# Symmetry

Quantum Diaries. We’ve been discussing the Higgs (its interactions, its role in particle mass, and its vacuum expectation value) as part of our ongoing series on understanding the Standard Model with Feynman diagrams.

Now I’d like to take a post to discuss a very subtle feature of the Standard Model: its chiral structure and the meaning of “mass.” This post is a little bit different in character from the others, but it goes over some very subtle features of particle physics and I would really like to explain them carefully because they’re important for understanding the entire scaffolding of the Standard Model. My goal is to explain the sense in which the Standard Model is “chiral” and what that means. In order to do this, we’ll first learn about a related idea, helicity, which is related to a particle’s spin. We’ll then use this as an intuitive step to understanding the more abstract notion of chirality, and then see how masses affect chiral theories and what this all has to do with the Higgs.

Helicity. Symmetry. Sphere symmetrical group o representing an octahedral rotational symmetry.

The yellow region shows the fundamental domain. Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement")[1] has two meanings. The first is a vague sense of harmonious and beautiful proportion and balance.[2][3] The second is an exact mathematical "patterned self-similarity" that can be demonstrated with the rules of a formal system, such as geometry or physics. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.[3] List of cycles. Lattice (group) In mathematics, especially in geometry and group theory, a lattice in is a discrete subgroup of which spans the real vector space .

Tessellation. Ceramic Tiles in Marrakech, forming edge-to-edge, regular and other tessellations A periodic tiling has a repeating pattern.

Some special kinds include regular tilings with regular polygonal tiles all of the same shape, and semi-regular tilings with regular tiles of more than one shape and with every corner identically arranged. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups. A tiling that lacks a repeating pattern is called "non-periodic". An aperiodic tiling uses a small set of tile shapes that cannot form a repeating pattern. History A temple mosaic from the ancient Sumerian city of Uruk IV (3400–3100 BC) showing a tessellation pattern in the tile colours. Tessellations were used by the Sumerians (about 4000 BC) in building wall decorations formed by patterns of clay tiles.[1] Etymology Overview Many other types of tessellation are possible under different constraints. Other methods also exist for describing polygonal tilings. Rotational symmetry. Generally speaking, an object with rotational symmetry, also known in biological contexts as radial symmetry, is an object that looks the same after a certain amount of rotation.

An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted. The degree of rotational symmetry is how many degrees the shape has to be turned to look the same on a different side or vertex. It can not be the same side or vertex. Formal treatment Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, so space is homogeneous, and the symmetry group is the whole E(m). Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. N-fold rotational symmetry The notation for n-fold symmetry is Cn or simply "n". The fundamental domain is a sector of 360°/n. Examples without additional reflection symmetry: Lorentz covariance. In physics, Lorentz symmetry, named for Hendrik Lorentz, is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space".[1] Lorentz covariance, a related concept, is a key property of spacetime following from the special theory of relativity.

Lorentz covariance has two distinct, but closely related meanings: This usage of the term covariant should not be confused with the related concept of a covariant vector. On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Confusingly, both covariant and contravariant four-vectors can be Lorentz covariant quantities. Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point. Examples Scalars Spacetime interval: Proper time (for timelike intervals): Rest mass: