Hamiltonian field theory

In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory.

The transition from Hamiltonian mechanics to Hamiltonian field theory can be thought of as generalizing the formalism from a finite number of particles to a field, which has an infinite number of degrees of freedom.

In Hamiltonian mechanics, you usually deal with a system of $N$ particles, and the state of the system is described by position $q = (q_{1}, q_{2}, \dots, q_{N})$ and momentum $p = (p_{1}, p_{2}, \dots, p_{N})$ coordinates. The Hamiltonian function $H (q, p, t)$ then generates the time evolution of the system via Hamilton's equations:

{\dot{q}}_{i} = \frac{\partial H}{\partial p_{i}}, {\dot{p}}_{i} = - \frac{\partial H}{\partial q_{i}}

for $i = 1, 2, \dots, N$ .

In Hamiltonian field theory, instead of having discrete $N$ particles, you have a field $ϕ (x, t)$ , which can be thought of as having a continuum of degrees of freedom indexed by the spacetime point $x$ . Analogous to positions and momenta, in field theory, you introduce field variables $ϕ (x)$ and their conjugate momenta $π (x)$ . The Hamiltonian density $H [ϕ (x), π (x), x]$ now plays a similar role to the Hamiltonian $H$ in mechanics. The equations of motion are then derived in a way analogous to Hamilton's equations but with partial derivatives replacing ordinary derivatives and integrals replacing sums.

The Hamiltonian for the entire field configuration is obtained by integrating the Hamiltonian density over all space:

H = \int d^{3} x H [ϕ (x), π (x), x] .

Using variational principles, one can derive the equations of motion for the field variables $ϕ (x)$ and $π (x)$ , which are often partial differential equations. The initial conditions for these equations would then indeed be the field and its conjugate momentum at $t = 0$ :

ϕ (x, t = 0) = ϕ_{0} (x), π (x, t = 0) = π_{0} (x) .

So we obtain evolution equations for the fields.

Example: Klein--Gordon field

Start with $N$ coupled harmonic oscillators arranged in a lattice, with positions $q_{i}$ and momenta $p_{i}$ . The Hamiltonian for this system can be written as:

H = \sum_{i = 1}^{N} [\frac{p_{i}^{2}}{2 m} + \frac{1}{2} m ω^{2} q_{i}^{2} + \frac{k}{2} (q_{i + 1} - q_{i})^{2}],

where $m$ is the mass of each oscillator, $ω$ is the natural frequency, and $k$ is the spring constant coupling adjacent oscillators.
For the continuum limit we introduce a continuous field $ϕ (x)$ such that $q_{i} \approx ϕ (x_{i})$ , where $x_{i} = i a$ , and $a$ is the spacing between oscillators. In the limit $a \to 0$ (with $x$ treated as continuous and the number of oscillators $N \to \infty$ ), the discrete differences $q_{i + 1} - q_{i}$ become derivatives:

\frac{q_{i + 1} - q_{i}}{a} \to \frac{\partial ϕ (x)}{\partial x} .

If we replace sums over $i$ with integrals over $x$ , and rewrite the Hamiltonian in terms of $ϕ (x)$ :

H = \int d x [\frac{π (x)^{2}}{2} + \frac{1}{2} {(\frac{\partial ϕ (x)}{\partial x})}^{2} + \frac{1}{2} m^{2} ϕ (x)^{2}],

where $π (x)$ is the conjugate momentum field and $m^{2}$ (related to $ω^{2}$ and $k$ ) plays the role of the "mass" term for the field.
The equations of motion for $ϕ (x)$ derived from this Hamiltonian correspond to the Klein--Gordon equation:

\frac{\partial^{2} ϕ}{\partial t^{2}} - \frac{\partial^{2} ϕ}{\partial x^{2}} + m^{2} ϕ = 0.

In this process:

The field $ϕ (x)$ replaces the discrete positions $q_{i}$ of oscillators.
The coupling between oscillators translates into the spatial derivative (gradient) term $(\partial ϕ / \partial x)^{2}$ in the Hamiltonian.
The Klein-Gordon equation naturally emerges as the field's equation of motion.