Role of mathematics in search of physical theories

Prof. Ngangkham Nimai Singh *

1. Introduction: A Philosophical Exercise

Relation between the human mind, the physical world and the mathematics is very subtle. It is generally described by an eternal triangle. The most fundamental question ever asked is whether mathematics is an invention of human mind or a reflection of physical reality itself which could exist independent of human mind. The mystery is the question of why it is so successful in dealing with the physical world, even at the scales well outside our personal experiences such as quantum mechanics. The debate started from the time of Plato till today and it remains as a philosophical exercise. The debate is very intense in case of the Big-Bang - theory of the origin of the Universe. Stephen Hwaking remarks "There is no need for invoking God as long as we have mathematics in our hands".

The branch of science subject called "Physics" deals with the laws of physical world. It has two attributes – experiment and mathematics (theory) which are considered as two sides of the same coin. Mathematics is the language of physics and it is therefore considered as the universal language to understand the Nature. It may be true that the same mathematics we have in our planet, may still be valid in other planets or universe. A good physics teacher therefore has a very strong and clear mathematical background. Many famous physicists came from mathematics stream, viz., Newton, Einstein, Maxwell, Dirac, Salam, Penrose, S. N. Bose, Maghanad Saha etc., to mention a few.

2. Classical era: Geometrical approach

Kepler's laws (1609) were the starting point in understanding physical world (Astronomia Nova). This was followed by the celebrated Newton's laws of motion and Newton's gravitational theory (Principia). Euclidean geometry was used as the mathematical background in Newton's picture of the physical world. Newton invented a very important branch of mathematics "Calculus" to calculate the gravitational field at a point due to the Earth.

This led to the birth of classical mechanics which deals with Lagrangian and Hamiltonian formalism etc. Classical mechanics can therefore be considered in short as a summation of a series of refinements in mathematical techniques over Newton's theory. Many mathematicians contributed in the process of mathematical refinements, viz., Euler, Lagrange, Hamilton, Poisson, Jacobi etc. Classical mechanics is non-relativistic, macroscopic and deterministic in nature.

Next came the empirical laws of electromagnetism which lead to lines of force due to Faraday (experimentalist). This was followed by Maxwell (theorist) with his four differential equations which are first order differential forms in symmetric space and time co-ordinates. Fundamental natural laws such as Lorentz invariance and gauge invariance are inbuilt in it.

This leads to the geometry of flat space-time (Minkowski space) and the birth of Special Theory of Relativity (STR) at the hand of Albert Einstein (1905). The theory is valid for inertial frames of reference only, and there is no gravity yet, i.e. no accelerating frame. Einstein's famous equation E=mc2 controls the domain of nuclear energy and Big-bang. The next major development is the Classical Field Theory - a continuous system in terms of classical fields, which led to the birth of classical electrodynamics.

A more general symmetry (General Covariance for all frames including non-inertial frame) combined with Equivalence Principle (Inertia=Gravity), leads to the development of Einstein's General Theory of Relativity (GTR) (1916). In GTR the 4-dimentional curved space-time embodies the gravitational force. General Covariance is a local symmetry principle and extra structure known as affine connection (which represents gravitational potential) is needed to satisfy geometry of curved space-time and concept of classical field (Riemannian Geometry).

There are many important Impacts of GTR on the development of theoretical physics, particularly the understanding of cosmology and early universe. Maxwell's theory of electromagnetism can be interpreted as curvature in an enlarged space-time of 5-dimensions where the extra dimension is a circle or phase. The way this twists relative to the 4 coordinates is interpreted as the Electromagnetic force. Kaluza-Klein approach rests on the assumption that this additional dimension is compact and very small, like a tightly curved circle. It predicts a new field (dilation) which is not a physical one – a problem to be accepted as correct model. The interpretation of general covariance as a local symmetry principle puts gravity on the same footing as the other three fundamental gauge interactions – a hope of unification of four fundamental interactions!

3. Quantum era: Operator approach

This is an un-geometrical approach unlike classical era, when the size of the particle is below the atomic size. It is based on operators in Hilbert space, functional analysis, linear algebra and infinite dimensional vector spaces, etc. Discrete quantities e.g., energy level, angular momentum etc. are allowed. There are global effects e.g., wave-functions not localized. The theory is probabilistic (hence wave function) and also needs quantization (because of Hermitian operator as observables).

Both topology and quantum theory study the global properties of various spaces which are unchanged by continuous variation, and so are discrete. Developments in algebraic geometry, topology, homology on manifold etc. lead to Quantum Cohomology. Topology - a study of knots in 3-dimensional space, can give explanation for various conserved quantities in fundamental Physics – knot-invariants – John-Witten invariants etc.

Some important equations in quantum mechanics are Schrodinger equation for non-relativistic dynamics, Heisenberg equation in operator form, Klein-Gordon equation for relativistic dynamics, Dirac equation for relativistic dynamics, Swinger-Dyson equation in QFT, The Breit equation for Helium atom, Bethe-Salpeter equation for bound state systems etc. These equations have profound applications in diverse areas of theoretical physics.

