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Quantum Theory, Elementary Particles

Limits to Divisibility

N. Mukunda

In any effort to compare or bring together ancient insights and current scientific knowledge, there is always the question: what do we hope to achieve or learn from the exercise, if it is to go beyond recounting interesting stories to one another? There is always the problem of communication across disciplines and specialisations. Within science itself this is well-known, but the difficulties are even greater in the present situation. In any case, in matters of detail and recently acquired knowledge of nature through experiment, it is quite unreasonable to look for exact parallels in past thinking. Questions about the nature of matter and the structure of the cosmos have been raised since time immemorial. And in answer there has always been a great deal of speculation based to some extent on experience but also largely on pictorial thinking. We may view this as the first stage in coming to terms with the physical universe. One can generally think of four motivating factors behind speculative thinking: (a) the desire to respond to psychological needs, and even fears, about the unknown; (b) the desire for economy in concepts and independent causes, to unify what appears so diverse at first sight; (c) the wish to inspire and uplift the imagination and create poetic imagery; and (d) to hypothesize about the unknown, in the sense of science, and suggest new experimental tests of one’s ideas. Naturally in ancient times there was a tendency for the first three factors to be more prominent; and there was also, apart from astronomy, a general lack of precise and quantitative expression of one’s ideas.

Each of these motivations behind speculation is legitimate and has value in its own sphere; problems in comparison arise when different motivations get confused with one another. May be therefore it is wise not to expect too much in advance in any case when comparing ancient and modern insights, and to be cautious.

Let me focus on ideas on the questions: are there such things as the smallest units of matter, out of which all that we see can be built up? Or is there no end to the divisibility of matter? Ultimately is matter discrete or continuous, lumpy or smooth? Schroedinger, in his book Nature and the Greeks, explains one line of argument that led some of the early Greek thinkers to the view that there must be an end to the process of dividing matter into smaller and smaller parts. It is worth emphasizing that among the Greek philosophers, and in the course of several centuries, there was certainly no single uniform view about fundamental questions, but several differing schools and points of view existed. Schroedinger traces the views of Thales, his disciple Anaximander and then his disciple Anaximenes, all of whom lived around the sixth century b.c., on this question. Each of them believed that there was one basic substance out of which everything else was made up. Thales is quoted by Aristotle as having declared that "Water is the material cause of all things". For Anaximander it was not water but something else unseen, infinite and everlasting, which assumes many forms that we then see. In Anaximenes’ view, however, the primary substance was air, subjected to the processes of rarefaction and condensation. Through the development of these successive views, the relationship between being and becoming was gradually highlighted and clarified. For Heraclitus the basic element was fire, combining the attributes of substance and process. Parmenides too clung to a single fundamental principle but there seems to be some vagueness in his statements. In contrast to all these monists — whether it be a primary substance or process or principle — we see a change, a preference for multiplicity, in the views of others. Thus Empedocles hypothesized four basic elements — earth, water, air and fire — acted upon by love and strife. On the other hand, Anaxagoras believed in an infinite variety of infinitely small seeds, undergoing processes of mixing and separation. And then through Leucippus and Democritus we finally arrive at the Greek concept of the atom — the smallest units of matter, indivisible, indestructible, of finite size and located in empty space. Different atoms were of different sizes and shapes, but all of the same substance; with no qualities, mathematically divisible but physically whole. In recalling the development of this chain of ideas leading ultimately to the atom of Democritus, Schroedinger draws attention to a very interesting aspect. It was, according to him, the Greek encounter with irrational numbers and the continuum, and even their fear of these, that led them to the view that matter must ultimately consist of certain smallest indivisible and unchanging parts. And the various changes in the states of matter arose by these parts coming together or going apart, by condensation and rarefaction, in otherwise empty space. I mention this account by Schroedinger mainly because he highlights here a compelling reason based on experience as the motivation for believing that there must be an end to the divisibility of matter.

Contrasted with such views, as expressed by Democritus, Plato had taken the position that the smallest units of matter were certain elementary geometrical forms, so that they had more of a mathematical than a material significance; while Aristotle reverted more or less to the idea that matter is continuous. It would be interesting and necessary to learn in depth about any related and similar, but independent, lines of thinking in those times, in our tradition.

