Quantum physics

http://www.scottaaronson.com/democritus/lec9.html

• Why p-norm?
  
• Why 2-norm?
  
• Why preserve norm?
  
• Why linear transformations?
  
• Why complex amplitudes? http://arxiv.org/abs/quant-ph/0104088
  

The other thing you should know is that Copenhagen is not any longer widely accepted amongst people who actually work on interpretations of QM or in foundational physics in general, it's just accepted outside the community as something that most people have at least some knowledge of.
Now, what's the problem with Copenhagen. The biggest one is the so-called measurement problem, that there is an ill-defined delineation between a system and its observer, or between "quantum" and "classical". There's may others though, e.g. that randomness is postulated rather than a result of the theory.

Pilot wave theory: the pilot wave, particle follows it. Bohm trajectory.
If we could determine position and velocity, we would be able to predict trajectory exactly.

Incompatible with relativity, non-local.

global hidden variables pilot wave?
pilot wave theory is both physical and deterministic

a |00> + b |01> + c |10> + d |11>
Measure of entanglement: ad - bc

For 1 < p < inf, L_p is reflexive

Riesz theorem: p ∈ (1, inf), phi ∈ L_p^*.
There exists a unique g ∈ L_q, s.t. phi(f) = \ip{g}{f}. Norms are equal.
T: L^q -> (L^p)*
T(u) f = \ip{Tu}{f}

Theorem: any quantum algorithm must take at least O(sqrt(N)) queries.
Lemma: 1 query > no algorithm can guarantee success prob > const / N.
t steps: P[success] = O(t²)/N

commutator with Hamiltonian => time derivative (todo what?)
V = I - i eps G (generator)
[H, V] = 0
[H, I - i eps G] = 0
[H, G] = 0

degenerate level. Degeneracy is a consequence of symmetry

P(r_a, r_b) = P_a(r_a) P_b(r_b)

P+-(r_a, r_b) = A (P_a(r_a) P_b(r_b) +/- P_b(r_a) P_a(r_b))

• for bosons
  
• for fermions
  

spin-statistics?

X : symmetry operator

X P(r_a, r_b) = P(r_b, r_a)
[X, H] = 0
complete set of simultaneous eigenvalues

P(r_a, r_b) = +- P(r_b, r_a): fundamental law; symmetrization requirement

two identical particles can't occupy same single-particle states. Pauli exclusion principle.

1. Sequential repetition
   
2. Parallel repetition
   
3. Sequential repetition (multi-prover)
   
4. Parallel repetition (multi-prover)
   

Левый игрок: получает \(a\), вводит \(c\). Второй игрок: получает \(b\), вводит \(d\). Победа – если \(c \otimes d = a \lor b\).

Оптимальная стратегия: один игрок вводит \(0\), другой – \(1\). В \(\frac34\) случаев \(a \lor b\) равен \(1\), соответственно, они будут выигрывать с вероятностью в \(p_C = \frac34\) процентов.

Как доказать, что нельзя больше:

• Перебрать все стратегии (рассмотреть 16 случаев, ни одна не дает больше \(\frac34\), значит, и смешанная не даст больше)
  
• Заметим, что \(c\)(\(d\)) можно выразить как линейную функцию от \(a\)(\(b\)): \(c = \lambda a \otimes \mu\), \(d = \lambda' b \otimes \mu'\), их xor тоже линеен, а функция «или» в правой части нелинейна, поэтому будет фейл хотя бы в одном из четырех случаев.
  

https://physics.stackexchange.com/questions/437/what-equation-describes-the-wavefunction-of-a-single-photon

In quantum field theory, the Dirac equation and the Schrödinger equation have very different roles. The Dirac equation is an equation for the field, which is not a particle. The time evolution of a particle, ie, a quantum state, is always given by the Schrödinger equation. The hamiltonian for this time evolution is written in terms of fields which obey a certain equation themselves. So, the proper answer is: Schrödinger equation with a hamiltonian given in terms of a massless vector field whose equation is nothing else but Maxwell's equation.

https://physics.stackexchange.com/questions/251673/can-particle-quantum-spin-be-described-with-a-wave-function

