Symmetry is something very intuitive to humans, as it appears to be rooted in our brains. For example, it is very straightforward for us to spot which one of the tree shapes shown below is symmetric. In this post, we'll explore how the seemingly intuitive symmetry leads to fundamental physical laws.

So, what exactly is symmetry? The term "symmetry" originates from the Greek word summetria, describing a relation of commensurability. Early on, symmetry was linked closely with harmony, beauty, and unity, which became very important in theories about nature. For example, in Plato's Timaeus, shapes like regular polyhedra (tetrahedron, cube, & dodecahedron) were seen to represent the natural elements (fire, earth, & the universe) because of the beauty and proportion of their shapes. Similar ideas continued into science, like in Kepler's 1596 Mysterium Cosmographicum, which used the five regular solids to explain planetary architecture.

From a modern science perspective, symmetry does not have to do with proportions; rather, it is defined in terms of the invariance of any object under specific mathematical operations, such as reflection, rotation, or translation. Thus, a shape/object is symmetric if it remains unchanged under certain operations, i.e., it is not possible to distinguish the object before and after performing the operation. The possible types of symmetries for a geometric object depend on which properties of the object remain unchanged after a transformation. Depending on the movement of the objects, there are different types of symmetry operations:

1. Point symmetry

   • Rotation -- The object appears unaffected when rotated by an angle $\alpha$ about a selected direction. The axis of rotation, known as the symmetry element, is referred to as the n-fold symmetry axis with n = 360 / $\alpha$. For example, the snowflake shown below appears the same when rotated by an angle 60$^\circ$, i.e., it exhibits 6-fold rotational symmetry.

   • Reflection -- The object appears unchanged when reflected across a specific plane, known as the mirror plane. In other words, one half of the object is a mirror image of the other half. For example, a butterfly has mirror symmetry because its left and right wings are mirror images of each other.

   • Inversion -- An object to have inversion symmetry, each of its points $(x,y,z)$ corresponds to a point $(-x,-y,-z)$ such that the object looks the same when viewed from the center, known as the center of inversion.

   • Combination of any above.

     

     Shown here, a snowflake that will appear the same if rotated by 60 deg. about an axis passing through the center pointing in a perpendicular direction to the plane of the object. Thus, it exhibits a 6-fold rotational symmetry. Additionally, it possesses six mirror planes (one illustrated) and a center of inversion symmetry. Credit: Wikimedia

2. Space symmetry

   • Translation -- An object appears the same after being translated (shifted) by a certain amount in space. This symmetry is widely employed in physics, especially in crystallography and quantum mechanics. The simplest example is the fact that the laws of physics are the same if you are in Chennai or Delhi.

   • Combination of translation and any kind of point symmetry operations.

In physics, there are additional types of symmetries beyond reflections, rotations, and translations, which include continuous, discrete, and Gauge symmetries that are not part of our usual experiences. Continuous symmetry, such as the rotation of a circle, can be described by Lie groups, while discrete symmetries, like the rotation of a rectangle, are described by symmetry group. Gauge symmetry is central to quantum field theory and the Standard Model of particle physics, which describes how certain physical laws remain invariant under local transformations of fields.

Symmetry and Conservation laws: Noether's Theorem

Conservation laws are not only fundamental principles in physics but steer practical applications across all disciplines of physics & engineering. These laws state that certain quantities -- quite importantly -- remain constant within a closed system, regardless of the processes occurring within that system. Laws like the "conservation of energy," "conservation of momentum," and "conservation of angular momentum" are crucial in theories including Newtonian mechanics, thermodynamics, and quantum physics, among others. Such laws apply to phenomena ranging from planetary motion all the way to nuclear reactions occurring in the world's powerful nuclear reactors or stars. These evident conservation laws are not merely empirical; rather, they are a direct consequence of specific symmetries intrinsic to the theories. The unavoidable connection between symmetry and conserved quantities is one of the most profound physical insights in science history and is mandated by Noether's Theorem.

A young woman named Emmy Noether summarized the content of her two theorems, published earlier in 1915, for presentation to the German Mathematical Society on July 23, 1918. Being too young and a woman (a disallowed gender to speak in sessions of the Royal Academy at that time), she wasn't allowed to present the work herself. So the renowned mathematician Felix Klein presented her results. These results, for the first time, established that if you have a physical system:

• with conservative forces (equal and opposite), and

• whose "action" has continuous symmetries,

then every independent continuous symmetry will lead to an associated "conserved quantity."

