Ciro Santilli @cirosantilli 34 

Derivation of the Dirac equation  Updated 2024-12-15  +Created 1970-01-01

The Dirac equation can be derived basically "directly" from the Representation theory of the Lorentz group for the spin half representation, this is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) 6.3 "Dirac Equation".

The Diract equation is the spacetime symmetry part of the quantum electrodynamics Lagrangian, i.e. is describes how spin half particles behave without interactions. The full quantum electrodynamics Lagrangian can then be reached by adding the

U (1)

internal symmetry.

As mentioned at spin comes naturally when adding relativity to quantum mechanics, this same method allows us to analogously derive the equations for other spin numbers.

Bibliography:

Video "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=743
www.youtube.com/watch?v=zM-Lc16nyho&list=PL54DF0652B30D99A4&index=66 "L3. The Dirac Equation" by doctorphys
Video "Dirac equation for the electron and hydrogen Hamiltonian by Barton Zwiebach (2019)"

Deriving The Dirac equation by Andrew Dotson (2019)

Source.

Derivation of the Klein-Gordon equation  Updated 2024-12-15  +Created 1970-01-01

The Klein-Gordon equation directly uses a more naive relativistic energy guess of

p^{2} + m^{2}

squared.

But since this is quantum mechanics, we feel like making

p

into the "momentum operator", just like in the Schrödinger equation.

But we don't really know how to apply the momentum operator twice, because it is a gradient, so the first application goes from a scalar field to the vector field, and the second one...

So we just cheat and try to use the laplace operator instead because there's some squares on it:

H = \nabla^{2} + m^{2}

But then, we have to avoid taking the square root to reach a first derivative in time, because we don't know how to take the square root of that operator expression.

So the Klein-Gordon equation just takes the approach of using this squared Hamiltonian instead.

Since it is a Hamiltonian, and comparing it to the Schrödinger equation which looks like:

H ψ = i \frac{\partial ψ}{\partial t}

(2)

taking the Hamiltonian twice leads to:

H^{2} ψ = - \frac{\partial ^{2} ψ}{\partial ^{2} t}

(3)

We can contrast this with the Dirac equation, which instead attempts to explicitly construct an operator which squared coincides with the relativistic formula: derivation of the Dirac equation.

Derivation of the quantum electrodynamics Lagrangian  Updated 2024-12-15  +Created 1970-01-01

Like the rest of the Standard Model Lagrangian, this can be split into two parts:

spacetime symmetry: reaches the derivation of the Dirac equation, but has no interactions
add the $U (1)$ internal symmetry to add interactions, which reaches the full equation

www.youtube.com/watch?v=IFRyN3fQMO8&lc=UgzNGkLXdwcSl7z8Lap4AaABAg

Deriving the qED Lagrangian by Dietterich Labs (2018)

Source.

As mentioned at the start of the video, he starts with the Dirac equation Lagrangian derived in a previous video. It has nothing to do with electromagnetism specifically.

He notes that that Dirac Lagrangian, besides being globally Lorentz invariant, it also also has a global

U (1)

invariance.

However, it does not have a local invariance if the

U (1)

transformation depends on the point in spacetime.

He doesn't mention it, but I think this is highly desirable, because in general local symmetries of the Lagrangian imply conserved currents, and in this case we want conservation of charges.

To fix that, he adds an extra gauge field

A_{μ}

(a field of

4 \times 4

matrices) to the regular derivative, and the resulting derivative has a fancy name: the covariant derivative.

Then finally he notes that this gauge field he had to add has to transform exactly like the electromagnetic four-potential!

So he uses that as the gauge, and also adds in the Maxwell Lagrangian in the same go. It is kind of a guess, but it is a natural guess, and it turns out to be correct.

Dirac equation  Updated 2024-12-15  +Created 1970-01-01

Adds special relativity to the Schrödinger equation, and the following conclusions come basically as a direct consequence of this!

Experiments explained:

spontaneous emission coefficients.
fine structure, notably for example Dirac equation solution for the hydrogen atom
antimatter
particle creation and annihilation

Experiments not explained: those that quantum electrodynamics explains like:

Lamb shift
TODO: quantization of the electromagnetic field as photons?

