2.1 BV Action for the Point Particle
11
number gh, with gh(φ, e, ¯
π ) = 0 and gh(c) = −gh(b) = 1, respectively. In this
chapter, we will identify this ghost number with the degree. The Lagrange multiplier
field ¯
π localizes the functional integral over e which is sufficient to remove the
degeneracy of the path integral measure. The role of the (b, c) ghost system is to
cancel the Jacobian arising from the reparametrization (2.2).
The constraint action (2.4) has a nilpotent (BRST) symmetry
δ BRST e = −i ˙
c , δ BRST φ = −i
˙
φc
e
, δ BRST b = ¯
π , δ BRST c = δ ¯
π = 0 ,
as can be most easily seen by noticing that
I [φ, e, b, c, ¯
π] = I [φ, e] −
δ BRST b(e − ˆ
e).
Clearly, δ 2
BRST = 0, so that δ BRST defines a cohomological vector field on
the graded space of trajectories (2.5). The addition of the BRST exact term in
I [φ, e, b, c, ¯
π] localizes the functional integral at the critical points of the BRST
exact piece. Indeed, upon elimination of ¯
π the constraint e ≡ 1 is enforced so that
the kernel
K(τ f , τ i , φ f , φ i ) ≡
D[φ, b, c] J e
i
¯
h
τ f
τ i
1
2 g( ˙
φ, ˙
φ)−
1
2 m 2 +ib ˙
c
dτ
(2.6)
is well defined. Here,
J =
1
det
(
i
¯
h ∂ τ )
is included for later convenience. The prime is to highlight the fact that the
constant mode corresponding to the translation in proper time has not been included
in the above path integral measure. It will be included below in Eq. (2.11). The
reduced action (2.6) still enjoys a residual symmetry
δ BRST φ = −i ˙
φc , δ BRST b =
1
2
g( ˙
φ, ˙
φ) + m
2
.
(2.7)
In the canonical (as opposed to path integral) quantization of the reduced action
(2.6), {p, φ, c, b} form a graded operator algebra with non-vanishing, graded, equal
proper-time commutation rules
[φ, p] = i1
and
[b, c] = 1 ,
(2.8)
11
number gh, with gh(φ, e, ¯
π ) = 0 and gh(c) = −gh(b) = 1, respectively. In this
chapter, we will identify this ghost number with the degree. The Lagrange multiplier
field ¯
π localizes the functional integral over e which is sufficient to remove the
degeneracy of the path integral measure. The role of the (b, c) ghost system is to
cancel the Jacobian arising from the reparametrization (2.2).
The constraint action (2.4) has a nilpotent (BRST) symmetry
δ BRST e = −i ˙
c , δ BRST φ = −i
˙
φc
e
, δ BRST b = ¯
π , δ BRST c = δ ¯
π = 0 ,
as can be most easily seen by noticing that
I [φ, e, b, c, ¯
π] = I [φ, e] −
δ BRST b(e − ˆ
e).
Clearly, δ 2
BRST = 0, so that δ BRST defines a cohomological vector field on
the graded space of trajectories (2.5). The addition of the BRST exact term in
I [φ, e, b, c, ¯
π] localizes the functional integral at the critical points of the BRST
exact piece. Indeed, upon elimination of ¯
π the constraint e ≡ 1 is enforced so that
the kernel
K(τ f , τ i , φ f , φ i ) ≡
D[φ, b, c] J e
i
¯
h
τ f
τ i
1
2 g( ˙
φ, ˙
φ)−
1
2 m 2 +ib ˙
c
dτ
(2.6)
is well defined. Here,
J =
1
det
(
i
¯
h ∂ τ )
is included for later convenience. The prime is to highlight the fact that the
constant mode corresponding to the translation in proper time has not been included
in the above path integral measure. It will be included below in Eq. (2.11). The
reduced action (2.6) still enjoys a residual symmetry
δ BRST φ = −i ˙
φc , δ BRST b =
1
2
g( ˙
φ, ˙
φ) + m
2
.
(2.7)
In the canonical (as opposed to path integral) quantization of the reduced action
(2.6), {p, φ, c, b} form a graded operator algebra with non-vanishing, graded, equal
proper-time commutation rules
[φ, p] = i1
and
[b, c] = 1 ,
(2.8)