10.2 Lie algebras in particle physics

Lie algebras are the backbone of particle physics, providing a mathematical framework for describing symmetries in quantum systems. They capture continuous transformations and are crucial for understanding modern particle theory and quantum field theory.

These non-associative algebraic structures have unique properties that make them ideal for representing symmetries. From defining structure constants to exploring roots and weights, Lie algebras offer powerful tools for classifying particles and their interactions in quantum field theories.

Fundamentals of Lie algebras

• Lie algebras form the mathematical foundation for describing symmetries in particle physics
• Non-associative algebraic structures capture continuous transformations in quantum systems
• Crucial for understanding the underlying principles of modern particle theory and quantum field theory

Definition and properties

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• Vector spaces equipped with a bilinear operation called the Lie bracket $[X,Y]$
• Satisfy antisymmetry $[X,Y] = -[Y,X]$ and Jacobi identity $[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$
• Closed under the Lie bracket operation, maintaining algebraic structure
• Generators of Lie algebras represent infinitesimal symmetry transformations

Structure constants

• Determine the commutation relations between Lie algebra elements
• Defined by $[T_a, T_b] = if_{abc}T_c$, where $f_{abc}$ are the structure constants
• Antisymmetric in all indices and satisfy the Jacobi identity
• Uniquely characterize the Lie algebra and its associated Lie group
• Play a crucial role in calculating physical observables and interaction strengths

Roots and weights

• Roots represent the eigenvalues of the adjoint representation
• Weights correspond to eigenvalues of representations acting on physical states
• Form lattices in the space of the Cartan subalgebra
• Determine the structure and properties of representations
• Essential for classifying particles and their interactions in quantum field theories

Lie groups vs Lie algebras

• Lie groups describe continuous symmetry transformations in physics
• Lie algebras provide a local, linearized description of these symmetries
• Understanding both structures essential for developing quantum field theories

Relationship and distinctions

• Lie groups are smooth manifolds with a group structure
• Lie algebras are tangent spaces to Lie groups at the identity element
• One-to-one correspondence between connected Lie groups and Lie algebras
• Lie algebras capture the infinitesimal structure of Lie groups
• Group elements can be generated by exponentiating Lie algebra elements

Exponential map

• Connects Lie algebra elements to Lie group elements
• Defined as $\exp(X) = \sum_{n=0}^{\infty} \frac{X^n}{n!}$ for Lie algebra element X
• Preserves the algebraic structure and commutation relations
• Allows calculation of finite transformations from infinitesimal generators
• Crucial for understanding gauge theories and particle interactions

Symmetries in particle physics

• Symmetries play a fundamental role in describing fundamental particles and their interactions
• Lie algebras provide the mathematical framework for describing these symmetries
• Understanding symmetries leads to conservation laws and selection rules in particle physics

Continuous vs discrete symmetries

• Continuous symmetries described by Lie groups and algebras (rotations, translations)
• Discrete symmetries include parity, time reversal, and charge conjugation
• Continuous symmetries lead to conserved quantities through Noether's theorem
• Discrete symmetries impose constraints on allowed particle interactions
• Combination of discrete symmetries (CPT) fundamental to quantum field theory

Noether's theorem

• Establishes connection between continuous symmetries and conserved quantities
• Each continuous symmetry corresponds to a conserved current and charge
• Energy conservation from time translation invariance
• Momentum conservation from spatial translation invariance
• Angular momentum conservation from rotational invariance
• Crucial for understanding particle interactions and conservation laws in quantum field theories

SU(3) flavor symmetry

• Describes approximate symmetry among light quarks (up, down, strange)
• Fundamental to understanding hadron spectroscopy and strong interactions
• Provides a classification scheme for mesons and baryons

Quark model

• Proposes that hadrons composed of quarks with fractional electric charges
• Three quark flavors (up, down, strange) form the basis of SU(3) flavor symmetry
• Quarks transform as the fundamental representation of SU(3)
• Antiquarks transform as the conjugate representation
• Explains the observed patterns in hadron masses and quantum numbers

Meson and baryon multiplets

• Mesons formed from quark-antiquark pairs, classified in SU(3) octets and singlets
• Baryons composed of three quarks, forming SU(3) octets and decuplets
• Gell-Mann-Nishijima formula relates charge, isospin, and hypercharge
• Explains mass splittings within multiplets due to SU(3) symmetry breaking
• Predicts existence of particles (Omega-minus baryon) before experimental discovery

SU(3) color symmetry

• Fundamental symmetry of the strong nuclear force
• Describes the color charge of quarks and gluons
• Forms the basis of Quantum Chromodynamics (QCD)

Quantum chromodynamics

• Quantum field theory of strong interactions based on SU(3) color symmetry
• Quarks carry color charges (red, green, blue)
• Gluons act as force carriers, transforming in the adjoint representation of SU(3)
• Color confinement explains why isolated quarks are not observed
• Running coupling constant leads to asymptotic freedom at high energies

Confinement and asymptotic freedom

• Confinement prevents observation of isolated color charges
• Quarks and gluons only exist in color-neutral bound states (hadrons)
• Asymptotic freedom describes weakening of strong force at high energies
• Allows perturbative calculations in high-energy collider experiments
• Explains quark-gluon plasma formation in heavy-ion collisions

Standard Model gauge groups

• Describes fundamental particles and their interactions
• Based on the product of three Lie groups: SU(3) x SU(2) x U(1)
• Unifies strong, weak, and electromagnetic interactions

SU(3) x SU(2) x U(1)

