On Complementarity

The following sections are included:

• Dedication
• Preface
• Contents

The ruminations in the preface show just a smattering of the applicability of complementarity. Clearly the subject is immense: the title of this monograph alone would indicate that! So, I hope you’re not too disappointed when I tell you that this book is not primarily about the things I mentioned above. Rather, it emphasizes something of a completely different nature but every bit as important if you believe the nature of the physical world is important. It’s mostly about fundamental physics with a very important early foray into elementary biochemistry, DNA, no less. Then it reintroduces what I have called an Alternative Model of the Elementary Particles, alternative, that is to the well-known, even iconic Standard model. That model was originally developed in a series of papers published in the journal JKTR (Avrin 2005, 2008, 2011, 2012a), then, as more or less collected in the journal Symmetry (Avrin 2012b), and finally, in the book alluded to above (Avrin 2015)…

The two main physical scales of interest here are that of the elementary particles as exemplified by the Alternative Model and that of microbiology as exemplified by DNA. At this point I must stipulate that in addition to being neither a Physicist nor a Mathematician, I am also not a Microbiologist. Nevertheless, there’s no harm in me repeating what is quite generally known, namely the famous double helix which consists of two potentially parallel strands winding about each other. When (or if) they are uncoiled, the two strands taken together would exhibit a ladder-like structure encompassing, as rungs, organic molecules known as “nucleotides” of which there are four varieties, or more accurately, two sets of two. Each member of one set is composed of three molecules, a phosphate, a sugar and a base known as a Pyrimidine, which has a hexagonal ring-like structure with a particular atom at each of the corners. Each member of the other set has the same phosphate and sugar and a base known as a Purine composed of two ring-like structures, one hexagonal and one pentagonal, linked together…

Little in the preceding says much about the purpose of DNA which, of course, is to implement the genetic makeup of a new individual, something accomplished by means of the information lodged in the genes, each gene being a particular organization of a particular, nominally-sized number of what we have characterized in the preceding as “rungs” of the DNA ladder…

As mentioned in the Preface, a fundamental mathematical resemblance between DNA and our Alternative Model (AM) was established in my Book of Reference (BoR) (Avrin 2015). Here, we want to show this in a more detailed way. Nevertheless, although there is a lot of Complementarity associated with the AM, for comparison to DNA, all we really need are the basic set of fermions, and it is clear in the preceding chapter that the development of the AM taxonomy is essentially just an enlarged reflection of the essence of that set. So, here’s what we know about them:

• In the first place, there are four of them.
• Each one has three features, the quirks.
• The quirks are labeled using a binary alphabet

The previous chapter brought out a gratifyingly elegant result, namely that we can write a simple algebraic entity, the matrix m, that, in essence can be viewed as a common signature for both DNA and the set of four particles that are basic to the taxonomy of the Alternative Model (AM) of the elementary particles, a result actually documented in the previous book. However, the “gratification” does not end with just the statement of the matrix; there’s more and to explore it, we begin by finding the associated eigenvalues. As usual, an eigenvalue equation is found by setting the associated determinant to zero, thus:

Since DNA is a biochemical entity, in fact a very complex biological molecule, discovering its structure would most generally be considered an exercise in biochemistry which, by definition, may be viewed as a branch of Chemistry. However, if you recall, a bit of structural engineering (featuring Complementarity!) was, in the end, found to be crucial to that discovery! Conversely, it may have occurred to you that all those geometric, structural illustrations we used in this book to describe the development of a taxonomy for our Alternative Model of the Elementary Particles would appear to have a distinctly chemical flavor! On the other hand, Chemistry itself, generally speaking, is closely involved with geometric, structural considerations; what we are usually concerned with is how a set of atoms are deployed and connected together to construct a particular molecule…

This is the place to be basic; maybe quite basic. But not too basic; for one thing, in keeping with the purpose of this book, I have an ulterior motive; I’m really mainly interested in the complementarity exhibited by the formalisms we deal with. Also, we ought to agree on some things to start with or else we shall never get to the end of this chapter. To begin with, we should know what to call this chapter. I, personally, got stuck on whether to call it Dynamics or Mechanics. So, I looked in the dictionary and guess what: it looks like the dictionary couldn’t make up its mind either — something vague about “forces” acting on “things”…

