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 $\mathbf{q} = (q_1, q_2, \ldots, q_N)$ and momentum $\mathbf{p} = (p_1, p_2, \ldots, p_N)$ coordinates. The Hamiltonian function $H(\mathbf{q}, \mathbf{p}, t)$ then generates the time evolution of the system via Hamilton's equations:
$$ \dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i} $$for $i = 1, 2, \ldots, N$.
In Hamiltonian field theory, instead of having discrete $N$ particles, you have a field $\phi(x, t)$, 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 $\phi(x)$ and their conjugate momenta $\pi(x)$. The Hamiltonian density $\mathcal{H}[\phi(x), \pi(x), x]$ 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:
$$ H = \int d^3x \, \mathcal{H}[\phi(x), \pi(x), x]. $$Using variational principles, one can derive the equations of motion for the field variables $\phi(x)$ and $\pi(x)$, which are often partial differential equations. The initial conditions for these equations would then indeed be the field and its conjugate momentum at $t = 0$:
$$ \phi(x, t=0) = \phi_0(x), \quad \pi(x, t=0) = \pi_0(x). $$So we obtain evolution equations for the fields.
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Author of the notes: Antonio J. Pan-Collantes
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