22
2 Relativistic Point Particle
Remark 2.9 One might worry that, since the action (2.16) was constructed such
as to reproduce an n-particle scattering in a specific background, the whole
construction may not be background independent . That this is not so follows from
the observation that the definition of Q in (2.10) is universal, i.e., independent of ψ.
In fact, the presentation of the ψ 3 -field theory in this section is “reversed.” Indeed,
the usual starting point is the action (2.16), which is background independent
and one then derives the scattering amplitudes (2.15). However, we chose this
presentation for pedagogical reasons as will hopefully become clear in sections on
string field theory.
Although we have elaborated here on the simple example of a point particle with
the trivial ghost and gauge sectors, most of the algebraic structure discussed in this
section will hold for a generic BV theory and in particular for string field theory. One
important modification is that, due to the non-trivial gauge structure of the string,
the nilpotent algebras we encountered for the point particles will be replaced by
homotopy Lie- or associative algebras which will be the relevant algebraic structure
for the string.
2.3
Summary, Comments, and Remarks Towards Part II
The path integral (2.11) has an interpretation as a symmetric monoidal functor E
from the one-dimensional bordism category to the category of vector spaces. This
functor associates to a point ∗ the Hilbert space E(∗) = V of functions on the
“mapping” space {∗ → M × R 0|1 } = M × R 0|1 , where R 0|1 is the odd line generated
by the ghost c. The functional integral associates to a bordism (interval Δτ ) between
two points a map V → V through the integration of the kernel K(Δτ, φ 1 , φ 0 )
against the wave functions ψ(φ 0 , c). We will not elaborate on this point of view any
further.
The point we wished to elucidate here is that the space V naturally carries the
structure of a nilpotent abelian differential graded (dg) algebra (V , ∗, Q) equipped
with an invariant odd symplectic form ω. In addition,
S[ψ] =
1
2
ω(ψ, Q ψ) +
1
3
ω(ψ, ψ ∗ ψ)
(2.38)
defines a BV action on V , i.e., it satisfies the classical BV master equation {S, S} =
0. Equivalently, (Q + ∗) 2 = 0. Furthermore, with ψ = f 0 + f 1 c, f 0 of degree 0 and
f 1 of degree −1, we have
ψ ∗ ψ = cf 0 f 0 , Qψ = 1/2c( + m
2 )f 0 and ψ, ψ
=
M
f 0 f
1 + f
0 f 1 .
2 Relativistic Point Particle
Remark 2.9 One might worry that, since the action (2.16) was constructed such
as to reproduce an n-particle scattering in a specific background, the whole
construction may not be background independent . That this is not so follows from
the observation that the definition of Q in (2.10) is universal, i.e., independent of ψ.
In fact, the presentation of the ψ 3 -field theory in this section is “reversed.” Indeed,
the usual starting point is the action (2.16), which is background independent
and one then derives the scattering amplitudes (2.15). However, we chose this
presentation for pedagogical reasons as will hopefully become clear in sections on
string field theory.
Although we have elaborated here on the simple example of a point particle with
the trivial ghost and gauge sectors, most of the algebraic structure discussed in this
section will hold for a generic BV theory and in particular for string field theory. One
important modification is that, due to the non-trivial gauge structure of the string,
the nilpotent algebras we encountered for the point particles will be replaced by
homotopy Lie- or associative algebras which will be the relevant algebraic structure
for the string.
2.3
Summary, Comments, and Remarks Towards Part II
The path integral (2.11) has an interpretation as a symmetric monoidal functor E
from the one-dimensional bordism category to the category of vector spaces. This
functor associates to a point ∗ the Hilbert space E(∗) = V of functions on the
“mapping” space {∗ → M × R 0|1 } = M × R 0|1 , where R 0|1 is the odd line generated
by the ghost c. The functional integral associates to a bordism (interval Δτ ) between
two points a map V → V through the integration of the kernel K(Δτ, φ 1 , φ 0 )
against the wave functions ψ(φ 0 , c). We will not elaborate on this point of view any
further.
The point we wished to elucidate here is that the space V naturally carries the
structure of a nilpotent abelian differential graded (dg) algebra (V , ∗, Q) equipped
with an invariant odd symplectic form ω. In addition,
S[ψ] =
1
2
ω(ψ, Q ψ) +
1
3
ω(ψ, ψ ∗ ψ)
(2.38)
defines a BV action on V , i.e., it satisfies the classical BV master equation {S, S} =
0. Equivalently, (Q + ∗) 2 = 0. Furthermore, with ψ = f 0 + f 1 c, f 0 of degree 0 and
f 1 of degree −1, we have
ψ ∗ ψ = cf 0 f 0 , Qψ = 1/2c( + m
2 )f 0 and ψ, ψ
=
M
f 0 f
1 + f
0 f 1 .