4. Quantum Field Theory

Some mathematical inconsistencies in the interpretation of relativistic quantum mechanics, lead to the development of Quantum Field Theory (QFT). In fact, a successful marriage of Quantum Mechanics and Theory of relativity leads to a new physical theory – Quantum Field Theory (QFT) where the wave function takes a new nomenclature – the quantum fields which are operators. Both Lagrangian and Hamiltonian functions are now operators which are expressible in terms of these fields as independent co-ordinates.

Since quantum fields are operators with infinite degrees of freedom, they require field quantization (second quantization) using "equal time canonical quantization relations" (CCRs). There are two different paths for achieving the QFT- the first one via classical fields and second one via quantum mechanics as depicted in the diagram. The QFT gives birth to the formulation of Gauge theory of fundamental interactions – local gauge (phase) transformations of the fields. In this theory Langrangian density is invariant under such transformation, leading to interesting consequences such as Spontaneous symmetry breaking (SSB) and Higgs mechanism of the origin of masses of fermions and gauge bosons. This gives the mathematical theory for basic forces such as electromagnetic, weak and strong interactions in gauge theory and their unification schemes – Electroweak unification and Grand Unified Theories (GUT) respectively.

5. Fiber Bundle: Topology and Geometry link

Historically, QFT evolves in different forms – Canonical, algebraic and functional Integral QFT. Topology and geometry linkage leads to a combined formulation of topological QFT and GTR known as fiber bundle formalism. Fiber bundle formulation could explain the conceptual similarity of the theories for all four fundamental interactions - Gauge interactions and GTR in a general framework. Fiber bundle is an important concept in differential geometry.

It includes manifold or differentiable structure, group or symmetry structures, and affine connection structure. It provides a comprehensive framework within which we can compare various theories. Different dynamical systems involve different types of fiber bundles but all systems share the general notion of the Fibrillation of a whole. For example Gravitational interaction of General Relativity is described by Orthonormal fiber bundle (FB); strong, weak, electromagnetic interactions in Gauge theory by principal FB; Lagrangian and Hamiltonian formalism in classical mechanics by Tangent and Cotangent FB respectively.

6. The Cube of Nature: A unified explanation of the evolution of physical theories

The conceptual evolution of physical theories with their underlying mathematical formalisms, can be demonstrated by a cube with its three perpendicular axes as three fundamental constants of Nature namely, Newton's gravitational constant (G), the velocity of light (c), and the Planck constant (h). In the cube, the point (G=0, h=0, c=0) which is the origin represents the Newtonian physics.

The axis line G represents the Newtonian gravitational theory, the axis h line presents the non-relativistic quantum mechanics, and the axis 1/c line represents Maxwell's electrodynamics and special theory of relativity. The surface described by (G, c) represents the General Theory of Relativity (GTR) while the surface (h, c) represents the Quantum Field Theory (QFT). Probably the surface described by (G, h, c) will represent the final theory of physical world! The mathematical structure of this final theory of physical world is yet to be deciphered.

Planck's astonishing hypothesis says that these fundamental three constants are just not only constants but units of conversion: (h, c, G) leading to Natural Units. At Planck's scale, the following new physics occur - Space and time cease to have meanings at Planck's scale. All classical ideas break down and quantum effects dominate at Planck energy scale: 1.2x1019 GeV ( ). Space-time continuum becomes discrete. Planck length:1.61x10-35 m ( ) is the smallest measurement of length with any meaning. Planck time: 5.38x10-44 sec ( ) is the smallest measurement of time that has any meaning.

7. Theory of Everything (TOE): String theory

The physical theory in the cube of nature involving three fundamental constants (h, c, G), is the final theory of Nature. This new theory will encompass both GTR and QFT under the same framework. String theory as a candidate for the theory of everything, had been proposed. In this theory fundamental entities are considered as one-dimensional objects (strings close or open) as to point particle concept in QFT.

Two-Dimensional conformal field theory background is needed for string propagations (vibrations) in 10 space-time dimensions which include extra six space dimensions. It predicts three gauge interactions and gravity. It also predicts a new fermion-boson symmetry known as SUPERSYMMETRY. Relevant mathematical structures have not yet discovered in string theory. At the moment it is very promising but it is still a good theoretical conjecture subject to experimental verification.

It has been conjectured that strings moving in the extra fifth dimension are represented in the everyday world by their projection onto the four-dimensional boundary of the five-dimensional space-time. The same string located at different positions along the fifth dimension corresponds to particles of different sizes in four dimensions: the further away the string, the larger the particle. The projection of a string that is very close to the boundary of the four-dimensional world can appear to be a point-like particle.

The theory that describes the interior of the five-dimensional space-time includes gravity and this strong gravitational field pulls objects away from the boundary. String theory is supposed to be the only way of dealing with infinities in the quantum theory of gravitation. It is indeed a big step in a new direction. It is beautiful mathematically, and it may really be the right answer. Physicists become mathematicians for a while. Famous physicist Edward Witten who is a pioneer in string theory, is a Fields Medalist.

8. Conclusion: In the beginning era, physicists learn mathematics to develop Physical theories. Nowadays, the physicists are themselves mathematicians. They invent many new mathematics. The high-energy physics community has witnessed an unprecedented divorce between "theory" and "phenomenology", where the former has virtually no contact with experiment. Overlapping between mathematics and physics is increasing in quest for the final theory of physical world.