# The Intriguing Relationship Between Conservation Laws and Symmetries

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## Chapter 1: The Essence of Conservation Laws

Can energy truly be created or lost within our Universe? This question leads us to explore a remarkable story about a woman who defied the odds to become one of the most renowned mathematicians of her era. She developed many modern theories that form the foundation of abstract algebra today. Her groundbreaking theorem elucidates the ultimate fate of our Universe, addressing a conundrum that stumped both Hilbert and Einstein!

Her extraordinary findings reveal the fundamental principles that govern nature, spanning from general relativity to the intricacies of quantum physics.

### Noether's Journey

Amalie Emmy Noether was born in 1882 in Germany, a time when women were largely excluded from mathematics classes. This challenge was particularly daunting for young Emmy, who was passionate about the subject. Fortunately, her father was a mathematics professor at the University of Erlangen, which provided her with the opportunity to study mathematics. After earning her doctorate in 1907 at the age of 25, she worked at the institute for seven years without pay.

In 1915, she was invited to teach at the University of Göttingen by the eminent David Hilbert and Felix Klein, although many of their colleagues opposed the idea of a woman instructing men. Hilbert famously defended her, 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 advocated for her talent, Noether was finally able to teach. However, her early years at the University were fraught with difficulties. She lacked an official position, was unpaid, and her lectures were often attributed to Hilbert. It wasn't until 1922 that she received the title of "professor with limited internal administrative rights and functions," yet still without compensation. At that time, her brilliance was widely recognized, and her talent was unquestioned.

A pivotal moment arrived in 1916 when she established a vital result: Noether's theorem, a fundamental principle in physics. While many mathematicians recognize her contributions to abstract algebra, physicists often remember her for this theorem.

### Understanding the Universe

The year 1915 marked a significant milestone in physics with Einstein's publication of his general theory of relativity. Hilbert, a leading mathematician, grasped the theory's implications but noticed inconsistencies within its mathematical framework. Specifically, it appeared to violate the principle of conservation of energy—a cornerstone of physics.

Conservation laws are among the most essential principles in physics, yet they arise from a deeper truth encapsulated in Noether's theorem. These laws allow us to solve otherwise insurmountable problems, making them seem almost magical.

But what exactly is a conservation law? According to Wikipedia, it states that a measurable property of an isolated physical system remains unchanged as the system evolves over time. For energy, this means that in a closed system, the total energy is conserved.

For instance, consider a billiard ball striking another on a pool table. The striking ball transfers its energy to the target ball, setting it in motion. While the source ball 'loses' energy, it is transformed into kinetic energy in the other ball and dissipated as heat and friction.

As per the laws of thermodynamics, energy can shift from one form to another but can never be created or destroyed. The conservation of energy law asserts that the total energy remains constant over time, even if it changes form, leading to increased entropy within the system.

This law implies that energy and mass are interchangeable, as demonstrated by Einstein's theory. In 1915, physicists were already aware of this conservation principle, which also revealed that space itself could warp and change, contradicting the long-held Newtonian view of a static universe.

Einstein's theory proposed a dynamic universe where space (or spacetime) could expand. This posed a challenge for Hilbert and his peers, as the stretching of light traveling through expanding space resulted in a phenomenon known as redshift. This process increased light's wavelength, decreasing the energy of photons and raising questions about the conservation of energy.

Thus, Hilbert and Klein turned to Noether, whose expertise in abstract algebra and group actions made her the ideal candidate to tackle this mystery. Noether not only clarified this apparent inconsistency but also unveiled a profound truth: every conservation law in physics stems from an underlying fundamental theory.

Her theorem states:

**Noether's Theorem**: For every continuous symmetry of the Universe, there exists a conserved quantity.

To grasp this result, we need to delve into the concept of symmetry.

### Symmetry Explained

What do we mean by symmetry? Simply put, symmetry refers to a transformation that results in an object appearing unchanged. For example, an equilateral triangle exhibits rotational symmetry when rotated by 120 degrees, while a playing card has reflectional symmetry when flipped.

