Editing The Road to Reality Study Notes (section)

=== 5.5 Some Relations To Modern Particle Physics ===
Penrose rounds out the chapter with some examples of complex concepts in the world of particle physics.  Additive quantum numbers were briefly introduced in section 3.5, and here we are introduced to multiplicative quantum numbers, which are quantified in terms of nth roots of unity.

The notion of [https://en.wikipedia.org/wiki/Parity_(physics) parity] is introduced as approximately a multiplicative quantum number with n=2, and an example is the family of particles called [https://en.wikipedia.org/wiki/Boson bosons].  Penrose notes that [https://en.wikipedia.org/wiki/Fermion fermions] could also be considered a parity group but it is not the normal convention.   The distinction between these two particles are that bosons are completely restored to their original states under a <math>2π</math> rotation, whereas fermions require <math>4π</math> (two rotations).  Thus a multiplicative quantum number of <math>-1</math> can be assigned to a fermion and <math>+1</math> to a boson.

An example of a multiplicative quantum number with <math>n=3</math> relates to quarks, which have values for electric charge that are not integer multiples of the electron’s charge, but in fact <math>\frac{1}{3}</math> multiples.  If <math>q</math> is the value of electric charge with respect to an electron (<math>q=-1</math> for electron charge), then quarks have q=<math>\frac{2}{3}</math> or <math>-\frac{1}{3}</math> and antiquarks q=<math>\frac{1}{3}</math> or <math>-\frac{2}{3}</math>.  If we take the multiplicative quantum number <math>e^{-2qπi}</math>, then we find the values <math>1,ω,ω^2</math> from section 5p4, which constitute the cyclic group Z<sub>3</sub>.