Paul Adrian Maurice Dirac

In my last two years of Bachelors' programme in the department, I had to read many science papers, some for a literature survey, some for more detailed study and some even for continuing the research further. Many papers written by many authors; some physicists, some mathematicians. Some were easy to read and understand even for a “lay-man”. Some were really meant for people pursuing the subject fiercely. But there were papers by one man that were different. Its like among many dates in your life, you remember one or two special ones, the day you proposed to your girl or some such day! It had a really different aspect to it. Its like you hide the author's name and ask me to read the paper and I will tell you “its his isn't it?”. I was asked to do a term paper on magnetic monopoles and I came across his paper while searching for the literature. I fell in love with his style, instantly. Well, not instantly, really, because I had to read it thrice to digest the maths; more so because of his style. With due respect to all physicists, he never repeats a idea, never tries to put a single idea in as many words as possible. He is a master at making concise statements. A genius in being precise. Thats how I tried to know more about him. Thats how he became one of my three favourite physicists. I was reading “The Theory of Magnetic Poles” by Dirac, Phys. Rev. 74, 817-830, 1948.

Paul Adrian Maurice Dirac

We (physics students) use Dirac's products so many times: Dirac's delta distribution(it would be bad to call it a function after the recent campaign by a maths prof.), Fermi-Dirac statistics, Dirac's Bra-Ket notation (that was one beautiful thing), Dirac's wave equation, Dirac's quantization equation and many more. So its only just to know some more about him.

Dirac is one of the giants on whose shoulders my areas of interest stand (firmly so). He is the founder of quantum electrodynamics and worked extensively on quantum mechanics; the work is published in his book “Principles of Quantum Mechanics” (1930); I am afraid it is out of print, (atleast so in India) but I know someone who has a copy and I am sure he is proud of his possession. Dirac introduces the Bra-Ket notation and the Delta distribution in this book. Dirac, being one of the founders of Quantum Mechanics, is one of the few physicists who truly understood the whole business of quantum physics. After all, he formulated a complete mathematical description of quantum mechanics. Dirac noticed an analogy between the old Poisson brackets of classical mechanics and the then recently proposed quantization rules in Werner Heisenberg's Matrix formulation of quantum mechanics, which led to formulation of quantization rules in a novel and more illuminating manner. For this work, published in 1926, he received a Ph.D. from Cambridge. Dirac predicted the existence of positron, the electron's anti-particle, which he interpreted in terms of what is known as Dirac sea (of particles). The famous and important Dirac Equation, not mentioning its other properties, integrates two (opposite) frameworks: the quantum theory and the relativity theory. Quantum spin turns out to be obvious and necessary from this equation.

I would like to digress a little to explain in some detail the essentials of the theory of magnetic poles by Dirac.

“If one supposes that a particle with a single magnetic pole can exist and that it interacts with charged particles, the laws of quantum mechanics lead to the requirement that the electric charges shall be quantized—all charges must be integral multiples of a unit charge e connected with the pole strength g by the formula eg=1 / 2ℏc. Since electric charges are known to be quantized and no reason for this has yet been proposed apart from the existence of magnetic poles, we have here a reason for taking magnetic poles seriously. The fact that they have not yet been observed may be ascribed to the large value of the quantum of pole.”

That is an extract from the abstract of his paper. In other words, Dirac showed that existence of a magnetic monopole is sufficient (and necessary?) to explain the observed quantization of charge. Some of you might be wondering whats so difficult in finding a magnetic pole. The problem is all magentic poles you and I have seen so far exist as dipole. A bar magnet has two poles (named north and south). No matter how many times you cut the magnet in half, you always end up with a dipole. Infact, no “scientific” evidence of monopole exists till date. I insist on scientific, because there have been rumours of evidence. However, the very nature of the experiment is non-scientific. The experiment consists of setting up a superconducting ring at some place where you think the monopole might pass through. Let a small current go round and round in the ring and wait for the monopole to shower its grace on you. If the monopole indeed passed through the ring, according to Faraday's laws, you would notice a increase in amplitude of current. So you see, the experiment relies on chance and therefore is non-scientific (Q.E.D!). Added to this is catch in Dirac's statement. Existence of even one monopole is enough. And that one monopole can be existing just anywhere in this vast vast universe.

