Genady

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

Yes, this is the "hypothesis 2" above. (I guess we cross posted.)

Let it be "hypothesis 1". Here is "hypothesis 2": From an everyday physical experience, air needs to be blown in order to make it move, i.e., its direction is determined by its source. Water, OTOH, flows by itself, e.g., downhill or towards the ocean, i.e., its direction is determined by its destination.

Winds but not currents, AFAIK. For example, (Sea current direction @ Windy Community)

Why are the directions of wind and current designated in an opposite way, as in "Northern wind" vs. "Northern current"?

There is no such distinction in calculus. BTW, there is no thing called "Einstein calculus".

It does not seem to be correct. For example, the dimension "height" does not contain within it the dimensions "length" and "width".

Unimaginable. If it consists of individual particles, it is not homogeneous.

The two statements above are contradictory. The first says that information is an increase in entropy: \(I=\Delta S\). The second says that information is entropy: \(I=S\). Before continuing with the argument, this contradiction needs to be cleared out. Which one is true, \(I=\Delta S\) or \(I=S\)?

I am now in the first chapters of Penrose's The Road to Reality. I know that he has his ideas on this topic, and I am curious but not there yet.

How does it relate to Astronomy and Cosmology?

When we look at the image of, for example, a spiral galaxy, we do not look at it in the disk plane, but close to perpendicular to the plane. The thickness of Milky Way's disk is about 1000 ly. We don't see through the disk, so we see about 500 ly thick image. The maximal difference in "past times" is thus about 500 years.

We can take in account the motion of stars in a galaxy and estimate their positions at one instant of time. As the past times you mention are only different by several hundred years, we find that on a galaxy scale this does not have any visible effect on the image we get.

We receive at present the signals from different times in the past and we separate them if we know relative distances to the sources. For example, we can see simultaneously the Sun and the Moon, but we know that we see the Moon as it was one second ago and we see the Sun as it was 8 minutes ago. The same with galaxies and all.

And then, there is Freeman Dyson (F. Dyson, “The World on a String,” New York Review of Books, May 13, 2004)

From Susskind & Cabannes, General Relativity: The Theoretical Minimum (p. 227):

The direction of spin is continuous. The spin of the electrons can have any one of continuous 3D directions.

Generally, an observable value can be uncertain AND conserved - there is no contradiction. For a simple example, consider an entangled Bell pair of two electrons with opposite spins. The spins of the electrons are maximally uncertain but they certainly sum to zero, i.e., are certainly opposite to each other.

Exactly. This was my message:

We were discussing this example:

Right. The fluctuations are above and below a positive average, but not below zero average.

Please explain: if the average is zero, then a slightly smaller value would be negative, wouldn't it? Energy is conserved in every possible outcome, not only on average. In no quantum or particle experiment conservation of energy was ever violated.

Momentum is conserved on every line and at every vortex of every Feynman diagram. The path integral calculation sums up all available paths, not only straight ones. There is an inconsistency in this claim: energy is non-negative, so if it becomes positive even temporarily, its average cannot be zero.

In QM, as well as in QFT, momentum is conserved. Lagrangians are translationally invariant. Momentum conservation follows by the Noether's theorem.

I've had an experience with the nitrogen narcosis, some myself and more in other divers (students and those I was guiding.) I would not describe it as being drunk, but rather as being euphoric and very unfocused. As the divers ascend and the effect disappears, they often don't have any recollection of what they were doing and what happened while they were affected, similarly to what happens immediately after waking from being sedated by a drug.