To be used in future
 Section A.8: Anticommutators Up Appendix A: Useful Mathematics

A.9 Standard Matrices

Pauli matrices
The Pauli matrices are defined as↓
These are Hermitean ( ) and satisfy↓ This can be summarised as This gives the useful result↓
Dirac Matrices
The standard choice for the four Dirac matrices is↓
or more explicitly
Spin in Dirac equation
↓↓
Gamma matrices
The standard choice of the four matrrices is↓ Anticommutation relations↓ Lowered indices↓ Thus
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Index

2D electron gas: ↑

Aharonov-Bohm effect|(: ↑

Aharonov-Bohm effect|): ↑

Bohr magneton: ↑

Boosts: ↑

Cartesian coordinates: ↑

Completeness: ↑

Cramer’s rule: ↑

Dalembertian: ↑

Dirac equation:

Hamiltonian: ↑

covariant form: ↑, ↑

spherical symmetry|(: ↑

spherical symmetry|): ↑

spin operator: ↑

standard non-covariant form: ↑

Dirac equation|(: ↑

Dirac equation|): ↑

Dirac matrices:

explicit form: ↑

Dirac spinor:

upper and lower components: ↑

Dirac spinors:

completeness: ↑

plane wave: ↑

Discrete symmetries|(: ↑

Einstein summation convention:

full: ↑

naive: ↑, ↑

Euler angles: ↑

Ga: ↑

GaAs: ↑

Galilean invariance|(: ↑

Galilean invariance|): ↑

Gaussian integral: ↑

Graphene|(: ↑

Graphene|): ↑

Hall current: ↑

Hermitean conjugate:

of a matrix: ↑

Klein paradox:

Dirac equation|(: ↑

Dirac equation|): ↑

Klein-Gordon equation:

covariance: ↑

explicit form: ↑

external potentials: ↑

minimal coupling: ↑

plane wave: ↑

probability current: ↑

Klein-Gordon equation|(: ↑

Klein-Gordon equation|): ↑

Kronecker \delta: ↑

Kronecker delta: ↑

Landau gauge: ↑

Landau level: ↑

Levi-Civita tensor: ↑

Lie group: ↑

Lorentz invariance|(: ↑

Lorentz invariance|): ↑

Lorentz transformation: ↑

Pauli equation: ↑

Pauli matrices:

dot product with vector: ↑

explicit form: ↑

pairwise multiplication: ↑

Pauli-Schrödinger equation: ↑

Poincaré invariance|(: ↑

Poincaré invariance|): ↑

Schrödinger picture: ↑

Space inversion: ↑

Stern-Gerlach experiment: ↑

Stokes theorem: ↑

Symmetry generators:

translation: ↑

Symmetry generators|(: ↑

Symmetry generators|): ↑

Time reversal: ↑

Translational invariance|): ↑

Unitary operators|(: ↑

Unitary operators|): ↑

Wigner D matrix: ↑

Wigner d:

J=1/2: ↑

L=1: ↑

\gamma matrices:

anticommutation relations: ↑

explicit form: ↑

lowering indices: ↑

angular momentum:

total: ↑

anticommutator: ↑

compound operators: ↑

carbon nanotube: ↑

charge conjugation: ↑

classical action: ↑, ↑

co-tangent bundle: ↑

commutator: ↑

compound operators: ↑

compound operators:

anticommutator: ↑

commutator: ↑

continuity equation: ↑

continuous symmetries: ↑

coordinates:

Cartesian: ↑

cylindrical: ↑

spherical: ↑

correspondence principle: ↑

covariant: ↑

curl: ↑

cylindrical coordinates: ↑

discrete symmetries: ↑, ↑

divergence: ↑

dot product: ↑

electron gas:

2D: ↑

energy operator:

relativistic: ↑

exponent of operator: ↑

four vector: ↑

gauge:

Landau: ↑

gauge potential: ↑

gauge symmetry: ↑

gradient: ↑

group: ↑

helicity basis: ↑

inertial frame: ↑

inner product: ↑

integral:

Gaussian: ↑

invariance:

of transformed operator: ↑

ladder operators:

angular momentum: ↑

linearity: ↑

lower component:

Dirac spinor: ↑

lowering indices: ↑

magnetic field:

homogeneous: ↑

matrix representation:

of \hat{L}_{x}: ↑

metric: ↑

metric tensor: ↑

minimal action principle: ↑

negative energy solutions: ↑

parity: ↑

parity operator: ↑

positive energy solutions: ↑

probability current:

Klein-Gordon equation: ↑

Schrödinger equation: ↑

propagator: ↑

quantum Hall effect:

integer: ↑

relativistic energy momentum relation: ↑

relativistic tensors: ↑

representation matrices: ↑

rotation:

unitary operator: ↑

rotational invariance|(: ↑

rotational invariance|): ↑

scalar product: ↑

simultaneous diagonalisation: ↑

space groups: ↑

spherical coordinates: ↑

spin: ↑

spin operator:

Dirac equation: ↑

symmetry generators:

boosts: ↑

commutation: ↑

rotation: ↑

time translation: ↑

time evolution operator: ↑

time reversal operator: ↑

time translation:

unitary operator: ↑

time translations: ↑

total angular momentum: ↑

transformations:

active: ↑

passive: ↑

transition amplitude: ↑

translation:

unitary operator: ↑

translational invariance|(: ↑

transpose:

of a matrix: ↑

triple product:

scalar: ↑

vector: ↑

unitary operator:

definition: ↑

rotation: ↑

time translation: ↑

translation: ↑

unitary transformation:

for gauge change: ↑

upper component:

Dirac spinor: ↑

vector calculus: ↑

view|active vs. passive: ↑

von Klitzing constant: ↑