At left, a picture of Amalie Emmy Noether is shown, who proved Noether's theorem connecting the symmetries of a physical system with an associated conservative law. On the right is an illustration of how our sun and earth wrap space and time (or spacetime). In his theory of general relativity, Albert Einstein revealed that the Earth's revolution around the Sun is not because of gravitational pull; rather, the Earth falls freely through curved space wrapped due to the dominating mass of the Sun. Credit: Mathematical Association of America, T. Pyle/Caltech/MIT/LIGO Lab

In other words, Noether's theorem establishes that if a physical system possesses symmetry, there must be a corresponding physical quantity associated with the system that will remain conserved. Here are some scenarios validating the theorem:

• If a system has time translation symmetry, i.e., the laws of physics do not change over time, then the system's total energy will remain conserved. A pendulum movement in a frictionless environment is an example of energy conservation.

• If a system is symmetric under rotations, irrespective of its orientation, then this leads to the conservation of angular momentum in the system.

• If your system is space-translation invariant, meaning that laws of physics are the same everywhere in space, then it leads to the conservation of momentum.

Noether's theorem is one of the most profound insights in all of physics. Consider the conservation of energy. Historically, mechanics evolved through trial and error, with scientists making educated guesses about which quantities might be useful to measure and which might be conserved. Even the very fundamental law of conservation of energy was initially an empirical observation -- a useful construct but its clear origin was not comprehended. For example, the idea that potential and kinetic energy could be converted into each other when objects fall or rise was largely recognized because it was useful. There was no known fundamental reason that energy should be conserved; it was just observed that if one properly accounted for all the different forms of energy, one couldn't create or destroy it, only transform it from one form to another. Noether's theorem, however, reveals that energy conservation stems from a plausible origin: the laws of nature should be independent of time. This means that what seemed like empirical regularities can actually be derived from symmetry principles.

In 1918, an intriguing problem attracted the attention of mathematicians, just a few years after Einstein's General Relativity revolutionized our understanding of gravitation, spacetime, and the Universe. Mathematicians and physicists began to puzzle over a curious issue: General Relativity did not inherently include energy conservation as a natural consequence of spacetime itself; instead, it appeared only as an identity. This raised significant questions, particularly for David Hilbert, one of the leading mathematicians of the time. He wondered how it was possible to constrain physical quantities, like energy, within the framework of General Relativity.

The resolution to this problem lies in the nature of spacetime symmetry. Energy conservation is tied to a specific symmetry known as time-translation invariance, which means the laws of physics remain unchanged over time. However, this invariance only holds in static spacetimes, where spacetime itself does not change as time progresses. For example, in the spacetime around a non-rotating black hole, which is static, energy is conserved because time-translation invariance is preserved. In contrast, our Universe is not static; it is expanding. The description of this expansion is given by the FLRW spacetime, which changes over time -- either expanding or contracting. In such a non-static spacetime, time-translation invariance is absent, and consequently, the associated conservation law for energy does not apply. To understand this, imagine a photon -- a quantum of light -- traveling between two galaxies. In static spacetime, the photon's wavelength and frequency would remain unchanged as it moves from one point to another, preserving its energy. However, in our expanding Universe, the space between the galaxies stretches as the photon travels, causing its wavelength to increase and its energy to decrease. This lost energy does not transfer elsewhere; it simply reflects the fact that, in an expanding spacetime, there is no requirement for energy conservation due to the lack of time-translation invariance.

Thus, even one of the most fundamental laws -- the conservation of energy -- is not upheld in an expanding Universe. One must, therefore, be cautious to consider additional symmetries or hypothesize conservation laws that feel intuitively right but may not accurately reflect the true nature of the Universe. A deep understanding of the fundamental structure of a theory and its valid implication is crucial to discerning what is truly conserved in reality.

Symmetry Breaking

The study of symmetry breaking dates back to Pierre Curie, according to whom, for a phenomenon to occur in a medium, the medium's original symmetry must be broken. The breaking of symmetry doesn't mean that the system loses all symmetry; rather, some of the original symmetries are no longer present, and the system's present symmetry is reduced. Thus, a system transitions from a state of higher symmetry to one of lower symmetry, resulting in the emergence of new phenomena. For example, in particle physics, symmetry breaking explains how particles acquire mass through mechanisms like the Higgs field, while in condensed matter systems, it underpins phase transitions such as the formation of crystals. Symmetry breaking can be distinguished into two forms: explicit and spontaneous.

Explicit symmetry breaking indicates a circumstance when the system's dynamical equations -- Lagrangian/Hamiltonian formulations -- are not invariant under considered symmetry. This means that the system's governing equations contain some terms that explicitly break the symmetry. Such terms are imposed externally, for example, by applying a magnetic field to a ferromagnet that induces a favorable alignment along the applied field, breaking the symmetry of the system.

