simplest case of fields taking values inR^2 orC, one gets a theory of charged
relativistic particles, with antiparticles now distinguishable from particles (they
have opposite charge).
44.1 Internal symmetries
The relativistic real scalar field theory of chapter 43 lacks one important fea-
ture of the non-relativistic theory, which is an action of the groupU(1) by phase
changes on complex fields. This is needed to provide a notion of “charge” and
allow the introduction of electromagnetic forces into the theory (see chapter 45).
In the real scalar field theory there is no distinction between states describing
particles and states describing antiparticles. To get a theory with such a distinc-
tion we need to introduce fields with more components. Two possibilities are to
consider real fields withmcomponents, in which case we will have a theory with
SO(m) symmetry, or to consider complex fields withncomponents, in which
case we have a theory withU(n) symmetry. IdentifyingCwithR^2 using the
standard complex structure, we findSO(2) =U(1), and two equivalent ways
of getting a theory withU(1) symmetry, using two real or one complex scalar
field.
44.1.1 SO(m) symmetry and real scalar fields
Starting with the casem= 2, and taking as dual phase spaceMthe space
of pairsφ 1 ,φ 2 of real solutions to the two-component Klein-Gordon equation,
elementsg(θ) of the groupSO(2) will act on the fields by
(
Φ 1 (x)
Φ 2 (x))
→g(θ)·(
Φ 1 (x)
Φ 2 (x))
=
(
cosθ sinθ
−sinθ cosθ)(
Φ 1 (x)
Φ 2 (x))
(
Π 1 (x)
Π 2 (x))
→g(θ)·(
Π 1 (x)
Π 2 (x))
=
(
cosθ sinθ
−sinθ cosθ)(
Π 1 (x)
Π 2 (x))
Here Φ 1 (x),Φ 2 (x),Π 1 (x),Π 2 (x) are the continuous basis elements for the space
of two-component Klein-Gordon solutions, determined by their initial values at
t= 0.
This group action onMbreaks up into a direct sum of an infinite num-
ber (one for each value ofx) of identical copies of the case of rotations in a
configuration space plane, as discussed in section 20.3.1. We will use the cal-
culation there, where we found that for a basis elementLof the Lie algebra of
SO(2) the corresponding quadratic function on the phase space with coordinates
q 1 ,q 2 ,p 1 ,p 2 was
μL=q 1 p 2 −q 2 p 1
For the case here, we take
q 1 ,q 2 ,p 1 ,p 2 →Φ 1 (x),Φ 2 (x),Π 1 (x),Π 2 (x)