The science of Thermodynamics is a voluminous subject, wide and deep and at times not easily traversed by those used to straightforward, well-defined technical description. I am reminded of the Introduction to this book wherein I talked about The Meaning of “is” as a barrier, a thicket that bars our way to ultimate understanding. Nevertheless, we shall proceed, wielding our Machete of Meaning to eliminate the polysemic (!) thickets that impede our progress to thermodynamic understanding (using flowery words and phrases to bolster our courage!)…

Believe it or not — you be the judge — this chapter is a logical follow-on to the two preceding chapters because, as you’ll see, in essence, it’s all about energy. The way the subject came up goes back to a dinner my wife and I were invited to share with a very nice couple across the street. The wide-ranging conversation that ensued had somehow turned to (the subject of) exercise and when I started to relate how I used to be able to perform multiple “Muscle-ups” on the high-bar in days long gone, Tom remarked that it reminded him of a well-known weight lifting procedure. As it turns out, I was also acquainted with that procedure (which, by the way is, descriptively, known as “Clean and Jerk”!) and I quite agreed with him even though the relationship between the two activities might seem far-fetched at first thought (to the uninitiated of course!). At second thought, however, clearly both activities involve hoisting a weight, an actual dead weight in one case and body weight in the other, and both involve grasping a long, horizontal bar with both hands and shifting arm and hand orientation thereto at some phase during each procedure in order to accommodate a change in the method of hoist…

There is a long historical background associated with what has come to be called Electromagnetism. Electric and magnetic phenomena were well-known separately long before but the 18th and 19th centuries saw a greatly increased activity in both experimental and theoretical activity in both areas. The names Franklin, Volta, Coulomb, Poisson and Gauss are prominent in accounts of the period (Krider 2006, APS 2006a, APS 2016a). Then in 1820 Hans Christian Oersted (APS 2008) discovered that an electric current can deflect a compass needle and shortly thereafter Andre Ampere showed that two parallel electrically conducting wires would attract or repel each other depending upon whether the associated currents were in the same or opposite directions (NIST 2018). And finally, Michael Faraday (APS 2001) showed that a changing current in one circuit would induce a current in a neighboring circuit as a result of the transient magnetic flux induced thereby and thus showing the way to the development of transformers, electric motors and generators…

Nowadays a lot of people are giving a lot of thought to the nature of space and time. It’s really nothing new; so did the philosophers of the classical ancient world and in fact, in the 18th and 19th centuries that saw an explosion of both theoretical and experimental activity. Most influential in that regard was Isaac Newton who regarded “absolute” space as remaining “immutable” without “reference to any external object” and “absolute time” as flowing uniformly and similarly constrained (Newton 1729)…

Don’t be alarmed; this chapter is not a diversion; it is indeed pertinent to our main story line! The previous section featured a comparison of DNA and the basic ideas of the Alternative Model; that was what, as documented in the previous book, initiated my original focus on complementarity, a concept whose generalization as a principle promised to have fundamental significance. In contrast, here we begin with two subjects that are already so regarded, beginning with Noether’s Theorem, which has been described as forming “an organizing principle for all of Physics” (Neuenscwhander 2011). The theorem pairs the invariance of an entity governed by a symmetry principle to changes of reference with the conservation of an associated dynamic entity within a given reference system. More explicitly, each member of the pair implies the other: finding an entity that is invariant to change in reference implicates the existence of an associated entity whose value never changes as measured within a particular, associated reference. Conversely, awareness of an entity whose value is constant within a given reference implies that there exists an associate entity invariant to change of reference…

In the first place, whereas, in the preceding, discussion was primarily algebraic in nature (as it was for Einstein), the General Theory is intimately concerned with the essential geometry of Spacetime, an entity with Character and possessing attributes. You may recall the bit of “arcanity” presented in the chapter on Noether’s theorem and Gauge theory, to wit: “— as Utiyama and Byers point out, since General Relativity is a gauge theory, the symmetry associated with it is that of a Lie group and in particular the group of ‘all continuous coordinate transformations with continuous derivatives’ otherwise referred to as the ‘group of general coordinate transformations’ for which the symmetry associated with Special Relativity is a subgroup also known as the Poincare Group”. It has been said that what Minkowski discovered was that Einstein’s Special Relativity “— was nothing other than the theory of invariants of a definite group of linear transformations of R⁴, namely, the Lorentz group.” (my emphasis). As I said above, a bit of a put down that Einstein apparently justifiably resented…

