11.1 Lie groups in particle physics and gauge theories

Lie groups are the backbone of particle physics, describing symmetries that govern fundamental forces. They connect abstract math to real-world phenomena, helping us understand how particles interact and conserve quantities like charge and energy.

Gauge theories, built on Lie groups, explain how particles exchange forces through special fields. These theories have successfully described electromagnetic, weak, and strong interactions, forming the basis of the Standard Model of particle physics.

Lie Groups in Particle Physics

Symmetries and Conserved Quantities

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• Lie groups are continuous symmetry groups that play a fundamental role in describing the symmetries of physical laws in particle physics
• The transformations of a physical system that leave the equations of motion invariant form a Lie group, which captures the symmetries of the system
• Lie groups provide a mathematical framework to study both global and local (gauge) symmetries in particle physics
• The generators of a Lie group correspond to conserved quantities in the physical system, such as charge, isospin, and color charge

Structure and Interactions

• The structure of a Lie group determines the allowed particle interactions and conservation laws in a given theory
• The most important Lie groups in particle physics are the unitary groups U(1), SU(2), and SU(3), which describe the symmetries of the electromagnetic, weak, and strong interactions, respectively
  • U(1) symmetry is associated with the conservation of electric charge in quantum electrodynamics (QED)
  • SU(2) symmetry underlies the weak interaction and is related to the conservation of weak isospin
  • SU(3) symmetry is the basis for quantum chromodynamics (QCD) and is connected to the conservation of color charge

Gauge Theories from Lie Groups

Gauge Principle and Local Symmetry

• Gauge theories are constructed by promoting global symmetries described by Lie groups to local symmetries, which leads to the introduction of gauge fields
• The gauge principle states that the Lagrangian of a physical system should be invariant under local gauge transformations, which belong to a Lie group
• The requirement of local gauge invariance necessitates the introduction of gauge fields, which mediate the interactions between particles

Geometry of Gauge Fields

• The gauge fields are represented by connection 1-forms on a principal bundle, where the structure group is the Lie group describing the local symmetry
• The field strength tensor, which describes the curvature of the gauge fields, is constructed from the connection 1-forms and is invariant under gauge transformations
• The action of a gauge theory is built from the field strength tensor and matter fields, which transform under representations of the gauge group
• The gauge fields acquire dynamics through the kinetic term in the action, which is proportional to the square of the field strength tensor

Lie Groups, Gauge Fields, and Interactions

Gauge Fields as Interaction Mediators

• The gauge fields introduced to maintain local gauge invariance mediate the interactions between particles in a gauge theory
• The Lie group structure determines the number and properties of the gauge fields, as well as the allowed couplings between the gauge fields and matter fields
  • In QED, the U(1) gauge field is the photon, which mediates the electromagnetic interaction
  • In the weak interaction, the SU(2) gauge fields are the W and Z bosons
  • In QCD, the SU(3) gauge fields are the gluons, which mediate the strong interaction
• The generators of the Lie group correspond to the charges that the gauge fields couple to, such as electric charge for U(1), weak isospin for SU(2), and color charge for SU(3)

Non-Abelian Gauge Theories and the Higgs Mechanism

• The coupling constants in a gauge theory are determined by the structure constants of the Lie group, which encode the commutation relations between the generators
• The non-Abelian nature of SU(2) and SU(3) gauge theories leads to self-interactions among the gauge fields, which are absent in the Abelian U(1) theory of electromagnetism
  • The self-interactions of the W and Z bosons in the weak interaction lead to triple and quartic gauge boson vertices
  • The self-interactions of gluons in QCD result in asymptotic freedom and confinement
• The gauge fields can acquire masses through the Higgs mechanism, which involves the spontaneous breaking of the gauge symmetry by a scalar field transforming under a representation of the Lie group
  • The Higgs mechanism is responsible for the masses of the W and Z bosons in the electroweak theory, while keeping the photon massless

Lie Group Representations in Physics

Particle Classification and Charges

• Particles in a gauge theory transform under representations of the Lie group describing the local symmetry
• The representations of a Lie group determine the charges carried by the particles and their transformation properties under gauge transformations
• The fundamental representation of a Lie group is the smallest non-trivial representation and often corresponds to the basic building blocks of matter, such as quarks in SU(3) and leptons in SU(2)
  • Quarks transform under the fundamental representation of SU(3) and carry one unit of color charge (red, green, or blue)
  • Leptons transform under the fundamental representation of SU(2) and carry weak isospin
• Higher-dimensional representations, such as the adjoint representation, describe composite particles or excited states

Tensor Products and Interaction Terms

• The tensor product of representations can be decomposed into irreducible representations, which correspond to the allowed combinations of particles that can form bound states or undergo interactions
• The Clebsch-Gordan coefficients, which arise from the decomposition of tensor products, determine the relative strengths of particle interactions
  • The Clebsch-Gordan coefficients for SU(2) determine the coupling strengths of the weak interaction vertices involving fermions and W bosons
  • The SU(3) Clebsch-Gordan coefficients, known as the Gell-Mann matrices, specify the allowed color combinations in QCD interactions
• The invariant tensors of a Lie group, such as the Levi-Civita symbol for SU(2) and the structure constants for SU(3), play a crucial role in constructing gauge-invariant interaction terms in the Lagrangian
  • The Levi-Civita symbol appears in the self-interaction terms of the W and Z bosons in the electroweak theory
  • The structure constants of SU(3) determine the self-interaction terms of the gluons in QCD