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July 26, 2020

Causality: What It Means to Me

How and what we know about an object in a system is through the object's relationship with the other things in the system. In the physical world, especially the world of quantum mechanics, the objects we're trying to understand are inter-dependent on each other. The world is a dance with each member object dependent on the other. However, each member of this material reality dependents on some universal law without exception.

As a student studying classical optics, I learned that you can compute the propagation of light by using the idea that light moves through space in the most efficient manner possible. The method of computation uses optimization techniques, namely, the action principle of Fermat. From Fermat's principle, with the help of Huygen's principle, you can get Snell's law.

But Fermat's principle, to me, had a little bit of quantum mechanics' mystery (or problem) in the way light propagated. I remember clearly wondering if Fermat's principle was dependent on local causality. When I had thought about this for awhile, I realized that there was a paradox because light waves had to "know" the path it would take, before it took it, to satisfy Fermat's principle. This paradox could only be solved if I used Huygen's principle of wave interference which happened instantaneously and globally in this particular case which is Fermat's principle (like a principle enfolded in a principle.) I came to the conclusion that Fermat's principle could not be derived using "local" causality. I think Fermat's principle works for all waves: classical and quantum mechanical. Throughout the years, I never depreciated this idea.

Later as a graduate student, I studied the math of Fermat's principle further by investigating the total internal reflection of light, and light propagation as evanescent waves which is characterized by a coherent flow of a lateral wave along the reflection boundary layer. The only explanation for this important phenomena I found to be satisfactory consistent was in Max Born's book on optics [1]. The explanation for this was that the lateral waves along the boundary layer was caused by destructive wave interference along the reflection boundary layer. This destructive wave interfence kept the wave energy traveling as transverse wave in a duct along the boundary layer. This explanation is based purely on Huygen's principle, and in some sense is a "global" versus local kind of action at a distance.

For me, Fermat's principle which uses temporal optimization is a kind of action at a distance principle. It's a kind of "global" causality constraint. It seems to me that optimization methods necessarily (maybe magically) grind out globally constrained results. (But I also have to admit that I wonder if I don't understand.) The point is that back then, I felt I could stop thinking about this problem if I considered light propagation as a purely wave phenomena based on the superposition principle.

If you think of the interaction of objects in the physical world in terms of classical causality in the way ancient philosophers did, then I think global "instantaneous" causality is possible if objects are always dependent on each other when any interaction occurs. Objects may seems to interact "instantaneously" because the objects which seems separated are not truely separate objects; they are connected in some sense. For example, a pair of Pauli electrons, are coupled together somehow by the atomic "system" in which they exist. Statistical mechanics considers the interaction of objects in the physical world as an ensemble. This is the way I like to think about the world.

There were a few times when I tried to think about the possibility of describing the world without using probability, but I was not really successful. But I'll keep trying. I have the sense that we use probability because the enfolding of an event is innately dependent on chance. This is very important. It's "innate". Something about our universe is fundamentally random. In particularily, it tells you how likely an event will happen in the future and so has a temporal feeling with its use to me. Obviously, we usually never speak of the probability of an event happening in the past.

Hence, for me the causal occurrance of an event, the causality, is associated with probability. I suppose I've come to this conclusion mainly from conditioning. My awareness is limited living the macro "big" world, the "classical" (versus quantum) world, which "seems" naively to unfold with complete certainty. It was quite uncomfortable at times to learn about the strange the sub-atomic world. (Similarily, I've frequently thought that I never really felt comfortable with some mathematical concepts. I only got use to math, like the math of Hilbert, by repeated use of it.)

Finally, this model of causality to me works with how neurons interact or communicate with each other. Single neurons send signals to each other based on the idea that the signal is to be evaluated by network of other neurons. The brain is a network in which a single neuron is like a quantum in the field. The neuron necessarily behaves as an entity in an ensemble.

[1] Principles of Optics, 4th Edition, 1970. An explanation for this evanescent wave by a lateral phase shift of interfering waves is given by Goos-Hanchen. The original paper describing this effect was written by Goos and Hanchen in Germany in 1947.