Symmetry and the Law: A Hidden Connection

If you ask someone to name a woman scientist, they will probably name Marie Curie. If you ask them to name another, most will not know what to say. And if you ask someone to name a woman mathematician, most people will not be able to name a single one. Why is that? Have there not been any woman mathematicians? Or have people not bothered to find about woman mathematicians. I think it is important for our children to know that there have been great woman mathematicians and scientists and that science, and more specifically mathematics is not just a province of men. In today’s VERITAS we will talk about a great woman mathematician who made huge contributions to physics and mathematics. Her work led to a new insight into the fundamental laws of Physics and resulted in major developments in modern physics. The name of this mathematician was Emmy Noether and this is what Einstein said of her: “Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

Emmy Noether was born to a Jewish family in the German town of Erlangen in 1882. Her father was a professor of mathematics at the University of Erlangen. At that time middle class European girls were expected to learn languages, arts, music, dancing and also household activities. When she passed out of high school she got a certificate which would allow her to become a language teacher. But Emmy had other plans. At the age of 18 she decided to study mathematics at her father’s university. At that time women were not allowed to take classes at the university. She was, however, given permission to “audit” classes. So, she sat in the classes but not as a regular student. But she did so well in her studies that she passed out 7 years later(in 1907) with a doctorate in mathematics. From 1908 to 1915 she worked without pay and without title at the Mathematical Institute of Erlangen. This was because the university had a policy that did not allow hiring women professors. In 1915 she was invited by David Hilbert and Felix Klein to join them at the university of Gottingen. Hilbert and Klein are very famous names in the history of mathematics and physics. Though she was allowed to lecture there, it was many years later that she was formally recognised as a teacher at the university. It was at the university of Gottingen that she formulated a very important theorem of physics that is now known as Noether’s theorem. We will learn more about this theorem after finishing our story about Emmy’s life.

In 1920s Emmy did a lot of work on abstract Algebra. This work is considered to be of fundamental importance by mathematicians. She worked on group theory, ring theory and number theory. During this time she also lectured throughout Europe and was recognized as a mathematician of the first rank. In 1932, The International Mathematical Congress in Zurich asked her to give a plenary lecture. This was the high point of her mathematical career. But, in 1933 the rise of Hitler changed all that. The Nazis removed Jews from all government jobs. Emmy lost her right to lecture at the university. Many Jewish scientists left Germany and migrated to other counties. Even Einstein had to leave Germany and he joined Princeton University in USA. Emmy too left Germany and joined Bryn Mawr College in USA. She died two years later from complications following a surgery. She was 53.

Now, lets look at Noether’s theorem and understand why it is of fundamental importance to Physics. When I first read this theorem I was amazed at what this theorem showed. This theorem showed that two of the deepest concepts in Physics and Mathematics are related at a very deep level. The two ideas are symmetry and conservation laws. Emmy Noether showed that every symmetry implies the presence of a conservation law and vice versa. It is a beautiful idea and a very surprizing one too. To understand Emmy’s wonderful theorem, lets first define what we mean by symmetry and conservation laws.

We all know about symmetry. However, the definition of symmetry in everyday life is different from the definition in mathematics. In everyday life we typically call an object symmetric if we can draw an imaginary line through it and the two parts on either sides of the line look exactly the same. For example a circle is symmetric because if we draw a line through the centre, the two parts look exactly the same. We humans are attracted to symmetry. We find it visually appealing and we try to create objects that exhibit some form of symmetry. We ourselves are symmetric to a large extent – if we draw a vertical line through the centre of our nose, the two sides look nearly the same. But this is one kind of symmetry- mirror symmetry. Mathematicians generalized it and formed more kinds of symmetry. For example, if you take and object and rotate it by an angle and if the rotated object looks the same as the original object, it is said to have rotational symmetry. Take a square, and rotate it about its centre by 90 degrees(clockwise or anticlockwise), the result is a square that looks the same as the original square. So a square possesses rotational symmetry( of course, it also possesses mirror symmetry). So the general mathematical definition of symmetry is this: take an object or system; do some transformation on it; if the transformed object or system is the same as the original one then we can say that the system is symmetric with respect to the transformation. I will give you some examples later when we discuss Noether’s theorem.

