$#*! I unlearned #2: causality and antiparticle

In the $#*! I unlearned #1, we saw that the propagator for a real Klein-Gordon scalar field is finite for two space-like separated points, which seems to violate causality. But not really. In the question of causality, we should not ask whether particles can propagate over space like intervals, but whether a measurement performed in one point can affect a measurement at another space-like separated point.

\[ \begin{aligned} [\phi(x), \phi(y)] &= \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2 p^0}} \int \frac{d^3q}{(2\pi)^3} \frac{1}{\sqrt{2 q^0}} \left[a_{\mathbf{p}} e^{-i p \cdot x} + a_{\mathbf{p}}^{\dagger} e^{i p \cdot x} ,\; a_{\mathbf{q}} e^{-i q \cdot y} + a_{\mathbf{q}}^{\dagger} e^{i q \cdot y} \right] \\ &= \int \frac{d^3p}{(2\pi)^3} \frac{1}{2 p^0} \left( e^{-ip(x-y)} - e^{ip(x-y)} \right) = D(x-y) - D(y-x) \end{aligned} \]

When \( (x - y)^2 < 0 \), we can perform a Lorentz transformation from \( (x-y) \) to \( - (x-y) \)--going around the light cone--so that the two propagators are actually equal and therefore exactly cancel; causality if preserved. When \( (x - y)^2 > 0 \), there is no such Lorentz transformation. In this case, the amplitude is nonzero, and roughly \( ( e^{-imt} - c.c ) \) for the special case \( \mathbf{x} - \mathbf{y} = 0 \), which we also calculated. Therefore: no measurement in the Klein-Gordon theory can affect another measurement outside the light-cone.

$#*! I unlearned #1: scalar field theory

The Hamiltonian:

\[ H = \frac12 \int d^3 x \left[ \pi^2 + (\nabla \phi)^2 + m^2\phi^2 \right] \]

where \( \pi(\mathbf{x}) \) is the conjugate momentum density of field \( \phi(\mathbf{x}) \) is quadratic. In particular by going to momentum space with:

\begin{align} \phi(\mathbf{x}) &= \int \frac{d^3p}{(2\pi)^3} \sqrt{\frac{1}{2\omega_{\mathbf{p}}}} \left(a_{\mathbf{p}} + a_{-\mathbf{p}}^{\dagger} \right) e^{i \mathbf{p} \cdot \mathbf{x}} \\ \pi(\mathbf{x}) &= \int \frac{d^3p}{(2\pi)^3} (-i) \sqrt{\frac{\omega_{\mathbf{p}}}{2}} \left(a_{\mathbf{p}} - a_{-\mathbf{p}}^{\dagger} \right) e^{i \mathbf{p} \cdot \mathbf{x}} \end{align}

\( H \) become just sum of uncouple oscillators:

\[ H = \int \frac{d^3p}{(2\pi)^3} \omega_{\mathbf{p}} \left( a_{\mathbf{p}}^{\dagger} a_{\mathbf{p}} + \frac12 [a_{\mathbf{p}}, a_{\mathbf{p}}^{\dagger} ] \right) \]

with well-known spectrum and stuff. More important are these commutation relations:

\[ [ H, a_{\mathbf{p}} ] = - \omega_{\mathbf{p}} a_{\mathbf{p}} \,,\quad [ H, a_{\mathbf{p}}^{\dagger} ] = \omega_{\mathbf{p}} a_{\mathbf{p}}^{\dagger} \]