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“In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians” – Albert Einstein

Out of the pantheon of Gods ^[1] involved in both the development of Group Theory and its application to Standard Model of Particle Physics, I have seen fit to take just two biographical detours to cover specific individuals. In Chapter 12 we met the “Father of Group Theory”, here we will spend some time on someone who could lay claim to being the “Mother of Symmetry’s role in Physics”. While Symmetry had been a guiding force in developing an understanding of the Natural World for many years, if not centuries, this person put these endeavours on a rigorous basis, unveiling a fundamental truth about reality. The remarkable woman we are talking about is Amalie Emmy Noether, always known as Emmy.

We will cover Emmy’s highly significant contributions to Physics in this Chapter, but she was first and foremost a Mathematician. While she can certainly be described as the greatest female Mathematician of all time, the adjective is superfluous. Emmy was simply one of the greatest Mathematicians of all time, with no need for gender-based modification. To build on my “Mother of…” theme, in 1972, acclaimed abstract algebraist Irving Kaplansky had this to say about Emmy:

It surely is not much of an exaggeration to call her the mother of modern algebra.

And for a perspective from the overlapping worlds of Mathematics and Theoretical Physics, we can turn to the incomparable, Herman Weyl in 1935:

Her strength lay in her ability to operate abstractly with concepts. It was not necessary for her to allow herself to be led to new results on the leading strings of known concrete examples. She was sometimes but incompletely cognizant of the specific details of the more interesting applications of her general theories. She possessed a most vivid imagination, with the aid of which she could visualize remote connections; she constantly strove toward unification. In this she sought out the essentials in the known facts, brought them into order by means of appropriate general concepts, espied the vantage point from which the whole could best be surveyed, cleansed the object under consideration of superfluous dross, and thereby won through to so simple and distinct a form that the venture into new territory could be undertaken with the greatest prospect of success.

Bringing things more up to date, Physicist, author and Science populariser, Brian Greene, tweeted the following:

Before looking at Noether’s work, it is instructive to cover some elements of her life and – in particular – the challenges she faced; ones that were exclusively due to her parents’ neglect. First, they unaccountably did not provide her with a Y chromosome, a prerequisite for any career in academia at the time. Second, they failed to ensure that Emmy was not of Jewish extraction, another oversight that is hard to understand in people who otherwise appeared as competent and caring parents.
 
 
Emmy’s Life and Hard Times

For a fuller biography of Emmy Noether, I would refer readers to a range of books, either dedicated biographies ^[2] or more general works in which she is featured ^[3]. Here I will simply sketch some salient details.

Emmy was born in 1882 in the German town of Erlangen, the oldest of four children. Her Father, Max, was an algebraic geometer, with (somewhat confusingly) both his own eponymous and shared-credit theorems. Her Mother, Ida Amalia, was born into a wealthy Cologne family. Both parents were supportive of all their children (regardless of gender) having the benefit of an advanced education. Two of her brothers achieved this goal. Alfred received a doctorate in Chemistry, though he unfortunately died just nine years later. Fritz became an Applied Mathematician, but left Germany for Russia when – as a Jew – he was unable to work. Unfortunately, while in Russia, he fell victim to one of the pogroms and was executed; though later formally exonerated of any crime. Perhaps most tragically in this family of academic achievers, her youngest brother Gustav suffered from intellectual disability and died before his fortieth birthday.

From one perspective, it could be argued that Emmy’s path ran smoother than her brothers during this turbulent time in European history. However, as a woman, it was much harder for Emmy to pursue her parents’ academic dreams in 1900s Germany. She showed no special aptitude for Mathematics early in life. Indeed her talents seemed rather to lie in English and French. Given this, Noether initially prepared herself for a career as a language teacher; suitable employment for a young lady. However, thankfully for the disciplines of Mathematics and Physics, she changed her mind and, in 1900, enlisted in an undergraduate course in Mathematics at her hometown University of Erlangen.

