Noether’s Challenges

Amalie Emmy Noether entered the world in 1882 in Germany, a time when women were barred from attending mathematics classes, which made her academic journey quite challenging since this was her passion.

Fortunately, her father was a mathematics professor at the University of Erlangen, which provided her a pathway to pursue her studies. After obtaining her doctorate in 1907 at the age of 25, she spent seven years working at the institute without a salary.

In 1915, she received an invitation from David Hilbert and Felix Klein to teach at the University of Göttingen. However, many faculty members opposed the idea of a woman instructing male students.

Hilbert staunchly defended her, famously stating:

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

Once Hilbert persuaded the faculty of her capabilities, Noether was finally able to teach in Göttingen. The initial years were incredibly tough, as she lacked an official position and did not receive any pay.

Moreover, her lectures were often credited to Hilbert. It wasn't until 1922 that she earned the title of “professor with limited internal administrative rights and functions”, which still came without any remuneration.

Despite her well-recognized talent, the obstacles she faced were immense. In 1916, she proved a vital theorem, now known as Noether’s theorem, which is pivotal in physics; while many mathematicians recognize her contributions to abstract algebra, physicists often acknowledge her for this theorem.

We’ll delve into the theorem shortly, highlighting some intriguing examples along the way.

The Enigmatic Universe

The year 1915 marked a pivotal moment in physics, as Einstein unveiled his general theory of relativity. Hilbert, a leading mathematician of the era, grasped this theory's implications thoroughly.

However, there appeared to be a discrepancy within the mathematical framework of the theory, revealing a potential violation of a fundamental physical principle—the conservation of energy.

Conservation laws are foundational in physics, but they stem from a deeper reality—Noether’s theorem! One reason these laws hold such significance is their ability to help solve otherwise insurmountable problems, making them seem almost magical.

So, what exactly is a conservation law?

According to Wikipedia:

> In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time.

In terms of energy, this means that in a “closed” system, the total energy remains constant. This is intuitive; energy can be transferred from one object to another, but the overall amount in the system remains unchanged.

For instance, consider a billiard ball striking another on a pool table, transferring its energy and setting the second ball into motion.

In this process, the “lost” energy from the first ball manifests as kinetic energy in the second ball, along with friction and heat from the table.

According to the laws of thermodynamics, energy can transform from one type to another but only in one direction (if we consider the overall increase in entropy). Thus, certain forms of energy are deemed “finer” and of higher “order” than others.

In the previous example, some of the kinetic energy converts to heat, but you won’t witness a “warm” ball suddenly take off like a rocket, reversing this process!

Nonetheless, the total energy within the system should remain constant, even if it changes forms, resulting in increased entropy (disorder). Therefore, energy conservation means energy cannot be created or destroyed.

When discussing energy, we also refer to mass, as Einstein demonstrated that energy can convert into mass, implying that total energy encompasses total mass-energy.

Physicists were aware of this conservation principle when Einstein introduced his theory in 1915, which accurately predicted that space itself could change shape, thereby dispelling the Newtonian static universe that had dominated physics for three centuries.

Einstein's theory posited a dynamic Universe, where the very fabric of spacetime could bend and stretch like a rubber sheet.

This posed a challenge for Hilbert and his contemporaries, as the Universe’s expansion would stretch light traveling through it.

This phenomenon, known as redshift, means the light's wavelength increases, causing a decrease in photon energy—so where does that energy go?

The conservation principle seemed compromised!

Einstein's equations hinted at other scenarios where energy conservation was not upheld.

This discrepancy was precisely why Hilbert and Klein sought Noether's expertise. She was well-versed in abstract algebra and group actions, and Hilbert believed she held the key to this mystery.

Noether not only clarified this apparent inconsistency but also revealed a more fundamental layer of nature underpinning every conservation law in physics!

Her theorem can be encapsulated as follows: Noether’s Theorem: ================================ > For every continuous symmetry of the Universe, there exists a conserved quantity.

To grasp this result, we first need to explore the concept of symmetry.

Symmetry

What is symmetry? The simplest definition is:

A symmetry is an action that transforms an object such that it appears or behaves identically after the transformation.

For example, an equilateral triangle has rotational symmetry when rotated by 120 degrees, while a playing card exhibits reflectional symmetry when flipped.

Continuous symmetry refers to a symmetry where the action can vary in magnitude. A circle displays continuous rotational symmetry because it can be rotated by any angle without changing its appearance. A line possesses continuous translational symmetry since it can shift along its direction without alteration (it remains the same line).

In physics, symmetry pertains to the equations that describe motion laws for a given system.

