Symmetries And Group Invariants

Spinor Fermions & Vector Gauge Bosons of Field Theories The notion of the chirality of a particle is both clarified by studying all the (Group of) possible transformations that do no materially affect a representative object in either real or mathematical abstract spin space. In this sense it turns out that the chirality of a particle is determined by whether it transforms as either right or left-handed representations of the Poincaré group. This is a connected set of orientation preserving Lorentz boosts and rotations with the addition of translational invariance.

Massive Dirac Spinors & Massless Weyl Spinors Different representative mathematical objects are rooted from a common hierarchy and represent certain particles. Some representations, such as Dirac spinors used to describe massive fermions -like electrons, have both right and left-handed components, others such as the Weyl-Spinor describing (essentially) massless neutrinos are left or right-handed.

A non-chiral (i.e. parity symmetric preserving) theory is called a vector theory. The terms chiral or vector derive from the types of invariant objects that arise from the Representation of the underlying theory`s Group of symmetries. In this sense the familiar "vectors" of three-dimensional space are the objects that are invariant -stay the same- when the underlying basis (co-ordinate axis) set is rotated.

Quantum Chromo Dynamics (QCD), the quantum field theory that describes the (non-linear) theory of the strong interaction binding together the quarks of a nucleon is an example of a vector theory since both left and right-handed chiralities of all the quarks appear in the theory, and they couple the same way.

The electroweak theory as part of QED controlling radioactive decay is a chiral theory despite one of its invariant objects having both right- and left hands. The object being the mass-less neutrino is described by a so-called Weyl spinor that is invariant under the (double cover) of the Lorentz transformations of Einstein`s Special theory of Relativity.

Rotations & Group of proper Lorentz Transformations Such Lorentz transformations generalise the Galilean transformations that enable you to equivalently observe Newtonian motion from any inertial reference frame (essentially an oriented co-ordinate axis with clock) with the added requirement that the speed of light remains both the same and a constant between each reference frame.

It is helpful to thinks of rotations of objects being implemented by matrix operations such as the rotation matrix applied to a (column) vector object (say of velocity or force) rotating it anti-clockwise by an angle theta in 2 dimensional x-y space.

The set of the relativistic 4-dimensional transformations that represents equivalent ways to observe an experiment is called the (Special) Orthogonal group of Lorentz boosts and rotations.The "special" here refers to the set of transformations that do not include inversion of space or parity transforming ones. The "double cover" is more arcane referring to the spinor object needing to rotate 720 degrees to return to the same orientation and position versus the 360 degrees of a vector. We see to the right its reverse orientation when only rotated 360 degrees around a Möbius strip.

Developing the mathematics of Group theory and Calculus is crucial to understanding the symmetries of fundamental physics.