a
v x
u
+
v u
x
b
u=v x
+
u=v x
Figure 9: Schematic representation of Lu, v ; x for a u 6= v and b u = v. Here, the tilted arrows
denote spatial bonds, while the short double line segments at u in Case a denote unspecified bonds that could be spatial or temporal.
5.1 Constructions: I
First, in Section 5.1.1, we introduce several constructions that will be used in the following sections to define bounding diagrams on relevant quantities. Then, in Section 5.1.2, we show that these
constructions can be used iteratively by studying the effect of applying constructions to diagram functions. Such iterative bounds will be crucial in Sections 5.2–5.3 to prove Proposition 5.1.
5.1.1 Definitions of constructions
For b = u, v with u = u, s and v = v, s + ǫ, we will abuse notation to write pb or pv − u
for p
ǫ
v − u, and Db or Dv − u for Dv − u. Let ϕx − u = p
⋆
τx − u, 5.5
and see Figure 9
Lu, v ; x = ϕx − u τ
⋆
λǫDx − v + ϕ
⋆
λǫDx − u τx − v u 6= v,
λǫD
⋆
τx − u τ
⋆
λǫDx − u + λǫD
⋆
τ
⋆
λǫDx − u τx − u u = v,
5.6 where
ϕ for u 6= v corresponds to λǫD
⋆
τ for u = v. We call the lines from u to x in Lu, v ; x the
L-admissible lines. Here, with lines, we mean
ϕx − u and ϕ
⋆
λǫDx − u when u 6= v. If u = v, then we define both lines from u to x in each term in Lu, u; x to be L-admissible. We note that
these lines can be represented by 2-point functions as, e.g., ϕ
⋆
λǫDx − u =
X
b=u,
·
X
b
′
= · ,x
spatial
τb − b τb
′
− b τb
′
− b
′
. 5.7
Thus, below, we will frequently interpret lines to denote 2-point functions. We will use the following constructions to prove Proposition 5.1:
841
Definition 5.2 Constructions B, ℓ, 2
i
and E. i Construction B. Given any diagram line
η, say
τx − v, and given y 6= x , we define Construction B
η spat
y to be the operation in which τx − v is replaced by
τy − v λǫD
⋆
τx − y =
x
v y
, 5.8
and define Construction B
η temp
y to be the operation in which τx − v is replaced by
X
b= · ,y
τb − v λǫDb Pb, b
+
−→ x =
v y
x
, 5.9
where {b −→ x } = {b is occupied} ∩ {b −→ x } and v
+
= v, t
v
+
ǫ for v = v, t
v
. Con- struction B
η
y applied to τx − v is the sum of τx − vδ
x ,y
and the results of Construc- tion B
η spat
y and Construction B
η temp
y applied to τx − v. Construction B
η
s is the opera- tion in which Construction B
η
y, s is performed and then followed by summation over y ∈ Z
d
. Constructions B
η spat
s and B
η temp
s are defined similarly. We omit the superscript η and write,
e.g., Construction By when we perform Construction B
