Dirac equation Puns

If I'm not mistaken, the wave-function of any particle has right and left handed components, even if at one point in time it was purely right or left-handed the other component would arise as it moves

Now let's take this to neutrinos which "are all left handed," what's going on with the wave-function?

What I suspect is happening is that the wave-function of neutrinos also has right and left handed components, which means that neutrinos should be able to spontaneously become antineutrinos and viceversa, without having to emit or interact with any other particle, which is similar to what already happens with neutrino oscillations, so it wouldn't be that crazy

But then I've also heard that maybe right-handed neutrinos and left-handed anti neutrinos exist, in which case the question would be: why are those components of the wave-function so astronomically small all the time?

How was Paul Dirac able to use his equations to prove the energy levels in a hydrogen atom if it supposedly only describes how relativistic electrons behave? Or is there more to the equation?

Also, are there any other equations in quantum mechanics whereby the existence of antimatter can be proven?

• From a 17 year old who has never formally studied quantum mechanics, but is curious regardless.

I'm not quite sure how to make partial derivative in text so I'll just use d for the partial derivative operator.

I've been teaching myself General Relativity/Field Theory so bare with me. For the Dirac equation, the Covariant Derivative operator is

D^a = d^a - ieA^a

I was messing around today and thought, what if I replaced every partial with this operator in the Riemann tensor, even the ones in the Cristofel symbols. Neglecting the terms quadratic in the Cristofel symbol, and contracting twice, this gives a scalar curvature

R' = R - e^2 A^2

where R is the original Ricci scalar.

I was surprised at how nice this came out in the end and that there were no i's left. Is there any use to doing this? I'm not quite sure what the physical significance would be, if there is any. I know that adding the (- ieA) term in the Dirac equation adds the current term.

Any helpful thoughts/comments are welcomed.

I'm fascinated with Paul Dirac. I've watched as many documentaries as I can about him on YouTube. I was hoping someone can remember the documentary where his equation was on a plaque outside his old class room door? It was in red and looked almost 3-D. I want to get a tattoo of it on my arm and can no longer locate the documentary.

I've been taking the time to self-teach myself a lot of stuff about quantum physics that I hadn't really understood and I've been going down a rabbit hole.

Thus far I have a much better understanding of wave functions, Plank constant and h bar, operators, the laplacian, shrodinger's equation, heisenberg uncertainty, spin, angular momentum, and total angular momentum as well as the hermitian matrix stuff for dirac and pauli matrices, and how ultimately the the dirac equation is derived from E2=M2C4 +P2C2 and momentum operators with the 4 component wave function and the expansion of the 4 component matrix. I've had to learn and relearn a lot of math to confirm how you get a lot of this stuff. (Can't say I understand Bra Ket stuff all that much, but I'm getting there)

I think I get it for the most part. My understanding has gotten a lot better.

Do I need to understand GR in full to understand QED with the addition of the dirac stuff? Or can I just know the famous equation and the general vibe of relativity and mass (rather, energy) bending spacetime such that things in an inertial reference frame travel along a new inertial spacetime geodesic without sensing any new force being applied or acceleration? Or is the rabbit hole that is GR really super important to understanding the math in QED.

I get the lagrangian = KE - PE

Looks like i need to understand what the dirac adjoints is, what gauge covariants or gauge covariance is, as well as electric field tensors and probably a bunch more, right?

My end goal is just to understand how they got to all that QED stuff right now so that hopefully all the other things become a bit easier to understand.

Does anyone have any suggestions of what to look into next?

So the question asks for a Gaussian packet where at time t=0 we have

Equation (1)

Where u is Equation (2) in the Dirac representation. We have to find the temporal evolution for it.

Then we gotta show the expansion coefficients of psi(t,r) (r is a vector) corresponding to states of negative energy are only significant if d <~ hbar/mc.

This class kinda dropped this on us to figure out on our own without any sort of experience with the Dirac equation. How do I start it at least?

So I'm reading intermediate quantum mechanics by H.A. bethe for relativitsic QM and saw a derivation for the Dirac equation, it starts by demaning certain characteristic for a relativistic wave equation

• Must have time derivatives with order <=1 (to prevenr -rho)
• It must be first order in all four coordinates (cause x,y,z and ct are treated symmetrically)
• Must be linear (to satisfy superposition)
• Must correspond to classical relativity at large quantum numbers (correspondence principle)

They then suppose that the "wavefunction" (at this point idk if to call it that) has N components (i believe they call it a spinor?)

psi = [psi_0 psi_1 psi_2 ... psi_N]^T

And concludes from one instant to the next that THIS monstrosity is the most general equation that satisfies this. My question is, where tf did this come from? Is it even important to understand so? I thought so cause the dirac matrices come from this no? Forgive me if I'm being silly, I'm new to this formalism.

