String Theory Demystified

116 String Theory Demystifi ed
BRST Operators and Introductory Remarks
We begin by considering our old friend, Lie algebra. This is the algebra that is
obeyed by the familiar spin angular momentum operators of ordinary quantum
mechanics. In general, let some physical theory contain a gauge symmetry with
operators K i . These operators satisfy the Lie algebra:
k
[ K , K ] = f K
(6.1)
i j ij
k
where the fij are called the structure constants of the theory (note we are using
the Einstein summation convention, so repeated indices are summed over). The
structure constants satisfy
k
m l m l m l
f f + f f + f f =0 (6.2)
ij
mk
jk
mi
The BRST quantization procedure begins with the following. We introduce two
ghost fi elds that are denoted by bi and c which satisfy an anticommutation relation
j
given by
ki
mj
i
i
{ c , b } = δ (6.3)
j j
i j
Furthermore { c , c } = { bi, bj}
=0. Here the c i
are ghost fi elds and the bj are
“ghost momenta.”
Notice that since an anticommutation relation is satisfi ed by the ghost fi elds,
these fi elds are fermionic. Now, recall that a fi eld φ( z, z ) has conformal dimension
( h, h ) provided that it transforms under some conformal transformation z→ w() z
as follows:
h h
w w
φ(, zz)
= ϕ(
ww , )
z z
∂ ⎛ ⎞ ⎛ ∂ ⎞
⎝
⎜
∂ ⎠
⎟
⎝
⎜
∂ ⎠
⎟ (6.4)
The fi elds b and c are chosen such that they have conformal dimension 2 and
1, as we will see later.
There are two operators that are constructed out of the ghost fi elds and the K . i
The fi rst of these is the BRST operator which is given by
i
k i j
Q= c Ki − fij cc b
1
2
k
(6.5)