rivera351]

Zoltán Kürti What is your definition of quantum mechanics? Because my understanding of the term is that there is no obvious reason this should not be called quantum mechanics. It can be derived from the axiomatic postulates of a quantum theory, so it is quantum. And, there is an operator called the d'Alembert involved.

[@angelmendez-rivera351]

d'Alembert's operator is the 4-wavevector operator squared and multiplied by a constant.

[@miguelamaral9642]

I don't think the d'Alembert Operator should be squared, it is simply equal to exactly what you derived

[@waqarali9166]

I enjoyed 😍

[@williamwalker39]

The speed of light is not a constant speed as once thought, and this has now been proved by Electrodynamic theory and by Experiments done by many independent researchers. The results clearly show that light propagates instantaneously when it is created by a source, and reduces to approximately the speed of light in the farfield, about one wavelength from the source, and never becomes equal to exactly c. This corresponds the phase speed, group speed, and information speed. Any theory assuming the speed of light is a constant, such as Special Relativity and General Relativity are wrong, and it has implications to Quantum theories as well. So this fact about the speed of light affects all of Modern Physics. Often it is stated that Relativity has been verified by so many experiments, how can it be wrong. Well no experiment can prove a theory, and can only provide evidence that a theory is correct. But one experiment can absolutely disprove a theory, and the new speed of light experiments proving the speed of light is not a constant is such a proof. So what does it mean? Well a derivation of Relativity using instantaneous nearfield light yields Galilean Relativity. This can easily seen by inserting c=infinity into the Lorentz Transform, yielding the GalileanTransform, where time is the same in all inertial frames. So a moving object observed with instantaneous nearfield light will yield no Relativistic effects, whereas by changing the frequency of the light such that farfield light is used will observe Relativistic effects. But since time and space are real and independent of the frequency of light used to measure its effects, then one must conclude the effects of Relativity are just an optical illusion. Since General Relativity is based on Special Relativity, then it has the same problem. A better theory of Gravity is Gravitoelectromagnetism which assumes gravity can be mathematically described by 4 Maxwell equations, similar to to those of electromagnetic theory. It is well known that General Relativity reduces to Gravitoelectromagnetism for weak fields, which is all that we observe. Using this theory, analysis of an oscillating mass yields a wave equation set equal to a source term. Analysis of this equation shows that the phase speed, group speed, and information speed are instantaneous in the nearfield and reduce to the speed of light in the farfield. This theory then accounts for all the observed gravitational effects including instantaneous nearfield and the speed of light farfield. The main difference is that this theory is a field theory, and not a geometrical theory like General Relativity. Because it is a field theory, Gravity can be then be quantized as the Graviton. Lastly it should be mentioned that this research shows that the Pilot Wave interpretation of Quantum Mechanics can no longer be criticized for requiring instantaneous interaction of the pilot wave, thereby violating Relativity. It should also be noted that nearfield electromagnetic fields can be explained by quantum mechanics using the Pilot Wave interpretation of quantum mechanics and the Heisenberg uncertainty principle (HUP), where Δx and Δp are interpreted as averages, and not the uncertainty in the values as in other interpretations of quantum mechanics. So in HUP: Δx Δp = h, where Δp=mΔv, and m is an effective mass due to momentum, thus HUP becomes: Δx Δv = h/m. In the nearfield where the field is created, Δx=0, therefore Δv=infinity. In the farfield, HUP: Δx Δp = h, where p = h/λ. HUP then becomes: Δx h/λ = h, or Δx=λ. Also in the farfield HUP becomes: λmΔv=h, thus Δv=h/(mλ). Since p=h/λ, then Δv=p/m. Also since p=mc, then Δv=c. So in summary, in the nearfield Δv=infinity, and in the farfield Δv=c, where Δv is the average velocity of the photon according to Pilot Wave theory. Consequently the Pilot wave interpretation should become the preferred interpretation of Quantum Mechanics. It should also be noted that this argument can be applied to all fields, including the graviton. Hence all fields should exhibit instantaneous nearfield and speed c farfield behavior, and this can explain the non-local effects observed in quantum entangled particles. *KZbin presentation of above arguments: kzbin.info/www/bejne/qZazlX1tq7iErLM *More extensive paper for the above arguments: William D. Walker and Dag Stranneby, A New Interpretation of Relativity, 2023: vixra.org/abs/2309.0145 *Electromagnetic pulse experiment paper: www.techrxiv.org/doi/full/10.36227/techrxiv.170862178.82175798/v1 Dr. William Walker - PhD in physics from ETH Zurich, 1997

[@miguelaphan58]

how natural units works..?,...besides...nice job

[@alexbaykov9221]

Basically, it is just measuring velocities in fractions of c and measuring action (or angular momentum) in hbars.

[@Excidium96]

In the beginning, you talk about not being comfortable with non-relativistic quantum or special relativity then this video isn't for you. Buuuuuut I still watch all your math-deriving videos even though I don't even know calculus yet......It's just interesting to see what mathematical skillsI could potentially have If I just study as hard as you have....And that's what I am aiming for with my physics degree :)

[@non-inertialobserver946]

Pi=e=3

[@samgeribo]

Ben Affleck

[@PetraKann]

Physics is the least complicated and simple of all the scientific disciplines. Complexity increases markedly when moving from the field of Physics to Chemistry to BioChemistry to Biology. The level of complexity increases once again when doing Psychology and economics. That's one of the advantages of Physics - it's simplicity.

[@DeltaXGamerPT]

In what sense is physics simple? It is extremely complex... Even compact equations, which are certainly not the rule, are notation heavy and many are notoriously long even with compact notation. And to be honest, although great mathematical prowess is absolutely necessary, I think the greatest challenge - and also beauty - of physics is gathering a sensible intuitive interpretation of said mathematical jargon to explain the world around you. I really don't understand where the triviality begins.

[@PetraKann]

DeltaX Gamer PT physics is the most rudimentary and least complex of all the scientific disciplines. It’s the pre-school of science. Basic idealism and simple assumptions. Nice geometric forms and convenient system behaviour. Even when you examine Quantum Mechanics, it’s mathematical foundations are simpler than Newtonian Mechanics. I am not trivialising they field of Physics just stating an obvious fact about its nature and the functional limits. The predictions made by theories in physics are orders of magnitude easier to validate than all other scientific fields Ever wondered why? Physicists often take it personally which is the wrong attitude. Their field is a basal fundamental footing for other scientific disciplines to build upon.

[@DeltaXGamerPT]

@@PetraKann It is that way because it is our own interpretation of the language of our universe. I understand where you are coming with this and agree with most of it. I just don't like how you say it is not a complex field. Its assumptions and implications are far from the whole truth of the theories, which is why it so rare to find someone who can communicate them diligently and in a simple manner. It is simpler in the sense that it is the science where you skip the least logical steps apart from mathematics. It is still, however, one of the hardest sciences to be able to grasp.

[@Prxwler]

Yeah, because it's more difficult to learn a bunch of long and complicated names than learning how to reason.

[@meetghelani5222]

del squared >>> laplacian

[@jesther2575]

Cursed squared square

[@finthechat7134]

Duh, I knew all of that.