Quantum Mechanics for Mathematicians

and magnetic field strength components as in theU(1) case, the equations for
matrix-valued electric and magnetic fields are:

Ej(x) =

∑^3

a=1

Eja(x)

σa

2

=−

∂Aj

∂t

+

∂A 0

∂xj

−ie[Aj,A 0 ] (45.3)

and

Bj(x) =

∑^3

a=1

Bja(x)

σa

2

=jkl

(

∂Al

∂xk

−

∂Ak

∂xl

−ie[Ak,Al]

)

(45.4)

The Yang-Mills theory thus comes with electric and magnetic fields that
now are valued insu(2) and can be written as 2 by 2 matrices, or in terms of
the Pauli matrix basis, as fieldsEa(x) andBa(x) indexed bya= 1, 2 ,3. These
fields are no longer linear in theAμfields, but have extra quadratic terms. These
non-quadratic terms will introduce non-linearities into the equations of motion
for Yang-Mills theory, making its study much more difficult than theU(1) case.

45.6 For further reading

Most electromagnetism textbooks in physics will have some discussion of the
vector potential, electric and magnetic fields, and gauge transformations. A
textbook covering the geometry of connections and curvature as it occurs in
physics is [29]. [30] is a recent textbook aimed at mathematicians that covers
the subject of electromagnetism in detail.