Coming back to Dirac, I would like to iterate that he is precise. You might want to argue that one should try to explain any concept in as many words as possible. In reality it does not work with science concepts. I have tried it myself. The more you try to explain, the farther you move away from the truth. Dirac's brevity, I think, should be attributed to his BA in Applied Mathematics. Yes, Dirac is as much a mathematician as he is a physicist. We all know, maths is the language of physics, and he writes poetry with maths. Unfortunately (and this is where I don't agree with him), his stand on poetry itself is best expressed in his own words:
“In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it's the exact opposite.”
Nevertheless, Dirac believed in “beauty”. Quoting his words, “This result is too beautiful to be false; it is more important to have beauty in one's equations than to have them fit experiment.” and “If one is working from the point of view of getting beauty into one's equation, ... one is on a sure line of progress.”

Here is where I would like to make another digression. Dirac had one regret. He was deeply dissatisfied with the renormalization approach to dealing with infinites in Quantum Field Theory. In his own words (again): “I am very disturbed by the situation because the so-called good quantum theory does involve neglecting infinities in an arbitrary way. This is not sensible. Sensible Mathematics involves neglecting a quantity when it's small; not because it's infinitely great and we do not want it”. It is expected of a man who saw beauty in all the work. But this dissatisfaction is something that he shares with Albert Einstein (and may be other physicists, too). Einstein after his fame in coming up with theories of relativity, spent later years of his life trying to formulate a unified theory of universe; one theory to explain everything. So did Dirac spend his later years trying to do away with the renormalization approach. In both the cases, the physics community claims, they “digressed from the mainstream”. I think its a polite way of saying they have gone crazy. But I would not agree with that view. History is full of reknown artists and scientists whose work from their later years of life was not understood and appreciated. I would like to think, these people moved on to a higher plane, leaving us to our ignorance.

Coming back again, Dirac brought to his personal life, the logical directness he used in his scientific work. A testimonial to this is a story of his typical attitude to a questioner at the end of a lecture. The man indicated that he did not follow a particular point of Dirac's argument. A silence followed and at last the chairman asked if Professor Dirac would deal with the question. The inevitable reply that followed was, “It was a statement, not a question”. Talking about his attitude, Dirac was noted for his modesty. He called the equation for the time-evolution of a quantum-mechanical operator, which Dirac was in fact the first to write down, the "Heisenberg equation of motion". Most physicists speak of Fermi-Dirac statistics for half integer spin particles and Bose-Einstein statistics for integer spin particles. While lecturing later in life, Dirac always insisted on calling the former "Fermi statistics". He referred to the latter as "Einstein statistics" for reasons, he explained, of "symmetry".

Paul Dirac (August 8, 1902 – October 20, 1984) was a British theoretical physicist born in Bristol. He held the Lucasian Professor of Mathematics at the University of Cambridge. He shared the Nobel Proze in Physics, 1933, with Erwin Schrodinger for “the discovery of new productive forms of atomic theory”. He was elected a Fellow of the Royal Society in 1930, being awarded the Society's Royal Medal and the Copley Medal. He was elected a member of the Pontifical Academy of Sciences in 1961.

I would like to end this article by mentioning again his unique style: There is a standard folklore of Dirac stories, revolving around Dirac saying exactly what he meant and no more. When Niels Bohr complained that he did not know how to finish a sentence in a scientific article he was writing, Dirac replied, "I was taught at school never to start a sentence without knowing the end of it”.

So thats Dirac to me, my favourite physicist.

In this context I would like to mention one commonly occuring flaw when students write mathematical equations, something that I often notice and hate. Mathematics is a language and without relevant grammar and punctuation it is meaningless. Which is why it is important to always mention the domain, when you write a function!

Jade.

Science and Magic

Outrageous? Blasphemy? Hillarious?

