248
R. A. Iseppi
which acts on X 0 as follows:
G × X 0 −→ X 0
(u, ϕ) → uϕu ∗ + u[D, u ∗ ].
The proof of the above classical Proposition is a straightforward checking. We
remark that a similar construction can be performed also in the infinite dimensional
case. To conclude the section, we recall that there is another notion of action
for spectral triples, which will play a key role in our construction: the so-called
fermionic action.
Definition 5 For a finite spectral triple (A, H, D) (finite real spectral triple
(A, H, D, J )) the fermionic action on H f ⊆ H is given by
S ferm [ϕ] =
1
2
ϕ, Dϕ
S ferm [ϕ] =
1
2
J ϕ, Dϕ
;
for ϕ ∈ H f .
3 The BV Construction in the Algebraic Context
The Batalin–Vilkovisky (BV) formalism (cf. [1, 2]) can be viewed as the end point
of a long path, which had its motivation in the problem of defining the path integral
(cf.[16]) for gauge theories and its origin in the introduction of the concept of ghost
field by Faddeev and Popov in 1967 [14]. As suggested by the name, the ghost fields
are non-existing particles, whose function is to compensate the presence of local
symmetries and hence the appearance of divergences in the path integral. Moreover,
next to the ghost fields, the BV formalism requires also the introduction of all the
corresponding anti-fields/anti-ghost fields.
Definition 6 A field/ghost field ϕ is a graded variable characterized by two integers:
deg(ϕ) ∈ Z and (ϕ) ∈ {0, 1}, with deg(ϕ) = (ϕ) (mod Z/2).
deg(ϕ) is the ghost degree, while (ϕ) is the parity, which distinguishes between the
bosonic case, where = 0 and ϕ behaves as a real variable, and the fermionic
case, where = 1 and ϕ behaves as a Grassmannian variable:
ϕψ = −ψϕ,
and
ϕ
2
= 0,
if = = 1.
The anti-field/anti-ghost field ϕ ∗ corresponding to a field/ghost field ϕ satisfies
deg(ϕ
∗ ) = − deg(ϕ) − 1,
and
(ϕ
∗ ) = (ϕ) + 1, (mod Z/2).
Then, given an initial gauge theory (X 0 , S 0 ), the BV construction associates with
that a new pair (
X, S), where the extended configuration space
X and the extended
action S are given by:
R. A. Iseppi
which acts on X 0 as follows:
G × X 0 −→ X 0
(u, ϕ) → uϕu ∗ + u[D, u ∗ ].
The proof of the above classical Proposition is a straightforward checking. We
remark that a similar construction can be performed also in the infinite dimensional
case. To conclude the section, we recall that there is another notion of action
for spectral triples, which will play a key role in our construction: the so-called
fermionic action.
Definition 5 For a finite spectral triple (A, H, D) (finite real spectral triple
(A, H, D, J )) the fermionic action on H f ⊆ H is given by
S ferm [ϕ] =
1
2
ϕ, Dϕ
S ferm [ϕ] =
1
2
J ϕ, Dϕ
;
for ϕ ∈ H f .
3 The BV Construction in the Algebraic Context
The Batalin–Vilkovisky (BV) formalism (cf. [1, 2]) can be viewed as the end point
of a long path, which had its motivation in the problem of defining the path integral
(cf.[16]) for gauge theories and its origin in the introduction of the concept of ghost
field by Faddeev and Popov in 1967 [14]. As suggested by the name, the ghost fields
are non-existing particles, whose function is to compensate the presence of local
symmetries and hence the appearance of divergences in the path integral. Moreover,
next to the ghost fields, the BV formalism requires also the introduction of all the
corresponding anti-fields/anti-ghost fields.
Definition 6 A field/ghost field ϕ is a graded variable characterized by two integers:
deg(ϕ) ∈ Z and (ϕ) ∈ {0, 1}, with deg(ϕ) = (ϕ) (mod Z/2).
deg(ϕ) is the ghost degree, while (ϕ) is the parity, which distinguishes between the
bosonic case, where = 0 and ϕ behaves as a real variable, and the fermionic
case, where = 1 and ϕ behaves as a Grassmannian variable:
ϕψ = −ψϕ,
and
ϕ
2
= 0,
if = = 1.
The anti-field/anti-ghost field ϕ ∗ corresponding to a field/ghost field ϕ satisfies
deg(ϕ
∗ ) = − deg(ϕ) − 1,
and
(ϕ
∗ ) = (ϕ) + 1, (mod Z/2).
Then, given an initial gauge theory (X 0 , S 0 ), the BV construction associates with
that a new pair (
X, S), where the extended configuration space
X and the extended
action S are given by: