Aharonov–Bohm effect does not Prove Nonlocality

I heard the suggestion that the Aharonov–Bohm effect proves a form of quantum nonlocality.

The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (φ, A), despite being confined to a region in which both the magnetic field B and electric field E are zero.[1] The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, and the Aharonov–Bohm effect is accordingly illustrated by interference experiments.

The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally.[2]

So the effect depends on the potential, and not just the fields.

The potential and fields are all locally defined, so what is the problem?

The problem is that only the fields are directly observable, and there is considerable discretion in defining the potential. Sometimes the potential is defined to satisfy a distant condition. This is allowed, because gauge symmetry means it has the same physical effect.

From the viewpoint of differential geometry, the potential is a connection on a complex line bundle, and is a purely local object. It is more fundamental than the fields.

The paradox is that an electron can interfere with itself after going around a non-null-homotopic loop with a flat complex line bundle. Arguably there is something nonlocal about that. I don't think so. It is not like action-at-a-distance at all.

Electrons Are Spinning

Scientific American reports:

Quantum Particles Aren’t Spinning. So Where Does Their Spin Come From?

A new proposal seeks to solve the paradox of quantum spin ...

But despite appearances, electrons don’t spin. They can’t spin; proving that it’s impossible for electrons to be spinning is a standard homework problem in any introductory quantum physics course. If electrons actually spun fast enough to account for all of the spinlike behavior they display, their surfaces would be moving much faster than the speed of light (if they even have surfaces at all). Even more surprising is that for nearly a century, this seeming contradiction has just been written off by most physicists as yet another strange feature of the quantum world, nothing to lose sleep over.

No, this is wrong. Electrons do spin. You only get that paradox if you assume that electrons are very tiny spheres or point particles, but quantum mechanics teaches that electron are non-classical entities with wave-like properties.

The article goes on to give the history of quantum spin, and how crucial it is for understanding chemistry and many other things.

But all of these fabulous discoveries, applications, and explanations still leave Goudsmit and Uhlenbeck’s question on the table: what is spin? If electrons must have spin, but can’t be spinning, then where does that angular momentum come from? The standard answer is that this momentum is simply inherent to subatomic particles, and doesn’t correspond to any macroscopic notion of spinning.

Yet this answer is not satisfying to everyone. “I never loved the account of spin that you got in a quantum mechanics class,” says Charles Sebens, a philosopher of physics at the California Institute of Technology.

No, this is silly. The QM textbooks teach that position, momentum, energy, angular momentum, and spin are observables that correspond to the classical variables, but cannot be taken literally about electrons as point particles, as the uncertainty principle prevents such a literal treatment. There is not really any difference between spin and the other variables in this respect.

I previously posted Electrons do spin.

Peter Woit explains:

Despite what Sebens and Carroll claim, it has nothing to do with quantum field theory. The spin phenomenon is already there in the single particle theory, with the free QFT just providing a consistent multi-particle theory. In addition, while relativity and four-dimensional space-time geometry introduce new aspects to the spin phenomenon, it’s already there in the non-relativistic theory with its three-dimensional spatial geometry.

Asking whether electrons really spin is a like asking whether they orbit the nucleus of an atom. A century ago, physicists tried to model an atom as classical electron orbits, and figured out that it doesn't work. You need a quantum model. But it is still correct to say that the electrons orbit the nucleus.

TV Show on Zero and Infinity

I just watched the latest PBS TV Nova on Zero to Infinity:

Discover how the concepts of zero and infinity revolutionized mathematics.

It was stupid and boring.

A Black woman professor narrated. They always seems to find Blacks and women for these shows. Not sure why. Does PBS have a lot of Black viewers? Is it trying to get more?

I doubt it. My guess is that the typical WHite liberal PBS viewer gets a good feeling of social justice when a Black woman is lecturing.

Much of the show was about the invention of the Zero. It attributed it to India, and said that Persians and other middle easterners brought it to Europe.

But what was the invention? The use of 0 as a placeholder, or as a counting number, or as a number on the same footing as other numbers?

I looked for some statement from India or Persia saying something like: The counting numbers are { 0, 1, 2, ... }, and for any such numbers, A + B = B + A.

That would show that the author considered 0 to be a number just like 1 and 2.

But on the contrary, the show itself did not even do that. The moderator kept referring to the counting numbers as 1, 2, 3, ..., and not including 0.

Any who says that has still not grasped the invention of 0. 0 is a counting number. If I ask you how many apples you have, and you have none, then you answer 0. Maybe you answer -2, if you owe 2 apples. If you say there is no answer, then you have not accepted the 0.

Even the business pages of a typical newspaper rarely treat zero as a number. It will often avoid it with various euphemisms.

The show eventually moved on to infinity, but that was not any better. It gave a faulty version of Cantor's diagonal proof of the uncountability of the reals.

Suppose you list .4, .49999..., .009, .0009, ... . It said to add 1, mod 10, to each diagonal digit. That gives .50000. That is the same real number as the 2nd item on the list, with a different decimal representation.

A good proof must somehow take into account that real numbers can have two decimal representations.

Cantor's orginal proof did not use diagonalization.

The show went on to Zeno's paradoxes and Hilbert's hotel. It was all fairly trivial.

Since it tried to trace the origin of the zero, I thought that it might tell us who invented infinity?

It did talk about approximating π as a limit of an infinite sequence. I guess that idea goes back to the ancient Greeks. The invention of infinitesmal calculus required limits. Those ideas were made rigorous centuries later. Cantor introduced the concept of different infinite cardinals. I am not sure who really first had the modern concept.

These PBS shows appears to be expensively produced, but you can find lots of free YouTube videos that explain the math much better, and are more entertaining.

Making Finitary Deductions About Infinities

That is my 5-word definition of Mathematics. It is what distinguishes Math from every other field.

Some say that Math is the study of numbers, or the use of symbolic notation. But Music uses symbolic notation, and numbers are used by all the hard and soft sciences.

None of the empirical sciences ever encounter infinities. Cosmologists may talk about the universe having infinite extent, but there is no reason to believe that, and we cannot observe that. We only observe finite quantities.

And the sciences never make a finitary deduction either. An experiment might convince us of some fact, but it is really just evidence that makes an outcome 99% likely, or something like that. The experiment has to be refined and redone to become more and more sure of it.

Math has infinities all over the place. This is obviously true about work on limits, but it is also true about elementary statements like the Pythagorean Theorem. There are infinitely many possible right triangles, and the theorem gives a formula about all of them.

Even with all the infinities, the proofs always use a finite set of steps from a finite number of axioms. The proofs about the infinities are always strictly finitary.

Here is the Wikipedia definition of Mathematics:

Mathematics (from Ancient Greek μάθημα; máthēma: 'knowledge, study, learning') is an area of knowledge that includes such topics as numbers (arithmetic and number theory),[2] formulas and related structures (algebra),[3] shapes and the spaces in which they are contained (geometry),[2] and quantities and their changes (calculus and analysis).[4][5][6] Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or — in modern mathematics — entities that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and (in case of abstraction from nature) some basic properties that are considered as true starting points of the theory under consideration.

Mathematics is used in science for modeling phenomena, which then allows predictions to be made from experimental laws. The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. Inaccurate predictions, rather than being caused by incorrect mathematics, imply the need to change the mathematical model used.

Here is Britannica:

mathematics, the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter. Since the 17th century, mathematics has been an indispensable adjunct to the physical sciences and technology, and in more recent times it has assumed a similar role in the quantitative aspects of the life sciences.

Here are some dictionaries:

The abstract science of number, quantity, and space. -- Oxford

An abstract representational system studying numbers, shapes, structures, quantitative change and relationships between them. -- Wiktionary

The science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations. -- Merriam-Webster

The study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. -- dictonary.com

The science that deals with the logic of shape, quantity and arrangement. -- Live science

The study of numbers, shapes, and space using reason and usually a special system of symbols and rules for organizing them. -- Cambridge

These are pretty good, but do not distinguish Math from science well. Yes, Math is an area of knowledge that includes numbers, but the mathematicians do proofs, with infinite numbers and finite arguments.

Sean M. Carroll Goes Woke on Sex

Sean M. Carroll has become one of the leading expositors of advanced Physics, but he has a lot of strange views that will make you skeptical of whatever he tells you.

The biggest is that he believes in the many-worlds alternative to quantum mechanics. This is a belief that anything is possible, and that nothing is more likely than anything. It is a complete rejection of all modern science.

He has his own rationalization that is mostly circular reasoning.

In his latest Ask Me Anything podcast, he says that he does not see a moral justification for parents spending money on their children's education. He says all children should get the same education.

He is married with no kids.