Well, whatever you may think, I have believed for several years that science and magic are connected. You may want to call me a madman, but I am a student of science discipline myself; so I know what I am talking about, which makes me sane! There are many examples, many of them self-realised that can justify my claim. But I am a little too tired right now to go into a duel with you over our philosophies. And yet, I can not resist it. I can not resist telling you this story.

I have read many stories about stones. You might have, too. There is this one story that I remember with all the details: The Moonstone; its the first detective story in english written by Wilkie Collins, in which the moonstone is associated with ill-fate in the hands of a mortal. There are many more stories in which precious stones are associated with sorcery. This is one such story. However the stone involved is not a well known precious stone such as a moonstone or a diamond or an emerald.

It began almost eight years back, on the day of my thirteenth birthday. I remember it so well, partly because of the following story and partly because the birthday fell on a 13th with a particularly bright yellow full moon. I have this rather immature fascination for trinkets and ornaments. The odd ones especially. I am allergic to gold, but all other metals suit me well. That day, I had been to a jewellery store and the moment I entered, my eyes were set on a beautiful iridescent Opal. Nobody else seems to be interested in it at all. But then again not many share such a liking for odd trinkets! So I decided to buy it; it was costly, but not as much as the stone is worth which I came to know only later. Had any one known its true value, it would not have been on sale in the first place. How it ended up in the jewellery store and later around my neck is a mystery I am yet to solve till date. So far so good.
That night (oh, I am definitely not divulging the details of the party; thats for private listening) my life took such a dramatic change, I am still recovering from the effects, even as I am writing this article. (You must be thinking, I am writing this out of delirium, a side effect of that dramatic change. I know what you are thinking and your sarcasm hurts me!) I was taking a stroll in my garden, watching the bats flit by, listening to the cicadas' haywired chello orchestra, when all of a sudden the illusion broke down. Oh, I forgot to mention, this optical physicality that you see and force your brains to experience is an illusion and I dont want to argue trying to describe reality to someone who is so used to this illusion. So when the illusion broke down, I was in a secluded place. Everything was so different. Dynamic! The shape of the stones and boulders changed all the time. The ground under my feet itself was moving; grumbling rather. For a long time I stood trying to understand when an old woman appeared. She said, "Stop moulding your thoughts so much. You are wasting your energy!". That shocked me so much, I was thoughtless for a brief moment and during that time, everything around was still and I realised that the reality took the shape of my thoughts. Far fetched, eh? Does not matter. That is not the point of the story anyway. We, myself and the old woman, started talking and she explained to me what happened and that explanation is the point of the story.

If you have been reading carefully, I mentioned earlier that this story is not about diamonds and emeralds. I also mentioned that stones are associated with magic. I also mentioned about opal. I take it that you all have drawn your obvious conclusions. Yes, not all stones are associated with sorcery. Diamonds and emeralds are kept in the King's treasury. Wizards and witches carry opals and other stones. Whats so special about opals and those other stones? They are naturally occuring "photonic crystals".
Photonic crystals are optical equivalents of semiconductor electronic devices like diodes and transistors. Just like transistors are used to route a stream of electrons to desired locations in desired amounts, photonic crystals can be used to control stream of photons. While the speed of electrons depend on the voltage under which they accelerate, the speed of photons is .. well, you know. Needless to say, replacing the billions of transistors with an equivalent circuit made of photonic crystals (and using a photon source) will increase the speed of processing of a computer, for that matter, any electronic device. Since light is the fastest travelling entity known to muggle (non-magical) folk, this transformation is like achieving the pinnacle of a technology. But ofcourse, there are difficulties in implementing this. To understand them, it is essential to understand what photonic crystals really are.
The easiest way to understand the behaviour of light in a photonic crystal is to compare it to the movement of electrons and holes in a semiconductor. In a silicon crystal, for example, the atoms are arranged in a diamond-lattice structure, and electrons moving through this lattice experience a periodic potential as they interact with the silicon nuclei via the Coulomb force. This interaction results in the formation of allowed and forbidden energy states. Consider photons moving through a block of transparent dielectric material that contains a number of tiny air holes arranged in a lattice pattern. The photons will pass through regions of high refractive index - the dielectric - interspersed with regions of low refractive index - the air holes. To a photon, this contrast in refractive index looks just like the periodic potential that an electron experiences travelling through a silicon crystal. Indeed, if there is large contrast in refractive index between the two regions then most of the light will be confined either within the dielectric material or the air holes. This confinement results in the formation of allowed energy regions separated by a forbidden region - the so-called photonic band gap. Since the wavelength of the photons is inversely proportional to their energy, the patterned dielectric material will block light with wavelengths in the photonic band gap, while allowing other wavelengths to pass freely. We can also have a different dielectric material in place of air holes. By tweaking the parameters of the dielectric materials, such as the refractive index, the size and periodicity of the material, we can control the device. And just like doping introduces new energy levels in band gap in semiconductors, breaking the periodicty by changing size of the interspersed air holes introduces new levels in photonic band gap.