Now, lets briefly talk about Conservation laws. In Physics there are some quantities that never change if the system is closed. Such quantities are said to be conserved and the laws that “assure” this are called conservation laws for these quantities. Lets take an example: The law of conservation of energy says that the total energy of a closed system always remains the same. Energy can change form- Kinetic energy can change to potential energy. Energy can change to mass and mass can turn into energy. But the sum total of all the energy remains constant. When we use the word closed system, we mean that no energy comes in or flows out of the system. Like the conservation of energy, there are many other conservation laws- law of conservation of momentum, law of conservation of angular momentum, law of conservation of charge etc. Many years back( in the year 2000), I wrote a VERITAS series on the conservation laws. You can see the series here: https://unvarnishedveritas.wordpress.com/tag/conservationlawsweek/

Emmy noticed a very deep connection between conservation laws and symmetry and this is what Noether’s theorem is all about. Emmy showed that every symmetry in nature results in a conservation law of physics and vice versa.

But from a physics point of view, what does it mean when we say that a system is symmetric with respect to a transformation. So, in other words, when we compare a system with respect to the transformed system, how do we check if they are the “same”? From the point of view of Physics, the two systems are the same if their Lagrangian is equal. The Lagrangian is a quantity named after the great mathematician Joseph-Louis Lagrange. It is the difference between the kinetic and potential energy. So L= T-V where T is kinetic energy and V is potential energy. If a system’s Lagrangian stays the same when it is transformed then we can say that the system is symmetric with respect to the transformation.

Lets take some examples of conservation laws and find out which symmetry results from each of them. The simplest case to understand is that of translational symmetry. A system is said to have translational symmetry if we can change its coordinates by a fixed amount( by addition or substraction) without changing the equations of physics that govern it. Or, in other words, if we take a system and then add a fixed amount to its x, y and z coordinates and if the Lagrangian stays the same, then we sat that the system has translational symmetry. And Noether’s theorem tells us that translational symmetry implies law of conservation of linear momentum. Isn’t it remarkable? Lets look at it this way: The reason why linear momentum is conserved in our universe is because particles or systems of particles display translational symmetry.

Lets now look at the law of conservation of angular momentum. What symmetry is it based on? Noether was able to show that rotational symmetry implies the law of conservation of angular momentum. In more precise terms, if you take a system and then rotate it and the Lagrangian of the system does not change due to the rotation, then the system will exhibit the law of conservation of angular momentum.

There is one symmetry-conservation law relationship that I find particularly interesting. And very surprising. Lets take a system and calculate its Lagrangian. Let’s now wait for some time and calculate its Lagrangian again. When I say wait for some time, I mean any length of time. So if the Lagrangian is the same in both the cases then we can say that the system was symmetrical with respect to time change. In this case the quantity that is conserved is Energy. Isn’t that amazing! The very familiar conservation of Energy is due to symmetry with respect to time changes. This symmetry is different from the usual notion of symmetry which is spatial in nature. This particular symmetry has nothing to do with space and is based on time alone. So Noether was able to show a deep relationship between Energy and Time. And later when Quantum Mechanics was discovered( or rather formulated), we again found that Energy and Time are related through Heisenberg’s uncertainty relationship.

Another conservation law that we are familiar with is the law of conservation of electric charge. What symmetry is at the root of this? Quantum Field theory tells us that charge conservation is due to gauge invariance( or symmetry) of the electrostatic potential and vector potential. It is not easy to explain this in simple terms. But it is interesting to note that this relationship between conservation of charge and gauge symmetry was discovered several decades after Noether proposed her famous theorem.

There are other conservation laws too: CPT( charge, parity and time), colour charge etc and each of these conservation laws has been shown to be related to a symmetry of some transformation. Note that some of these do not follow directly from Noether’s theorem. There is also a Quantum version of Noether’s theorem. It is called Ward-Takahashi identity.

Noether did not just give us the relationship between a few conservation laws and the underlying symmetry of nature. She gave us a recipe to find the symmetry if the conservation law is known and vice versa. So, if you discover a previously unknown symmetry in nature, you have not just discovered a symmetry. You have also discovered a new law of conservation. You can use Noether’s theorem to find the conserved quantity.

So we see that Emmy Noether made tremendous contributions to Physics. We have just explored a simplified version of her famous theorem. We have not been able to even briefly look at her contributions to mathematics. The next time some kid mentions the names of Isaac Newton and Albert Einstein as great scientists, please do name Emmy Noether also. It is important that women mathematicians get recognized for their great contributions.

Many years back, in 2002 , I wrote an article about another great woman mathematician, Sophie Germain: https://unvarnishedveritas.wordpress.com/2002/11/27/women-of-mathematics-sophie-germain/
Kanwar

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Go wondrous creature, mount where science guides
go measure earth, weigh air, state the tides,
instruct the planets in what orbs to run
correct old time, regulate the sun
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