Quantum Field Theory

The following section is included:

• Contents

The following sections are included:

• From Classical Physics to Modern Physics
• Quantum Mechanics
• Quantum Electrodynamics
• Gauge Theories
• Quantum Chromodynamics
• Standard Model of Particle Physics
• Beyond the Standard Model

The following sections are included:

• Classical Mechanics
• Relativistic Mechanics

The following sections are included:

• Scalar Fields
• Electrodynamics

The following sections are included:

• Nonrelativistic Quantum Mechanics
• Schrödinger Picture, Heisenberg Picture and Interaction Picture
• Relativistic Quantum Mechanics

Group theory is a field of mathematics that has become important for physics, especially for the understanding of symmetries. A group is a set of elements (a, b, c, …), which can be multiplied, following a specific set of rules:

• The multiplication of two elements a and b gives another element c:
  $$a\in G,b\in G,\Rightarrow a\times b=c\in G.$$
• The multiplication is associative:
  $$(a\times b)\times c=a\times (b\times c).$$
• The group has a unit element e:
  $$a\times e=e\times a=a.$$
• For each element there is an inverse element. The multiplication of an element with the inverse element gives the unit element :
  $$a\times (a{}^{-1})=(a^{-1})\times a=e$$
  .

The following sections are included:

• Real Scalar Fields
• Complex Scalar Fields
• Covariant Commutation Relations

Electrons have an internal angular momentum, the spin, which is a quantum phenomenon. There is no classical analogue. The wave function of the electron consists of two functions:

Vector fields are important for the description of the fundamental interactions. These are generated by the exchange of vector particles. The electromagnetic interaction between charged particles is due to the exchange of photons, which are vector particles. The strong interactions between the quarks inside a proton are generated by the exchange of vector gluons. The weak interaction is due to the exchange of weak vector bosons.

A scattering process is the transition from an initial state, e.g. an electron and a proton with fixed momenta, to a final state. If the final state is also an electron and a proton, it is an elastic scattering process. But if the energy of the electron is large enough, the proton can be excited to a delta resonance, which afterwards decays e.g. into a neutron and a charged pion. The final state is now a state with three particles — it is an inelastic scattering process.

The theory of quantum electrodynamics (QED) combines electrodynamics, quantum mechanics and relativity theory. It describes the interaction of charged particles with the electromagnetic field.

If the physical properties of a system do not change under a transformation, one has a symmetry. There are discrete symmetries, for example the symmetry of a reflection of space, and continuous symmetries, for example the symmetry of the rotations of space or of the translation in time.

In quantum electrodynamics the field equations do not change, if the electron field is changed by a local phase transformation:

The following sections are included:

• The Colors of the Quarks
• Gauge Theory of Colored Quarks
• Asymptotic Freedom
• Electron–Positron Annihilation
• Deep-Inelastic Scattering
• Violation of Scaling
• The Masses of the Quarks

The following sections are included:

• Scalar Fields
• Spontaneous Mass Generation
• The Gauge Group SU(2)

The beta decay of the neutron, e.g. the decay into a proton, an electron and an antineutrino, is caused by the weak interactions. It can be described by an interaction given by the product of two weak currents multiplied with a coupling constant, the Fermi constant. This interaction was proposed by Enrico Fermi in 1934:

In the Standard Model of particle physics there are 25 fundamental constants that cannot be calculated, but have to be determined by experiments: the three coupling constants of the electromagnetic, weak and strong interactions, the six masses of the leptons, the six masses of the quarks, the eight mixing parameters for the flavor mixing of the quarks and leptons, the mass of the weak boson and the mass of the Higgs boson.

The following sections are included:

• Appendix
• Bibliography