Within the context of first quantization (the Schrodinger equation), spin itself is not characterized by the spatial or momentum space wave function (the function that defines a complex quantity over all space). It is instead treated with a separate Hilbert space that quantifies the spin states 'tacked on' to this wavefunction by an outer product. It is a matter of semantics whether you consider 'wavefunction' to refer to the complex field alone, or the composite field/spin hilbert spaces.
More advanced field equations incorporate spin more naturally. The Dirac equation, for example, no longer describes the evolution of a complex scalar field, but the evolution of a four-vector, that naturally describes spin states. For this system, spin isis characterized in the wavefunction itself.

https://en.wikipedia.org/wiki/Anyon

So it can be seen that the topological notion of equivalence comes from a study of the Feynman path integral.[3]:28

https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem

Excellent questions - the answers are very subtle.

First of all, let's look at your first question, why exchange has to be symmetric or antisymmetric. You may be surprised to know that It's actually a peculiarity of having 3 (actually, just of having more than 2) spatial dimensions.

To see why this is, let's think first about how to implement particle exchange. A careful way to do this is to imagine that you start with two particles and slowly drag one of them around the other one until it's on the other side, then slide both particles over so that they are in the initial positions. You should try to draw this if my description is confusing. The statistics shouldn't depend on *how* you drag one particle around the other, just that you complete one half loop.

Now, suppose you exchange the particles twice. That's the same as dragging one particle halfway around the other, and then doing it again, that is, having one particle make a full loop around the other one. Now here's the key point: again, the result here shouldn't depend on the exact shape or size of the loop, only that the particles have been exchanged twice. In two dimensions, this loop must contain the other particle, and you cannot shrink the loop to zero size without crashing the first particle into the second at some point. However, in three dimensions, you can push the loop out of the original plane - in fact, you can shrink the loop to zero size while always keeping the first particle far away from the second particle. (Sorry I don't have pictures! Try drawing some.) This means that two exchanges *must* have the same effect as doing nothing at all, since the statistics shouldn't depend on what the loop looks like! That is, a single exchange can *only* either give you a minus sign or no change at all, so that you get no change upon doing it twice. In 2D, this is not true - in fact, you can get [anyons](http://en.wikipedia.org/wiki/Anyon) in 2D.

This exhausts my expertise for lay explanations to your questions - unfortunately I don't have a non-technical answer to the others. Hopefully someone else can fill in. But I'll give a couple of starting points.

Your second question, of why a particle can't have other spins. ~~It's a result of putting special relativity and quantum mechanics together, actually, in 3 spatial dimensions. For the mathematically inclined, it has to do with the fact that 1/2-integer and integer spins are the only allowed representations of the Lorentz group (which is the set of symmetries of spacetime in special relativity).~~ (EDIT: Important correction given as a response.)

And finally, the connection between the two - the [spin-statistics theorem](http://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem) tells you that bosons have integer spin and fermions have half-integer spin. This again relies on special relativity and having 3 spatial dimensions.

http://twitter.com/statuses/536870238781194240

https://quantumcomputing.stackexchange.com/questions/2030/what-exactly-are-anyons-and-how-are-they-relevant-to-topological-quantum-computi

https://quantumcomputing.stackexchange.com/questions/2030/what-exactly-are-anyons-and-how-are-they-relevant-to-topological-quantum-computi

> Your second question, of why a particle can't have other spins. It's a result of putting special relativity and quantum mechanics together, actually, in 3 spatial dimensions. For the mathematically inclined, it has to do with the fact that 1/2-integer and integer spins are the only allowed representations of the Lorentz group (which is the set of symmetries of spacetime in special relativity).

Actually, plain old rotational symmetry in 3D will give you the restriction to half-integer and integer spins. Adding special relativity will give you the notions of chirality and helicity, as well as the spin-statistics theorem.

http://philsci-archive.pitt.edu/223/1/Objectivity.pdf

The best summary can be found starting at page 7 in http://philsci-archive.pitt.edu/223/1/Objectivity.pdf
Especially: "local gauge invariance in quantum theory does not imply the existence of an external electromagnetic field"!

