In Section 2.2 of their QFT textbook, Peskin & Schroeder introduce the Lagrangian and Hamiltonian field theories of a classical scalar field. While defining the action $S[\phi]$ and deriving the Euler-Lagrange equation for the classical scalar field $\phi$, the classical scalar field $\phi$ is considered to be a function of a position $4$-vector $x = (x^0, x^1, x^2, x^3) = (ct, x, y, z)$.
Then on page no. 16, suddenly they start writing their classical scalar field $\phi$ as function of only position vector $\textbf{x} = (x, y, z)$ and write Hamiltonian $H$ as (Eq. (2.5))
$$H = \int d^3x \left[\pi(\textbf{x}) \dot{\phi}(\textbf{x}) – \mathcal{L}\right].\tag{2.5}$$
After that they again switch back to using the position $4$-vector $x$ in the discussion of Klein-Gordon equation and write equations where $\phi$ is a function of $x$; e.g., Eq. (2.8) on page no. 17:
$$H = \int d^3x \, \mathcal{H} = \int d^3x \left[\frac{1}{2}\pi^2 + \frac{1}{2} (\nabla\phi)^2 + \frac{1}{2} m^2\phi^2\right]. \tag{2.8}$$
My Questions
Why is the position $4$-vector $x$ changed to the position 3-vector $\textbf{x}$? What is the motivation for that?
Best Answer
When you transform from the Lagrangian to the Hamiltonian picture, you necessarily must choose a particular foliation of spacetime -- that is, you must single out a particular time direction, and consider surfaces of constant time. One simple way to understand this is that while the Lagrangian is a Lorentz scalar, the Hamiltonian density is the timelike component of the four-momentum density, so its value depends on your choice of space and time axes.
Of course, with that said, there's no reason I can think of that you can't let your classical fields include their time dependence once you pick a time direction. My best guess is that Peskin and Schroeder initially eliminated the time dependence to emphasize that they're considering one particular time slice. They could also have in mind that they're about to quantize their field theory in the Schrodinger picture, where the fields certainly do not depend on time. But they could also just be a bit sloppy here.