Emmy Noether’s beautiful theorem

The number of women who have excelled in mathematics is lamentably small. Many reasons may be given, foremost being that the rules of society well into the twentieth century debarred women from any leading role in mathematics and indeed in science. But a handful of women broke through the gender barrier and made major contributions. [TM070: search for “thatsmaths” at irishtimes.com ]

Death of the philosopher Hypatia, by Louis Figuier
[Wikimedia Commons].

Perhaps the earliest was Hypatia, who taught philosophy and astronomy at the Neoplatonic school in Alexandria. In 415 AD she became embroiled in a feud and was murdered by an angry mob. Other noteworthy female mathematicians include Sophie Germain and Sonya Kovalevskaya. Probably the most brilliant of all was Emmy Noether, born in Erlangen, Germany in 1882.

Emmy Noether

Noether was the daughter of a professor of mathematics at the University of Erlangen. She must have learned maths with the help of her father, for she was excluded from access to any higher level teaching. Through personal study and research, she became an expert in the theory of invariants, quantities that retain their value under various transformations.

Emmy Noether, 1882 – 1935

The conservation of energy is a fundamental principle of science. Energy may take different forms and may be converted from one to another, but the total amount of energy remains unchanged. Around 1915, when Albert Einstein was putting the final touches on his theory of general relativity, two mathematicians in Göttingen, David Hilbert and Felix Klein, became concerned about a problem in the theory: energy was not conserved. They felt that, given her knowledge, Noether might be able to solve the problem, so they invited her to come to Göttingen.

She accepted with enthusiasm: Göttingen was the leading mathematical centre and Hilbert the leading mathematician at that time. Hilbert made efforts to persuade the university authorities to hire Noether, but got her only an unpaid teaching post. However, she had greater success, coming up with a truly remarkable theorem that relates conserved quantities and symmetries.

Symmetry and Conservation

It is usually surprising and occasionally delightful when apparently unrelated concepts or quantities are found to be intimately connected. Energy is usually conserved in physical systems. Under certain circumstances, so is angular momentum, roughly the spin of a body. And there are several other conserved quantities.

The mathematical expression that encapsulates the dynamics of a system is called the Lagrangian, after the outstanding French mathematician Joseph Louis Lagrange. If a change of a basic variable, such as the position of the system or a shift of the time origin, leaves the Lagrangian unchanged, we have a symmetry. Noether found a totally unexpected connection between conserved quantities and symmetries of the Lagrangian.

Proving Noether’s Theorem

We can prove the theorem simply in just a few lines. This is not the most general form but it illustrates the process. Let ${L(q_\rho,\dot q_\rho)}$ be the Lagrangian and consider a coordinate transformation depending on a parameter ${s}$ :

$\displaystyle q_\rho(s,t) = q_\rho(0,t) + \delta q_\rho(s,t)$

For example, we could have a simple translation along one coordinate direction ${q_\sigma}$ so that ${q_\rho(s,t) = q_\rho(0,t) + s\,\delta^\sigma_\rho}$ . Suppose the Lagrangian is unchanged for this transformation. Then

$\displaystyle \frac{d L}{d s} = 0 \qquad\mbox{or}\qquad \sum_{\sigma}\left[ \frac{\partial L}{\partial q_\sigma}\frac{d q_\sigma}{d s} + \frac{\partial L}{\partial \dot q_\sigma}\frac{d \dot q_\sigma}{d s} \right] = 0$

The equations of motion are

$\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot q_\sigma} = \frac{\partial L}{\partial q_\sigma}$

Combining these results to remove ${{\partial L}/{\partial q_\sigma}}$ , we get

$\displaystyle \sum_{\sigma}\left[ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot q_\sigma}\right) \frac{d q_\sigma}{d s} + \frac{\partial L}{\partial \dot q_\sigma}\frac{d \dot q_\sigma}{d s} \right] = 0$

which can be written as a total time derivative

$\displaystyle \frac{d}{d t} \sum_{\sigma}\left[ \frac{\partial L}{\partial \dot q_\sigma} \frac{d q_\sigma}{d s} \right] = 0$

Thus we see that the following quantity is conserved following the motion:

$\displaystyle N = \sum_{\sigma}\left[ \frac{\partial L}{\partial \dot q_\sigma} \frac{d q_\sigma}{d s} \right]$

This is the constant corresponding to the symmetry. For a much more general treatment see for example ( Neuenschwander, 2011).

Noether’s Theorem does much more than simply establish a relationship between symmetries and conserved quantities. It provides an explicit formula by means of which, knowing a symmetry, we can actually calculate an expression for the quantity that is conserved. Moreover, the theorem was not confined to the classical mechanics of Newton, but found its true potential when used in the context of quantum mechanics.

Abstract Algebra

Emmy Noether has been called “the mother of modern algebra.” The renowned algebraist Saunders MacLane wrote that abstract algebra started with Noether’s 1921 paper “Ideal Theory in Rings” and Hermann Weyl said that she “changed the face of algebra by her work.” Noether’s algebraic work was truly ground-breaking and hugely influential.

With the rise of the National Socialist Party, Noether, along with many others, was dismissed in 1933 from Göttingen. She emigrated to America, taking a position at Bryn Mawr College in Pennsylvania. Sadly, she died just two years later at the height of her creative powers. Her remarkable standing and reputation can be seen from the obituary written by Einstein in The New York Times: “Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”

Sources:

Neuenschwander, Dwight E., 2011: Emmy Noether’s Wonderful Theorem. Johns Hopkins Univ. Press., 243pp. ISBN: 978-0-8018-9694-1.

Kostmann-Schwarzbach, Yvette, 2011: The Noether Theorems. Springer, 205pp. ISBN: 978-0-3878-7867-6.

Emmy Noether: Her Life, Work and Influence. Video from Perimeter Institute.

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