Quantum Mechanics for Mathematicians

definition of the curvature, where new terms with different behavior arise, due
to the non-commutative nature of the group.
The diagonalU(1)⊂ U(m) subgroup is treated using exactly the same
formalism for the vector potential, covariant derivative, electric and magnetic
fields as above. It is only theSU(m)⊂U(m) subgroup which requires a separate
treatment. We will do this just for the casem= 2, which is known as the Yang-
Mills case, since it was first investigated by the physicists Yang and Mills in

1954. We can think of theiφ(x) in theU(1) case as a function valued in the Lie
      algebrau(1), and replace it by a matrix-valued function, taking values in the
      Lie algebrasu(2) for eachx. This can be written in terms of three functionsφa
      as

iφ(x) =i

∑^3

a=1

φa(x)

σa

2

using the Pauli matrices. The gauge group becomes the groupGY M of maps
from space-time toSU(2), with Lie algebra the mapsiφ(x) from space-time to
su(2). Unlike theU(1) case, this Lie algebra has a non-trivial Lie bracket, given
by the point-wisesu(2) Lie bracket (the commutator of matrices).
In theSU(2) case, the analog of the real-valued functionAμwill now be
matrix-valued and one can write

Aμ(x) =

∑^3

a=1

Aaμ(x)

σa

2

Instead of one vector potential function for each space-time directionμwe now
have three (theAaμ(x)), and we will refer to these functions as the connection
or “gauge field”. The complex fieldsψare now two-component fields, and the
covariant derivative is

DμA

(

ψ 1

ψ 2

)

=

(

∂

∂xμ

−ie

∑^3

a=1

Aaμ(x)

σa

2

)(

ψ 1

ψ 2

)

For the theories of complex fields withU(m) symmetry discussed in chapters 38
and 44, this is them= 2 case. Replacing derivatives by covariant derivatives
yields non-relativistic and relativistic theories of matter particles coupled to
background gauge fields.
In the Yang-Mills case, the curvature or field strengths can still be defined
as a commutator of covariant derivatives, but now this is a commutator of
matrix-valued differential operators. The result will as in theU(1) case be a
multiplication operator, but it will be matrix-valued. The curvature can be
defined as
∑^3

a=1

Fμνa

σa

2

=

i

e

[DμA,DAν]

which can be calculated much as in the Abelian case, except now the term in-
volving the commutator ofAμandAνno longer cancels. Distinguishing electric