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10 Free BRST String Field Theory
10.3 Path Integral Quantization
The string field theory can be quantized with a path integral:
Z =
d cl e
−S[ cl ]
=
d cl e
−
1
2 cl |Q B | cl .
(10.57)
An index has been added to the field to emphasize that it is the classical field
(no spacetime ghosts). The simplest way to define the measure is to use the
expansion (9.4) such that
Z =
s
dψ s e
−S[{ψ r }] .
(10.58)
10.3.1 Tentative Faddeev–Popov Gauge Fixing
The action can be gauge fixed using the Faddeev–Popov formalism. The gauge
fixing condition is
F (( cl ) := b 0 | cl = 0.
(10.59)
Its variation under a gauge transformation (10.29) reads
δF = b 0 Q B | cl ,
(10.60)
which implies that the Faddeev–Popov determinant is
det
δF
δδ cl
= det b 0 Q B .
(10.61)
This determinant is rewritten as a path integral by introducing a ghost C and an
anti-ghost B string fields (the prime on B will become clear below):
det b 0 Q B =
dB
dC e
−S FP ,
S FP = − −B
|b 0 Q B |C .
(10.62)
The ghost numbers are attributed by selecting the same ghost number for the C
ghost and for the gauge parameter and then requiring that the Faddeev–Popov action
is non-vanishing:
N gh (B
) = 3,
N gh (C) = 0.
(10.63)