TODO: does it use full blown QED, or just something intermediate?
www.youtube.com/watch?v=NtnsHtYYKf0 "Mercury and Relativity - Periodic Table of Videos" by Periodic Videos (2013). Doesn't give the key juicy details/intuition. Also mentioned on Wikipedia: en.wikipedia.org/wiki/Relativistic_quantum_chemistry#Mercury
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: 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.
Do electrons spontaneously jump from high orbitals to lower ones emitting photons?
Explaining this was was one of the key initial achievements of the Dirac equation.
A critical application of this phenomena is laser.
Can be easily seen from the solution of Equation "Expanded Dirac equation in Planck units" when the particle is at rest as shown at Video "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)".
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.
Can produce two entangled particles.
Described for example in lecture 1.
TODO, including why the Schrodinger equation is not.
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".
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.
- www.youtube.com/watch?v=Fu1BGGeyqHQ&list=PL54DF0652B30D99A4&index=63 "K6. The Pauli Equation" by doctorphys
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:
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.
- 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)
The Klein-Gordon equation directly uses a more naive relativistic energy guess of squared.
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...
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.
Predicts fine structure.
Possible values are half integer numbers: 0, 1/2, 1, 3/2, and so on.
The approach shown in this section: Section "Spin comes naturally when adding relativity to quantum mechanics" shows what the spin number actually means in general. As shown there, the spin number it is a direct consequence of having the laws of nature be Lorentz invariant. Different spin numbers are just different ways in which this can be achieved as per different Representation of the Lorentz group.
Video 1. "Quantum Mechanics 9a - Photon Spin and Schrodinger's Cat I by ViaScience (2013)" explains nicely how:
- incorporated into the Dirac equation as a natural consequence of special relativity corrections, but not naturally present in the Schrödinger equation, see also: the Dirac equation predicts spin
- photon spin can be either linear or circular
- the linear one can be made from a superposition of circular ones
- straight antennas produce linearly polarized photos, and Helical antennas circularly polarized ones
- a jump between 2s and 2p in an atom changes angular momentum. Therefore, the photon must carry angular momentum as well as energy.
- cannot be classically explained, because even for a very large estimate of the electron size, its surface would have to spin faster than light to achieve that magnetic momentum with the known electron charge
- as shown at Video "Quantum Mechanics 12b - Dirac Equation II by ViaScience (2015)", observers in different frames of reference see different spin states
Originally done with silver in 1921, but even clearer theoretically was the hydrogen reproduction in 1927 by T.E. Phipps and J.B. Taylor.
The hydrogen experiment was apparently harder to do and the result is less visible, TODO why: physics.stackexchange.com/questions/33021/why-silver-atoms-were-used-in-stern-gerlach-experiment
Needs an inhomogenous magnetic field to move the atoms up or down: magnetic dipole in an inhomogenous magnetic field. TODO how it is generated?
Best mathematical explanation: Section "Spin comes naturally when adding relativity to quantum mechanics".
Physics from Symmetry by Jakob Schwichtenberg (2015) chapter 3.9 "Elementary particles" has an amazing summary of the preceding chapters the spin value has a relation to the representations of the Lorentz group, which encodes the spacetime symmetry that each particle observes. These symmetries can be characterized by small integer numbers:As usual, we don't know why there aren't elementary particles with other spins, as we could construct them.
Leads to the Klein-Gordon equation.
Leads to the Dirac equation.
Leads to the Proca equation.
Theorized for the graviton.
More interestingly, how is that implied by the Stern-Gerlach experiment?
physics.stackexchange.com/questions/266359/when-we-say-electron-spin-is-1-2-what-exactly-does-it-mean-1-2-of-what/266371#266371 suggests that half could either mean:
- at limit of large
lfor the Schrödinger equation solution for the hydrogen atom the difference between each angular momentum is twice that of the eletron's spin. Not very satisfactory.
- it comes directly out of the Dirac equation. This is satisfactory. :-)
Initially a phenomenological guess to explain the periodic table. Later it was apparently proven properly with the spin-statistics theorem, physics.stackexchange.com/questions/360140/theoretical-proof-of-paulis-exclusion-principle.
And it was understood more and more that basically this is what prevents solids from collapsing into a single nucleus, not electrical repulsion: electron degeneracy pressure!
The name actually comes from "any". Amazing.
Key physical experiment: fractional quantum Hall effect.
Exotic and hard to find experimentally.
Video "The Biggest Ideas in the Universe | 17. Matter by Sean Carroll (2020)" at youtu.be/dQWn9NzvX4s?t=3707 says that no one has ever been able to come up with an intuitive reason for the proof.
The basic idea is that there is a field for each particle particle type.
