PHY431 Lecture 8
1
4/30/2000
11.
Gauge Theories
All true present-day theories are based on local gauge (or phase) invariance, i.e. the phase of the wave-
functions can be varied according to an almost arbitrary space-time dependent function, and the La-
grangian
L
=
L
(
ψ
,
∂
µ
ψ
) that describes the system keeps the same form under phase (historically named
“gauge-”) transformations of the type:
( , )
( , )
'( , )
( , )
i
t
t
t
e
t
α
ψ
ψ
ψ
→
=
r
r
r
r
(11.1)
i.e. if
ψ
(
r
,
t
) is a solution, then
ψ
’
(
r
,
t
) is a solution too! Phase transformations of this type are called
“local”, because the phase angle is dependent on the local space-time coordinates (
r
,
t
). Transforma-
tions where the phase angle is a constant (i.e. the same for all space-time points) are instead called
“global”. The term “gauge” is inherited from early, unsuccessful, attempts by Hermann Weyl to con-
struct theories with invariance under local scale (“eich” in German) transformations as a basis for a
theory of electromagnetism.
Local Gauge Invariance has been proven to be a very powerful and guiding requirement for a theory,
and is a necessary condition for renormalizability of the calculations. It is strongly believed that all
theories realized in nature are locally gauge invariant theories. You may ask what other local phase
transformations exist apart from the one shown in equation (11.1); indeed, one can construct more
complicated versions of the phase rotation, e.g. exp{
i
σ⋅
b
(
x
)} which is a two-by-two matrix, as can be
seen from the series expansion of the exp function:
2
3
3
1
2
1
1
2
2
3
3
1
2
3
(
)
(
)
...,
with
2!
3!
i
b
b
ib
i
i
e
i
b
b
b
b
ib
b
σ
σ
σ
⋅
−
⋅
⋅
=
+
⋅
+
+
+
⋅
=
+
+
=
+
−
σ
b
σ
b
σ
b
1
σ
b
σ
b
(11.2)
with the standard form of the Pauli matrices. Before going any further, we present a quick overview of
notations for the kinematics of special relativity.
11.1 Intermezzo: Special Relativity
Four-vectors are denoted by
x
µ
, with the greek index
µ
=0,1,2,3; a single time-like component 0, fol-
lowed by the three space-like components 1,2 and 3:
x
µ
= (
x
0
,
x
1
,
x
2
,
x
3
). The space-like indices 1,2, and 3
are habitually represented by lower case roman characters: e.g.
x
i
=
x
= (
x
1
,
x
2
,
x
3
). In (special) relativity
a space-time distance (four-length)
t
2
−
x
2
−
y
2
−
z
2
is constant and invariant under Lorentz transforma-
tions: i.e.
t
’
2
−
x
’
2
−
y
’
2
−
z
’
2
=
t
2
−
x
2
−
y
2
−
z
2
, a direct consequence of the constancy of the speed of light for
all observers.
The space-time fourvector coordinate is denoted by
x
µ
= (
x
0
,
x
1
,
x
2
,
x
3
) = (
x
0
,
x
i
) = (
t
,
x
) = (
t
,
x
,
y
,
z
). The
space-time components mix between themselves by Lorentz transformations from one inertial system
to another. A Lorentz transformation from an (unprimed) system to another (primed) system moving at
relative velocity
β//
x
with respect to the first, can be expressed in terms of speed
β
and relativistic fac-
tor
γ
≡
(1
−
β
2
)
−
½
as:
2
2
2
0
0
0
1
0
0
,
with
det(
)
1
0
0
1
0
0
0
µ
µ
ν
ν
γ
βγ
γ
β γ
βγ
γ
−
Λ
=
Λ
=
−
=
−
(11.3)