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In another word, all other spins 0 particles have odd wave functions on the spin basis. It subjects the choice of basis of the vector space. However, when two representations differ only by the choice of basis, we call them equivalent representations. However, when two representations map to two different vector spaces, they remain no longer equivalent. Yang and Lee then proposed a theoretical explanation that the two particles must be the same while implying that parity is not conserved under the decays. To further distinguish scalar and vector function, physicists call even scalar functions and vector functions just as scalar and vector function, while calling odd scalar function and odd vector functions pseudoscalar function and pseudovector function. For example, velocity changes its direction by 180 degrees on the other side of the mirror, while angular momentum remained unchanged by the mirror. 1 as “right-handed”, the discovery is that neutrinos are always “left-handed” on both sides of the mirror. So what leads to the discovery of the broken mirror image of the neutrinos? Now let’s define helicity as the quantity that determines whether a particle’s spin angular momentum and velocity are in the same direction, that is, when helicity equals to 1, spin angular momentum and velocity are in the same direction, and -1 vice versa, then we know that a particle’s helicity must change its sign under a mirror reflection.
As a result, particles with non-zero spin are vector/pseudovector particles. Despite there are many spin 0 particles, there is only one known scalar particle so far: the famous Higgs Boson. Note that condition 1 implies that for each vector space, there exists only one representation for the group as there exists an unique identity operator in the vector space. V. Note that the Algebra is defined over a field implies that the vector space is also equipped with addition, multiplication, and scalar multiplication from the field. We revisit concepts in introductory group theory courses that are important for the understanding of Lie groups and Lie Algebra. The area of understanding groups by their representations and Algebra is called representation theory, which is also the area we will dive into when studying Lie Groups in the following chapters. Our goal is to prepare undergraduate and perhaps even some upper-level high school students with the appropriate background in classical mechanics, quantum mechanics, and Linear Algebra for advanced physical studies such as Quantum Field Theory and Particle Physics. Besides, we provide examples in particle physics that illustrate these concepts.
We will introduce here some useful concepts which we will be using in the later chapters in the process of proving an important Theorem of the finite groups and their representations. In turns out that there is a rule to generate these regular representations. It turns out that because neutrinos are not massless and thus may not always travel at the speed of light chirality is a better way to define neutrino’s “handedness”. POSTSUPERSCRIPT. It turns out that we can also represent symmetrical transformations in a group as linear operators in a vector space. ARG unit rotation transformations represent the shape symmetry of a unit square. ARG unit rotation transformations can be achieved by only applying one of the 4 transformations described above, on any 2D Euclidean-Space geometrical objects. ARG unit rotation transformations we described previously. ARG unit rotation transformations group. To understand why a group is related to symmetry, let’s start with a unit square.