He is welcome to his opinions, but he does not describe the American situation accurately. In California, his home state until recently, the schools get about 50% of the state budget, and the poor districts get at least as much as the rich districts. The rich are not getting any better educational opportunities.

Some rich parents do send their kids to expensive schools, but the educational opportunities are not much different from public schools.

He has also joined the sex-deniers who say that biological sex is not binary. Biology professor Jerry Coyne is a big fan of Carroll, because of what he says in favor of determinism and against libertarian free will, but schools him on biological sex.

In reality, what they are trying to do is the reverse: adjust scientific reality so that it aligns with social justice. That is, if sex is a spectrum and not binary, then people of different genders can somehow feel that they are in harmony with biological reality. But that’s an example of the “appeal to nature.” The rights of people of different genders, including transsexual people, do not depend on the developmental biology of sex, or of any observations in nature about sex dichotomies.

I’m not going to discuss my claim that sex is binary; I’ve talked about it at length, as did Luana Maroja in her piece at Substack. I’ll just put it out there that the going biological definition of sex is that there are two sexes in vertebrates: males (who produce small mobile gametes) and females (who produce large, immobile gametes). There is no group that produces intermediate types of gametes that can unite with other gametes, so there is nothing beyond these two sexes.

Carroll is not a biologist. He is not to be confused with the somewhat more accomplished scientist, Sean B. Carroll, who really is an expert biologist.

I assume that Sean M. Carroll is smart enough to know the difference between male and female. But it appears that he is willing to recite nonsense in order to please his Leftist Woke fans.

I suggest keeping this in mind when listening to him. He sometimes gives pretty good explanations of textbook physics, but his opinions on big picture physics are dubious, and his moral and politcal opinions are garbage.

Einstein and the Equivalence Principle

New paper:

Einstein's Happiest Moment: The Equivalence Principle
Paul Worden, James Overduin

Einstein's happiest thought was his leap from the observation that a falling person feels no gravity to the realization that gravity might be equivalent to acceleration. It affects all bodies in the same way because it is a property of spacetime -- its curvature -- not a force propagating through spacetime (like electromagnetic or nuclear forces). When expressed in a way that is manifestly independent of the choice of coordinates, this idea became General Relativity. But the ground for what is now known as the "equivalence principle" was laid long before Einstein, affording a fascinating example of the growth of a scientific idea through the continuous interplay between theory and experiment.

As this article and Eikipedia explain, the equivalence principle goes back centuries. Einstein was very happy about using it in a 1907 paper, but it was not because gravity was a realization of curvature, as he did not even know what curvature was at the time.

It is my understanding that what Einstein was actually happy about was that he figured out a way to use the principle to use special realtivity to show gravitational time dilation.

Special relativity is often described as a theory about constant velocity, but back during the early days, say 1995-2010, it was widely understood to cover accelerating particles also. Poincare proposed a couple of relativistic gravity theories, but the geometry was not understood.

Einstgein figured out how to sidestep having a gravity theory, by saying that gravity was like non-gravitational acceleration. That was enough to figure out clocks in a gravitational field.

People often say that general relativity is needed for GPS navigation, but I don't think that is true. It only needs special relativity, and this trick of Einstein.

As far as I know, this idea of Einstein was his own, and not plagiarized from anyone else. Maybe that is why he was so happy about it.

Probability is Subjective

Ulrich J. Mohrhoff writes:

With Mermin, I also hold this truth to be self-evident (though it took me some time to get there), that probabilities are intrinsically subjective. ...

Mermin invokes the celebrated probabilist Bruno de Finetti, who wrote: “The abandonment of superstitious beliefs about the existence of Phlogiston, the cosmic ether, absolute space and time. . . , or Fairies and Witches, was an essential step along the road to scientific thinking. Probability too, if regarded as something endowed with some kind of objective existence, is no less a misleading misconception, an illusory attempt to exteriorize or materialize our actual probabilistic beliefs.”

Taking the mind-independent existence of the external world for granted, de Finetti holds that there is no place for probability in such a world, any- more than there is for Phlogiston and the rest.

I agree with this, but do not deny the importance of probability.

All scientific theories are inherently probabilistic. Even classical celestial mechanics, the textbook example of the clockwork deterministic universe, was always probabilistic in practice. Observations in the sky always had errors, and predictions had uncertainty. Linear regression was invented to make probabilistic predictions about celestial orbits.