C
Quantum Field Theory
In this appendix, we gather useful information on quantum field theories. The first
section describes how to compute with path integral with non-trivial measures,
generalizing techniques from finite-dimensional integrals. Then, we summarize the
important concepts from the BRST and BV formalisms.
C.1
Path Integrals
In this section, we explain how analysis, algebra and differential geometry are
generalized to infinite-dimensional vector spaces (fields).
C.1.1 Integration Measure
In order to construct a path integral for the field , one needs to define a notion of
distance on the space of fields. The distance between a field and a neighbouring
field + δδ is
|δδ|
2
= G(()(δδ, δδ),
(C.1)
where G is the (field-dependent) metric on the field tangent space (the field
dependence will be omitted when no confusion is possible). This induces a metric
on the field space itself
||
2
= G(()((, ,),
(C.2)
from which the integration measure over the field space can be defined as
d
det G(().
(C.3)
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7
393