The aim of physics is to understand the world we live in. Given its myriad of objects and phenomena, understanding means to see connections and relations between what may seem unrelated and very different. Thus, a falling apple and the Moon in its orbit around the Earth. In this way, many things “fall into place” in terms of a few basic ideas, principles (laws of physics) and patterns.

As with many an intellectual activity, recognizing patterns and analogies, and metaphorical thinking are essential also in physics. James Clerk Maxwell, one of the greatest physicists, put it thus: “In a pun, two truths lie hid under one expression. In an analogy, one truth is discovered under two expressions.”

Indeed, physics employs many metaphors, from a pendulum’s swing and a coin’s two-sidedness, examples already familiar in everyday language, to some new to itself. Even the familiar ones acquire additional richness through the many physical systems to which they are applied. In this, physics uses the language of mathematics, itself a study of patterns, but with a rigor and logic not present in everyday languages and a universality that stretches across lands and peoples.

Rigor is essential because analogies can also mislead, be false or fruitless. In physics, there is an essential tension between the analogies and patterns we draw, which we must, and subjecting them to rigorous tests. The rigor of mathematics is invaluable but, more importantly, we must look to Nature as the final arbiter of truth. Our conclusions need to fit observation and experiment. Physics is ultimately an experimental subject.

Physics is not just mathematics, leave alone as some would have it, that the natural world itself is nothing but mathematics. Indeed, five centuries of physics are replete with instances of the same mathematics describing a variety of different physical phenomena. Electromagnetic and sound waves share much in common but are not the same thing, indeed are fundamentally different in many respects. Nor are quantum wave solutions of the Schroedinger equation the same even if both involve the same Laplacian operator.

Along with seeing connections between seemingly different phenomena, physics sees the same thing from different points of view. Already true in classical physics, quantum physics made it even more so. For Newton, or in the later Lagrangian and Hamiltonian formulations that physicists use, positions and velocities (or momenta) of the particles involved are given at some initial instant and the aim of physics is to describe the state at a later instant. But, with quantum physics (the uncertainty principle) forbidding simultaneous specification of position and momentum, the very meaning of the state of a physical system had to change. A choice has to be made to describe the state either in terms of positions or momenta.

Physicists use the word “representation” to describe these alternatives that are like languages in everyday parlance. Just as with languages, where one needs some language (with all equivalent) not only to communicate with others but even in one’s own thinking, so also in physics. One can use the “position representation” or the “momentum representation” (or even some other), each capable of giving a complete description of the physical system. The underlying reality itself, and most physicists believe that there is one, lies in none of these representations, indeed residing in a complex space in the mathematical sense of complex versus real numbers. The state of a system in quantum physics is in such a complex “wave function”, which can be thought of either in position or momentum space.

Either way, the wave function is not directly accessible to us. We have no wave function meters. Since, by definition, anything that is observed by our experimental apparatus and readings on real dials, is real, these outcomes access the underlying reality in what we call the “classical limit”. In particular, the step into real quantities involves a squared modulus of the complex wave functions, many of the phases of these complex functions getting averaged (blurred) out. Many so-called mysteries of quantum physics can be laid at this door. It is as if a literary text in its ur-language is inaccessible, available to us only in one or another translation.

What we understand by a particle such as an electron, defined as a certain lump of mass, charge, and spin angular momentum and recognized as such by our electron detectors is not how it is for the underlying reality. Our best current understanding in terms of quantum field theory is that there is a complex electron field (as there is for a proton or any other entity), a unit of its excitation realized as an electron in the detector. The field itself exists over all space and time, these being “mere” markers or parameters for describing the field function and not locations where the electron is at an instant as had been understood ever since Newton.

Along with the electron, nearly all the elementary particles that make up our Universe manifest as particles in the classical limit. Only two, electrically neutral, zero mass bosons (a term used for particles with integer values of spin angular momentum in terms of the fundamental quantum called Planck’s constant) that describe electromagnetism and gravitation are realized as classical electric and magnetic or gravitational fields. The very words *particle* and *wave*, as with *position* and *momentum*, are meaningful only in the classical limit. The underlying reality itself is indifferent to them even though, as with languages, we have to grasp it in terms of one or the other representation and in this classical limit.

The history of physics may be seen as progressively separating what are incidental markers or parameters used for keeping track through various representations from what is essential to the physics itself. Some of this is immediate; others require more sophisticated understanding that may seem at odds with (classical) common sense and experience. As long as that is kept clearly in mind, many mysteries and paradoxes are dispelled, seen as artifacts of our pushing our models and language too far and “identifying” them with the underlying reality, one in principle out of reach. We hope our models and pictures get progressively better, approaching that underlying reality as an asymptote, but they will never become one with it.

*Headline Image credit: Milky Way Rising over Hilo by Bill Shupp. CC-BY-2.0 via shupp Flickr*

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Patterns in physicsas of 11/13/2014 5:25:00 AM