The Dirac equation is a set of 4 partial differential equations on 4 complex valued wave functions. The full explicit form in Planck units is shown e.g. in Video 1. "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=1010:

i \partial_{t} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} ψ_{3} ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤ = - i \partial_{x} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{4} ψ_{3} ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ + \partial_{y} ⎣ ⎢ ⎢ ⎢ ⎡ - ψ_{4} ψ_{3} - ψ_{2} ψ_{1} ⎦ ⎥ ⎥ ⎥ ⎤ - i \partial_{z} ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{3} - ψ_{4} ψ_{1} - ψ_{2} ⎦ ⎥ ⎥ ⎥ ⎤ + m ⎣ ⎢ ⎢ ⎢ ⎡ ψ_{1} ψ_{2} - ψ_{3} - ψ_{4} ⎦ ⎥ ⎥ ⎥ ⎤

Equation 1.

Expanded Dirac equation in Planck units

Then as done at physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600 from why are complex numbers used in the Schrodinger equation?, we could further split those equations up into a system of 8 equations on 8 real-valued functions.

Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)

Source.

Video 2.

PHYS 485 Lecture 14: The Dirac Equation by Roger Moore (2016)

Source.

Dirac equation solution for the hydrogen atom  Updated 2024-12-15  +Created 1970-01-01

Predicts fine structure.

Bibliography:

Video "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)" youtu.be/tR6UebCvFqE?t=549

Dirac equation for the electron and hydrogen Hamiltonian by Barton Zwiebach (2019)

Source. Uses perturbation theory to get to the relativistic corrections of fine structure! Part of MIT 8.06 Quantum Physics III, Spring 2018 by Barton Zwiebach

Video 2.

How To Solve The Dirac Equation For The Hydrogen Atom | Relativistic Quantum Mechanics by Dietterich Labs (2018)

Source.

Dirac equation vs quantum electrodynamics  Updated 2024-12-15  +Created 1970-01-01

TODO: in high level terms, why is QED more general than just solving the Dirac equation, and therefore explaining quantum electrodynamics experiments?

Also, is it just a bunch of differential equation (like the Dirac equation itself), or does it have some other more complicated mathematical formulation, as seems to be the case? Why do we need something more complicated than

Advanced quantum mechanics by Freeman Dyson (1951) mentions:

A Relativistic Quantum Theory of a Finite Number of Particles is Impossible.

Bibliography:

physics.stackexchange.com/questions/101307/dirac-equation-in-qft-vs-relativistic-qm
physics.stackexchange.com/questions/44188/what-is-the-relativistic-particle-in-a-box/44309#44309 says:
By several reasons explained in textbooks, the Dirac equation is not a valid wavefunction equation. You can solve it and find solutions, but those solutions cannot be interpreted as wavefunctions for a particle
physics.stackexchange.com/questions/64206/why-is-the-dirac-equation-not-used-for-calculations
www.physicsforums.com/threads/is-diracs-equation-still-useful-after-qed-is-developed.663994/

Dirac Lagrangian  Updated 2024-12-15  +Created 1970-01-01

L = \overset{ˉ}{ψ} (i ℏ c \partial / - m c^{2}) ψ

where:

$\partial$ : Feynman slash notation
$\overset{ˉ}{ψ}$ : Dirac adjoint

Remember that

ψ

is a 4-vetor, gamma matrices are 4x4 matrices, so the whole thing comes down to a dot product of two 4-vectors, with a modified

ψ

by matrix multiplication/derivatives, and the result is a scalar, as expected for a Lagrangian.

Like any other Lagrangian, you can then recover the Dirac equation, which is the corresponding equations of motion, by applying the Euler-Lagrange equation to the Lagrangian.

Fine structure  Updated 2024-12-15  +Created 1970-01-01

Split in energy levels due to interaction between electron up or down spin and the electron orbitals.

Numerically explained by the Dirac equation when solving it for the hydrogen atom, and it is one of the main triumphs of the theory.

Hyperfine structure  Updated 2024-12-15  +Created 1970-01-01

Small splits present in all levels due to interaction between the electron spin and the nuclear spin if it is present, i.e. the nucleus has an even number of nucleons.

As the name suggests, this energy split is very small, since the influence of the nucleus spin on the electron spin is relatively small compared to other fine structure.

TODO confirm: does it need quantum electrodynamics or is the Dirac equation enough?

The most important examples:

hydrogen line useful in astronomy, and also the simplest possible case between 1s
caesium standard, which is used to define the second in the International System of Units since 1967.