E.g. in QED, one for the electron and one for the photon: physics.stackexchange.com/questions/166709/are-electron-fields-and-photon-fields-part-of-the-same-field-in-qed.
And then those fields interact with some Lagrangian.
One way to look at QFT is to split it into two parts:
Then interwined with those two is the part "OK, how to solve the equations, if they are solvable at all", which is an open problem: Yang-Mills existence and mass gap.
- deriving the Lagrangians of the Standard Model: why do symmetries such as SU(3), SU(2) and U(1) matter in particle physics?s. This is the easier part, since the lagrangians themselves can be understood with not very advanced mathematics, and derived beautifully from symmetry constraints
- the qantization of fields. This is the hard part Ciro Santilli is unable to understand, TODO mathematical formulation of quantum field theory.
TODO holy crap, even this is hard to understand/find a clear definition of.
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.
The term and idea was first introduced initialized by Hermann Weyl when he was working on combining electromagnetism and general relativity to formulate Maxwell's equations in curved spacetime in 1918 and published as gravity and electricity by Hermann Weyl (1918). Based on perception that symmetry implies charge conservation. The same idea was later adapted for quantum electrodynamics, a context in which is has even more impact.
A random field you add to make something transform locally the way you want. See e.g.: Video "Deriving the qED Lagrangian by Dietterich Labs (2018)".
Yup, this one Focks you up.
Second quantization also appears to be useful not only for relativistic quantum mechanics, but also for condensed matter physics. The reason is that the basis idea is to use the number occupation basis. This basis is:
Basically a synonym for second quantization.
This one might actually be understandable! It is what Richard Feynman starts to explain at: Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979).
The difficulty is then proving that the total probability remains at 1, and maybe causality is hard too.
The path integral formulation can be seen as a generalization of the double-slit experiment to infinitely many slits.
Feynman first stared working it out for non-relativistic quantum mechanics, with the relativistic goal in mind, and only later on he attained the relativistic goal.
TODO why intuitively did he take that approach? Likely is makes it easier to add special relativity.
This approach more directly suggests the idea that quantum particles take all possible paths.
As mentioned at: physics.stackexchange.com/questions/212726/a-quantum-particle-moving-from-a-to-b-will-take-every-possible-path-from-a-to-b/212790#212790, classical gravity waves for example also "take all possible paths". This is just what waves look like they are doing.
Mentioned for example in quantum field theory in a nutshell by Anthony Zee (2010) page 8.
www.youtube.com/watch?v=WB8r7CU7clk&list=PLUl4u3cNGP60TvpbO5toEWC8y8w51dtvm by Iain Stewart. Basically starts by explaining how quantum field theory is so generic that it is hard to get any numerical results out of it :-)
But in particular, we want to describe those subtheories in a way that we can reach arbitrary precision of the full theory if desired.
- www.youtube.com/watch?v=-_qNKbwM_eE Unsolved: Yang-Mills existence and mass gap by J Knudsen (2019). Gives 10 key points, but the truly hard ones are too quick. He knows the thing though.
While Ciro acknowledges that QED is intrinsically challenging due to the wide range or requirements (quantum mechanics, special relativity and electromagnetism), Ciro feels that there is a glaring gap in this moneyless market for a learning material that follows the Middle Way as mentioned at: the missing link between basic and advanced. Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979) is one of the best attempts so far, but it falls a bit too close to the superficial side of things, if only Feynman hadn't assumed that the audience doesn't know any mathematics...
The funny thing is that when Ciro Santilli's mother retired, learning it (or as she put it: "how photons and electrons interact") was also one of her retirement plans. She is a pharmacist by training, and doesn't know much mathematics, and her English was somewhat limited. Oh, she also wanted to learn how photosynthesis works (possibly not fully understood by science as that time, 2020). Ambitious old lady!!!
Experiments: quantum electrodynamics experiments.
Combines special relativity with more classical quantum mechanics, but further generalizing the Dirac equation, which also does that: Dirac equation vs quantum electrodynamics. The name "relativistic" likely doesn't need to appear on the title of QED because Maxwell's equations require special relativity, so just having "electro-" in the title is enough.
Before QED, the most advanced theory was that of the Dirac equation, which was already relativistic but TODO what was missing there exactly?
As summarized at: youtube.com/watch?v=_AZdvtf6hPU?t=305 Quantum Field Theory lecture at the African Summer Theory Institute 1 of 4 by Anthony Zee (2004):
- classical mechanics describes large and slow objects
- special relativity describes large and fast objects (they are getting close to the speed of light, so we have to consider relativity)
- classical quantum mechanics describes small and slow objects.