So why don't we have photo-chip fabs and opto-electro computers? (Don't mind those fancy names) The problem is the lattice spacing. A rough estimate of the lattice spacing is the wavelength. In case of the semiconductor devices, the electron wavelength and for photonic crystals its the electromagnetic wavelength (divided by the refractive index of the material, ofcourse). Now, you realise difference. The problem in making small structures is compounded because it is more favourable for a photonic band gap to form in dielectrics with a high refractive index, which reduces the size of the lattice spacing even further. For example, suppose we wanted to create a photonic crystal that could trap near-infrared light with a wavelength of 1 µm in a material with a refractive index of 3.0. We would have to create a structure in which the air holes were separated by about 0.3 µm - an extremely difficult task. If the scale was 1000 times smaller, we could build the structure atom-by-atom using a chemical reaction; and if it was 1000 times larger we could build the structure mechanically.

In 1991, Eli Yablonovitch and co-workers at Bell Communications Research in New Jersey produced the first photonic crystal by mechanically drilling holes a millimetre in diameter into a block of material with a refractive index of 3.6. The material, which became known as "Yablonovite", prevented microwaves from propagating in any direction - in other words, it exhibited a 3-D photonic band gap. Other structures that have band gaps at microwave and radio frequencies are currently being used to make antennae that direct radiation away from the heads of mobile-phone users.

So much for an introduction to photonic crystals. Opal is a naturally occuring photonic crystal. The very reason behind the iridescence of Opal is because its a photonic crystal, unlike diamonds which sparkle because of cut edges that reflect light totally internally. Apparently the opal around my neck had the optical signal that is the key to the real world out there. Thats how I ended up talking to the old woman. She told me how they exploited the applicability of these crystals to store energy. That is how iridescent stones became essential to using magic. They use the stored electromagnetic energy to do that spooky action at a distance. Thats a simple way of explaining things. Oh for those who of you who did not know, the word magic is derived from the root mage (I have an unsual interest in etymology, too; very essential for mages). Rings a bell?? Image! Imagination. Magic is science raised to its zenith with the help of imagination. Afterall, did not Albert say, "Imagination is more powerful than knowledge"!

Jade.

that sync' ing feelin'!

I am proud to belong to a species that holds many an enigmatic myth among the biped monkeys. The legends are not merely myths. They are true. Atleast some of them, I have experienced myself. This myth about the full moon that you are trying to investigate is one of those few experiences. The transformation into a human? Well, I have not experienced it. But why would I? Its so clumsy to walk on two feet. How do you manage it? Anyway, coming back to your question, yes it is true that we derive strength and energy, physical and spiritual, from the full moon. We literally drink it off the pool! My mamma taught me to feed off the moon when I was young. And when I grew of age, 3 months old, if you were wondering, I had to leave home as all wolf cubs do to find my own pack. Since then, I have fed on the moon myself and its really relishing, you know without all that hassle of tearing flesh from bones.