Although the usual scale of application of General Relativity is the cosmos, at some point in this chapter we shall employ it in so unorthodox a manner as to violate all rules of technical etiquette: we are going to use it to show how our Alternative Model (AM) particle model can arise out of spacetime and, in the process, to justify a statement we made earlier concerning the time-honored action integral. I was going to request that you don’t tell anyone but, recently, I came across a paper published, in 1919 by Albert Einstein, no less, who investigated the possibility that gravitational effects might have some bearing on the nature of the elementary particles. He concluded that “— there are reasons for thinking that the elementary forces that make up the atom are held together by gravitational forces (Einstein 1919)”. Albert was actually thinking of the interior of the electron as some kind of localization of electro-magnetic effects held together by gravitational forces!!…

In April 1950, Albert Einstein wrote “On the Generalized Theory of Gravitation” (Scientific American 1990). Here’s a snippet of what he had to say about electrodynamics and Special Relativity: speaking of Michael Faraday “unencumbered by the traditional way of thinking, he felt that the introduction of the ‘field’ as an independent element of reality helped him to coordinate the experimental facts”…

It seems to me that the Elementary Particles that appear in Chap. 9 above are just exactly what the good Professor is talking about! Perhaps you recall the phrase I quoted from one of my earlier publications describing my elementary particles as solitons existing “in-and-of the fabric of spacetime”. Precisely! And each such “particle” moves through space as a wave which is what a soliton does — continually forming and reforming of that fabric of spacetime! Not the usual concept of particulate motion. Furthermore, the set of elementary particles epitomize the Principle of complementarity, so we don’t have to worry about their legitimacy!…

This chapter is a condensed version of the notion discussed in the previous book (Avrin 2015) that each basic AM fermion experiences “a situation reminiscent of the symmetry-breaking topology that preconditions the Higgs mechanism. The complete Higgs mechanism is, necessarily, rather complicated; without it the Standard Model (SM) would not be able to attribute mass to the elementary particles of the model without violating some of its basic theoretical underpinnings, primarily SU(2) gauge symmetry and the efficacy of renormalization. However, it is not necessary to invoke the entire mechanism here because, in the first place, SU(2) symmetry in the AM is inherent to particle structure and, furthermore, renormalization is not an issue. What is basic to the SM’s Higgs mechanism, however, is the need to introduce a symmetry-breaking version of the potential energy that each particle encounters. That potential is also of interest here because of the way in which symmetry breaking was introduced into the AM, that is, by way of the initial implicit assumption of toroidal topology and the reduction, as per the above, of the value of linear energy density, A, of spacetime in the neighborhood of the MS…

About the title: according to the verb “undulate” is defined as “to move with a wavelike motion”. Also, as used in this book, “Complementarity” is basically a discussion of the verb to “complement”. So, can “Undulatority”, which is hereby defined to mean discussions of wavelike motion, pair with “undulate” in the same way that “complementarity” pairs with “complement”? Why not?…

In 1992, in a question — and — answer book intended to explain the highlights of Superstring Theory, the immensely knowledgeable physicist Edward Witten began his response to the question “What are the essential problems that the superstring theory claims to address?” thusly: “In twentieth century physics there are two really fundamental pillars, one of them is general relativity which is Einstein’s theory of gravity and the other is quantum mechanics, which is the theory of everything that goes on in the microscopic domain. In other words it’s the theory of atoms, molecules and smaller objects called elementary particles. The basic problem in modern physics is that these two pillars are incompatible (Witten 1992)”…

The 20th century was an amazing epoch. My mother was born in January of 1900 and lived to the age of 95, something of great significance to me personally. Of more interest to this monograph and indeed to this chapter is a paper I downloaded from the Internet entitled “One Hundred years of Quantum Physics” by Daniel Kleppner and Roman Jackiw, both of the Massachusetts Institute of Technology (Kleppner and Jackiw 2000). The opening paragraphs rank, for me, among the most effective, informative such I’ve ever read, and I would like to quote them as follows:

“An informed list of the most profound scientific developments of the twentieth century is likely to include general relativity, quantum mechanics, big bang-cosmology, the unraveling of the genetic code, evolutionary biology, and perhaps a few other topics of the reader’s choice. Among these, quantum mechanics is unique because of its profoundly radical quality. Quantum mechanics forced physicists to reshape their ideas of reality, to rethink the nature of things at the deepest level, to revise their concepts of position and speed, their notions of cause and effect”…