Internal and spacetime symmetries  Updated 2024-12-15  +Created 1970-01-01

physics.stackexchange.com/questions/106392/internal-and-spacetime-symmetries

The different only shows up for field, not with particles. For fields, there are two types of changes that we can make that can keep the Lagrangian unchanged as mentioned at Physics from Symmetry by Jakob Schwichtenberg (2015) chapter "4.5.2 Noether's Theorem for Field Theories - Spacetime":

spacetime symmetry: act with the Poincaré group on the Four-vector spacetime inputs of the field itself, i.e. transforming $L (Φ (x), \partial Φ (x), d x)$ into $L (Φ^{'} (x^{'}), \partial Φ^{'} (x^{'}), x^{'})$
internal symmetry: act on the output of the field, i.e.: $L (Φ (x) + δ Φ (x), \partial (Φ (x) + δ Φ (x)), x)$

From defining properties of elementary particles:

spacetime:
- mass
- spin
internal

From the spacetime theory alone, we can derive the Lagrangian for the free theories for each spin:

Then the internal symmetries are what add the interaction part of the Lagrangian, which then completes the Standard Model Lagrangian.

Klein-Gordon equation  Updated 2024-12-15  +Created 1970-01-01

A relativistic version of the Schrödinger equation.

Correctly describes spin 0 particles.

The most memorable version of the equation can be written as shown at Section "Klein-Gordon equation in Einstein notation" with Einstein notation and Planck units:

\partial_{i} \partial^{i} ψ - m^{2} ψ = 0

Has some issues which are solved by the Dirac equation:

it has a second time derivative of the wave function. Therefore, to solve it we must specify not only the initial value of the wave equation, but also the derivative of the wave equation,
As mentioned at Advanced quantum mechanics by Freeman Dyson (1951) and further clarified at: physics.stackexchange.com/questions/340023/cant-the-negative-probabilities-of-klein-gordon-equation-be-avoided, this would lead to negative probabilities.
the modulus of the wave function is not constant and therefore not always one, and therefore cannot be interpreted as a probability density anymore
since we are working with the square of the energy, we have both positive and negative value solutions. This is also a features of the Dirac equation however.

Bibliography:

Video "Quantum Mechanics 12a - Dirac Equation I by ViaScience (2015)" at youtu.be/OCuaBmAzqek?t=600
An Introduction to QED and QCD by Jeff Forshaw (1997) 1.2 "Relativistic Wave Equations" and 1.4 "The Klein Gordon Equation" gives some key ideas
2011 PHYS 485 lecture videos by Roger Moore from the University of Alberta at around 7:30
www.youtube.com/watch?v=WqoIW85xwoU&list=PL54DF0652B30D99A4&index=65 "L2. The Klein-Gordon Equation" by doctorphys
sites.ualberta.ca/~gingrich/courses/phys512/node21.html from Advanced quantum mechanics II by Douglas Gingrich (2004)
- sites.ualberta.ca/~gingrich/courses/phys512/node23.html gives Lorentz invariance

Lamb-Retherford experiment  Updated 2024-12-15  +Created 1970-01-01

Published as "Fine Structure of the Hydrogen Atom by a Microwave Method" by Willis Lamb and Robert Retherford (1947) on Physical Review. This one actually has open accesses as of 2021, miracle! journals.aps.org/pr/pdf/10.1103/PhysRev.72.241

Microwave technology was developed in World War II for radar, notably at the MIT Radiation Laboratory. Before that, people were using much higher frequencies such as the visible spectrum. But to detect small energy differences, you need to look into longer wavelengths.

This experiment was fundamental to the development of quantum electrodynamics. As mentioned at Genius: Richard Feynman and Modern Physics by James Gleick (1994) chapter "Shrinking the infinities", before the experiment, people already knew that trying to add electromagnetism to the Dirac equation led to infinities using previous methods, and something needed to change urgently. However for the first time now the theorists had one precise number to try and hack their formulas to reach, not just a philosophical debate about infinities, and this led to major breakthroughs. The same book also describes the experiment briefly as:

Willis Lamb had just shined a beam of microwaves onto a hot wisp of hydrogen blowing from an oven.

It is two pages and a half long.

They were at Columbia University in the Columbia Radiation Laboratory. Robert was Willis' graduate student.

Previous less experiments had already hinted at this effect, but they were too imprecise to be sure.

Lamb shift  Updated 2024-12-15  +Created 1970-01-01

2s/2p energy split in the hydrogen emission spectrum, not predicted by the Dirac equation, but explained by quantum electrodynamics, which is one of the first great triumphs of that theory.

Note that for atoms with multiple electrons, 2s/2p shifts are expected: Why does 2s have less energy than 1s if they have the same principal quantum number?. The surprise was observing that on hydrogen which only has one electron.

