Wu–Yang dictionary

The Wu-Yang dictionary relates terms in particle physics with terms in mathematics, specifically fiber bundle theory. Many versions and generalization of the dictionary exist. Here is an example of a dictionary, which puts each physics term next to its mathematical analogue:[7]

Aharonov-Bohm experiment: electrons pass around a cylinder where there is a non-zero magnetic field. Outside the cylinder the field strength is 0.

Wu and Yang considered the description of an electron traveling around a cylinder in the presence of a magnetic field inside the cylinder (outside the cylinder the field vanishes i.e. ${\displaystyle f_{\mu \nu }=0}$). According to the Aharonov–Bohm effect, the interference patterns shift by a factor ${\displaystyle \exp(-i\Omega /\Omega _{0})}$, where ${\displaystyle \Omega }$ is the magnetic flux and ${\displaystyle \Omega _{0}}$ is the magnetic flux quantum. For two different fluxes a and b, the results are identical if ${\displaystyle \Omega _{a}-\Omega _{b}=N\Omega _{0}}$, where ${\displaystyle N}$ is an integer. We define the operator ${\displaystyle S_{ab}}$ as the operator that brings the electron wave function from one configuration to the other ${\displaystyle \psi _{b}=S_{ba}\psi _{a}}$. For an electron that takes a path from point P to point Q, we define the phase factor as

${\displaystyle \Phi _{PQ}=\exp \left(-{\frac {i}{\Omega _{0}}}\int _{P}^{Q}A_{\mu }\mathrm {d} x^{\mu }\right)}$,

where ${\displaystyle A_{\mu }}$ is the electromagnetic four-potential. For the case of a SU₂ gauge field, we can make the substitution

${\displaystyle A_{\mu }=ib_{\mu }^{k}X_{k}}$,

where ${\displaystyle X_{k}=-i\sigma _{k}/2}$ are the generators of SU₂, ${\displaystyle \sigma _{k}}$ are the Pauli matrices. Under these concepts, Wu and Yang showed the relation between the language of gauge theory and fiber bundles, was codified in following dictionary:[2][8][9]

Wu–Yang dictionary (1975)

Gauge field terminology	Bundle terminology
Gauge (or global gauge)	Principal coordinate fiber bundle
Gauge type	Principal fiber bundle
Gauge potential ${\displaystyle b_{\mu }^{k}}$	Connection on principal fiber bundle
${\displaystyle S_{ba}}$	Transition function
Phase factor ${\displaystyle \Phi _{QP}}$	Parallel displacement
Field strength ${\displaystyle f_{\mu \nu }^{k}}$	Curvature
Source ${\displaystyle J_{\mu }^{k}}$	?
Electromagnetism	Connection in a U1(1) bundle
Isotopic spin gauge field	Connection in a SU2 bundle
Dirac's monopole quantization	Classification in a U1(1) bundle according to first Chern class
Electromagnetism without monopole	Connection on a trivial a U1(1) bundle
Electromagnetism with monopole	Connection on a nontrivial a U1(1) bundle