58
2. Electromagnetism as a Gauge Theory
one-dimensional matrix is of course a single number – in this case a complex
number. Condition (2.68) limits this to being a simple phase: the set of phase
factors of the form e
iα , where α is any real number, form the elements of a
U(1) group. These are just the factors that enter into our gauge (or phase)
transformations for wavefunctions. Thus we say that the electromagnetic
gauge group is U(1). We must remember, however, that it is a local U(1),
meaning (cf (2.54)) that the phase parameters α, β, . . . depend on the space–
time point x.
The transformations of the U(1) group have the simple property that it
does not matter in what order they are performed: referring to (2.65)–(2.67),
we would have got the same final answer if we had done the β ‘rotation’ first
and then the α one, instead of the other way around; this is because, of course,
exp(iα) · exp(iβ) = exp[i(α + β)] = exp(iβ) · exp(iα).
(2.69)
This property remains true even in the ‘local’ case when α and β depend
on x. Mathematicians call U(1) an Abelian group: different transformations
commute. We shall see later (in volume 2) that the ‘internal’ symmetry spaces
relevant to the strong and weak gauge invariances are not so simple. The
‘rotations’ in these cases are more like full three-dimensional rotations of real
space, rather than the two-dimensional rotation of (2.64). We know that, in
general, such real-space rotations do not commute, and the same will be true
of the strong and weak rotations. Their gauge groups are called non-Abelian.
Once again, we shall have to wait until chapter 7 before understanding
how the symmetry represented by (2.63) is really related to the conservation
law of charge.
Comment (iv)
The attentive reader may have picked up one further loose end. The vector
potential A is related to the magnetic field B by
B = ∇ × A.
(2.70)
Thus if A has the special form
A = ∇f
(2.71)
B will vanish. The question we must answer, therefore, is: how do we know
that the A field introduced by our gauge principle is not of the form (2.71),
leading to a trivial theory (B = 0)? The answer to this question will lead us
on a very worthwhile detour.
The Schr¨ odinger equation with ∇f as the vector potential is
1 (−i∇ − q∇f )
2 ψ = Eψ.
(2.72)
2m
We can write the formal solution to this equation as
( ∫
)
x
ψ = exp iq
∇f · dl · ψ(f = 0)
(2.73)
−∞
2. Electromagnetism as a Gauge Theory
one-dimensional matrix is of course a single number – in this case a complex
number. Condition (2.68) limits this to being a simple phase: the set of phase
factors of the form e
iα , where α is any real number, form the elements of a
U(1) group. These are just the factors that enter into our gauge (or phase)
transformations for wavefunctions. Thus we say that the electromagnetic
gauge group is U(1). We must remember, however, that it is a local U(1),
meaning (cf (2.54)) that the phase parameters α, β, . . . depend on the space–
time point x.
The transformations of the U(1) group have the simple property that it
does not matter in what order they are performed: referring to (2.65)–(2.67),
we would have got the same final answer if we had done the β ‘rotation’ first
and then the α one, instead of the other way around; this is because, of course,
exp(iα) · exp(iβ) = exp[i(α + β)] = exp(iβ) · exp(iα).
(2.69)
This property remains true even in the ‘local’ case when α and β depend
on x. Mathematicians call U(1) an Abelian group: different transformations
commute. We shall see later (in volume 2) that the ‘internal’ symmetry spaces
relevant to the strong and weak gauge invariances are not so simple. The
‘rotations’ in these cases are more like full three-dimensional rotations of real
space, rather than the two-dimensional rotation of (2.64). We know that, in
general, such real-space rotations do not commute, and the same will be true
of the strong and weak rotations. Their gauge groups are called non-Abelian.
Once again, we shall have to wait until chapter 7 before understanding
how the symmetry represented by (2.63) is really related to the conservation
law of charge.
Comment (iv)
The attentive reader may have picked up one further loose end. The vector
potential A is related to the magnetic field B by
B = ∇ × A.
(2.70)
Thus if A has the special form
A = ∇f
(2.71)
B will vanish. The question we must answer, therefore, is: how do we know
that the A field introduced by our gauge principle is not of the form (2.71),
leading to a trivial theory (B = 0)? The answer to this question will lead us
on a very worthwhile detour.
The Schr¨ odinger equation with ∇f as the vector potential is
1 (−i∇ − q∇f )
2 ψ = Eψ.
(2.72)
2m
We can write the formal solution to this equation as
( ∫
)
x
ψ = exp iq
∇f · dl · ψ(f = 0)
(2.73)
−∞