16.2. Discretization
153
that we shall still want to formulate the theory in terms of an action of the
form
∫
˙
S = d
4 x L(φ, ∇φ, φ).
(16.1)
It seems plausible that it might be advantageous to treat space and time
as symmetrically as possible, from the start, by formulating the theory in
‘Euclidean’ space, instead of Minkowskian, by introducing t = −iτ ; further
motivation for doing this will be provided in section 16.3. In that case, the
action (16.1) becomes
∫
S → −i d
3
∂φ
xdτ L(φ, ∇φ, i )
(16.2)
∂τ
∫
≡ i d
3
xdτ L E ≡ iS E .
(16.3)
A typical free scalar action is then
∫
1
[
]
S E (φ) =
d
3
xdτ (∂ τ φ)
2 + (∇φ)
2 + m
2 φ
2 .
(16.4)
2
We now represent all of space-time by a finite-volume ‘hypercube’. For
example, we may have N 1 lattice points along the x-axis, so that a field φ(x)
is replaced by the N 1 numbers φ(n 1 a) with n 1 = 0, 1, . . . N 1 − 1. We write
L = N 1 a for the length of the cube side. In this notation, integrals and
differentials are replaced by the finite sums and difference expressions
∫
∑
∂φ
1
dx → a
,
→ [φ(n 1 + 1) − φ(n 1 )],
(16.5)
∂x
a
n1
so that a typical integral (in one dimension) becomes
∫
( ) 2
∂φ
∑ 1
2
dx
→ a
[φ(n 1 + 1) φ(n 1 )] .
(16.6)
∂x
a 2
n1
−
As in all our previous work, we can alternatively consider a formulation in
momentum space, which will also be discretized. It is convenient to impose
periodic boundary conditions such that φ(x) = φ(x + L). Then the allowed
k-values may be taken to be k ν1 = 2πν 1 /L with ν 1 = −N 1 /2+1, . . . 0, . . . N 1 /2
(we take N 1 to be even). It follows that the maximum allowed magnitude of
the momentum is then π/a, indicating that a
−1 is (as anticipated) playing
the role of our earlier momentum cut-off Λ. We then write
∑
1
φ(n ) =
e
i2πν1n1/N1 ˜
1
1
φ(ν 1 ),
(16.7)
2
ν (N 1 a)
1
which has the inverse
( ) 1
a
2 ∑
˜
φ(ν 1 ) =
e
−i2πν1n1/N1 φ(n 1 ),
(16.8)
N 1
n1
153
that we shall still want to formulate the theory in terms of an action of the
form
∫
˙
S = d
4 x L(φ, ∇φ, φ).
(16.1)
It seems plausible that it might be advantageous to treat space and time
as symmetrically as possible, from the start, by formulating the theory in
‘Euclidean’ space, instead of Minkowskian, by introducing t = −iτ ; further
motivation for doing this will be provided in section 16.3. In that case, the
action (16.1) becomes
∫
S → −i d
3
∂φ
xdτ L(φ, ∇φ, i )
(16.2)
∂τ
∫
≡ i d
3
xdτ L E ≡ iS E .
(16.3)
A typical free scalar action is then
∫
1
[
]
S E (φ) =
d
3
xdτ (∂ τ φ)
2 + (∇φ)
2 + m
2 φ
2 .
(16.4)
2
We now represent all of space-time by a finite-volume ‘hypercube’. For
example, we may have N 1 lattice points along the x-axis, so that a field φ(x)
is replaced by the N 1 numbers φ(n 1 a) with n 1 = 0, 1, . . . N 1 − 1. We write
L = N 1 a for the length of the cube side. In this notation, integrals and
differentials are replaced by the finite sums and difference expressions
∫
∑
∂φ
1
dx → a
,
→ [φ(n 1 + 1) − φ(n 1 )],
(16.5)
∂x
a
n1
so that a typical integral (in one dimension) becomes
∫
( ) 2
∂φ
∑ 1
2
dx
→ a
[φ(n 1 + 1) φ(n 1 )] .
(16.6)
∂x
a 2
n1
−
As in all our previous work, we can alternatively consider a formulation in
momentum space, which will also be discretized. It is convenient to impose
periodic boundary conditions such that φ(x) = φ(x + L). Then the allowed
k-values may be taken to be k ν1 = 2πν 1 /L with ν 1 = −N 1 /2+1, . . . 0, . . . N 1 /2
(we take N 1 to be even). It follows that the maximum allowed magnitude of
the momentum is then π/a, indicating that a
−1 is (as anticipated) playing
the role of our earlier momentum cut-off Λ. We then write
∑
1
φ(n ) =
e
i2πν1n1/N1 ˜
1
1
φ(ν 1 ),
(16.7)
2
ν (N 1 a)
1
which has the inverse
( ) 1
a
2 ∑
˜
φ(ν 1 ) =
e
−i2πν1n1/N1 φ(n 1 ),
(16.8)
N 1
n1