How does symmetry lead us to seek truth？

　　An aesthetic term

　　In the world we live in, symmetry seems to be everywhere. It exists not only in our bodies, but also in art, architecture and music. In addition, symmetry is also very important in the fields of science and mathematics. A theory will be beautiful because it contains symmetry, and the study of these symmetries in turn leads the development of the theory. ???????

　　As an aesthetic term, symmetry can be a sense of balance or harmony, just like the arrangement of characters in Raphael's mural Athens College. It can also be a perfect or nearly perfect reflection about an axis of symmetry, just like most humans and other vertebrates are symmetrical on both sides.

　　Athens college (photo /Public Domain)

　　Mathematicians and physicists use this term more in line with the latter description: formally speaking, if a system still looks the same after transformation, then it has symmetry. For example, snowflakes look the same after rotating one-sixth of a turn (60) or turning around the central axis; Another example is that the circle will not change after any degree of rotation, or about any linear reflection passing through the center of the circle.

　　Rotate 60, and it looks just like before. (Figure/Principle)

　　finite and infinite

　　Symmetry is also related to group theory in mathematics. Taking snowflake as an example, we can create a new symmetry by rotating it for 1/6 turn and reflecting it. All these symmetries are combined to form the symmetry group of snowflake. This group is limited, because only a limited number of operations can accurately maintain the appearance of snowflakes.

　　The complete catalogue with finite symmetry is one of the most important mathematical achievements in the last century. This catalogue classifies finite simple groups (the basic building blocks of all finite groups) in the form of thousands of pages of proofs. The classification of finite simple groups covers objects such as squares or snowflakes, and their symmetry is a discrete set of rotation and reflection. Many finite groups do not cover the symmetry of any tangible three-dimensional shape, but describe the symmetry of abstract, high-dimensional objects.

　　An object that can rotate at any angle like a circle has an infinite symmetry group. These groups correspond to continuous transformation families and are directly related to some of the most important relationships between symmetry and physics in the last century, such as Einstein's theory of relativity.

　　Mathematicians are also inspired by physics in their search for new fields of exploration. This relationship is most obvious in the field of coherent mirror symmetry. Coherent mirror symmetry belongs to the field of mathematics, but it is developed from string theory. In string theory, particles are considered as tiny, vibrating strings. Because of the symmetry of the equations of control theory, chords not only occupy the familiar three-dimensional space, but also have six additional dimensions coiled into a structure called Karabi-Hill Manifold.

　　Karabi-Hill manifold is a mathematical structure used by string theorists to add extra dimensions to space-time. These objects have beautiful symmetry: pairs of seemingly completely different Karabi-Chueh manifolds actually encode the same physical characteristics. (Photo /Wikipedia)

　　When mathematicians carefully observe these manifolds, they notice a beautiful symmetry: the Karabi-Hill manifold pairs that look completely different actually encode the same physical principles. These mirror pairs have opened up a brand-new road for mathematical research, and this road is booming as researchers find similar "mirror pairs" in other manifolds.

　　From symmetry to new theory

　　Although symmetry has led researchers to find new theories and relationships, physics may be able to surpass it one day. One of the main driving forces of modern physics is to seek a theory of everything (string theory is generally regarded as theory of everything's main candidate theory) that can combine the quantum theory that dominates the microscopic world with the general relativity that describes gravity. Quantum theory provides a framework for understanding other basic forces in nature (namely electromagnetic force, weak force and strong force), but so far, there is no recognized method to describe gravity in the language of quantum theory.

　　In string theory, the AdS/CFT duality discovered by physicist juan maldacena in 1997 provides a promising way forward. Marda Sina connects the gravitational force in a space-time region called anti-de Sitter, AdS) space with the quantum description (Conformal Field Theory, CFT) of particles moving around the boundary of this region. But this road has a price, and this price is symmetry. Researchers have always speculated that the unified theory of quantum gravity needs to break one of the two symmetries, that is, global symmetry (the other is gauge symmetry).

　　Studies have shown that if this unified theory comes from AdS/CFT duality, the overall symmetry will indeed be sacrificed: no quantum gravity formula consistent with string theory and AdS/CFT correspondence can have overall symmetry in describing its mathematical framework.

　　Does this result mean that we must put symmetry behind us in order to make future scientific progress? Although global symmetry must be abandoned, many local symmetries still exist. Perhaps the clue revealed by past studies is that the concept of symmetry may need to be revised and made more accurate if it is to continue to lead researchers to make new discoveries.

　　# Creative Team:

　　Compile: Xiaoyu

　　Typesetting: Wenwen

　　# Reference source:

　　https://www.simonsfoundation.org/2020/02/06/how-symmetry-guides-our-search-for-truth/

　　# Image source:

　　Cover picture & the first picture: J.F. Podevin via princeton.edu