A well known professor of analog circuit design and theory once said "all models are wrong". Analog design is particularly rewarding and challenging, I believe, for this very reason. Those words have echoed in my mind during many junctures in my journeys to learn something about the physical world. Whether you are a theorist, phenomenologist, experimentalist or some combination of these, this fact is inescapable.

I am fascinated by how concepts of everyday life leak into all scientific thought, whether it be an elementary concept in engineering or within the confines of some rarefied physical theory. And this is a topic I hope to write much about.

A good example where models creep in is to capture the notion of locality vs. non-locality. I use the term "local" here loosely to capture the idea of something confined within a region of physical space (local) vs. something spread out through all space.

The local vs. non-local behavior of physical things becomes particularly important when special relativity is brought into the picture. Among many other things, special relativity tells us that there is a speed limit to the physical propagation of information in the universe and that speed limit is the speed of light (approximately 186,000 miles/sec or 3*10^8 meters/sec). This raises question about action at a distance. In electrodynamics, which is compatible with special relativity, a change in the electromagnetic field at one point can not act instantaneously on another distant point, but rather the disturbance must propagate (at the speed of light).

Two everyday ideas that express locality vs. non-locality might be respectively the trajectory of a billiard ball on a billiards table vs. the propagation of water waves.

Water waves and all wave phenomena for that matter have an interesting property - superposition. Multiple waves may co-exist with one another, each wave distributed throughout all space. The aggregate effect of these multiple waves manifests as interference. That is, at any point in space, we can add up the contributions of the all the waves in our medium of interest and this sum expresses the net effect of the waves. This interference may occur in either a constructive fashion or destructive fashion such as the peaks and troughs observable with ordinary water waves.

People often talk about the "wave particle duality" of quantum mechanics. What does this mean? It turns out the neither the billiard ball model nor the wave model is quite right to capture the essence of nature. The physical phenomena we observe sometimes act in a local particle-like way (i.e., like billiard balls), but in other contexts in a non-local wave-like way that is distributed throughout all of space.

The upshot of this is that things that we call particles, which we think of as being local in nature such as electrons, protons and the like, in fact also behave in a non-local wave-like way. How so? Particles like electrons and protons can exhibit interference behavior in the same way that waves can. That mathematical object invented by physicists to capture this wave-like nature is appropriately called the wave function. The wave function for an electron, for example, is embedded with all of the physics associated with the electron. What does the wave function consist of? It is simply a function that assigns a complex number to each coordinate in space. And, it obeys one of the most famous equations in physics called the Schrodinger equation, which describes the dynamical behavior of the wave function (i.e., how it evolves over time). Without straying too far from the focus of this discussion, the wave function, by the way, also has an interpretation. The square of the magnitude of it's value at any particular point is interpreted to be the probability that the particle, upon a measurement, will be found at that point.

The dual nature of all physical phenomena is often cited famous double slit experiment (which I plan to discuss in a later column). Without delving into any details here, physicists typically describe matter as sometimes acting like a particle (local) and other times like a wave (non-local). I prefer the position that neither the wave nor the particle analogy is appropos. In fact, both are wrong, but we haven't an object to draw upon from our ordinary experience that somehow captures the local and non-local features simultaneously.

Particle physicists tell us that certain particles such as the electron are "point particles". What does this mean? In essence, a point particle has no dimension. It behaves and is modeled like an abstract 0 dimensional point in mathematical space. I plan to talk about this notion of point particles at a later time as this is a peculiar but physically interesting statement. And there are some interesting physical reasons necessitating the point particle treatment such as special relativity, which would present problems if these particles such as the electron had some finite physical extent.

Another area where the local and non-local viewpoints appear is in classical mechanics. An elementary problem in classical mechanics is to determine the trajectory of a classical object such as a billiard ball through space. And, there exist two equivalent theories - a local one and a non-local one - to do this.

The first theory, which is familiar to every high school or freshman physics student is known as Newtonian mechanics expressed in the famous equation f=ma, where f is the force, m the inertial mass of the object and a the acceleration. Acceleration simply means how quickly the velocity is changing in time. Newtonian mechanics asserts effectively that at every point in space an object experiences a force. The force causes an acceleration of the object in proportion to the inertial mass.

We can then determine the trajectory of an object such as a billiard ball if we have some initial conditions on the objects position and velocity. What I am talking about is calculus - in particular integration, but this can be understood without any mathematical baggage. One way to do this is to begin at the initial position where the object lies. It's initial velocity is known. From this, we can determine with where the object will be a very short (in fact infinitesimal) time later. And, similarly, we can also determine what the velocity of the object will be a short time later since we know the initial acceleration. We can now move the particle to its new position we just determined and assign the new velocity we just computed to it. Then we can repeat this procedure an arbitrary number of times with an arbitrary degree of precision to track the trajectory of the particle.

We can think of the Newtonian formulation as in some sense a local theory. The billiard ball feels a local force at each point and moves along accordingly in an iterative fashion.

There is another way to solve this problem, which is known as Lagrangian mechanics. The idea here is to imagine all possible paths that the billiard ball could take from a starting point to an ending point. There will be an infinite number of them of course. We can assign each possible path a value and then seek to find the path that has an extreme value (i.e., minimum or maximum). Will this work? It turns out yes, but the interesting question is what is the quantity we need to use to generate the appropriate weights so that the path having the extreme value is in fact the classical trajectory. The necessary quantity is called the Lagrangian and it is related to the energy of the object, L=T-V where T is something called the kinetic energy of the billiard ball (or whatever we are tracking) and V is the corresponding potential energy. What then is the weight we need to minimize. There is also a term for this. It is called the action and turns out to be the integral of the Lagrangian over time. So, for each possible path, we can compute the action and then at the end, we find the classical trajectory is the path which has the minimum action.

The Lagrangian formulation expresses a non-local idea. The billiard ball somehow "senses" the entirety of possible path and their corresponding actions over each entire path and chooses the path of least action.

It turns out that the Newtonian and Lagrangian formulation are entirely equivalent and in fact Newton's law (f=ma) can be directly derived from the Lagrangian formulation (of course it had to be that way). But, the point to be made here is that these two theories, the Newtonian and the Lagrangian formulations of classical mechanics express respectively a local and non-local theory but both are equivalent in predicting the trajectory of the billiard ball.

The ideas of local and non-local phenomenon are present all around us in the macroscopic world. We draw upon these concepts in the theories and mathematical objects we devise to understand and predict physical phenomena. In an uncanny and amazing way, these models are enormously successful.

The Role Of Everyday Ideas In Physical Theories (Locality Vs. Non-Locality)

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