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The Mechanics of the Fermionic and Bosonic Fields: an Introduction to the Standard Model and Particle Physics

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The Mechanics of the Fermionic and Bosonic Fields: An Introduction to the Standard Model and Particle Physics Evan McCarthy Phys. 460: Seminar in Physics, Spring 2014 Aug. 27,! 2014 1.Introduction 2.The Standard Model of Particle Physics 2.1.The Standard Model Lagrangian 2.2.Gauge Invariance 3.Mechanics of the Fermionic Field 3.1.Fermi-Dirac Statistics 3.2.Fermion Spinor Field 4.Mechanics of the Bosonic Field 4.1.Spin-Statistics Theorem 4.2.Bose Einstein Statistics !5.Conclusion ! 1. Introduction While Quantum Field Theory (QFT) is a remarkably successful tool of quantum particle physics, it is not used as a strictly predictive model. Rather, it is used as a framework within which predictive models - such as the Standard Model of particle physics (SM) - may operate. The overarching success of QFT lends it the ability to mathematically unify three of the four forces of nature, namely, the strong and weak nuclear forces, and electromagnetism. Recently substantiated further by the prediction and discovery of the Higgs boson, the SM has proven to be an extraordinarily proficient predictive model for all the subatomic particles and forces. The question remains, what is to be done with gravity - the fourth force of nature? Within the framework of QFT theoreticians have predicted the existence of yet another boson called the graviton. For this reason QFT has a very attractive allure, despite its limitations. According to !1 QFT the gravitational force is attributed to the interaction between two gravitons, however when applying the equations of General Relativity (GR) the force between two gravitons becomes infinite! Results like this are nonsensical and must be resolved for the theory to stand. Unfortunately no means to purge these anomalies from the theory have yet been developed. For this reason many physicists have turned to String theory as a means of resolving these anomalies, as it successfully describes how two gravitons interact without these unwanted infinities. In this paper I present independent discussions on six of the foundational topics of modern and particle physics, which are then related and presented as a consequence of each other in the conclusion. I focus on the structure of the standard model Lagrangian and present a concise overview of how it is derived and constructed, followed by a discussion on gauge invariance and how it presented a small crisis for the standard model. Then there will be discussions on the the statistical distribution of fermions and bosons, and the nature of there interactions; namely, exploring the concept of spinor fields and the spin-statistics theorem using these fields. ! 2. The Standard Model of Particle Physics The Standard Model (SM) is a model of particle physics designed to explain how the particles of matter - known as fermions - and the force carrying particles - known as bosons - interact according to the four forces of nature. In creating this model, physicists have divided all the particles of the SM into a few categories to sort them by mass, spin, and type. Fermions (quarks and leptons) all have a fractional spin and quarks take part in the strong nuclear interaction so they exist in the nucleus of the atom and thus have not been directly observed. !2 Experiments done in places like the Large Hadron Collider (LHC) at CERN, FermiLab in Chicago, and other accelerator laboratories around the world have repeatedly confirmed the existence of quarks from the theoretical predictions of the SM. The most significant of these experiments was the discovery of the J/ψ particle. Before 1974 the quark model had remarkable success as a mathematical theory, and demonstrating the existence of the charm quark would confirm the validity of the theory. In particle physics there are two sorts of composite particles called mesons and baryons; all baryons are fermions and all mesons are bosons. Baryons are composed of three quarks and mesons are composed of two quarks. The J/ψ particle is a meson and is composed of a charm quark and a Generation Generation Generation I II III Q u a r k s charm anti-quark, and when its discovery was Name UP CHARM TOP Spin 1/2 1/2 1/2 F e r m i o n s announced by two independent research Charge 2/3 2/3 2/3 Mass 2.4 MeV 1.27 GeV 171.2 GeV groups - at Brookhaven National Laboratory Name DOWN STRANGE BOTTOM Spin 1/2 1/2 1/2 and at Berkeley - on November 10, 1974 the Charge -1/3 -1/3 -1/3 Mass 4.8 MeV 104 MeV 4.2 GeV charm quark was no longer a mathematical Name ELECTRON MUON TAU L e p t o n s Spin 1/2 1/2 1/2 speculation. This would come to be known as Charge -1 -1 -1 Mass 0.511 MeV 105.7 MeV 1.777 GeV ELECTRON MUON TAU the November