Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

6 gauge theories
Invariance of the Lagrangian requires that the covariant derivative D µ φ transforms
like the field φ itself under gauge transformations, which justifies its name.
This dictates the transformation law for the vector potential,
A µ (x) → e −ieθ(x) [A µ (x) − i e ∂ µ]e ieθ(x) = A µ (x) + ∂ µ θ(x). (2.5)
The scalar field φ is parallel transported along a spacetime path by demanding that
its covariant derivative vanishes. Infinitesimally, one adds a correction,
ieA µ (x)φ(x), (2.6)
when φ(x) is transported from x to x + δx . Transportaton along a finite path γ to
some point y is an iterative process that compounds the infinitesimal corrections.
They exponentiate,
∫
φ(y) = (P exp i eA µ dx µ )φ(x), (2.7)
γ
where the path ordering P reminds us that the infinitesimal transportations are
performed successively.
If parallel transportation around a closed loop C is non-trivial,
∮
P exp i eA µ dx µ ̸= 1, (2.8)
C
then the connection has curvature. This object is called a loop holonomy. It gives
the Aharonov-Bohm phase seen by a quanta of φ when it traverses the loop C.
The field strength tensor F µν (x) is an infinitesimal measure of the curvature. It
measures the extent to which parallel transport depends on the path taken. Transporting
φ around a small square with sides of length ɛ in the (µ, ν) -plane gives the
holonomy,
where the field strength tensor is the 2-form defined by,
≈ 1 + ieɛ 2 F µν , (2.9)
F µν = − i e [D µ, D ν ] = ∂ µ A ν − ∂ ν A µ + ie[A µ , A ν ], (2.10)
and the commutator is trivial for electrodynamics. The addition of the gauge invariant
term − 1 4 F µνF µν to the Lagrangian encodes the contribution of the curvature
of the connection specified by A µ , the photon field.
Generally, the vector space of matter fields that form the standard fibers may
be in any representation of a gauge group. The quarks of QCD belong to the fundamental
representation of SU(3). Each Dirac spin component of a quark field ψ
transforms as a Grassman valued 3-vector under SU(3) matrix multiplication. Since
the covariant derivative acts infinitesimally on the quark field, the gluon vector potential
A µ lives in the Lie algebra, su(3). It is a 3 × 3 matrix spanned by 8 traceless,
hermitian generators,
A µ (x) = A A µ (x)T A ,
with
[T A , T B ] = i f ABC T C , tr T A T B = 1 2 δAB .
(2.11)