140
H. Wittig
This definition includes all basic ingredients: starting from the functional integral
itself avoids any particular reference to perturbation theory. This is what we mean
when we call lattice QCD an ab initio method. The Euclidean formulation, which is
obtained by rotating to imaginary time, reveals the close relation between Quantum
Field Theory and Statistical Mechanics. In particular, the Euclidean functional
integral is equivalent to the partition function of the corresponding statistical system.
This equivalence is particularly transparent if the field theory is formulated on a
discrete space-time lattice. Via this relation, the whole toolkit of condensed matter
physics, including high-temperature expansions, and, perhaps most importantly,
Monte Carlo simulations, are at the disposal of the field theorist.
Many of the basic concepts introduced in this section are discussed in several
common textbooks on the subject [1–4], which can be consulted for further details.
5.2.1 Euclidean Quantization
The generic steps in the Euclidean quantization procedure of a lattice field theory
are the following:
1. Define the classical, Euclidean field theory in the continuum;
2. Discretize the corresponding Lagrangian;
3. Quantize the theory by defining the functional integral;
4. Determine the particle spectrum from Euclidean correlation functions.
We shall now illustrate this procedure for a simple example, namely the theory for
a neutral scalar field.
Step 1 Consider a real, classical field φ(x), with x = (x 0 , x 1 , x 2 , x 3 ), whose time
variable x 0 is obtained by analytically continuing t to −ix 0 . The Euclidean action
S E [φ] is defined as
S E [φ] =
d
4 x
1
2
∂ μ φ(x)∂ μ φ(x) + V (φ)
,
∂ μ ≡
∂
∂x μ
,
(5.1)
where
V (φ) =
1
2
m
2 φ(x)
2
+
λ
4!
φ(x)
4 .
(5.2)
Step 2 In order to discretize the theory, a hyper-cubic lattice, E , is introduced as
the set of discrete space-time points, i.e.
E =
x ∈ R
4
x
0 /a = 1, . . . , N t ; x
j /a = 1, . . . , N s , j = 1, 2, 3
,
T = N t a, L = N s a.
(5.3)
H. Wittig
This definition includes all basic ingredients: starting from the functional integral
itself avoids any particular reference to perturbation theory. This is what we mean
when we call lattice QCD an ab initio method. The Euclidean formulation, which is
obtained by rotating to imaginary time, reveals the close relation between Quantum
Field Theory and Statistical Mechanics. In particular, the Euclidean functional
integral is equivalent to the partition function of the corresponding statistical system.
This equivalence is particularly transparent if the field theory is formulated on a
discrete space-time lattice. Via this relation, the whole toolkit of condensed matter
physics, including high-temperature expansions, and, perhaps most importantly,
Monte Carlo simulations, are at the disposal of the field theorist.
Many of the basic concepts introduced in this section are discussed in several
common textbooks on the subject [1–4], which can be consulted for further details.
5.2.1 Euclidean Quantization
The generic steps in the Euclidean quantization procedure of a lattice field theory
are the following:
1. Define the classical, Euclidean field theory in the continuum;
2. Discretize the corresponding Lagrangian;
3. Quantize the theory by defining the functional integral;
4. Determine the particle spectrum from Euclidean correlation functions.
We shall now illustrate this procedure for a simple example, namely the theory for
a neutral scalar field.
Step 1 Consider a real, classical field φ(x), with x = (x 0 , x 1 , x 2 , x 3 ), whose time
variable x 0 is obtained by analytically continuing t to −ix 0 . The Euclidean action
S E [φ] is defined as
S E [φ] =
d
4 x
1
2
∂ μ φ(x)∂ μ φ(x) + V (φ)
,
∂ μ ≡
∂
∂x μ
,
(5.1)
where
V (φ) =
1
2
m
2 φ(x)
2
+
λ
4!
φ(x)
4 .
(5.2)
Step 2 In order to discretize the theory, a hyper-cubic lattice, E , is introduced as
the set of discrete space-time points, i.e.
E =
x ∈ R
4
x
0 /a = 1, . . . , N t ; x
j /a = 1, . . . , N s , j = 1, 2, 3
,
T = N t a, L = N s a.
(5.3)