revolution amongst the physics Name NEUTRINO NEUTRINO NEUTRINO community [1]. Spin 1/2 1/2 1/2 Charge 0 0 0 There are the quarks and their anti- Mass <2.2 eV <0.17 MeV <15.5 MeV Name PHOTON GLUON Z B o s n particles that are categorized by ascending Spin 1 1 1 Charge 0 0 0 mass into three generations known as Mass 0 0 91.2 GeV Name W+ W- HIGGS generation I, II, and III. In generation I there Spin 1 1 0 Charge +1 -1 0 are the lightest quarks - the up and down Mass 80.4 GeV 80.4 GeV ≈126 GeV !3 quarks, in generation II there are the charm and strange quarks, and in generation III the heaviest quarks - the top and bottom quarks. The up, charm, and top quarks all have a positive 2/3 charge, and the down, strange, and bottom quarks all have a -1/3 charge [2]. There are also leptons which are particles that do not have any part of the strong nuclear interaction and so have been able to be directly observed. Most people are to some extent familiar with leptons, as they are electrons, muons, tau particles, and their corresponding neutrinos. Leptons are categorized into the same three generations of ascending mass; in generation I there is the electron an electron neutrino, in generation II the muon and muon neutrino, and in generation III the tau and tau neutrino. The electron, muon, and tau all have a -1 charge, and the electron neutrino, muon neutrino, and tau neutrino all have a 0 charge, and as all leptons are fermions they all have a fractional spin. Forces are experienced between particles of matter, like the gravitational force between two massive bodies. The SM accounts for this supposed ‘action at a distance’ by assigning a boson to each force. When two matter particles called fermions interact in this model they exchange force carrier particles called bosons. All fermions “transfer discrete amounts of energy by exchanging bosons with each other” and this is perceived as a force acting on those fermionic particles [3]. The fundamental forces of electromagnetism, the strong nuclear force, the weak nuclear force, and the gravitational force are theorized to be carried by the photon, gluon, W+, W- and Z bosons [4], and graviton respectively. Though the graviton is yet to be observed, the recent discovery of the Higgs boson is a good indicator that there is a graviton.1 ! 1 The Higgs boson is necessary in the SM to preserve gauge invariance. This is discussed in greater detail in 2.2. !4 2.1. The Standard Model Lagrangian The Lagrangian of a system is effectively the difference between the system’s kinetic and potential energy. This term is used to describe the mechanics of a system in a very succinct and efficient mathematical expression. The Lagrangian may be used to trace the path of the system over some period of time with Hamilton’s variational principle, expressed as the action, t (1) " S = ∫ L dt t0 As the system evolves over time the sum of all the changes in potential to kinetic energy represent the totality of the different paths the system could have taken, and the maximized solution is the path actually taken by the system’s evolution [5], such that t (2) "δ ∫ L dt = δS = 0 t0 The SM Lagrangian, LSM, is known for it’s accuracy in predicting the properties and behavior of all the fundamental particles. The Standard Model Lagrangian comes out of the unification of the electromagnetic and weak nuclear forces, known as the electroweak theory. While it is true that the unitary group U(1) defines the electromagnetic gauge field, it is not accurate to say the SU(2) group defines the weak nuclear bosons, because the weak nuclear force exhibits particularly massive gauge bosons. Through a method known as spontaneous symmetry breaking the scalar Higgs field was introduced to give the W± and Z bosons their large mass [6,8] as an attempt to rectify these necessarily massive gauge bosons with quantum theory, and thereby maintain gauge invariance. In this new electroweak theory these gauge bosons are naturally massless and are !5 only given mass by the Higgs field. The electromagnetic and weak nuclear forces are naturally intertwined in a single SU(2) × U(1) gauge symmetry, and when this interacts with the scalar Higgs field the result is the electromagnetic and the massive, short range, weak nuclear bosons that manifest as the photon and weak nuclear observables. The electroweak theory has allowed the long accepted SM Lagrangian to have been such a tremendous success because it has all three generations of quarks and leptons, one Higgs boson, and it describes the SU(3) × SU(2) × U(1) gauge group [7] which corresponds with the SU(3) unitary group the defines the eight massless gauge fields known as gluons, and the SU(2) × U(1) unitary group that defines the three massless gauge fields known as the W± and Z bosons and the one massless gauge field known as the photon [8].
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