Thanks in advance for any help!

I understand what Schrödinger's, Dirac's, and Maxwell's equations are but don't see how the latter two are derivable from the first, without additional methods and ansatz. I'm unsure what Penrose means in the below quote, which seems to indicate you can simply "substitute" a photon in for Psi (or "substitute" an electron in for Psi, especially when Psi already describes a non-relativistic electron).

https://i.imgur.com/xrIR0Oq.png (from Emperor's New Mind)

Can someone please take me through, step by step, with equations what he means? Thank you dearly.

I'm struggling with understanding the Dirac equation from the perspective of the 2-component, massless Weyl fields. Usually the $m \bar{\Psi_L} \Psi_R$ (and the conjugate) term is talked about in reference to how it violates Gauge symmetry, but it seems like it also violates 3-momentum conservation, as well?

By this I mean $\bar{\Psi_L} \Psi_R \neq 0$ only when the spinors before and after are pointing in the same direction (angular momentum conservation). But the Weyl spinors are both helicity and chirality eigenstates, meaning that their momentum has to be antiparallel in this term. This seems to violate 3-momentum conservation.

Looking at the 4-component spinors as a guide, the other resolution to this would be to utilize the negative energy Weyl plane wave solution, which would make the left handed field have its spin and momentum aligned. However, this term seems to also violate charge conservation, since we are mixing the positive and negative energy solutions.

Here, I'm going off of P&S and saying the u+(0) = $\sqrt(E)$(1,0,1,0) = $(1/\sqrt(2))( u+(p>>m) + v+(p>>m))$.

I think that I'm getting confused switching between the different bases, but I'm somewhat stuck trying to conceptualize the Dirac mass terms.

https://preview.redd.it/kdc2hhhp5ge51.png?width=1257&format=png&auto=webp&s=b35cc6ced0900cdaec066513a6e9951e938b589c

This always fascinate me as a noob who's intrigued about QM.

How do we get the Dirac equation in matrix notation when we split the Dirac spinor into two Weyl spinors? I know that since we have a -m term in the equation, this is going to correspond to the diagonal entries being equal to -m. But, how do we get the off-diagonal parts? Thanks!

Image of the math I am having trouble with:

https://drive.google.com/file/d/1hXFOvg5aeRVUfHPnoohmRnkg6ICFWe4z/view?usp=sharing

How does the first line simplify to the second line? For some background, η = diag(-1,1,1,1), h is metric perturbations, and ξ are killing vectors. This is part of a GR question regarding gravitational waves and we are supposed to calculate all relevant quantities to the first words (where applicable).

The Dirac equation is usually shown as a first order differential equation with 4 components, necessary to describe the two possible projections of spin and the two possible signs for energy. But I can as well represent it with two components obeying a second order differential equation, necessary to descibre the two possible projections of spin with both possible signs for the energy in the same component. How are these connected?

Edit: grammar.

Hello everyone,

I'm going through MA Thomson lectures. I'm in QED part about chilarity chirality, Thomson tells that it's straightforward to show that for spinors written in terms of their left/right handed chiral components that [;\overline{\Psi}_{R} \gamma^{\mu}\Phi_{L}=0;] (and so for R->L ; L->R)

I would like to find this result but I don't see how to start. It's written to use [;\gamma^{5};] matrix properties but I don't see how to start. I tried to insert this matrix squared (since it's equal to identity) but I don't really see how to go further without going back to the initial form.

Thanks for your reading

edit: word

The wave function in the Shrödinger equation must fall out of the Dirac equation in the non relativistic limit, right? Or will it only fall out of the Klein Gordon equation?

Can anyone explain the relationship between the "Wavefunction" (Bispinor) in the Dirac equation and the scalar wavefunction as seen in Shrödingers equation.