• SU(3) describes strong interactions (quantum chromodynamics)
• SU(2) x U(1) describes electroweak interactions
• Gauge bosons (gluons, W and Z bosons, photon) arise as force carriers
• Fermions (quarks and leptons) transform under various representations
• Higgs boson provides mass generation through spontaneous symmetry breaking

Electroweak unification

• Unifies electromagnetic and weak interactions
• Based on SU(2) x U(1) gauge symmetry
• Weak mixing angle relates coupling constants of SU(2) and U(1)
• Predicts existence of W and Z bosons, later confirmed experimentally
• Spontaneous symmetry breaking generates masses for W and Z bosons, leaving photon massless

Representations of Lie algebras

• Describe how Lie algebra elements act on vector spaces
• Essential for understanding particle multiplets and their transformations
• Different representations correspond to different particle types in physics

Fundamental representations

• Lowest-dimensional non-trivial representations of a Lie algebra
• Quarks transform in the fundamental representation of SU(3) color
• Leptons and quarks form doublets under SU(2) weak isospin
• Determine the transformation properties of elementary particles
• Crucial for constructing gauge-invariant Lagrangians in quantum field theories

Adjoint representation

• Representation of a Lie algebra on itself
• Gauge bosons transform in the adjoint representation
• Gluons form an octet under SU(3) color
• W bosons form a triplet under SU(2) weak isospin
• Determines the self-interactions of gauge bosons in non-Abelian theories

Casimir operators

• Commute with all generators of a Lie algebra
• Invariant under all transformations of the algebra
• Crucial for classifying representations and determining particle properties

Quadratic Casimir

• Constructed from the sum of squares of generators
• Eigenvalues label irreducible representations
• Determines the mass splittings in hadron multiplets
• Related to the strength of color interactions in QCD
• Used in calculating scattering amplitudes and decay rates

Higher-order Casimirs

• Exist for algebras of rank greater than one
• Provide additional invariants for classifying representations
• Important in grand unified theories and supersymmetry
• Used in constructing effective field theories
• Play a role in understanding the structure of exceptional Lie algebras

Lie algebra applications

• Lie algebras find numerous applications in particle physics and quantum field theory
• Provide a framework for understanding symmetries and their consequences
• Essential for developing predictive theories of fundamental interactions

Conservation laws

• Noether's theorem connects continuous symmetries to conserved quantities
• Energy conservation from time translation invariance
• Momentum conservation from spatial translation invariance
• Angular momentum conservation from rotational invariance
• Charge conservation from U(1) gauge invariance in electromagnetism
• Baryon and lepton number conservation in the Standard Model

Selection rules

• Determine allowed and forbidden transitions in particle interactions
• Based on conservation of quantum numbers associated with symmetries
• Isospin selection rules in strong interactions
• Weak interaction selection rules (Cabibbo-allowed vs Cabibbo-suppressed decays)
• Parity and angular momentum selection rules in electromagnetic transitions
• Crucial for predicting decay rates and branching ratios in particle physics

Grand Unified Theories

• Attempt to unify strong, weak, and electromagnetic interactions
• Based on larger Lie groups containing the Standard Model gauge group as a subgroup
• Predict new phenomena beyond the Standard Model (proton decay, magnetic monopoles)

SU(5) and SO(10) models

• SU(5) smallest simple group containing Standard Model gauge group
• Unifies quarks and leptons in common multiplets
• Predicts proton decay with lifetime ~10^31 years
• SO(10) incorporates right-handed neutrinos naturally
• Explains neutrino masses through see-saw mechanism
• Provides a framework for understanding matter-antimatter asymmetry

Proton decay predictions

• Proton decay mediated by new heavy gauge bosons in GUTs
• Typical decay modes: p → e+ π0, p → μ+ K0
• Current experimental limits exceed predictions of simplest GUT models
• Motivates development of more sophisticated unification schemes
• Drives construction of large underground detectors (Super-Kamiokande, Hyper-Kamiokande)

Supersymmetry algebras

• Extend Lie algebras to include fermionic generators
• Relate bosonic and fermionic degrees of freedom
• Provide a framework for addressing hierarchy problem in particle physics

Supercharges and superfields

• Supercharges Q generate transformations between bosons and fermions
• Satisfy anticommutation relations {Q, Q†} ~ P (momentum generator)
• Superfields unify bosonic and fermionic fields in superspace formalism
• Chiral superfields describe matter particles and their superpartners
• Vector superfields describe gauge bosons and gauginos

SUSY particle spectrum

• Each Standard Model particle has a superpartner with opposite statistics
• Squarks and sleptons (spin-0 partners of quarks and leptons)
• Gauginos (spin-1/2 partners of gauge bosons)
• Higgsinos (spin-1/2 partners of Higgs bosons)
• Neutralinos and charginos (mixed states of electroweak gauginos and higgsinos)
• Lightest supersymmetric particle (LSP) potential dark matter candidate

Exceptional Lie algebras

• Five exceptional simple Lie algebras: G2, F4, E6, E7, E8
• Unique mathematical structures with potential relevance to fundamental physics
• Arise in attempts to construct unified theories beyond the Standard Model

E6, E7, and E8

• E6 used in some grand unified theories and superstring models
• E7 appears in certain supergravity theories
• E8 largest and most complex of the exceptional Lie algebras
• E8 x E8 gauge group in heterotic string theory
• Provide rich mathematical structure for exploring higher-dimensional theories

String theory connections

• Exceptional Lie algebras naturally arise in various string theory constructions
• E8 x E8 heterotic string theory one of the five consistent superstring theories
• Compactification of extra dimensions can lead to E6 grand unified theories
• F-theory uses E6, E7, E8 singularities to describe gauge theories
• Exceptional Lie algebras play a role in understanding dualities between different string theories