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 particle 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 conjugate momentum $π (x)$ associated with the field $ϕ (x)$ is defined as:

π (x) = \frac{\partial L}{\partial \dot{ϕ} (x)}

where $L = L (ϕ, \dot{ϕ}, x)$ is the Lagrangian density of the field theory. This calculation can be duly formalized in the context of jet bundles.

Once $π (x)$ is defined, the Hamiltonian density is obtained via the Legendre transformation:

H = π (x) \dot{ϕ} (x) - L .

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

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

The equations of motion for $ϕ (x)$ and $π (x)$ are the field-theoretic versions of Hamilton's equations:

\dot{ϕ} (x) = \frac{δ H}{δ π (x)}, \dot{π} (x) = - \frac{δ H}{δ ϕ (x)} .

Here, $\frac{δ H}{δ ϕ (x)}$ and $\frac{δ H}{δ π (x)}$ are functional derivatives, which are the field-theoretic analogs of partial derivatives.

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 [ϕ, π] = \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.
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.

Let's calculate the equations of motion for $ϕ (x)$ derived from this Hamiltonian. First, let's compute $\frac{δ H}{δ π (x)}$ . From Hamilton's equations, this will give us $\dot{ϕ} (x)$ . I know that $\frac{δ H}{δ π (x)}$ is interpreted as the measure of the variation of the functional $H$ when we uniquely modify the value of the function $π$ at the particular value $x$ . We can write

\frac{δ H}{δ π (x)} = \frac{d}{d ϵ} H [ϕ, π + ϵ δ_{x}] |_{ϵ = 0}

where $δ_{x} (y) = δ (y - x)$ is the Dirac delta. Observe that

H [ϕ, π + ϵ δ (x - y)] = \int d x [\frac{(π (x) + ϵ δ (x - y))^{2}}{2} + (terms independent of π)] .

Expanding the square and keeping only terms linear in $ϵ$ (since we're taking a derivative), we get:

H [ϕ, π + ϵ δ (x - y)] \approx H [ϕ, π] + ϵ \int d x π (x) δ (x - y) + O (ϵ^{2}) .

The functional derivative is then:

\frac{δ H}{δ π (y)} = π (y) .

Thus, the first Hamilton's equation gives:

\dot{ϕ} (y) = π (y) .

Now, let's compute $\frac{δ H}{δ ϕ (x)}$ . This will give us $- \dot{π} (x)$ . We have:

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

Expanding and keeping only terms linear in $ϵ$ , we get:

H [ϕ + ϵ δ (x - y), π] \approx H [ϕ, π] + ϵ \int d x [\frac{\partial ϕ (x)}{\partial x} \frac{\partial δ (x - y)}{\partial x} + m^{2} ϕ (x) δ (x - y)],

where $\frac{\partial δ (x - y)}{\partial x}$ is symbolic. But using integration by parts, we get:

\int d x \frac{\partial ϕ (x)}{\partial x} \frac{\partial δ (x - y)}{\partial x} = - \int d x \frac{\partial^{2} ϕ (x)}{\partial x^{2}} δ (x - y) = - \frac{\partial^{2} ϕ (y)}{\partial y^{2}} .

Thus, the functional derivative is:

\frac{δ H}{δ ϕ (y)} = - \frac{\partial^{2} ϕ (y)}{\partial y^{2}} + m^{2} ϕ (y) .

and Hamilton equations are

\begin{aligned} \dot{ϕ} (y) & = π (y), \\ \dot{π} (y) & = \frac{\partial^{2} ϕ (y)}{\partial y^{2}} - m^{2} ϕ (y) . \end{aligned}

Which is nothing other than the Klein--Gordon equation:

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