12
2 Relativistic Point Particle
acting on a graded vector space V over C spanned by the quantum-mechanical states
ψ of our spinless particle. Here and in what follows we set ¯
h = 1. To construct an
irreducible module for the above canonical commutation relations, we start with
representing the operator φ on functions as the multiplication by the cartesian
coordinates φ i and p as the derivative
p =
1
i
∇ φ .
The degree zero part of the module V is then naturally identified with the vector
space of test functions on M with suitable regularity and integrability properties.
To continue, we realize the anticommuting (b, c)-system on V in analogy to the
(p, φ)-system by taking c to act through multiplication by c and
b =
∂
∂c
.
(2.9)
This completes the construction of V as a module over the Grassmann numbers
C Z 2 = C 0 ⊕ C 1 , where C 0 resp. C 1 represent the commuting, respectively,
anticommuting numbers. Altogether, V is the tensor product of the space of wave
functions of a scalar particle with the two-dimensional Fock space of the (b, c)system. A generic vector ψ ∈ V is of the form f 0 (φ) + f 1 (φ)c = Ψ i e i , where {e i }
is a homogeneous basis of the vector space V . f 0 and f 1 are C- and Grassmannvalued functions, respectively, interpreted as space-time fields. The degree of Ψ i is
such that Ψ has total degree 0.
The BRST symmetry (2.7) is then generated through commutation with
Q = c H , H =
1
2
g
−1 (p, p) + m
2
.
(2.10)
Remark 2.1 In the BV formalism which we will review below, fields of degree zero
are interpreted as classical fields, while fields of degree −1 are referred to as antifields, see also Sect. 2.3. Also, sometimes it will be convenient to write δ(c) instead
of c. This is possible due to c being of odd degree.
Remark 2.2 To derive the correct BRST transformations in this way, one has to use
the equation of motion g( ˙
φ, ·) = p. Alternatively, one can work in the phase space
where
I [φ, b, c, ¯
π] =
τ f
τ i
p ˙
φ −
1
2
(g
−1 (p, p) + m
2 ) + ib ˙
c
dτ ,
which, in turn, is invariant under
δ BRST φ = −ip(c) ,
δ BRST b =
1
2
p
2
+ m
2
, and δ BRST c = δ BRST p = 0 ,
now generated by (2.10) without using equations of motion.
2 Relativistic Point Particle
acting on a graded vector space V over C spanned by the quantum-mechanical states
ψ of our spinless particle. Here and in what follows we set ¯
h = 1. To construct an
irreducible module for the above canonical commutation relations, we start with
representing the operator φ on functions as the multiplication by the cartesian
coordinates φ i and p as the derivative
p =
1
i
∇ φ .
The degree zero part of the module V is then naturally identified with the vector
space of test functions on M with suitable regularity and integrability properties.
To continue, we realize the anticommuting (b, c)-system on V in analogy to the
(p, φ)-system by taking c to act through multiplication by c and
b =
∂
∂c
.
(2.9)
This completes the construction of V as a module over the Grassmann numbers
C Z 2 = C 0 ⊕ C 1 , where C 0 resp. C 1 represent the commuting, respectively,
anticommuting numbers. Altogether, V is the tensor product of the space of wave
functions of a scalar particle with the two-dimensional Fock space of the (b, c)system. A generic vector ψ ∈ V is of the form f 0 (φ) + f 1 (φ)c = Ψ i e i , where {e i }
is a homogeneous basis of the vector space V . f 0 and f 1 are C- and Grassmannvalued functions, respectively, interpreted as space-time fields. The degree of Ψ i is
such that Ψ has total degree 0.
The BRST symmetry (2.7) is then generated through commutation with
Q = c H , H =
1
2
g
−1 (p, p) + m
2
.
(2.10)
Remark 2.1 In the BV formalism which we will review below, fields of degree zero
are interpreted as classical fields, while fields of degree −1 are referred to as antifields, see also Sect. 2.3. Also, sometimes it will be convenient to write δ(c) instead
of c. This is possible due to c being of odd degree.
Remark 2.2 To derive the correct BRST transformations in this way, one has to use
the equation of motion g( ˙
φ, ·) = p. Alternatively, one can work in the phase space
where
I [φ, b, c, ¯
π] =
τ f
τ i
p ˙
φ −
1
2
(g
−1 (p, p) + m
2 ) + ib ˙
c
dτ ,
which, in turn, is invariant under
δ BRST φ = −ip(c) ,
δ BRST b =
1
2
p
2
+ m
2
, and δ BRST c = δ BRST p = 0 ,
now generated by (2.10) without using equations of motion.