Bell's theorem is most significant because its conclusion is so striking and its assumptions so innocuous that it requires us to radically change how we think about the world (and not just about quantum theory).He then goes on to argue that the nonlocality conclusion can only be avoided by either accepting superdeterminism or rejecting probability theory.

Before Bell's theorem, the picture we have about the world is like this: physical things interact only locally in space. For example, a bomb dropped on the surface of Mars will produce immediate physical effects (chemical reactions, turbulences, and radiations) in the immediate surroundings; the event will have (much milder) physical effects on Earth only at a later time, via certain intermediate transmission between Mars and the Earth. More generally, we expect the world to work in a local way that events arbitrarily far apart in space cannot instantaneously influence one another. This picture is baked into classical theories of physics such as Maxwellian electrodynamics and (apparently) in relativistic spacetime theories. After Bell's theorem, that picture is untenable. Bell proves that Nature is nonlocal if certain predictions of quantum mechanics are correct.

He doesn't even mention that Bell only showed that a classical theory of hidden variables would have to be nonlocal. His theorem says nothing about non-classical theories.

He also doesn't mention that merely rejecting counterfactual definiteness resolves the problem.

I don't know if these authors are misguided or dishonest. They can believe in action-at-a-distance if they want, but if they claim to survey the opposing views and leave out the mainstream explanation, then they are not telling the truth.

The expectation being kept from the intermediary is the crucial element here.

ReplyDeleteClassical mechanics---rather, NM---works through direct contact.

First, spatially discrete objects.

If you move the billiard-table cue in the air without touching the intermediate ball, there is no way that the target ball is going to take notice of your existence. Changes in objects must ride on an actually displacing object(s). There is no other way to communicate changes. If there are many objects in the intermediate region, then the change gets communicated from one object to another in the same order in which they appear on the path of communication of changes. (In NM of point-particles, it means: on a straight line path, in the order of increasing straight-line distance from the initial mover.)

Classical mechanics thus induces an unstated expectation, if you objectify the change too: A change displaces in space only as if it itself were an NM object, albeit, a massless one. (Mass remains the property of the objects being changed.)

Now, spatially continuous NM objects.

A splash in water at one end of the pool progresses exactly as if all intermediate fluid parcels successively transmitted the change in an orderly manner. The change rides on displacing local fluid parcels. The abstractly objectified change is the wave. It expands (i.e., the region enclosed by the crest of a pulse of a wave expands). It also attenuates, etc. But that's not relevant here. Relevant is: change displaces as if it were an NM object.

There is another *classical* theory, viz. Fourier theory, which doesn't work this way.

In the Fourier theory, if you heat a very long thin rod at one end, then the other end also gets warm due to the *same* heat, and in the *same* instant. (The basis function has support over the entire domain; it doesn't have a compact support.)

How do we make sense of it, on the basis of the classical expectation? Well, we *don't*!

In the very act of using the *classical* Fourier theory, we agree to abide by a *new* expectation---one which is not borne out by the *classical* *NM* mechanics: There is no such thing as heating only one end of the rod, while leaving the other end at the original temperature. You cannot apply heat to only one end, even if you have held the bunsen burner near one end only. The only way to heat up any rod is to apply the thermal energy at all points of the rod. However, the mechanics of the rise in the temperature is such that the portions closer to the heat source (applied only at one end) get hotter much, much faster. Etc.

The only way to make sense of QM is to assume that there is a material but non-massive medium in between the Mars and the Earth, and that this medium follows the Fourier-theoretical expectation, not the Newtonian mechanical.

People have been debating the issues in simply wrong terms. There have been global theories in classical mechanics too---every theory that uses the Fourier theory is global. That is to say, every theory that uses fields, a Laplacian, and a first-order differential in time, is global: whether heat, diffusion, or QM. (I need to check how they deal with changes propagating in a Helmoltzian field---i.e., not standing waves, but a change in standing waves. I am willing to bet that they use the Fourier theory.)

Bell didn't use the right terms. "Classical" is too vague. You need to qualify "Classical Newtonian" or "Classical Fourier-theoretical". The difference in Classical vs. Quantum is easier: It is the Classical Fourier-theoretical with full complex field for the solution (i.e. a system of two real-valued solutions coupled to each other via complex-algebraic algebra.)

Best,

--Ajit

Correction: I typed it wrong near the end. Here is what I meant (and now, given opportunity, I am going to expand!):

ReplyDeleteThe difference in Classical vs. Quantum is easier to state, assuming you have digested the nature of the Fourier theory.

If a classical theory uses the Fourier theory, then the field is real-valued. Even if you use complex algebra for intermediate manipulations, in the beginning and in the end, you take only one part (real or imaginary).

The nature of QM is that it too has the *classical* Fourier-theoretical nature. The difference from the classical theory, however, is only this much: We take the full complex field for the solution (i.e. a system of two real-valued solutions coupled to each other via complex-algebraic algebra.) That's because the first-order derivative of time (a steady angular velocity) still has to generate a wave phenomenon. It does so by putting two space components to work. In classical waves, the time-derivative is second-order. So, two space-components is an overkill. Using both in the end would destroy unique-ness of solution.

Best,

--Ajit