Well, there are some other important things that emerged about this time before the big quantum revolution of the mid-twenties that we’ll get to shortly. These include Bose-Einstein and Fermi-Dirac statistics, the identification of spin, Pauli exclusion and more, some of which well also get to later but for now, I want to sketch out what is perhaps the salient feature of this introductory phase, the Bohr model of the atom, something Niels Bohr put together (literally) in 1913. Here’s a thumb nail version of what he did, taken mainly from what I put together in my previous book:

As we know, a hundred years or so ago, Bohr formulated a model of the Hydrogen atom that predicts that element’s discrete line spectra. It had already been established (by Rutherford in 1911 and recognized with a Nobel Prize) that the known atoms consisted of a tiny, centrally located nucleus with a positive charge and negatively charged electrons whose disposition in the atom where still mysterious (APS 2006b). As a young man, Bohr a Dane had spent some time in Rutherford’s Manchester laboratory and had become interested in the mystery himself. In characteristic fashion, he boldly decided to investigate the case of electrons circling about the nucleus even though it seemed obvious to many that they would eventually just end up there due to electromagnetic attraction. So he wrote down the attractive force,

Case closed, right? Well, sort of. Unfortunately, trying to fit that model to anything more complex than Hydrogen turned out to be infuriatingly difficult, in fact impossible, as was realized and acted upon primarily by four heroes of the quantum revolution over a span mainly of just a few years of the second decade of the twentieth century. The first was Werner Heisenberg at that time a researcher in a physics group headed up by Professor Max Born, once a graduate student of Minkowski (whom we met earlier in connection with Relativity)…

This Chapter has a special meaning for me; for one thing, in a way it represents another example (see Chap. 4) of the identification, seemingly, of two quite different sets of phenomenological considerations. As per the title of this chapter, one of the two is radar, my association with which began almost 70 (I think) years ago! Actually it began with a combination of radar and missile guidance because the company I had started working for was in the missile business, specifically the development of a “beam-rider” surface-to-air missile. What that means is that the missile continually corrects its location in the cross section of the beam transmitted by a radar that tracks a target the missile is supposed to intercept and demolish — old-time technology!…

As we saw in the foregoing, convolution and its partner the Fourier Transform are fundamental ways to understand what’s going on in both Radar (as an example of a sensorial system) and quantum mechanics. I don’t know for sure, but I think the Fourier Transform (FT) is better known to students just getting used to the ideas involved than is the idea of convolution. Historically, the FT provides one with the occupancy in a mathematical frequency space of things in real space, but it is often moot which space is the more real! Here’s one way to express the FT; in one dimension we have

We begin this chapter with a quotation: “No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the ‘square root’ of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.” The speaker, a mathematician of highest repute in many areas including the fields of particle physics and its relationships, shall be nameless, mainly because I seem to have mislaid the reference for this rather gloomy prognostication. Were he to become aware of it, I should hope he would forgive me for saying so, but I think we may be able to do better than that; I don’t think I can afford to wait centuries and perhaps the rest of this chapter will indicate why we don’t really need to…

As a species, we have a concept of time as well as of space and to one extent or another, as the current manifestation of the species, we run our lives accordingly, even many of our scientific pursuits. However, with Einstein, Minkowski and Lorentz, we have graduated to a more sophisticated concept, the Relativistic concept of Spacetime that led to fundamental advances in Physics. Now, however, we are faced with a relatively new but related concept, the empirical notion of nonlocality, a phenomenon that apparently depends on a requirement for what is known as entanglement. Actually, rather than “now” I should really say “still” because the ideas involved have been with us and debated for on the order of eighty years and are still evoking fresh discussion!…

The preface to a small but most illuminating monograph on the life and work of Dirac (Pais, Jacob, Olive and Atiyah 1998) begins “Paul Adrien Maurice (PAM) Dirac was one of the founders of quantum mechanics and the author of many of its most important developments. He is numbered alongside Newton, Maxwell, Einstein and Rutherford as one of the greatest physicists of all time.” And, we might add, his equation, universally known as the Dirac equation marks a revolutionary milestone in the story of Quantum Mechanics (QM) for several remarkable reasons; not only does it unite QM with Special Relativity, but in the process it introduces the inevitable existence of antiparticles and illuminates the fundamental particle attribute of spin…

