|
|
|
The role of maths in physical sciences
MATHEMATICS, as E. Wigner has defined, is “a fusion of skilfull operations with concepts and rules invented just for this purpose.” The principal emphasis is on invention of concepts, which go beyond those contained in the axioms. Without concepts, a mathematician would not go far.
As we shall see the role which mathematics plays in physical sciences, where one is concerned to understand basic mysteries of nature, has also beauty of its own and a source of joy and excitement to its practitioners.
Science is a fusion of philosophical thinking, which supplies concepts and skilled crafts, which supply tools. The two are intimately connected. Concepts are needed to explain old things (in the form of empirical data) in new ways. Tools are needed to discover new things that have to be explained (in terms of concepts) or to discover things predicted by a concept-driven theory so as to verify or discard that theory. The concept-driven revolutions have been rare. Taking quantum mechanics as a prime example of a concept-driven revolution, Thomas Kuhn, in his book The Structure of Scientific revolution, lists, in addition to quantum mechanics, only six major concept driven revolutions in the last 500 years, associated with the names of Copernicus, Newton, Darwin, Maxwell, Freud and Einstein.
According to F.J. Dyson, during the same period, there have been about twenty tool-driven revolutions, some in physics itself, but mostly in biology and astronomy, using tools created by physics. Physics has had great success in creating new tools that have started revolution in biology, computer science, engineering, astronomy and medicines. Two prime examples are the Galilean revolution resulting from the use of the telescope in astronomy and the Crick-Watson revolution (1950) resulting from the use of X-ray crystallography to determine the structure of DNA in biology. Another tool driven revolution having a great impact on society was based in the invention of transistor resulting in the advent of computers and memory banks in the 1960s.
Electronic data processing and simulation revolutionized every branch of science, increasing the power of scientific theories to interpret and predict new phenomena. Computers, becoming cheaper and smaller, have become personal and are used for variety of purposes, from toys to highly sophisticated scientific work. They have revolutionized the communication, the mode of information and finance.
First, there is a mundane role which is to facilitate for the physicists the numerical calculation of certain constants or the integration of certain differential equations. Mathematics does, however, play a more sovereign role in which we will be concerned and bring out how higher mathematics found applications in subtle empirical problems.
The laws of nature: These are written in the language of mathematics. “All laws are deduced from experiment, but to enunciate them, a special language is needful, ordinary language is too poor, it is besides too vague, to express relations so delicate, so rich and so precise”[H. Poincare]. Let us discuss some examples:
The basic axioms of Newtonian: Mathematical physics is stated in the preface of the first edition of the Principia: rational mechanics ought to address “motion” with the same precision as geometry handles the size and shape of idealized objects. The association of “motion” (particularly the change in motion) with “maths” was a stroke of genius. The mathematical language in which it was formulated contained the concept of second derivative, not a very immediate concept.
The act of writing down a fundamental law is a rather singular and rare event. It is a miracle that in spite of the baffling complexity in the world, certain regularities in the events could be discovered. A monumental example of such a law is Newton’s law of gravitation — a single law which explained everything from planetary motion to the terrestrial motion of pendulums and which appears simple to the mathematicians and which proved accurate beyond all reasonable expectations but still it is a law of limited scope.
The concepts of modern physics are abstract: As P.A.M. Dirac stated: “Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit in its power in this field”. In this context, let us consider two theories: Relativity and Quantum Mechanics, both of which involve mathematics of transformations. This is because the important quantities in nature appear as the invariant or having simple transformation properties under these transformations. In many cases mathematical concepts were independently developed by the physicist and recognized then as having been conceived before by the mathematician.
Quantum mechanics is a good example of this where Dirac invented his own mathematics in his formulation of quantum mechanics. Einstein, on the other hand, recognized Remiann Geometry as tailor-made for implementing his view of gravitational force.
Symmetry and analogies: Let us consider this role of mathematics by discussing Maxwell’s equations. The laws of electrodynamics are described by four basic equations which expressed all known facts at the time Maxwell began his work. Maxwell noticed that two of these equations lack symmetry. He removed this asymmetry by modifying one of these equations by adding a new term. It was not a new experiment, which came to invalidate the equations. But in looking at them under a new perspective, Maxwell saw that the equations become more symmetrical with his modification. In that Maxwell was twenty years ahead of experiment since his “a priori” views awaited twenty years for an experimental verification. He formulated these views because he “was profoundly steeped in the sense of mathematical symmetry”.
Maxwell unified electricity and magnetism and as a result the electromagnetic radiation in the form of light, radiowaves and X-rays provide many of the conveniences of modern life, electrical lights, television, telephones, etc. Furthermore the requirement of mutual compatibility of Newtonian mechanics and Maxwellian electrodynamics leads to the foundation of special theory of relativity.
Maxwell’s equations give much more than what was put in, purely from mathematical symmetry and analogies. Moreover they reveal to a physicist the hidden symmetry of things in making him see them in a new way.
Symmetry Principles and Group Theory: Until the twentieth century group theory played a little role in theoretical physics. This had a background. With priority accorded to the differential equations of Newtonian mechanics, it was widely believed that infinitesimal analysis must be the way mathematics enters microphysics. Partly it may also be due to new and unfamiliar mathematics comprising group theory. It was Wigner, whose early experience in X-ray crystallography led him to a programme of applying the theory of group representations to atomic and molecular spectra as well as nuclear physics. He laid foundation both for the application of group theory to quantum mechanics and for the role of symmetry in microphysics.
In the last three decades Lie groups and Lie algebras played a major role in applying symmetry principles in containing allowable dynamical laws mainly for the reason that in many cases in subatomic phenomena, the dynamical laws were not known a priory. Here symmetry dictates dynamics. Further its formulation in terms of a sophisticated concept in differential geometry gives it a deep and beautiful foundation.
Modern trends
We have two great theories of the last century: the quantum mechanics and the theory of relativity. The two theories have their roots in mutually exclusive groups of phenomena. Quantum Mechanics provide a theoretical framework for understanding the universe on the smallest of scales: molecules, atoms and all the way to subatomic particles like electrons and quarks. General relativity provides a theoretical framework for understanding the universe on the largest of scales: stars, galaxies, clusters of galaxies, and beyond to the immense expanse of the universe itself. The two theories operate with different mathematical concepts, infinite dimensional Hilbert space and the four dimensional Riemann space, respectively. In most situations their union is not even required. This is because in most situations as mentioned above the domains (like atoms and their constituents) in which quantum mechanics is interested and domains like (stars and galaxies) in which general theory of relativity is relevant have no overlap.
These are, however, situations where both theories become relevant. For instance, in a black hole an enormous mass is crumped to a very small size. At the moment of big bang the whole of the universe erupted from a microscopic nugget, compared to which even the grain of sand looks enormous. These are domains that are tiny and yet incredibly massive and as such require Quantum Theory and General Theory of Relativity simultaneously.
Until recently the two theories could not be united i.e. no mathematical formulation exists to which both of these theories are approximations. It turns out that to achieve this one needs (i) a higher dimensional space-time for bringing together different mathematical concepts mentioned above (ii) supersymmetry to avoid tachyons (that is, the particles which move faster term the speed of light) (iii) going beyond point field theory i.e. the most fundamental entities are not point-like but extended one dimensional objects. The above three ingredients are incorporated in superstring theory.
First a word about supersymmetry. Supersymmetry incorporates boson-fermion symmetry. Such theories predict a new kind of matter in the form of supersymmetric partners of all observed elementary particles. Observation of these partners would provide the first experimental evidence for supersymmetry. But there is so far no experimental evidence for such particles. The experimental situation will become clear in about five years time when the world’s largest accelerator being developed at CERN, Geneva become operational.
The mathematics of supersymmetry involves use of Clifford algebra and Grassmann numbers which unlike ordinary numbers, anticommute. It turned-out that dynamics of superstring theory can be formulated in 10-dimensional space-time: four familiar space-time dimensions and six extra dimensions. The extra-spatial dimensions of string theory are to be “crumped” up (to microscopic dimensions) in a particular class of 6-dimensional geometrical shapes known as Calabi, Yau shapes. The mathematics of Calabi Yau shapes is studied in a field called Algebraic Geometry - a relatively new field that combines algebra and geometry. Towards the end of last century, it led to some of the crowning achievements of pure mathematics, including the solution of Fermat’s last theorem, Mordell conjecture and the Weil conjectures. It is now being used by string theorists leading to a new branch of Physics and Mathematics, which may be called Quantum Geometry. Let me end this article by quoting from E.P. Wigner’s thought provoking article: “Unreasonable Effectiveness of Mathematics”:
“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning”.
The writer
|
|
|
|
