1Classical field theory
III Quantum Field Theory
1.4 Hamiltonian mechanics
We can now talk about the Hamiltonian formulation. This can be done for field
theories as well. We define
Definition
(Conjugate momentum)
.
Given a Lagrangian system for a field
φ
,
we define the conjugate momentum by
π(x) =
∂L
∂
˙
φ
.
This is not to be confused with the total momentum P
i
.
Definition (Hamiltonian density). The Hamiltonian density is given by
H = π(x)
˙
φ(x) − L(x),
where we replace all occurrences of
˙
φ(x) by expressing it in terms of π(x).
Example. Suppose we have a field Lagrangian of the form
L =
1
2
˙
φ
2
−
1
2
(∇φ)
2
− V (φ).
Then we can compute that
π =
˙
φ.
So we can easily find
H =
1
2
π
2
+
1
2
(∇φ)
2
+ V (φ).
Definition (Hamiltonian). The Hamiltonian of a Hamiltonian system is
H =
Z
d
3
x H.
This agrees with the field energy we computed using Noether’s theorem.
Definition (Hamilton’s equations). Hamilton’s equations are
˙
φ =
∂H
∂π
, ˙π = −
∂H
∂φ
.
These give us the equations of motion of φ.
There is an obvious problem with this, that the Hamiltonian formulation is
not manifestly Lorentz invariant. However, we know it actually is because we
derived it as an equivalent formulation of a Lorentz invariant theory. So we are
safe, so far. We will later have to be more careful if we want to quantize the
theories from the Hamiltonian formalism.