164
16. Lattice Field Theory, and the Renormalization Group Revisited
‘Euclidean’ space-time arises. If we make the replacement t → −iτ , (16.60)
becomes
[
[
f
i 2
]
]
∫
2
‹q
f
1
m(q − q )
∞
(τ
τ )p
′
|
›
−
i
e
− ˆ
H(τ f −τi)
|q
i
f
=
exp
dp
′ exp
−
2π
2(τ f − τ i )
−
2m
−∞
(16.61)
and the integral is a simple convergent Gaussian. Using the result
∫
/
∞
dξe
−bξ
2
π
=
(16.62)
b
−∞
we finally obtain
[
] 1
[
]
2
ˆ
f
H(τ f τi) i
m
m(q
f
− q
i )
2
‹q |e
−
−
|q › =
exp −
.
(16.63)
2π(τ f − τ i )
2(τ f − τ i )
We must now understand how the result (16.63) can be represented in the
form (16.49). In Euclidean space, (16.49) is
( ∫
∑
τ f
( )
)
2
1
dq
exp
paths
−
m
dτ
(16.64)
τi 2
dτ
in the free-particle case. We interpret the τ integral in terms of a discretization
procedure, similar to that introduced in section 16.2. We split the interval
τ f − τ i into N segments each of size ∈, as shown in figure 16.3. The τ -integral
in (16.64) becomes the sum
∑ N (q
j
q
j−1 )
2
m
−
,
(16.65)
2∈
j=1
and the ‘sum over paths’, in going from q
0
≡ q
i at τ i to q
N
≡ q
f at τ f , is
now interpreted as a multiple integral over all the intermediate positions
q
1 , q
2 , . . . , q
N −1 which paths can pass through at ‘times’ τ 1 , τ 2 , . . . , τ N −1 :
[
]
∫ ∫
∫
1
∑ N (q
j
− q
j−1 )
2
exp |
d
. . .
. . .
A(∈)
−m
| dq
1 dq
2
q
N −1
, (16.66)
2∈
A(∈) A(∈)
A(∈)
j=1
where A(∈) is a normalizing factor, depending on ∈, which is to be determined.
The integrals in (16.66) are all of Gaussian form, and since the integral
of a Gaussian is again a Gaussian (cf the manipulations leading from (16.57)
to (16.60), but without the ‘i’ in the exponents), we may perform all the
integrations analytically. We follow the method of Feynman and Hibbs (1965),
section 3.1. Consider the integral over q
1 :
∫
{
I
1
m [
] }
≡ exp −
(q
2
− q
1 )
2 + (q
1
− q
i )
2
dq
1 .
(16.67)
2∈