 https://physics.stackexchange.com/questions/81041/a-hermitian-operator-with-imaginary-eigenvalues?rq=1
 For your model, the classical hamiltonian H=12(q3p+pq3)=q3pH=12(q3p+pq3)=q3p produces the Hamilton equations
⎧⎩⎨p˙=−∂H∂q=q˙=∂H∂p=3q2p,q3.
{p˙=−∂H∂q=3q2p,q˙=−∂H∂p=q3.
These are fairly easy to solve, and the solutions are not well behaved:
⎧⎩⎨⎪⎪12q2p20p2=t0−t,=(t0−t)3,
{12q2=t0−t,p02p2=(t0−t)3,
where t0t0 and p0p0 are constants of integration. Note, in particular, that there are no (real) solutions after a certain time t0=tin+12q(tin)2t0=tin+12q(tin)2. How, then, are you expecting reasonable physics out of the quantized version of this?

String theorists like anti-de Sitter spacetime... but it's very weird.  In 2d it's the hyperboloid shown here!  Time loops around in a circle, while space extends infinitely far left and right.   (The blue cone is just for fun, not part of the spacetime.)
(continued) https://t.co/hxTWwjiH6a

https://twitter.com/johncarlosbaez/status/1052634350695501824

https://physics.stackexchange.com/questions/52768/infinite-square-well-in-momentum-space/289368#289368

space is a bounded region of the real axis, so no translation group can be defined

huh, that's interesting…

https://physics.stackexchange.com/questions/232831/conservation-of-momentum-in-infinite-square-well

Physics students always say that the Schrodinger equation doesn't hold for SR, but it does. It's not the Schrodinger equation that fails though, it's the non-relativistic Hamiltonian that fails. The Dirac equation *is* the Schrodinger equation, but with the Dirac Hamiltonian. The Schrodinger equation is valid.

The point is, it has made you think about the issue. Whereas we all might agree a hydrogen atom is not an observer and a human is an observer, the case of a cat is not so clear. The point of the thought experiment is to expose problems with the Copenhagen interpretation - which it does very successfully.

As you open the box, this quantum state again evolves unitarily (i.e. without any kind of "collapse") into

Wigner's friend

   Wigner puts his friend in with the cat. The external observer believes the system is in state ( | dead ⟩ + | alive ⟩ ) / 2 {\displaystyle (|{\text{dead}}\rangle +|{\text{alive}}\rangle )/{\sqrt {2}}} (|{\text{dead}}\rangle +|{\text{alive}}\rangle )/{\sqrt {2}}. However, his friend is convinced that the cat is alive, i.e. for him, the cat is in the state | alive ⟩ {\displaystyle |{\text{alive}}\rangle } |{\text{alive}}\rangle . How can Wigner and his friend see different wave functions?

   The Copenhagen interpretation: The answer depends on the positioning of Heisenberg cut, which can be placed arbitrarily. If Wigner's friend is positioned on the same side of the cut as the external observer, his measurements collapse the wave function for both observers. If he is positioned on the cat's side, his interaction with the cat is not considered a measurement.

studying spinors – try to translate dirac equation solution from moving reference frame (only p_x component) to still

make some flashcards from this video..

https://physics.stackexchange.com/questions/75363/how-is-the-schroedinger-equation-a-wave-equation/75374#75374

Actually, a wave equation is any equation that admits wave-like solutions, which take the form f(x⃗ ±v⃗ t)f(x→±v→t). The equation ∂2f∂t2=c2∇2f∂2f∂t2=c2∇2f, despite being called "the wave equation," is not the only equation that does this.

hmm, so if you do variable transform and substitute in pde, the substitution behaves like a wave… interesting

right, so it's unclear whether neutrinos get their mass from Higgs or something else

So far most particles we discovered seem to be described by Dirac equation? We haven't seen evidence of Majorana particles so far?

We can even define a ‘metric’ that is antisymmetric: a two-dimensional space called spinor space has such a metric, and it turns out to be of fundamental importance in physics.

hmm, interesting

https://twitter.com/hamish_todd/status/1288637024203833344
youtu.be/kvQuuFB8fpk?t=179

@hamish_todd: The Schrödinger equation describes the wave function or state function of a quantum-mechanical system. It states that, over time, the humps will move in a direction perpendicular to the colored stripes

Vector potential is important because it is impossible to state action principle in a Lorentz invariant way directly in terms of E and B fields.

