6.4 Operator Formalism and Radial Quantization
137
Computation: Equation (6.160c)
: e
A(z)
: : e
B(w)
: =
m,n
1
m!n!
: A(z)
m
: : B(w)
n
:
=
m,n,k
k!
m!n!
m
k
n
k
A(z)B(w)
k : A(z)
m−k
: : B(w)
n−k
:
=
m,n,k
1
k!(m − k)!(n − k)!
A(z)B(w)
k : A(z)
m−k
: : B(w)
n−k
:.
The factorial k! counts the number of possible ways to contract the two
operators.
The general properties of normal ordered expressions are identical for both
vacua: what differs is the precise computation in terms of the operators (or modes).
Hence, the energy normal ordering can be defined in parallel with (6.157), but
changing the definitions of creation and annihilation operators:
AB(z)
=
m
AB(z)
n
z m+h A +h B
,
(6.162a)
AB
m =
n≤0
A n B m−n +
n>0
B m−n A n .
(6.162b)
To simplify the definition we assume that A 0 is a creation operator and it is thus
included in the first sum (this must be adapted in function of which vacuum state is
chosen if the latter is degenerate).
The relation between the normal ordered modes is
: AB : m =
AB
m +
h A −1
n=0
[B m+n , A −n ].
(6.163)
Computation: Equation (6.163)
: AB : m =
n≤−h A
A n B m−n +
n>−h A
B m−n A n
=
n≥h A
A −n B m+n +
n>0
B m−n A n +
h A −1
n=0
B m+n A −n