Spontaneous symmetry breaking indicates a situation where, given a symmetry of equation of motion, the solutions are not invariant under the action of this symmetry. In other words, the system naturally settles into a state opposing the symmetry of its underlying laws and hence the attribute "spontaneous." A simple analogy to explain spontaneous symmetry breaking was provided by Abdus Salam -- Imagine a dinner party where guests are seated around a circular table, with a napkin placed between each pair of neighbors. The table is perfectly symmetrical, as every guest has a napkin on both sides. Now, if one guest decides to pick up the napkin on their left or right, this results in the breaking of symmetry. Now, the other guests know to do the same, and the left-right symmetry is spontaneously broken. A large number of physical systems exhibit such spontaneous symmetry-breaking phenomenon that includes a second-order phase transition in ferromagnets occurring due to the discontinuous change in order parameter (magnetization), the appearance of a massless Nambu-Goldstone boson because of the spontaneous breaking of global gauge symmetry, and many more.

In condensed matter physics, spontaneous symmetry breaking is ubiquitous and often occurs when there is a phase transition from a high-temperature, symmetric phase to a low-temperature one in which the symmetry is spontaneously broken. One simplest example is the freezing of water. Observing a round bowl of water kept on a table, one notices that it looks the same in every direction; it has rotational symmetry. However, after freezing, the ice crystals form in specific orientations, breaking the rotational symmetry.

One classic example of spontaneously broken symmetries in physics is the temperature-dependent behavior of ferromagnetic materials that exhibit net magnetization in the absence of an external magnetic field. Pierre Curie discovered that these materials lose their magnetic properties above a specific temperature, the so-called critical temperature $T_c$ or the Curie point, giving rise to a transition from ordered to disordered state. The value of $T_c$ varies for different materials. In a ferromagnet, atoms interact through a spin-spin interaction: $H = -J \sum_{ij} \mathbf{S}_i\cdot~\mathbf{S}_j.$ This interaction is invariant under rotations. When the temperature is below the critical value ($T < T_c$), the magnetic moments of the atoms tend to align within magnetic domains. As the temperature increases, thermal fluctuations disrupt this alignment until the net magnetization vanishes at $T = T_c$. At temperatures $T > T_c$, the magnetic moments orient randomly in space with no preferred direction, resulting in a symmetric ground state in the system (see above figure). Thus, the system's governing equations remain symmetric under rotations, but the ground state at $T<T_c$ chooses a preferred orientation and does not maintain this symmetry. The ferromagnet thus demonstrates spontaneous symmetry breaking: the system's equations have rotational symmetry, but the chosen ground state does not.

In biology, symmetry breaking is a prevalent process and occurs at multiple scales. The survival of organisms relies heavily on the presence of well-defined structures and patterns, both at the microscopic and macroscopic scales. In fact, the distinctive patterns we see on a tiger, for example, are a direct result of broken symmetry. In the early stages of development, cells in an organism might be nearly identical, with no distinguishing features. However, as the organism grows, various factors such as chemical gradients, genetic expression, or environmental effects provoke certain cells to behave differently than others. This differentiation breaks the initial symmetry and leads to the formation of complex patterns.

A possible explanation for the tiger's stripes is a process known as reaction-diffusion. This is where two substances -- a chemical activator and an inhibitor -- interact and diffuse through the developing tissue at different rates. The interaction between these substances results in the formation of repeating patterns like the stripes on a tiger.

Another common occurring phenomenon in biological systems is the directional motility of the cells, which is vital in various processes, including embryonic development, wound healing, immune response, and organ formation, among others. This movement is significantly influenced by symmetry breaking in the cell's internal structure. Actin, a protein that forms long, thin fibers within the cell's cytoskeleton, plays a crucial role in these processes. These actin filaments give the cell its shape and play a key role in various cellular processes, including movement. These filaments are dynamic --they can rapidly grow and shrink by adding or removing actin molecules at their ends. In a perfectly symmetrical system, actin filaments would grow or shrink uniformly in all directions, resulting in no net movement of the cell. However, cells break this symmetry by regulating where and how actin filaments assemble. This control leads to an asymmetric arrangement of the filaments. When actin filaments assemble more on one side of the cell than the other, they generate a protrusive force that pushes the cell membrane outward in that direction. This asymmetric force drives the cell to move in a specific direction. Thus, breaking symmetry in actin filament assembly is crucial for creating directional movement, enabling the cell to crawl along surfaces or through tissues effectively. Symmetry provides stability and uniformity, but its breaking is equally crucial as it enables dynamic processes like cell movement, development, and adaptability.