Skipping many centuries, when we come to Descartes we learn that he was a very imaginative and speculative thinker, and also that he was a confirmed opponent of the atomic hypothesis. For him, we read, matter was definitely infinitely divisible. The atomic idea had to wait for Dalton, and the developments in chemistry, to be revived again in compelling and quantitative terms. By that period, of course, the experimental method and the use of mathematical reasoning had been established as the twin pillars of science. In Faraday’s experiments on electrolysis too there were hints that there was an ultimate "molecule of electricity", which later turned out to be the electron. But it is interesting to recall that Maxwell had some doubts about it as he felt it would not fit in with his electromagnetic theory! It is also well-known that even around the end of the last century, Ostwald and Mach were strongly against the atomic hypothesis, because in their opinion it had not been demonstrated that the idea was scientifically useful and that atoms could be ‘seen’. Indeed one of the important motivations behind Einstein’s 1905 work on the Brownian movement was to bring out the reality of the existence of atoms through observable effects.

One sees in these developments spanning many centuries a cyclical process in thoughts about the ultimate nature of matter — like swings of a pendulum, going from the doctrine of existence of ultimate individual entities to their non-existence. It is quite interesting that this tendency has continued even into this century, albeit in a new framework defined by relativity and quantum theory.

The discovery of the electron in the late 1890s by J.J. Thomson, the 1905 explanation of the Brownian movement bringing out the reality of atoms, and the 1911 discovery of the atomic nucleus by E. Rutherford — all these were important steps towards the modern form of atomic theory. In a sense they culminated in the 1913 Bohr-Rutherford model of the atom, which explained Mendeleev’s periodic table of chemical elements. We may imagine that we had come to the end of the road on which Democritus first set out. What yet remained was the mathematical and physical framework of quantum mechanics.

In Dirac’s justly famous book on the Principles of Quantum Mechanics, he explains why, on general philosophical grounds, when one wishes to account for the ultimate structure of matter, there is a need to depart from the ideas of classical physics. His argument is that classical physics is scale invariant and: "So long as big and small are merely relative concepts, it is no help to explain the big in terms of the small. It is therefore necessary to modify classical ideas in such a way as to give an absolute meaning to size". Then he goes on to say that quantum mechanics provides a resolution by bringing in the concept of disturbance caused by observation, thereby giving an absolute meaning to size: "If the object under observation is such that the unavoidable limiting disturbance is negligible, then the object is big in the absolute sense and we may apply classical mechanics to it. If, on the other hand, the limiting disturbance is not negligible, then the object is small in the absolute sense and we require a new theory for dealing with it".

Even after the discovery of quantum mechanics in 1925-26, though, Schroedinger attempted to go back to a classical continuous model of matter, based on waves and wave packets. His hope was to somehow avoid quantum discreteness and discontinuities. But the attempt did not succeed.

In a few years, by the early 1930s, the atom of chemistry was resolved into a nucleus made up of protons and neutrons, with electrons orbiting on the outside. It must have seemed for a while that all matter was ultimately built up out of three basic building blocks — protons, neutrons and electrons. But then an entirely unexpected and qualitatively new aspect was introduced into the whole problem — this was the prediction, and soon after the discovery of anti-matter. This came about through the understanding of the unexpected features of Dirac’s relativistic equation for the electron. It turns out that there is a partner to the electron, called the positron, which is similar in all respects except that the electric charge is exactly the opposite. Moreover, when an electron and a positron collide, with a certain probability they annihilate one another, leaving behind electromagnetic radiation. Thus matter can vanish in a flash. Conversely under suitable conditions matter in the form of an electron-positron pair can be created from radiation. So all these possibilities, which of course obey strictly the laws of energy and momentum and charge conservation, complicate the question of the ultimate nature of matter, by coupling it to radiation. It has turned out that not just the electron, but every other fundamental particle too has a partner anti-particle — in a few cases the two may even coincide! Thus anti-matter, a consequence of relativity and quantum theory, enters the picture in an unavoidable way.

With the steady advance in experimental methods, through the 1940s, 50s and 60s, and even to this day, many additional particles have been found to exist in nature. Some were discovered in the cosmic rays, others were created in man-made accelerators. But as these discoveries accumulated, beyond a certain point, it became rather embarrassing to have to deal with almost 200 or so fundamental particles. This was even more than the number of elements in Mendeleev’s table!

