Classical fields, initial data, and waves

Use $\hbar=c=1$, three spatial dimensions, and $\eta=\operatorname{diag}(1,-1,-1,-1)$. A real scalar transforms without an internal spacetime index. Its kinetic term is a time-derivative square minus spatial-gradient squares. A positive $m^2$ gives a restoring term. Keeping these signs together is essential: copying a formula from a mostly-plus convention without changing its partners can turn a stable oscillator into an apparent instability.

Vary the action and integrate by parts to obtain the Klein–Gordon equation. The canonical momentum is the time derivative of the field. A Legendre transform gives a Hamiltonian density built from three nonnegative terms when $m^2\geq0$. This checks the physical interpretation of our signs. The field amplitude is not a single-particle probability wavefunction; its classical energy, rather than an integral of its squared magnitude, is the immediate conserved quantity here.

Insert a plane wave $e^{-i\omega t+i\mathbf k\cdot\mathbf x}$ into the equation. Spatial derivatives turn into momentum factors and leave an ordinary oscillator equation for each Fourier mode. The positive-frequency branch obeys $\omega_{\mathbf k}=\sqrt{\mathbf k^2+m^2}$. Reality requires the negative-frequency part to be the complex conjugate of the positive-frequency part, so the two pieces cannot be independently assigned arbitrary classical amplitudes.

A narrow wave packet has a peak moving approximately at the group velocity $d\omega/dk$. A crest moves at the phase velocity $\omega/k$, which exceeds one for a massive mode. That is not superluminal communication: an infinite sinusoid has no localized beginning or independently addressable message. Front propagation and the domain of dependence are determined by the hyperbolic differential equation, whose characteristic speed is one in our units.

A second-order time equation needs both $\phi(0,\mathbf x)$ and $\dot\phi(0,\mathbf x)$. Specifying only the initial shape leaves the direction and speed of its evolution undetermined. For appropriate regular initial data, the free equation evolves uniquely, and disturbances propagate within the light cone. Boundary conditions also matter in a box: periodic walls give discrete momenta, whereas reflecting conditions produce standing-wave bases with a different spectrum.

The energy density is $\mathcal H=\tfrac12[\dot\phi^2+|\nabla\phi|^2+m^2\phi^2]$. For a closed region with suitable boundaries, its integral stays constant. If a potential has negative curvature around the chosen background, long-wavelength perturbations may grow exponentially; that diagnoses an unstable background, not an ordinary particle outrunning light. Nonlinear terms can stabilize a different background, which is why one should inspect the expansion point before interpreting a mass parameter.