While between 1861 and 1885 women had been admitted to full academic life in each of France, the United Kingdom and Italy, turn of the Century Germany was much less open to embracing the intellectual potential of half of its population. So, when I say “enlisted”, this was not entirely true. At this point, women were allowed to attend lectures, but not to be formally registered as students, or to be awarded degrees. Emmy was one of only two women amongst a University of 986 students. In 1903-04 Emmy spent a term at the famous University of Göttingen, seat of German Mathematical excellence. On returning to Erlangen, she found that miraculously the restrictions on at least undergraduate female students has been relaxed, allowing her to first graduate and then obtain her doctorate in 1907. However at this point the doors to an academic career were firmly flung shut in Noether’s face. Women in Germany were simply not allowed to progress to be full academics.

For the next seven years Emmy taught at her father’s University, sometimes standing in for him when he was ill, but she was never formally a member of staff and never paid. Despite her unofficial status, Noether’s work began to speak for itself. As a result, in 1915, David Hilbert and another renowned Mathematician Felix Klein, invited her back to Göttingen. These two gentlemen then fought a lengthy battle to have Emmy registered as a full member of the academic staff, one that they eventually won four years later in 1919. A famous quote from Hilbert dates to this time:

I do not see that the sex of the candidate is an argument against her admission as Privatdozent [teaching assistant]. After all, we are a university and not a bathing establishment.

Hilbert, never one for respecting authority, used to arrange courses to be taught in his name and got Emmy to deliver them herself, despite her having no actual standing at the University.

It was also around this period that Hilbert was taking a parallel path to Einstein towards the goal of General Relativity. The two men’s relationship was sometimes collaborative, sometimes combative, sometimes even acrimonious, but both recognised the influence that the other’s ideas had on their work. Hilbert wanted expert help and it was to Noether that he turned. One almost immediate result of this was Noether’s Theorem, filling an important gap in General Relativity, but also establishing a broader Scientific principle. This breakthrough work of almost endless influence was carried out by someone whose gender barred them from being a member of the Göttingen staff at the time.

I will neither pursue this biography further here, nor attempt to catalogue Emmy’s many other cutting-edge contributions to Mathematics. I will however add a couple of sad footnotes. In 1933, as part of the Nazi party’s purging of non-Aryan influence from accademia, Noether – like many others with a Jewish background – was dismissed from the position at Göttingen that she had spent so long achieving and cherished so much. Unlike many, she was able to relocate to the United States where she lectured for a short while until her untimely death in 1935 at the age of only 53. One of the great voices of both Mathematics and Science, a voice that had refused to be quieted by the social strictures of her time, finally fell silent. But, while remaining close to unknown to the general public, Emmy’s legacy has resonated ever more loudly in Scientific and Mathematical circles ever since.

It is the component of Emmy’s legacy which relates to both Symmetry and Physics that I plan to cover in the rest of this Chapter. As ever, we first need to lay some groundwork.
 
 
Lights… Camera… Action!

Note: The groundwork I refer to above is predominantly Mathematical apparatus relating to a variety of situations in Physics, e.g. the evolution of N-body systems under either non-relativistic or relativistic conditions, or the behaviour of fields (in the Physics sense of force fields rather than the Mathematical one of algebraic objects). This is not my area of expertise, so I am going to take a 30,000 foot approach with apologies to any Physicists reading (and indeed to the Mathematicians who created the apparatus in the first place, notably Euler and Lagrange).

Our first new concepts here are the Lagrangian of a system, the Action of the same system and the Principle of Least Action. The first two are related inasmuch as the second is the integral of the first between two points in time. The Principle of Least Action, when applied to – for example – a set of particles, enables the equations of motion of the particles to be determined by working out those equations which minimise the Action. Adjustments can allow the same approach to be used in General Relativity (indeed it was Hilbert who discovered this approach) and later Paul Dirac and then Julian Schwinger and Richard Feynman extended the Principle of Least Action into the realm of Quantum Mechanics. It has even been applied in String Theory.

I am not going to get into the details here, for those who are interested, I provide an extremely useful reference in the footnotes ^[4]. Instead I wanted to first provide an analogy that perhaps captures the essence of the area. This is not a novel analogy and indeed appears in many works. I am unclear as to who originated it.

So consider a lifeguard on a beach. Her tower is 10 m from the edge of the water. She spots a swimmer in trouble in the water 10 m from the shore and 20 m to the right of where her tower is situated. The lifeguard can run at 5 ms^-1 on the sand and can swim at 2 ms^-1 in the water. What is the optimum path for her to take to get to the swimmer most quickly? As ever a picture paints a thousand words:

It may seem that the obvious thing is for the lifeguard to move directly towards the swimmer, taking the straight line , which is the shortest distance between points and . However, this has the guard traversing as far in the water as on the land and she goes much slower in the water. Indeed, we can work out the time she takes to reach the swimmer as follows:

Distance travelled on land

Time spent on land

The distance travelled in the water is the same as that on land.