Initial experiment: Lamb-Retherford experiment.

On the return from the train from the Shelter Island Conference in New York, Hans Bethe managed to do a non-relativistic calculation of the Lamb shift. He then published as The Electromagnetic Shift of Energy Levels by Hans Bethe (1947) which is still paywalled as of 2021, fuck me: journals.aps.org/pr/abstract/10.1103/PhysRev.72.339 by Physical Review.

The Electromagnetic Shift of Energy Levels Freeman Dyson (1948) published on Physical Review is apparently a relativistic analysis of the same: journals.aps.org/pr/abstract/10.1103/PhysRev.73.617 also paywalled as of 2021.

TODO how do the infinities show up, and how did people solve them?

Lamb shift by Dr. Nissar Ahmad (2020)

Source. Whiteboard Lecture about the phenomena, includes description of the experiment. Seems quite good.

Video 2.

Murray Gell-Mann - The race to calculate the relativistic Lamb shift by Web of Stories (1997)

Source. Quick historical overview. Mentions that Richard Feynman and Julian Schwinger were using mass renormalization and cancellation if infinities. He says that French and Weisskopf actually managed to do the correct calculations first with a less elegant method.

www.mdpi.com/2624-8174/2/2/8/pdf History and Some Aspects of the Lamb Shift by G. Jordan Maclay (2019)

Video 3.

Freeman Dyson - The Lamb shift by Web of Stories (1998)

Source.

Mentions that he moved to the USA from the United Kingdom specifically because great experiments were being carried at Columbia University, which is where the Lamb-Retherford experiment was done, and that Isidor Isaac Rabi was the head at the time.

He then explains mass renormalization briefly: instead of calculating from scratch, you just compare the raw electron to the bound electron and take the difference. Both of those have infinities in them, but the difference between them cancels out those infinities.

Video 4.

Hans Bethe - The Lamb shift (1996)

Source.

Ahh, Hans is so old in that video, it is sad to see. He did live a lot tough. Mentions that the shift is of about 1000 MHz.

The following video: www.youtube.com/watch?v=vZvQg3bkV7s Hans Bethe - Calculating the Lamb shift.

Video 5.

Lamb shift by Vidya-mitra (2018)

Source.

Mathematical formulation of quantum field theory  Updated 2024-12-15  +Created 1970-01-01

TODO holy crap, even this is hard to understand/find a clear definition of.

The Dirac equation, OK, is a partial differential equation, so we can easily understand its definition with basic calculus. We may not be able to solve it efficiently, but at least we understand it.

But what the heck is the mathematical model for a quantum field theory? TODO someone was saying it is equivalent to an infinite set of PDEs somehow. Investigate. Related:

The path integral formulation might actually be the most understandable formulation, as shown at Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).

The formulation of QFT also appears to be a form of infinite-dimentional calculus.

Quantum electrodynamics by Lifshitz et al. 2nd edition (1982) chapter 1. "The uncertainty principle in the relativistic case" contains an interesting idea:

The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta,
polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process.

Particle creation and annihilation  Updated 2024-12-15  +Created 1970-01-01

en.wikipedia.org/wiki/Annihilation

Predicted by the Dirac equation.

We've likely known since forever that photons are created: just turn on a light and see gazillion of them come out!

Photon creation is easy because photons are massless, so there is not minimum energy to create them.

The creation of other particles is much rarer however, and took longer to be discovered, one notable milestone being the discovery of the positron.

In the case of the electron, we need to start with at least enough energy for the mass of the electron positron pair. This requires a photon with wavelength in the picometer range, which is not common in the thermal radiation of daily life.

Particle physics  Updated 2024-12-15  +Created 1970-01-01

Currently an informal name for the Standard Model

Chronological outline of the key theories:

Maxwell's equations
Schrödinger equation
- Date: 1926
- Numerical predictions:
  - hydrogen spectral line, excluding finer structure such as 2p up and down split: en.wikipedia.org/wiki/Fine-structure_constant
Dirac equation
- Date: 1928
- Numerical predictions:
  - hydrogen spectral line including 2p split, but excluding even finer structure such as Lamb shift
- Qualitative predictions:
  - Antimatter
  - Spin as part of the equation
quantum electrodynamics
- Date: 1947 onwards
- Numerical predictions:
  - Lamb shift
- Qualitative predictions:
  - Antimatter
  - spin as part of the equation