Me too. MIT has published many instructional videos. Go fill your boots.
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Personally, I hate titles like ādisproveāā¦ mostly because has dark matter even been evidenced?
Scientists have not yet observed dark matter directly. It doesnāt interact with baryonic matter and itās completely invisible to light and other forms of electromagnetic radiation, making dark matter impossible to detect with current instruments.Aug 7, 2020
So - using my consistent requirements for evidence, I withhold ābeliefā in the existence of dark matter. AND am open to explore other competing āhypothesesāā¦
MY BIGGEST BONE 
to pick with āscience-minded YouTube videos OR papers OR news OR google searchā¦OR scientific organizations ** IS the causal use of ātheoryā AND then educating lay-people that Theory means ā¦**
If theory is the āhunchā - an educated guess, AND
Theory is the fact based accepted explanation ā¦
THEN for fuckās sake START or consistently use āhypothesisā and āTheoryā. Itās less confusing for laypeople AND it will stop confusing the shit out of theists who say āevolution is only a Theoryā compared to ādark matter is a theoryā ā¦
Ummmā¦I donāt believe in atoms. They make up everything.
All I know is that the deeper I delve into quantum mechanics and such physics, I have to take more Tylenol for my headache. It is a mind-fuck that is not readily digestable.
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Weāre in good companyā¦Einstein felt the same wayā¦
Quantum demonstrates āeffectā before ācauseāā¦ hahahahaha (opens a beer and decides not to give a fuck)
Entanglement has a precise mathematical definition. Namely, that the commutator of the quantum operators in the system is nonzero. This pithy phrase, of course, requires further explanation, which I shall now provide.
Whenever any two quantum operators exist, coupled to any two given physical quantities, then those operators act upon the wave function in Hilbert space. Representing the wave function by Ļ, any two operators A and B, acting upon Ļ, can be written A(Ļ) and B(Ļ) respectively. In the classic case of position and momentum, these two operators are x and p respectively, and are defined thus:
x(Ļ) = xĻ
p(Ļ) = -iā(dĻ/dx)
Whenever two operators A and B exist, an entity called the commutator is defined, viz:
[A,B]Ļ = A(B(Ļ)) - B(A(Ļ))
For operators that commute, [A,B]Ļ = 0. Letās see what happens with the position and momentum operators, shall we?
The position operator is very simple - it simply multiplies the wave function by the position x. The momentum operator, on the other hand, differentiates the wave function with respect to the position, then multiplies it by the quantity -iā. The commutator is therefore:
[p,x]Ļ = p(x(Ļ)) - x(p(Ļ))
= p(xĻ) -x(-iā(dĻ/dx))
The first term in the above is obtained by applying the chain rule of differentiation, viz:
d/dx(uv) = v du/dx + u dv/dx
which gives us:
-iād/dx(xĻ) = -iā [ [dx/dx Ć Ļ ] + x dĻ/dx]
=-iā (Ļ + x dĻ/dx)
The second term is simply -x(-iā dĻ/dx) = iāx dĻ/dx
Therefore [p,x]Ļ is:
-iā (Ļ + x dĻ/dx) +iāx dĻ/dx
The terms involving the derivatives cancel out, and we are left with:
[p,x]Ļ = -iāĻ
The commutator of these two operators therefore has the same effect as multiplying the wave function by iā, and is clearly nonzero. Therefore position and momentum are entangled operators in Hilbert space, and the uncertainty relation applies to their real world measurement.
Classical operators always have a zero commutator, and therefore, entanglement is never observed with classical operators. As a corollary, if you have two classical operators U and V, then applying U followed by V to a physical quantity yields the same result as applying V followed by U. The operators are said to commute, in much the same way that two real numbers a and b commute under multiplication - it doesnāt matter which order you multiply them, you always yield the same result. An example is provided by two dimensional rotations: if A is a rotation about one angle (say, Īø), and B is a rotation about a different angle (say, Ļ), then AB = BA. You end up in both cases with a rotation about an angle of (Īø + Ļ). Note that this does NOT work for rotations in three dimensions about different axes, the reasons for this being rather complicated, but which are reflected in the fact that using quaternions to handle rotations in three dimensions involves multiplication of the quaternions, and multiplication of quaternions is not commutative.
Various quantum operators, on the other hand, as demonstrated above, may have a non-zero commutator, and when that happens, the operators will yield different results when the order of application to a physical quantity is changed. Precedents for this exist in pure mathematics - matrix multiplication being perhaps the canonical example. If you have two matrices A and B, and the product AB is defined, then the product BA may not even exist. Even if you restrict your attention to square matrices, for which AB and BA both exist, the two products are in general different from each other - matrix multiplication is not commutative. Iāve already mentioned quaternions above, which are interesting in that they can represent 3D rotations in a manner that avoids a problem known as the āgimbal lockā problem, but thatās a topic for its own post, or even its own thread. 
A similar commutator can be constructed for energy and time. I leave it as an exercise for the reader to do this.
āDavid - psssttttā¦Davidā¦ā (quiet, whispering voice)
āI got our Dick and Jane, See Spot Run Bookā¦ uh, letās, us skiddaddle outta this playgroundā¦ the big kids are hereā¦ā
My brain hurts.
Truthfully, I will have to do a lot of hard work to digest Caliās stuff, but it is relevant and worth learning.
Old Man, break the emergency glass and get the WD40, mechanical lubricants and cattle prod at the readyā¦ if Tin Man reads what Nyar and company have just put, he will completely break down!!!
Shakespearian aside
Beautiful work folks, thoroughly enjoyed reading that interaction.
Heh, that is pretty information dense. Probably sounds like Chinese to most.
What Iād offer to others is that when we read about this topic, we hear about all kinds of crazy, counter-intuitive results. What Iām says is that while the subject is kind of crazy, it isnāt as crazy as it first sounds. The āblameā for all of this craziness can always be tracked back to having something to do with how:
What is weird, is while that idea was very radical when it was suggested, the idea that order matters certainly isnāt new.
Consider making a pie out of 3 parts:
- Crust
- Filling
- Topping
Put them together in the wrong order and you get a horrible mess. In reality: order matters, and we kind of somehow already knew that. Life is weird.
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I tried to break the emergency glass but hit my thumb with the little hammer. Thatās because I have been digesting Calliās reply to the 2lt thread and cant see straight. My brain hurts.
I prefer historyā¦
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Another random thought to try to blow someones mind: this order mattering business is an attack on one of the pillars of classical logic: logical OR.
Actually, it warns us that the universe isnāt governed in toto by commutative processes.
Though if you want to have fun with some seriously counter-intuitive entities, dive head first into spinor calculus.
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