Do you know about Emmy Noether? The legendary mathematician was praised as “creative” by Einstein. A German mathematician who faced many struggles to learn mathematics, she was able to change our understanding of the Universe—and even make Einstein wonder. Let me explain.

Amalie Emmy Noether, born in 1882 in Erlangen, Germany. Her father, Max Noether, was a mathematician. She was first qualified as a language teacher. But mathematics didn’t leave her; it held her tightly.

She wanted to learn mathematics but faced immediate obstacles. German universities did not admit women students at that time. She asked individual professors for permission to attend their lectures, but she didn’t get any credits for those courses. Despite these barriers, she received a doctorate in mathematics with research on algebraic invariants.

Even though she received her doctorate, German universities didn’t allow her to get a professorship. Instead, she worked as an assistant to her father, Max Noether. Her brilliance was undeniable, but institutional sexism didn’t allow her to get formal academic positions.

And in 1915, David Hilbert and Felix Klein invited Noether to a prestigious mathematics center. They wanted her expertise in invariant theory to solve problems in General Relativity. The university administration and faculty members opposed giving her an academic position. But Hilbert argued:

“I do not see that the sex of the candidate is an argument against her admission. After all, we are a university, not a bathhouse.”

Despite his advocacy, Noether worked unpaid for years. Sometimes she took classes under the name of Hilbert. After Noether came up with a revolutionary theorem, Einstein recognized its importance and wrote:

“Yesterday I received from Miss Noether a very interesting paper on invariants. I’m impressed that such things can be understood in so general a way.”

What is that theorem? What did Einstein wonder about?

I first heard about Emmy Noether at a conference—that was the first time I heard about her. Later, in a Veritasium video titled “The Biggest Misconception in Physics”, they argued how the symmetry of the Universe results in conservation laws and explained why energy is not conserved. Later, I read about her and tried to read her research paper on the symmetries of the Universe. It took me more than three months to focus on and read the paper. It is an interesting paper. I recommend you read it.

Invariante Variationsprobleme

Before discussing the research paper, I will explain symmetry. By definition, the symmetry of a physical system is a physical or mathematical feature of the system that is preserved or remains unchanged under some transformation. If you perform an experiment in one place and then change the location or orientation and the results remain the same, then it has symmetry—meaning the laws of physics are the same in different locations or orientations. But if the laws of physics change, then we have asymmetry.

When I read the book The God Equation by Michio Kaku, I found a better example to explain symmetry. Think of a dam filled with water. If you rotate the water in the reservoir, the water remains the same—that’s symmetry. The water is in a stable state. If the dam bursts or is opened, then the symmetry is gone. The water will find its true state of rest in the deep valley or sea. He used this example to discuss the Higgs boson and the state of our Universe. I think this example suits well.

Now let’s explore the depth of Emmy Noether’s 1918 treatise. Noether begins by narrowing the scope of variational problems to those that admit a continuous group of transformations. In this, she distinguishes two types of groups. One is finite continuous groups, discussing transformations depending on a finite number of parameters. The second is infinite continuous groups, discussing transformations depending on arbitrary functions and their derivatives. The fundamental question of this article is: if an integral $I$ remains invariant under these transformations, what must be true about the underlying differential equations?

Where

$I = \int \dots \int f\left(x, u, \frac{\partial u}{\partial x}, \frac{\partial^{2} u}{\partial x^{2}}, \dots\right) dx = \int \dots \int f\left(y, v, \frac{\partial v}{\partial y}, \frac{\partial^{2} v}{\partial y^{2}}, \dots\right) dy$

represents the physical action.

Noether’s First Theorem

This theorem firmly argues how conservation laws are born, and it shows how the system works for finite groups. The statement of the first theorem is:

If the integral $I$ is invariant with respect to a finite continuous group $\mathfrak{G}_{\rho}$ (depending on $\rho$ parameters), then $\rho$ linearly independent combinations of the Lagrange expressions becomes divergences.

For a system to have a conservation law, there must be a quantity whose flux through a boundary is zero—meaning “what flows in must flow out.” If an integral $I$ is invariant under a group, $\rho$ linearly independent combinations of the Lagrange expressions result in divergences. The linear combination was written as

$\sum\psi_i \bar{\delta} u_i = Div (B)$

Where $\psi_{i}$ are the Lagrange expressions and $B$ represents the conserved quantity.

Mathematically, conservation laws are represented by

Noether’s Second Theorem

This theorem argues the idea of infinite continuous groups. The statement is:

If the integral $I$ is invariant with respect to an infinite continuous group $\mathfrak{G}_{\infty\rho}$ (depending on $\rho$arbitrary functions and their derivatives up to the $\sigma$-th order), then there subsist $\rho$ identity relationships between the Lagrange expressions and their derivatives up to the $\sigma$-th order.

This theorem proves that the arbitrary functions $p(x)$ in the Lagrange equations are actually consequences of the remaining equations of the Lagrange expressions. Noether identified this as a generalization of the General Theory of Relativity, explaining n dependencies between Lagrange expressions and their first derivatives found in that theory. The converse of both theorems works well.

These theorems worked well and were able to solve problems in General Relativity. Before these theorems, Hilbert found that energy in General Relativity is not conserved. Energy is conserved only in an empty Universe, but he couldn’t provide a good reason for that—perhaps he thought it was a wrong result. Noether’s theorem, together with Hilbert’s formulation of general relativity, suggests that in a universe filled with mass and dark energy, global time-translation symmetry is broken. This leads to time asymmetry and the absence of a globally conserved energy.
Noether’s ideas gave a major upgrade to conservation laws and solved a great confusion in the early days of General Relativity.

Noether clarified this by distinguishing proper and improper conservation laws. Proper conservation laws arise from finite groups, where conservation is a physical necessity. Improper laws arise from infinite groups, where the conserved quantities are actually linear combinations of the field equations themselves. Noether’s proofs involved sophisticated mathematics from differential geometry and the calculus of variations.

Later, they found the conserved quantities and their respective symmetries.

Time translation symmetry results in energy conservation

Space translation symmetry results in momentum conservation

Rotational symmetry results in angular momentum conservation

Gauge symmetry results in charge conservation

This revelation changed physics. Conservation laws were no longer arbitrary facts requiring experimental verification. They were mathematical necessities flowing from the symmetries of nature.

When I first heard about this article, I thought someone was defying the law of conservation of energy and shared the article on Reddit. One person said, “That is not defying the law of conservation; that’s the best explanation of the law of conservation of energy.” After I completed the article, I felt the same. It is a great explanation of conservation laws—the best explanation.

Mathematics truly deserves Noether, the great woman mathematician. She faced many barriers to reach this level. She lost her job because of the Nazi government of Germany, which she got after a lot of effort. But she didn’t stop teaching.

If God is eternal indestructible energy, then he exists in an empty Universe, which may contradict His powers and existence.

References

• Invariante Variationsprobleme – Arxiv
• The Biggest misconception in Physics – Veritasium
• Emmy Noether’s Theorem: How Symmetry Creates Conservation Laws – KroneckerWallis
• The 100-year-old symmetry theorem that is still changing physics today – New Scientist
• How Noether’s Theorem Revolutionized Physics – Quanta Magazine