10
2 Relativistic Point Particle
for the point particle is the world-line action functional
I [φ, e] =
1
2
[τ i ,τ f ]
1
e
g( ˙
φ, ˙
φ) − m
2 e
dτ ,
(2.1)
where e(τ ) is a non-dynamical einbein on the world-line and the vector field ˙
φ =
d
dτ φ ∈ T M is tangent to the curve φ. As a consequence of the reparametrization
invariance, φ(τ ) → φ(f (τ )), this action is invariant under
δφ =
˙
φ
e
(τ ) ,
δe = ˙
(τ ) ,
(2.2)
for f (τ ) = τ −
(τ )
e(τ ) + O(( 2 ), where is a smooth function with (τ f ) = i ) = 0.
In particular, with a suitable choice of f (τ ) we can set e(τ ) to be a constant. On
the other hand, elimination of e by solving the corresponding constraint equation
1
e 2 g( ˙
φ, ˙
φ) + m 2 = 0 reproduces the usual geodesic length
I [φ] = m
τ f
τ i
−g( ˙
φ, ˙
φ) dτ ,
which is the familiar action for a relativistic point particle.
For quantization, the action (2.1) is, however, more suitable since it is polynomial
in the world-line field φ. We first consider path integral quantization which is
formally defined by replacing classical trajectories by a functional integral over
paths with evolution kernel
K(τ f , τ i , φ f , φ i )
f orm
=
D[φ, e] e
i
¯
h I [φ,e] .
(2.3)
Here we assume fixed boundary conditions φ(τ i ) = φ i , φ(τ f ) = φ f . Due to
the invariance (2.2) the path integral measure in (2.3) is degenerate. This can be
remedied with the help of the Faddeev–Popov BRST procedure where one starts
with an action functional on a super manifold M
I [φ, e, b, c, ¯
π] =
τ f
τ i
1
2e
g( ˙
φ, ˙
φ) −
m 2 e
2
+ ib ˙
c + ¯
π(e − ˆ
e)
dτ ,
(2.4)
where ˆ
e is a reference einbein which can be set to be one, ˆ
e = 1, by rescaling τ .
The maps
{φ, ¯
π, b, c} : [τ i , τ f ] → M
(2.5)
from the world-line to a super manifold form a graded commutative algebra under
point-wise multiplication. In physics, the Z grading is often referred to as the ghost