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micromass submitted a new PF Insights post
Groups and Geometry
Continue reading the Original PF Insights Post.
Groups and Geometry
Continue reading the Original PF Insights Post.
strangerep said:The relationship becomes even more fascinating in elementary particle theory. I.e., the insight that the state spaces of elementary (quantum) particles can be constructed by finding representations of a particular group. Also advanced classical mechanics where symmetry groups for the dynamics, and associated structure of the symplectic phase space take center stage. The notion that Minkowski spacetime is "really" just a homogeneous space for the Poincare group is also intriguing.
One might even say that these relationships are now intrinsic to most (if not all) of modern theoretical physics (after one has generalized the concept of "geometry" to "representations").
BTW, what about semigroups? Is there a well developed theory of (some alternate version of) homogeneous spaces when one is dealing with a semigroup in which some elements have no inverses? The obvious example is the heat equation for which only forward time evolution is sensible.
[Edit: Is "Erlanger" a typo? I thought it was "Erlangen".]
There is a very deep link between group theory and geometry.
micromass said:But Klein had a very general method of generating geometries. Basically, he took projective space and he equipped it with a special "distance functions". Then a lot of very pathological but also natural geometries pop up. For example, of course euclidean, hyperbolic and elliptic geometry shows up this way. But also Minkowski geometry and Galilean geometry shows up in this way outlined by Klein. In this way, Klein discovered Minkowski geometry far before SR and GR, but he probably dismissed it for being not useful.
The relationship between groups and geometry can be understood through the concept of symmetry. A group is a mathematical structure that represents the symmetries of an object or system. Geometry, on the other hand, deals with the study of shapes and their properties. By studying the symmetries of an object or system, we can understand its underlying geometric structure.
Groups are used in geometry to understand and analyze the symmetries of geometric objects. They provide a powerful mathematical framework for studying the transformations and operations that preserve the shape and structure of an object. Groups are also used to classify and categorize geometric objects based on their symmetries.
Yes, groups have many real-world applications, particularly in the fields of physics, chemistry, and computer science. For example, group theory is used to study the symmetries of molecules and crystals in chemistry, and to understand the fundamental laws of physics in quantum mechanics. In computer science, groups are used in the development of algorithms for data encryption and compression.
Symmetry plays a crucial role in group theory as it is the fundamental concept that underlies the study of groups. Symmetry is the property of an object or system that remains unchanged under certain transformations or operations. In group theory, we use this concept of symmetry to study the properties and structure of groups.
Yes, there are many real-life examples of groups and geometry. Some common examples include the symmetries of crystals, the symmetries of a soccer ball, and the symmetries of a Rubik's cube. In each of these cases, we can use group theory to understand and analyze the underlying geometric structure and properties.