Continuous symmetry allows for transformations of any degree. A circle, for instance, possesses continuous rotational symmetry, as it looks the same regardless of how much it is rotated. Similarly, a line exhibits continuous translational symmetry because it can be shifted along its length without altering its identity.

In physics, symmetry pertains to the equations governing the motion of a system. Returning to our billiard ball example, the balls experience identical forces (gravity and friction) regardless of their positions on a flat table, demonstrating translational symmetry for that system.

According to Noether's theorem, a conserved quantity corresponds to this symmetry—momentum. In a collision between two balls, their total momentum remains constant. However, on a hilly surface, momentum might not be conserved due to gravitational influences, indicating a lack of translational symmetry.

Despite the unchanged shape of the hilly surface, the forces acting upon it remain consistent over time, establishing translational symmetry in time. Noether's theorem thus indicates that energy is conserved, reflecting the system's continuous symmetry over time.

For example, a satellite in orbit around a planet experiences the same forces, irrespective of its location in orbit, signifying rotational symmetry, where the conserved quantity is angular momentum.

Noether's theorem elegantly illustrates that conservation laws and motion symmetries are intertwined—two aspects of the same concept. Moreover, her theorem elucidates the apparent violation of energy conservation. When space undergoes deformation, continuous temporal symmetry is disrupted, allowing energy to be lost (as in cosmological redshift) or created from 'nothing' (such as dark energy).

### A Profound Insight

Conservation laws—and their potential breakdowns—derive from Noether's theorem and the various symmetries present in our Universe. But does Noether's theorem rest on an even more fundamental principle? Indeed, it does, known as the principle of least action.

This principle posits that the Universe selects the path that minimizes action between two states. It is a profound law that underpins much of physics, including relativity and quantum mechanics.

You may have heard that light travels along the path of least time; the principle of least action generalizes this concept to any evolving system, whether macroscopic or quantum mechanical.

This principle serves as a foundational axiom in physics, upon which Noether's theorem is built. It enables us to identify conserved quantities for any system conforming to this principle, provided we can determine its symmetries.

This universality extends across various fields of physics and scales of reality, revealing that symmetries are pervasive. For instance, electric charge and current are conserved quantities, which, according to Noether's theorem, indicate an underlying symmetry—specifically, the phase of a quantum field.

You can rotate the phase of a quantum field's oscillation by any amount, exhibiting continuous rotational symmetry that preserves observable properties, such as electrons. This symmetry gives rise to the conservation of electric charge, representing just one of many 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, which describes the interactions between forces and their corresponding particles, underpinned by group theory and Noether's theorem.

### Final Thoughts

In this discussion, I aimed to present the theory in an accessible manner, intentionally omitting certain technical aspects that interested readers may wish to explore further. To truly grasp Noether's theorem, one would need to delve into differentiable symmetries, Lie groups, Lagrangian mechanics, and stress-energy tensors, among other topics. However, I hope this overview provides insight into this remarkable theory.

Most contemporary physicists predict that the Universe will continue to expand at an accelerating rate due to the repulsive force of dark energy. This energy's existence and increase are permitted by nature, all thanks to Noether's theorem. Ultimately, the Universe may face a fate where everything—planets, stars, molecules, and even atomic nuclei—will be torn apart, resulting in a state of high entropy and minimal structure.

Emmy Noether stands as one of the greatest mathematicians in history, dedicating her career to advancing ring theory and other realms of abstract algebra, now regarded as the "language of modern mathematics." Despite her limited time in physics, her contributions were profound.

Einstein remarked,

"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.

Similarly, physicist Hermann Weyl stated,

"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 an inspiring figure for those battling against unfair systems and for anyone who enjoys thinking outside the box. She was known for allowing her students to take credit for her ideas, facilitating their advancement. Let us all aspire to embody her warmth and brilliance.

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Thanks for your attention.

The first video, "Symmetries & Conservation Laws: A (Physics) Love Story," explores the intricate relationship between symmetries and conservation laws in physics.

The second video, "The Most Beautiful Result in Classical Mechanics," delves into the significance of conservation laws and how they shape our understanding of classical mechanics.