Time spent in the water

Total time

A way to speed things up might be to spend the least possible time in the water. This approach is shown by the dog-leg line . Here are elapsed time calculations are as follows:

Distance travelled on land

Time spent on land

The distance travelled in the water is just

Time spent in the water

Total time

This is a small improvement, but can we do better?

Let’s consider a path which goes through some intermediate point, , forming the dog-leg line . Furthermore, let’s assume that the point is sideways from the lifeguard. That is the length of line is and, by the magic of subtraction, the length of the line is .

Applying Pythagoras we get:

Distance travelled on land

So the time spent on land

The distance travelled in the water

So the time spent in the water

Thus the total time

We want to find for what value of the time is least and so employ the time honoured approach of differentiating the expression. Without going through the gory details, we get:

We get the minimum (or maximum) value of our original expression when its derivative is equal to zero. Again sparing the reader the details, the answer we arrive at is which gives a minimum (and it is a minimum in this case) time of approximately .

Thus the minimum time for the lifeguard to get to the swimmer is a path somewhere in between the one that is of minimum distance and that in which she spends the minimum time in the water. Of interest here is that the way that light is refracted when it moves between two media of different densities (e.g.air and water) can be calculated in precisely the same way. That is – given that light travels at different speeds in different media ^[5] – the angle of refraction may be calculated by determining which angle yields the least time of transit. The lifeguard diagram could be a picture of light being refracted.

So that was meant to be an impressionistic painting of the Principle of Least Action, the elements don’t precisely line up, but hopefully you get the idea. If we now more realistically consider a system of particles and forces acting on them in Classical Mechanics. Then the Principle of Least Action tells us that the equations of motion of the particles are precisely those which minimise the Action of the system. Somehow Nature knows that – of all possible paths that the particles could take – she must select the one that minimises Action ^[6].

Well that’s great, what have we achieved. Well Newton’s approach would be to apply his famous equation many times to all the particles and forces in the system and then solve the resulting (differential) equations. Taking a Least Action approach, we would first of all define the system’s Lagrangian as its total Kinetic Energy (energy due to motion) less its total Potential Energy (energy due to forces ^[7]), noting that both of these will vary as a function of time ^[8]. The Action of the system is then the integral of the Lagrangian between the starting and ending times. We set the expression for the Action that we have calculated to zero and then work from there. This is not magic, we still end up calculating a bunch of (differential) equations. However often we can eliminate complexities inherent in Newton’s approach and we can also judiciously select a generalised coordinate system so as to reduce complexity, which is often a very valuable thing to do.

Here I have referred to Classical Mechanics, for Relativistic Mechanics or Quantum Mechanics there are modifications to the details (e.g.how the Lagrangian is defined and the presence of other functions in equations defining the Action), but the shape of the approach remains the same. The Principle of Least Action has proved to be an invaluable tool for Physicists and Mathematicians alike.

Having hopefully given you some ideas about these concepts, I’ll now move on to the next one, Conservation Laws. After some arduous work, the next section will go much quicker I promise!
 
 
Going Green

In Physics, a Conservation Law is not a regulation about pollution, but instead a statement that some property of a system remains invariant over time. One obvious example is that the total energy of a closed system remains constant; energy is neither created nor destroyed, though it may change form. A somewhat related example is the conservation of linear momentum. If two particles are both travelling with specified speeds and directions and then elastically collide, their post-collision speeds and directions will most likely be radically different. However the total linear momentum of the two particles will be precisely the same post-collision and pre-collision.

However there can be more recherché types of conservation, in the world of Particle Physics, certain types of particle interactions or transformations can conserve strangeness, or spin or parity (whereas other interactions or transformations may not conserve these). A Conservation Law tells us a lot about the system we are studying. In particular it restricts the system’s behaviour – certain things cannot happen as they would break the Conservation Law.

Well I told you that this would be short! The only other concept we need to cover before stating Noether’s Theorem is that of Symmetry. We may have covered some of this already in the preceding 23 Chapters, so let’s plunge in.
 
