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.