In his book, Fermi (1937) goes on to say how the original view of heat as some kind of fluid was eventually supplanted by the notion of the “ — equivalence of heat and dynamical energy” which is to be sought in the kinetic interpretation, which reduces all thermal phenomena to the disordered motion of atoms and molecules. — However, what’s involved is “ — the mechanics of an ensemble of such an enormous number of particles (atoms or molecules) that the detailed description of the state and the motion loses importance and only average properties of large numbers of particles are to be considered”…

At this point we interject a subject of some pertinent interest to complementarity as well as to both Alternative Model (AM) and Standard Model (SM) deliberations. A model was previously shown of Pions mediating Yukawa type exchanges between nucleons to maintain deuteron stability in what may be characterized as (strong) isospin space and we reproduce it here in Fig. 29.1. This is a dynamic process, as postulated to maintain that stability. The figure is pretty much self-explanatory: all four Pions of the AM are involved, two for the proton and two for the neutron. At each stage of the process what was a free proton becomes a neutron and conversely what was a free neutron becomes a proton. Two fusions and two fusions take place. Also, (in what we might arbitrarily call the first stage) what was a π⁻ becomes a π^0R and what was a π⁺ becomes a π^0L. Two fusions and two fusions take place, in each case. The process then reverses to recover the original pair of nucleons…

What’s known nowadays as the Standard Cosmological Model postulates the existence of invisible “Dark Matter“, the source of the gravitational influence that keeps the stellar constituents of our visible galaxies from being strewn about due to galactic rotation. Most literate people have probably at least seen or heard of it and are aware that it’s quite unlike “ordinary matter” with which it otherwise does not interact. And that there are numerous candidates for its constituents. And perhaps also that it serves a vital cosmological function: According to current expertise, dark matter seems to be distributed in a kind of grid-like fashion throughout space and many of our galaxies appear at the intersection of the grid lines. So, although we can’t see it, dark matter is a vital constituent of the cosmos we know about…

I really hadn’t intended to write another substantive chapter before the concluding section but here it is — my history repeating itself! The bit of history involved here is that when my Book of Reference (BoR) was almost complete, I realized that a simple 2 × 2 matrix can encapsulate the essence (as symbolized by the Principle of Complementarity) of both the elementary particles of physics and DNA, that marvelous, elementary molecule that enables both the inherent multiplicity of life forms on earth and their evolution with time. That realization was quite mind Boggling, a condition so dire, as to launch my quest into the range of that Principle’s applicability — that is to say, where and how fundamentally it applies; ergo the current book on the Principle of Complementarity itself…

If you read the Introduction, you may recall a quotation attributed to Professor John D. Barrow (2007) regarding his discussion of the quest for “The theory of everything” and repeated here as follows: “… If we are to arrive at a full understanding of complex systems, especially those that result from the haphazard workings of natural selection, then we shall need more than current candidates for the title “Theory of Everything” have to offer. We need to discover if there are general principles that govern the development of complexity in general which can be applied to a variety of different situations without becoming embroiled in their peculiarities.” (My emphasis)…

What follows takes a while to expound so pleased bear with me: With Spinoza and Clifford in mind I ask rhetorically, what can we say about The Meaning of “Is” and the way in which it relates to Complementarity? In that regard, I would like to present the following “semi-axiomatic” chain of statements:

• Let us assume that the word “is” implies the existence of something rather than nothing, words whose meaning we purport to understand. (Axiom 1).
• Assume further that Spinoza was right: there is only one “something”, a, universal, absolutely infinite, fundamental substance. (Axiom 2).
• Now identify that “substance” with the Einstein/Minkowski Spacetime.
• Further, identify Clifford’s local disturbance of that spacetime with the solitonic “particles” of the Alternative Model.
• Note that, as Solitons, these entities display both particle-like and wave-like characteristics and thus manifest their own “Complementarity” as per Bohr.
• Note, also that the existence of the question as to The Meaning of “Is” implies the concomitant existence of a live, sentient questioner, which, at least in our world, implies the associated existence of cells, composed of molecules, composed, in turn of atoms, and, ultimately, elementary particles.
• Which in the present context, implies the Solitonic particles of the Alternative Model for which Complementarity was shown to be a fundamental attribute.
• The implication is thus that the Principle of Complementary is a necessary, unavoidable partner of The Meaning of “Is” where the latter is defined as per Axiom #1 as the existence of something rather than nothing: the two are inextricably linked and at a cosmological level!

The following sections are included:

• Appendix: The Basics of Beta Decay
• Au Revoir
• References