 
The Amazing Theorem

Emmy’s theorem states the following (omitting some technical language):

For a given system, which has an Action, if there exists a Symmetry of the Action ^[9], then there will exist a related Conservation Law, and vice versa.

A Symmetry of the Action means that the Action is invariant under some transformation of the system. Let’s cover some basic examples:

• If we move the system in some direction and the Action is constant, then this is a spatial translation Symmetry. This implies that Linear Momentum will being conserved in the system.
   
• If a mirror version of the system has the same Action, then this is a spatial reflection Symmetry. This implies that Parity will being conserved in the system.
   
• If the whole system is rotated by an angle and the Action remains the same, this is a rotational Symmetry. In this case, Angular Momentum will be conserved.
   
• If the Action of a system is the same when time is run backwards, then this is a temporal reflection Symmetry and implies that Entropy is conserved ^[10].
   
• If the Action of a system is the same if we jump forward 10 seconds (or back 10 seconds), then this is a temporal translation Symmetry. This implies that Energy is conserved.

Why is this important? Well if we want to understand a system’s behaviour (for example a number of sub-atomic particles interacting) then any observable symmetries will lead us inevitably to some quantity that is conserved. Equally, if we want some new theory to conserve a given quantity, we can formulate Lagrangians that do this and use these to figure out if we are on the right track, e.g. that things remain consistent. Obviously, with the emphasis on symmetry as a fundamental part of Physical Law, we can see why Group Theory is of such interest to Physicists.

This is all thanks to Emmy Noether and her truly fundamental insights. As Brian Greene notes, maybe this remarkable woman should be more celebrated outside of technical circles than she is.
 
 
[Segue to next Chapter – To be completed when Chapter has been decided upon]
 
 

Concepts Introduced in this Chapter Lagrangian A function which captures the essential elements of the dynamics of a system and how it evolves. It thus generally contains information about the position of elements (coordinates) and how these are changing (derivative with respect to time). Action The integral of a Lagrangian between a start and end time. The Action distills down information inherent in the Lagrangain to generate a single Real Number. Principle of Least Action The actual paths taken by elements of a system as the system evolves will be precisely those that give the minimal value for the Action. Conservation Law A statement about physical quantities that remain constant as a system evolves over time. Noether’s Theorem Every differential Symmetry of the Action of a system corresponds one-to-one to a Conservation Law of that system.

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Chapter 24 – Notes

[1]	All of whom were obviously very human. An entirely non-exhaustive list of important early contributors to Group Theory might include (in alphabetical order): Abel Burnside Cartan Cauchy Cayley Frobenius Galois Hermite Jordan Killing Klein Kummer Lagrange Lie Poincaré Schur Sylow In Group Theory, it helps to have a surname that starts with with either a ‘C’ or a ‘K’.
[2]	For example: Emmy Noether’s Wonderful Theorem by Dwight E. Neuenschwander Emmy Noether: The Mother of Modern Algebra by M. B. W. Tent The Washington Post also provides a brief overview for those with less time on their hands. As do those twin bastions of the Scientific Community, Nature and Science.
[3]	These include: Fearful Symmetry by A. Zee. The Honors Class: Hilbert’s Problems and Their Solvers by B. Yandell.
[4]	A wonderful introduction to the area (though admittedly one that becomes more technical as it progresses) was given by Feynman himself in his famous lecture series at the California Institute of Technology.
[5]	The speed of light is only invariant when travelling through a media with a constant refractive index and is typically quoted for a vacuum.
[6]	It is not quite as mystical as that, indeed the link to Feynman’s lecture on the subject (see Note 4) above provides a proof of why this happens.
[7]	Readers are no doubt familiar with Potential Energy relating to Gravity, but the concept applies to any forces.
[8]	I should probably state that Kinetic Energy minus Potential energy is a Lagrangian, not the Lagrangian, other functions may be chosen, so long as they adhere to required properties.
[9]	Technically a differentiable Symmetry. Astute readers may begin to think of Lie Groups at this point.
[10]	And, given that entropy always increases, we have just demonstrated why there is an arrow of time!

Text: © Peter James Thomas 2016-18. Images: © Peter James Thomas 2016-18, unless stated otherwise. Published under a Creative Commons Attribution 4.0 International License.