Ecstasy of mathematics

IN EARLIER articles we emphasised that the ecstasy of mathematics can be experienced not only by an Euler or Ramanujan but also by an aspirant to a career in creative research.

We now discuss how a young entrant can assess himself to choose the area of research in mathematics and the type of unsolved problems he should aspire to solve.

Sir C. V. Raman told me frankly with his customary emphasis that the greatest deterrent to creative work is fame! By that he meant that all creative work is spontaneous which results in fame, but once a scientist becomes famous he tries to match his new work with the standard of his old work. This removes the spontaneity and therefore affects the quality of research, since no two discoveries are identical in their origin. But this assertion of Sir C. V. Raman should bother only famous scientists but should not deter a fresh entrant who takes up the challenge of creative work. The first level of achievement consists in proving the same result by a different method. In this case since the result is known, the discovery of a new method is just the first step toward creative research. But this helps the young scientists to choose the field of research in which he has demonstrated his originality. I encouraged my student S. K. Srinivasan to take to research since he proved that two different infinite series that arose as solutions of certain differential equation were actually equivalent. I took Mathews for research since he helped me solve a difference equation in a complex variable through first replacing the variable by an integer. I encouraged V. Devanathan in his research since he had a facile familiarity with angular momentum algebra. With considerable experience in foreign centers of learning, I encouraged my young son to take to research when he combined the Farey and Fibonacci sequences into another meaningful sequence in a very appealing way. All the four became full professors in leading institutions.

The second source of inspiration is to study the papers announcing the first discovery. Such papers reveal the `insurrection in the mind' of the discoverer and the reason he chose the new assumptions that led to the discovery. It is more purposeful to read the first papers of Einstein on Special Relativity than reading his biography eulogising his genius. In other words the first paper is an autobiography of the `discovery'.

It is a historical fact that Nobel prize work resulted from observing the properties of 2 x 2 matrices and interpreting them to suit the physical problem. The discoverer sees more than meets the eye and has the courage to assert his conclusions.

Pauli matrices are the roots of the unit matrix which any student can derive, but Pauli had the courage to identify them with `spin'. Likewise 2 x 2 circulant matrices are familiar to any student but Lorentz identified it with a transformation that preserves the difference of squares. Einstein had the imagination to interpret the transformation as relating to space and time and extend it to momentum and energy, a discovery which altered the course of civilisation! Dirac extended the anticommuting properties of 2 x 2 matrices to 4 x 4 matrices leading to the discovery of the equation of the electron which is the basis of quantum electro dynamics. He had the courage to interpret the positrons as holes in a sea of negative energy states while Feynman recognised that this was equivalent to a negative energy electron travelling back in time.

What is important is to recognise that all these discoveries are related but made by different scientists proving the Sir C. V. Raman dictum that the new extensions of a new work is not usually done by the original discoverer.

The following question remained open: How to derive Dirac matrices from Pauli's? I was able to obtain it by the operation which led to many problems good enough for doctorate degrees for more than six students of mine! Such situations arise over the whole domain of physics and it is a misleading claim that one can discover the `theory of everything' and reach the end of knowledge of natural phenomena. Actually extending the Churchillian phrase `We are not at the end, not even the beginning of the end, not even the end of the beginning, but at the beginning itself'. We have yet to understand the meaning of time, its origin and its reversal, the origin of creation, the expanding universe, the black holes and dark matter, the nature of three dimensional space and the extension to higher dimensions.

Articles are still published in established journals on the idle paradoxes of the ages of twins, or the properties of tachyons amounting to a distortion of clean and consistent theory of special relativity. This led me to the `Rod Approach to Special Relativity' confirming the pristine beauty of the impeccable logic of the Lorentz transform.

Very recently I enjoyed the thrill of discovery that the tangent to the hyperbola corresponds to an ellipse if the Cartesian coordinates are interpreted as polar coordinates! Our young students can do much more if they have faith in their powers, modesty to correct their faults and desire to reach the goal.

Our country abounds in such youthful talent which has to generate its own energy and momentum to reach the destination — the solution of an unsolved problem, or at least the derivation of an unnoticed result.

Lorentz transform in the 21st century

Here is an open question to a young entrant to theoretical physics. If is a Lorentz matrix so is — . If reverses momentum keeping energy constant, — reverses energy keeping momentum constant, both reversing the velocity. If is expressed as the square of a matrix L, the Lorentz matrix L brings the particle to the rest system and then to -v. If — is expressed as the square of a matrix, that matrix with imaginary elements brings the particle to rest but with imaginary mass and then to negative energy.

L* is a semi Lorentz matrix reversing the difference of squares.

What is the physical meaning of L* and iL*?

The properties of iL* and L* are connected to the striking features of the velocity transformation formula in which either all three velocities are less than unity or two of them greater than and one less than unity. This has been clarified by the author's New Rod Approach to Special Relativity which explains the distinction between space-like and time-like intervals. It was shown that a space-like interval is transformed into a space-like interval by a Lorentz transformation with velocity parameter less than unity. This implies that mathematically a time-like interval can be converted to a space-like interval or vice versa by a Lorentz transform with a velocity parameter greater than unity like iL*. Note that iL* preserves the difference of squares while L* reverses the difference. But both yield velocities greater than unity characteristic of space-like intervals! For iL* yields imaginary space-time co-ordinates while L* yields real coordinates both with ratios which are real and greater than unity! Mathematically s space-like interval implies that space and time may be pure real or pure imaginary while the ratio is greater than unity.

The Rod Approach deals with the transformation of space-like to space-like or time-like to time-like coordinates by a Lorentz transform involving velocity parameter less than unity.

Is this related to the Feynman propagator and intermediate states of Dirac "off the energy shell"?

This is an intrusion into the hallowed domain of Diract and Feynman. Nobel laureate Chandrasekhar told me frankly "Some simple problems which cannot be solved are either ignored or forgotten!"

A beginning was made by the author and independently by Shoenberg (Brazil) by splitting the propagator into two real positive and negative energy parts. Will a new entrant to theoretical physics of the 21st century take up the challenge?

Alladi Ramakrishnan

Alladi Centenary Foundation
Madras

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