66 2. THEORIES, LAGRANGIANS AND COUNTERTERMS
exists. Let us suppose, by induction, that these counterterms are local and
independent of L.
Then, we define the next counterterm by
Ii,k
CT
(L, ε) = Singε Wi,k
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT
(ε)⎠
⎞
.
The identity
Wi,k
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT
(ε) −
iIi,k CT
(L,
ε)⎠
⎞
=
Wi,k
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT
(ε)⎠
⎞
− Ii,k
CT
(L, ε)
shows that the limit
lim
ε→0
W≤(i.k)
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT
(ε) −
iIi,k CT
(L,
ε)⎠
⎞
exists.
To show locality of the counterterm Ii,k
CT
(L, ε), it suffices, as before, to
show that it is independent of L. If L L, we have
Ii,k
CT
(L , ε) = Singε Wi,k
⎛
⎝P
(ε, L ),I −
(r,s) (i,k)
rIr,s CT (ε)⎠
⎞
= Singε Wi,k
⎛
⎝P
(L, L ),W
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT
(ε)⎠⎠
⎞⎞
= Singε Wi,k
⎛
⎝P
(L, L ),W
(i,k)
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT (ε)⎠
⎞
+
iWi,k
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT
(ε)⎠⎠
⎞⎞
= Singε Wi,k
⎛
⎝P
(L, L ),W
(i,k)
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT (ε)⎠⎠
⎞⎞
+ Singε Wi,k
⎛
⎝P
(ε, L),I −
(r,s) (i,k)
rIr,s CT
(ε)⎠
⎞
Previous Page Next Page 
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown)
Copyright 2011 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org.
Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.