So there came a period — in the late 1950s and through the 1960s — when many physicists believed that we had come to the end of the idea of divisibility of matter, but in an entirely new way. It was no longer useful to seek for the ultimate constituents of matter — all the 200 or so known particles were made up of each other and could convert themselves into one another freely, subject only to the basic conservation laws. All of them were to be regarded as equally fundamental, their parts were themselves; equally well, they were equally non-fundamental. Thus if you were to ask what went to make up a proton, the answer might be — a neutron and a pi-meson. What about the neutron? Well, a proton and a pi-meson. But then, what is a pi-meson made of? Ah, it is a proton and an anti-neutron! Essentially, it was felt that the search for substructure had ended, because it was no longer a useful question to ask, or a useful view to take. This point of view was particularly strongly expressed by Heisenberg, as recently as in 1975, in the words: "The difference between elementary and composite particles has thus basically disappeared".

But then over the same period and in a few years the pendulum swung again — the concept of quarks, invented in 1962 or so, steadily gained credibility; and all the 200 odd particles, including protons and neutrons and mesons, could be regarded as built up out of a few quarks. It was like the story of Mendeleev’s periodic table all over again. The next swing of the pendulum, again an embarrassment, is that today the number of elementary entities at the level of quarks has slowly risen to some 40 or so, so one wonders ‘what next’? In this context, I cannot refrain from mentioning one qualitative difference between attitudes today and a few decades ago. At the time he predicted the existence of the positron, Dirac hesitated a great deal because he felt three basic building blocks of matter were good enough, and adding a fourth was somewhat excessive. No such hesitations constrain the speculations and theories of modern times, however. There is also the idea, actively pursued, that the ultimate entities in nature are strings, not particles at all, though of a very refined and unimaginable kind.

Against the background of such developments, it is hard to suppose that one will come to a definite conclusion — nature seems to be always hiding behind one more surprise, one more mystery. It is like peeling the layers of an endless onion! At each stage there are some things we have learnt from experiment and theory, and then some educated guesses and speculations which only future developments can decide. Added to these is the fact that relativity and quantum theory put together make the question of constituents and substructure energy dependent.

There is one other general point I would like to make relating to the use of mathematics in the quest for understanding nature. Its value was clearly enunciated by Galileo in his statement: "The book of nature is written in the language of mathematics". In the physical sciences, certainly, mathematics is an essential and powerful language and mode of thinking, which of course gives an abstract quality to the basic theories of physics. One can get a feeling for this abstractness by appreciating that, firstly, the equations of physics are more fundamental than the individual phenomena which they describe. This of course is to be expected. But secondly, going beyond this, one finds that the symmetries of the equations are more basic than the equations themselves! In Dirac’s words, "both relativity and quantum theory seem to show that transformations are of more fundamental importance than equations". So the ‘stuff’ out of which one imagines nature to be made, and the principles underlying our understanding, get more and more refined.

In Heisenberg’s writings, he goes back here to ideas of Plato and of Aristotle: ". . . . modern physics takes a definite stand against the materialism of Democritus, and for Plato and the Pythagoreans . . . . The elementary particles in Plato’s ‘Timaeus’ are finally not substance but mathematical forms". And in the context of quantum mechanics, which he himself invented, Heisenberg says we have gone back to the idea of ‘potentia’ in Aristotle’s philosophy. It is truly staggering that our most basic, most fundamental language for physics today is that of deterministic evolution of potentialities and not of actualities. One treats in a quantitative and deterministic way "a strange kind of physical reality just in the middle between possibility and reality".

However, all these comparisons and discovery of parallels between ancient thinking and modern physics must be understood properly. As Heisenberg carefully clarifies, it is the appeal to controlled experiment that gives to modern science a really solid and serious foundation and meaning. The statements of quantum theory are in a very specific context, and are not largely poetic imagery and speculation, however elevating these may be. For one ancient ‘insight’ corroborated today, may be there are many others not so corroborated. But there is no surprise in all this. It shows the power that speculative thinking can bring in. At the end of his discussion, Heisenberg concludes by saying: "All the same, some statements of ancient philosophy are rather near to those of modern science. This simply shows how far one can get by combining the ordinary experience of nature that we have without doing experiments with the untiring effort to get some logical order into this experience to understand it from general principles".

Coming back to modern physics, it is the reliance on mathematics that provides guidance and rigour in thinking, and helps avoid internal contradictions and inconsistencies. But here it is a quite startling fact to realise that the true impact of careful and rigorous thinking within mathematics itself is a matter of very recent origin. Namely it is no older than the later half of the last century, through the work of mathematicians like Weierstrass, Dedekind and others. Even in this century we have had the striking results of Hilbert and Godel — the former trying to prove the internal consistency and completeness of an axiomatic framework for mathematics, the latter then showing that these can never be achieved!

It is well, then, to keep these facts in mind when we judge the ideas and speculations of ancient times on the nature of matter and the universe.


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