To find a vector potential we write

1. Bz = ∂xAy − ∂yAx,

and try (r = p x^2 + y^2 )^(1/2):

2. Ax = −B ∂y[g(r)] = −B (y/r) g′ (r),

3. Ay = B ∂x[g(r)] = B (x/r) g′ (r).

Then if

4.(∂^2 x + ∂^2 y )g = δ^2 (x*),*

we get the right formula for Bz. This is the equation for the Coulomb potential in 2 dimensions, or equivalently for an infinite line source in 3 dimensions, so the solution is

5***. g = ln (r).***

Now

6***.*** Axdx + Aydy = Aθdθ + Ardr = B (xdy − ydx)/r^2

implies that Ar = 0 and Aθ = −B, so the line integral around the circle is −2πB. This follows from Stokes’s theorem, which says that the line integral around any curve is equal to the integral of the magnetic flux through any surface bounded by that curve.

So I am not sure where the equations Ax and Ay (1 and 2) were derived from. Is it a manipulation of the first equation, or some rule to be remembered?

I do get that the right hand side is the partial of a function g(r).

Why is g=ln(r) the solution to the Coulomb potential in 2 dimensions?

I do see how equation 4 resembles Laplaces equation, except the dirac delta means that it's equal to something non-zero at the origin right?

Thanks for any help!

This may be a bit odd, but recently a major gaming figure (known for being very cryptic) released a picture of a character for his new game (Norman Reedus) that character has "dog tag" like things on his necklace with equations on them here is a picture

The top equation is clearly The Schwarzschild Radius and the second appears to be The Dirac Equation.

I am no physicist and can only grasp so much without someone explaining it. Is there any significance to these equations? Either separate or when looked at together?

https://i.imgur.com/CHsEP58.jpg

Can someone check if I did this right?

Also I'm having a hard time visualizing heaviside functions when they're written out like this. Does it just mean that the function is 0 except from pi to 2pi the function looks like sin(2t)?

We're also supposed to plot this function and it's derivative. I thought the derivative of a heaviside is the dirac delta function, but I also read that it's not really a function? So what would the derivative graph look like? Just 2cos(2t) from pi to 2pi?

Also we have to graph a phase plane of the function and it's derivative. What would that look like?

Thanks!

Hi,

I'm studying Dirac equation and I want to prove that [;\alpha_{i};] and [;\beta;] matrices introduced by Dirac are hermitian.

I start with the hamiltonian, and make the equality [;H_{D}=H^{\dagger}_{D};] which give me : [;\frac{\hbar c}{i}( \overrightarrow{\alpha} \overrightarrow{\nabla}+\overrightarrow{\nabla}^{\dagger}\overrightarrow{\alpha}^{\dagger} )+ mc^2 (\beta-\beta^{\dagger})=0;]

By identification in the formula, I have [;\beta=\beta^{\dagger};] but for alpha matrices, I have [;\overrightarrow{\alpha} \overrightarrow{\nabla}+\overrightarrow{\nabla}^{\dagger} \overrightarrow{\alpha} ^{\dagger} = 0;]

And I have no idea how to end to [;\alpha_{i}=\alpha^{\dagger}_{i};]

Is the method correct ? Then how can I use this last equation to show that alpha matrices are hermitian ?

Thank you

You can see the lagrangian here. https://en.wikipedia.org/wiki/Dirac_equation#Dirac_Lagrangian

All of 1.45 https://imgur.com/a/M4XDz

I know how to solve it if its in the form that is shown above, but not in the other forms.

The problem I have been given is:

> Calculus Cal told Careful Carrie that if [;y_1(t);] solves [;y'(t) + 8 y(t) = \delta(t-1) ;], with [; y(0) = 3 ;], and if [; y_2(t) ;] solves [; y'(t) + 8y(t) = \delta(t - 1);] , with [; y(0) = 7 ;], then [; y1(t) + y2(t) ;] solves [; y'(t) + 8y(t) = 2 \delta(t - 1) ;], with [; y(0) = 10 ;]. What did Carrie say to Cal?​

Given the naming of the two characters, I'm assuming I'm supposed to find the flaw in Cal's logic, but I don't see anything. I literally have no idea why this might be false. Is it because the dirac delta function isn't really a function so you can't add it up? That doesn't make sense to me since [; \int {(\delta(x) + \delta(x)) dx} = \int {\delta(x)dx} + \int {\delta(x)dx} = 2\int {\delta(x)dx} = \int {2\delta(x)dx} ;], so I'm really stuck. I just have no idea what to do.

Any suggestions? Anything would help.