- QED describes objects that are both small and fast
That video also mentions the interesting idea that:
QED is the first quantum field theory fully developed. That framework was later extended to also include the weak interaction and strong interaction. As a result, it is perhaps easier to just Google for "Quantum Field Theory" if you want to learn QED, since QFT is more general and has more resources available generally.
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?
www.mdpi.com/2624-8174/2/2/8/pdf History and Some Aspects of the Lamb Shift by G. Jordan Maclay (2019)
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.
Previous less experiments had already hinted at this effect, but they were too imprecise to be sure.
Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979) mentions it several times.
Apparently first published at the Magnetic Moment of the Electron by Kusch and Foley (1948).
journals.aps.org/pr/abstract/10.1103/PhysRev.74.250, paywall as of 2021.
Advanced quantum mechanics by Freeman Dyson (1951) mentions:
A Relativistic Quantum Theory of a Finite Number of Particles is Impossible.
- 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
Note that this is the sum of the:
Note that the relationship between and is not explicit. However, if we knew what type of particle we were talking about, e.g. electron, then the knowledge of psi would also give the charge distribution and therefore
As mentioned at the beginning of Quantum Field Theory lecture notes by David Tong (2007):
TODO find/create decent answer.
I think the best answer is something along:
- local symmetries of the Lagrangian imply conserved currents. gives conserved charges.
- OK now. We want a local symmetry. And we also want:Given all of that, the most obvious and direct thing we reach a guess at the quantum electrodynamics Lagrangian is Video "Deriving the qED Lagrangian by Dietterich Labs (2018)"
A basic non-precise intuition is that a good model of reality is that electrons do not "interact with one another directly via the electromagnetic field".
A better model happens to be the quantum field theory view that the electromagnetic field interacts with the photon field but not directly with itself, and then the photon field interacts with parts of the electromagnetic field further away.
From Video "Lorenzo Sadun on the "Yang-Mills and Mass Gap" Millennium problem":
I think they are a tool to calculate the probability of different types of particle decays and particle collision outcomes. TODO Minimal example of that.
At Richard Feynman Quantum Electrodynamics Lecture at University of Auckland (1979), an intuitive explanation of them in termes of sum of products of propagators is given.
No, but why?
fafnir.phyast.pitt.edu/py3765/ Phys3765 Advanced Quantum Mechanics -- QFT-I Fall 2012 by E.S. Swanson mentions several milestone texts including:
We shall not develop straightaway a correct theory including many particles. Instead we follow the historical development. We try to make a relativistic quantum theory of one particle, find out how far we can go and where we get into trouble.Oh yes, see also: Dirac equation vs quantum electrodynamics.
By Richard Feynman.
Talk title shown on intro: "Today's Answers to Newton's Queries about Light".
6 hour lecture, where he tries to explain it to an audience that does not know any modern physics. This is a noble effort.
Part of The Douglas Robb Memorial Lectures lecture series.
Feynman apparently also made a book adaptation: QED: The Strange Theory of Light and Matter. That book is basically word by word the same as the presentation, including the diagrams.
According to www.feynman.com/science/qed-lectures-in-new-zealand/ the official upload is at www.vega.org.uk/video/subseries/8 and Vega does show up as a watermark on the video (though it is too pixilated to guess without knowing it), a project that has been discontinued and has has a non-permissive license. Newbs.
This talk has the merit of being very experiment oriented on part 2, big kudos: how to teach and learn physics
- Part 1: is saying "photons exist"
- Part 2: is amazing, and describes how photons move as a sum of all possible paths, not sure if it is relativistic at all though, and suggests that something is minimized in that calculation (the action)
- Part 3: is where he hopelessly tries to explain the crucial part of how electrons join the picture in a similar manner to how photons do.He does make the link to light, saying that there is a function which gives the amplitude for a photon going from A to B, where A and B are spacetime events.And then he mentions that there is a similar function for an electron to go from A to B, but says that that function is too complicated, and gives no intuition unlike the photon one.He does not mention it, but P and E are the so called propagators.This is likely the path integral formulation of QED.On Quantum Mechanical View of Reality by Richard Feynman (1983) he mentions that is a bessel function, without giving further detail.And also mentions that:
mis basically a scale factor. such that both are very similar. And that something similar holds for many other particles.And then, when you draw a Feynman diagram, e.g. electron emits photon and both are detected at given positions, you sum over all the possibilities, each amplitude is given by: Spacetime points.This is basically well said at: youtu.be/rZvgGekvHes?t=3349 from Quantum Mechanical View of Reality by Richard Feynman (1983).TODO: how do electron velocities affect where they are likely to end up? suggests the probability only depends on the spacetime points.Also, this clarifies why computations in QED are so insane: you have to sum over every possible point in space!!! TODO but then how do we calculate anything at all in practice?
- Part 4: known problems with QED and thoughts on QCD. Boring.