The most famous such equation is the position/momentum commutation relation mentioned earlier, which for our purposes is just the following matrix equation:
AB – BA = I.
This equation can’t be satisfied by any finite-dimensional matrices, since AB and BA have the same trace, so Tr(AB-BA)=0, but Tr(I) is nonzero.

https://youtu.be/VwGyqJMPmvE?t=469 – if you limit the time domain of a pure note, counterintuitively, you'll have to use more sine waves to approximate it, so widen the frequency domain

https://github.com/karldray/quantum

https://vankessel.io/disproving-quantum-immortality

https://twitter.com/enclanglement/status/1204556108561559552

Helping my students to see beyond equations.

https://www.scottaaronson.com/blog/?p=3975

https://twitter.com/InertialObservr/status/1157356874729046016

A changing Magnetic flux induces an Electric field. This can create a current in a wire to power a
No battery needed!

very nice visualisation!

I'm excited to announce the launch of a new essay, "How quantum teleportation works" (joint with Andy Matuschak). Teleportation is fun and surprising, and a fundamental primitive in quantum computation. This essay explains in detail how it works. It's available here:

https://quantum.country/teleportation

http://www.gregegan.net/BORDER/Soccer/Soccer.html

basic idea: draw the wavefunction in the orthogonal plane, then it's easy to see where the wavepacket is moving

nice derivation of strong force energy via E=mc², the uncertaincy principle, and radius of the nucleus

https://twitter.com/nonlocalpodcast/status/1340138479502692352

@nonlocalpodcast: Seasons greetings! We - @captainhamptons , William, @henryquantum - are excited (and nervous!) to announce a project we've been working on: a quantum computing podcast! We have two episodes posted, and a bunch more to come. Tune in @ https://t.co/Nwz2cbHD8W, Spotify, or iTunes!

https://www.youtube.com/watch?v=ePA1zq56J1I&t=838s
Einstein was given Nobel Prize for photoelectic effect; no one in the commitee believed in light quanta

https://www.youtube.com/watch?v=ePA1zq56J1I&t=838s
apparently, Planck still didn't believe in light quanta in 1913

For massless particles – photons, gluons, and (hypothetical) gravitons – chirality is the same as helicity; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.

However, Klein’s result showed that if the potential is of the order of the electron mass, V ∼ m c 2 {\displaystyle V\sim mc^{2}} V\sim mc^{2}, the barrier is nearly transparent. Moreover, as the potential approaches infinity, the reflection diminishes and the electron is always transmitted.

wow, pretty insane

Paul Dirac: the purest soul in physics – Physics World
in context

That is a weak interaction effect and not something mediated by the electromagnetic interaction, which was specifically what the OP was asking about.
To discuss that we'd really need to move to QFT and beyond just the Schrodinger equation.
Though I presume the next logical question is why the weak interaction doesn't induce electron capture with hydrogen...
In this case, it is because a free neutron is a higher energy, less stable system. Electron capture is only relevant in systems where the resulting neutron is stabilized by the rest of the nucleus.
In cases where the neutron is *not* stabilized, you get positron emission. It is no accident that positron emission is a *common* decay mode among neutron rich nuclei.

whoa, crazy detailed answer… something involving S-matrix..

• [2021-08-21] "It should be noted that spin > 2 MASSIVE particles exist."
  

The best explanation can be found here: http://gregnaber.com/wp-content/uploads/GAUGE-FIELDS-AND-GEOMETRY-A-PICTURE-BOOK.pdf at page 18ff

Подобно тому, как скалярный потенциал связан с понятием энергии, векторный потенциал обнаруживает тесную связь с понятием импульса
Так, в случае быстрого отключения магнитного поля частица, находившаяся в нём, получает дополнительный импульс qA.

The only scalable quantum solution
We've taken a scalable topological approach, harnessing superior qubits that perform more complex computations, at lower cost, with higher accuracy.

https://github.com/qg-cpt-marseille/sl2cfoam

Visualizing particles in an easy way is HARD.
To the point, I've founded a startup precisely for that: https://quantumflytrap.com/

"... there is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity among physicists and optical scientists."
Lamb, Anti-photon - quite an interesting read

https://twitter.com/lisyarus/status/1151504899907162112

derivation for effect of macroscopic current on photons and how it gives rise to radio waves
semiclassical model, classic current coupled to the quantum field