166
16. Lattice Field Theory, and the Renormalization Group Revisited
Hence, after N − 1 steps we shall have a factor
[
]
exp −m(q
f
− q
i )
2 /2(τ f − τ i ) ,
(16.72)
remembering that q
N
≡ q
f and that τ f − τ i = N ∈. So we have recovered the
correct exponential factor of (16.63), and all that remains is to choose A(∈) in
(16.66) so as to produce the same normalization as (16.63).
The required A(∈) is
/
2π∈
A(∈) =
,
(16.73)
m
as we now verify. For the first (q
1 ) integration, the formula (16.66) contains
two factors of A
−1 (∈), so that the result (16.68) becomes
( )
1
m π∈
1
[
]
2
I
1
m
=
exp
2
[A(∈)]
2
m
− (q
2
i
∈
− q )
2
π∈
4
(
)
[
]
m
1
2
m
=
exp
2 2∈
− (q
2
4∈
− q
i )
2 .
(16.74)
π
For the second (q
2 ) integration, the accumulated constant factor is
(
)
1
1 (
) 1
m
4π∈
1
2
(
)
2
m 2
=
.
(16.75)
A(∈) 2π2∈
3m
2π3∈
Proceeding in this way, one can convince oneself that after N − 1 steps, the
accumulated constant is
(
)
m
1
[
] 1
2
m
2
=
,
(16.76)
2πN ∈
2π(τ f − τ i )
as in (16.63).
The equivalence of (16.63) and (16.64) (in the sense ∈ → 0) is therefore
established for the free-particle case. More general cases are discussed in
Feynman and Hibbs (1965) chapter 5, and in Peskin and Schroeder (1995)
chapter 9. The conventional notation for the path-integral amplitude is
∫
‹
f
|
− ˆ
H τ −
D
∫ τ
(
τ )
|
i
›
− f L dτ
q e
f
i q =
q(τ )e
τ i
,
(16.77)
where the right-hand side of (16.77) is interpreted in the sense of (16.66).
We now proceed to discuss further aspects of the path-integral formulaˆ
tion. Consider the (Euclideanized) amplitude ‹q
f
|e
−H(τ f −τi)
|q
i
›, and insert a
complete set of energy eigenstates |n›
ˆ
such that H|n› = E n |n›:
∑
‹
ˆ
q
f
|e
−H(τ f −τi)
|q
i
› =
‹q
f
|n›‹n|q
i
16.
n
›e
−En(τ f −τi) .
( 78)
Equation (16.78) shows that if we take the limits τ i → −∞, τ f → ∞, then the