In the billiard balls example, the balls experience the same forces (gravity and friction) regardless of their positions on the table, indicating a translational symmetry for that system.

According to Noether’s theorem, there exists a conserved quantity for the system, which is momentum. When two balls collide, their total momentum remains unchanged.

However, if we play pool on a hilly surface, momentum may not be conserved, as it could be “lost” to the gravitational field. This stems from the absence of translational symmetry on such a surface, as the gravitational field's direction shifts relative to the hilly pool table.

Despite not altering the overall shape of the hilly table, the forces (the motion equations) remain unchanged over time. Therefore, as long as the table's shape is fixed, we maintain translational symmetry in time.

By Noether’s theorem, a conserved quantity exists, and that quantity is energy. Thus, energy conservation reflects the system's continuous symmetry over time.

For example, a satellite orbiting a planet experiences identical forces throughout its orbit, showcasing rotational symmetry. The conserved quantity in this scenario is angular momentum.

Noether’s theorem illustrates that conservation and motion symmetry are fundamentally intertwined—two facets of the same concept.

Her elegant theorem elucidated the apparent breach of energy conservation that baffled Hilbert and Einstein. This occurs because when space undergoes deformation, the continuous symmetry over time is disrupted.

Consequently, energy can be lost (for instance, through cosmological redshift) and created from nothing (such as the intriguing notion of dark energy).

A Closer Examination

Thus, conservation laws (or their violations) arise from Noether’s theorem and the various symmetries (and their disruptions) present in our Universe.

Is there a more foundational principle underlying Noether’s theorem? Indeed, the answer is yes, and it is known as the principle of least action.

The principle of least action postulates that the Universe always selects the path between two states that minimizes changes in action.

This law is profoundly significant in physics, applied across various domains, including relativity and quantum mechanics.

You may have heard that light always opts for the path that minimizes travel time. The principle of least action generalizes this notion to encompass paths through any system evolving between two states, whether macroscopic or quantum.

This principle stands as one of the most fundamental laws of physics.

Similar to axioms in mathematics that we accept as true, this principle serves as a physical axiom, with Noether’s theorem depending upon it.

This implies that we can employ the theorem to identify conserved quantities for any system evolving according to the principle of least action, provided we can ascertain that system's symmetries.

Consequently, it applies across diverse fields of physics and various scales of reality. Symmetry is omnipresent.

For instance, we know that electric charge and current are conserved quantities. Noether’s theorem allows us to infer that a corresponding symmetry of nature must exist. This symmetry is revealed as the phase of the quantum field.

Remarkably, one can rotate the phase of oscillation in a quantum field by any degree, demonstrating continuous rotational symmetry that preserves the field's observable properties, such as its particles (electrons, in this case). This symmetry leads to the conservation of electric charge.

This is merely one example within a vast array of symmetries known as gauge symmetries. Each of these symmetries corresponds to a conserved charge responsible for interactions among fundamental particles in the standard model.

This marks the inception of gauge theory.

Gauge theory elucidates how our forces and associated particles interact, with group theory (mathematical symmetry) and Noether’s theorem operating beneath the surface.

Final Thoughts

In this discussion, I aimed to present the theory in an accessible manner, intentionally omitting technical details that readers may wish to explore further.

To truly grasp Noether’s theorem, one must delve into differentiable symmetries, Lie groups of transformations adhering to specific technical criteria, time integrals of Lagrangians, stress-energy tensors, and more. However, I hope this overview provides insight into this remarkable theory.

Most contemporary physicists believe the Universe will continue to expand at an accelerating pace due to the repulsive force of dark energy. This energy is permitted to exist and grow by the natural laws described by Noether’s theorem.

Ultimately, it may lead to the disintegration of everything—from planets and stars to molecules and atoms—resulting in a Universe characterized by extreme entropy and minimal activity!

Emmy Noether ranks among the greatest mathematicians in history. Throughout her career, she focused on advancing the mathematical field of ring theory and other aspects of abstract algebra, which is now considered by some as the “language of modern mathematics.”

Although her time spent on physics was limited, its impact was monumental.

Einstein himself remarked about her:

> “In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant mathematical genius thus far produced since the higher education of women began.” > — Albert Einstein.

The esteemed physicist Hermann Weyl expressed:

> “I was ashamed to occupy such a preferred position beside her whom I knew to be my superior as a mathematician in many respects.” > — Hermann Weyl.

Noether serves as a genuine inspiration for all individuals battling against an unjust system and for those who dare to think unconventionally. She was known to allow her students to take credit for her ideas, facilitating their self-promotion.

We should all aspire to embody her warmth and brilliance.