There is one thing I have noticed during this process. They say the moon is a spherical heavenly body and that it rotates about its own axis and revolves around the earth. But I always only see the same spots and marks the disc. Could it be that the moon is symmetrical? Did you ever notice it? Ever wondered? Well, ofcourse my instincts taught me more about the universe as I grew older and if you will promise to keep this a secret, I can share the knowledge with you. The truth is, the moon takes roughly the same time, around 28 days, to complete one rotation around its own axis and revolution around the earth. Then its only common sense that from any one hill on earth, we can only see one side of the moon. But ofcourse this is not the whole truth. My ancestors did not have the chance to feed off the moon every month like us. Because in their days, the rotational and revolutional period was not the same, just as the case for earth now. And the other side of the moon is really dark and would not reflect so much even on full moon nights. So they could only feed on the moon when this side of the moon faced the earth.

That made me wonder how the rotational and revolutional time periods changed over time. Our folk legends say that the Moon Maiden is helping us with food ever since man has started making his mark disturbing the ecology. It is true. Nevertheless, there is another explanation! Its the process of "synchronisation".

Long time back, moon's rotational period was less than 28 days and the revolutional period was more than 28 days. The dynamics the many body problem, you know with that thing you call as gravity being the main element, brought these two time periods closer and closer to a same number which happened to be 28 days. Its happening to earth, too. But ofcourse at an unnoticeable rate and the difference being so large, its going to take many many years. But its happening.

This synchronisation is an ubiquiteous phenomenon. Oh, ubiquiteous means universal. Its funny how we have different words with the same meaning! Anyway, many events out there in the universe happen according to laws of synchronisation. Its the way the universe works. All living and non living things must respect those laws. Indeed have you seen the fireflies on the banks of japanese streams? Oh you have not? Well, you must atleast know that fireflies glow to attract their mate. I know this because I am a wolf. I know the ways of all animals and plants, except humans, ofcourse. The weird lot! Oh and unlike in your species, in all other animals, from lions and peacocks to fireflies, its the male that is attributed with beauty. So in case of fireflies, the more beautifully it glows, more its chances of attracting a mate. So when many male fireflies gather at a place, they try to compete with each other to attract the females flying invisibly in the dark. The competition almost always reaches its pinnacle with all the fireflies flashing together in burst. See whats happened? All the fireflies without realising it synchronised their flashing.

And there is this plant, thats called "kurinji". It grows in Kerala and TamilNadu. It mass blooms once in twelve years. It happened last in 2006. Do you know why it blooms only once in 12 years? Kurinji (Strobilanthes sp.) and a number of other plant species synchronise their flowering (reproductive phase) within large local populations at a particular site. Populations at different sites may have different calendars, but the length of the cycle is almost the same within a particular species. This is one of the survival mechanisms evolved to escape complete destruction of the population by seed/flower predators and is termed `predator satiation'. Synchronisation of reproduction by large populations leads to an abundance of `prey' such that the predators are simply out-numbered. Therefore, the percentage of population destroyed by predators is significantly reduced. Jungle fowl and small mammals are the chief seed predators of Kurinji seeds. Other plant species also have adopted this.

Oh and one late night while I was taking a walk in the city, out of curiosity, I stopped by this mesmerising toy shop. There were these two pendulum clocks, small ones placed on the window sill. Since I had no particular place to go, I sat there and watched them. They were swinging exactly out of phase always. The next morning one cub from our pack had strayed in to the city and I went past this shop looking for him. The shop keeper was dusting the items and apparently because he disturbed the clocks, they were now swinging very randomly. I went back that night again and to my amazement saw that the pendulums were swinging exactly out of phase again. It might have been spooky for you. But I realised that even non living things synchronise! By the way, the cub returned home ofcourse!

Oh oh! You asked about we wolves feeding on moon and I seemed to have given you a lecture on something else. And its time I got back to my cave. But you should believ me. We do feed on the moon. How do you think we manage to eat flesh only once in 14 days? Our diet is synchronised! We feed off the flesh only on new moon nights!

-the wolf lady Saki.