G. Rajasekaran

Mathematics and Physics
 

Galileo and Newton

Eugene Wigner, the famous theoretical physicist said: "The unreasonable effectiveness of Mathematics is surprising."  The surprise is that Wigner said this. For, many centuries before Wigner, Galileo had proclaimed:

 

 

 

 

 
 

​

Science and, in particular Physics, made tremendous progress only by following Galileo's dictum.

​

Newton, who came after Galileo, gave the laws of motion. According to them,

​

F = ma.

That is to say, if we multiply the mass m of a body by its acceleration a, one gets the force F acting on the body. So, if we divide force by mass, we get the acceleration.

​

Newton created a new mathematics called calculus which facilitates the calculation of the motion of any body. Leibnitz also independently discovered calculus. Who discovered it first? Debate is still going on.

​

In the centuries after Newton, the use of calculus increased many times. Without calculus, Physics would not have progressed. 

​

Faraday and Maxwell

By using a small hand-held gadget we are able to talk to our friend living hundreds or thousands of kilometers away. Scientists control spacecrafts travelling millions of kilometers away, while sitting in space laboratories on the Earth. There are many such wonders! All this is possible because of the electromagnetic waves born in the electrodynamics of Faraday and Maxwell. Electromagnetic field pervades all Universe.

​

Michael Faraday who studied electromagnetic phenomena through experiments had already created a complete conceptual framework about the unseen electromagnetic field. James Clerk Maxwell who came later gave a precise form for it through his mathematical equations.

​

Those famous Maxwell equations are:

​

​

​

​

​

​

​

​

​

​

​

​

​

For reading this essay, it is not necessary to understand these equations. These are given here only to emphasize that all the laws of Physics are given only in
Mathematics. There is no Physics without Mathematics. In particular, there are no Maxwell equations without calculus. Maxwell equations are actually differential equations.

​

Important consequences followed from Maxwell equations. From the equations, Maxwell proved the existence of electromagnetic waves and that their speed was 300,000 km per second. That was the speed of light already determined by scientists. So Maxwell concluded that light is an electromagnetic wave. This was a great discovery. For until then nobody knew what light was.

​

Einstein and Minkowski

In 1905, Einstein revolutionized Physics by creating Special Theory of Relativity. His concepts on space and time are revolutionary. Einstein's teacher Herman Minkowski said: " Hereafter space and time will disappear as mere mirages. Only their combination space-time will be a reality." Einstein did not like this. He accused that these people are converting his physics into mathematics. But it is this space-time that became the arena for all the subsequent developments in Physics. Infact Einstein's General Theory of Relativity itself is formulated in space-time only. That is the next story.

​

Einstein and Hilbert

Einstein was not satisfied with his special relativity. He had to make Newton's law of gravitation also to conform to the space-time concept. For this, Einstein had to work hard for ten years. He found that only by using the non-Euclidean geometry of Riemann one can succeed. With the help of his mathematician friend, Einstein learnt Riemannian geometry and finally in 1915 he created his General Theory of Relativity. Although he found the correct equations in 1915, in the earlier years he published wrong equations. Finally 1n 1915 he achieved complete success.

​

But the famous mathematician David Hilbert knew Riemannian geometry already and hence could arrive at the correct equations before Einstein. Hilbert was generous enough to accept that the whole theory was Einstein's. So the equations of the General Theory of Relativity are called Einstein-Hilbert equations.

​

Dirac and Mathematics

It was Paul Adrien Maurice Dirac that discovered truths in Physics from Mathematics. Others used Mathematics as the language of laws of Physics, but Dirac brought laws of Physics from Mathematics.

​

He said: "I used to play with mathematical equations." From this play was born Dirac equation which combined relativity and quantum mechanics. This equation controls all the dynamics of the electron. The spin of the electron was derived from this equation. The existence of positron as the antiparticle to the electron also was obtained from Dirac equation. We now know that every particle has an antiparticle.

​

Before Carl Anderson found the positron in cosmic rays, Dirac had a testing time for a few years. He was prepared to throw away his equation as wrong.

​

Therefore he created another new idea. That is the Dirac monopole. This paper of Dirac is a wonderful essay. At the beginning of this essay he describes his views on Mathematics very beautifully. He says in future new ideas in Physics will come from Mathematics.

​

Dirac's discovery about monopole also is very beautiful. If monopole has to be consistent with quantum mechanics a condition connecting the magnetic charge of the monopole and the electric charge of the electron must be satisfied. This is called Dirac's quantum condition.

​

To discover this condition Dirac did not use any new mathematics. He used only the mathematics of quantum mechanics known to everybody.

​

Later many people found the connection of Dirac's quantum condition to a branch of Mathematics called Topology. From Topology, one can prove the Dirac condition. Many interesting things in Topology were discovered.

​

Dirac's monopole paper shows very clearly the strength of physical ideas and that it is not necessary to go to any difficult mathematics.

​

Dirac's idea "Physics from Mathematics" has gained further strength from String Theory.

​

String Theory

Hundred years of Fundamental Physics has culminated in Standard Model which has been established as the correct Quantum Field Theory of weak, electromagnetic and strong interactions. But gravitational interaction has been left out. Infact gravity has refused to be incorporated in Quantum Field Theory. 

​

String Theory offers a solution. It automatically incorporates gravity within a quantum mechanical framework. That is its special beauty. But it has been bought
with a price, namely string theory works only if the number of space dimensions is increased from 3 to 9. The extra 6 dimensions are supposed to be curled up to
very tiny sizes.

​

In string theory a point particle is replaced by a one-dimensional object. The length of this string is very tiny. String Theory leads to the existence of higher dimensional objects, two dimensional membranes and in fact branes of all dimensions upto 9.

​

Imagine the relativistic quantum mechanics of all these objects of higher dimensions. Imagine such a Universe existing in 9 dimensions. That is String Theory.

​

What kind of new mathematics is needed to understand this strange world? That is being discovered now. New Mathematics is discovered first and then Physics is derived from that.

​

Therefore, most of the physicists find String Theory and its Mathematics very difficult to understand and so hate String Theory. This is wrong and unfortunate. The Fields Medal Committee recognized the importance of the Mathematics of String Theory and awarded the Fields Medal to the String Theorist Edward Witten.

​

Good News is that many bright students are attracted to String Theory and work in that area. In particular, India is at the forefront of String Theory research.

​

The famous mathematician G H Hardy said: "There is no practical use for Mathematics. Mathematicians work only for their enjoyment."  In contrast, one is tempted to say 

​

​

​

​

A Crisis

We must not forget the second part of Galileo's dictum: The Laws of Physics can be proved or disproved only by experiment or direct observation.

​

That is where String Theory meets its obstacle. Accelerators with enormous energy are needed to do experiments bearing on String Theory.

​

In order to verify Standard Model, an accelerator with 14 trillion electron volt energy was constructed. But to test any theory of Quantum Gravity, energy
16 orders of magnitude more is needed. Most physicists think this is impossible.

​

That is the Crisis.

​

But human ingenuity knows no bounds. New principles of acceleration will be discovered and this energy barrier will be crossed.