Klein–Gordon Plane-Wave Solutions

Plane waves diagonalize the free Klein–Gordon equation, but a wave packet requires two independent coefficient functions. They encode the initial amplitude and its time derivative. Solving for these coefficients makes frequency selection, reality, and wave-packet motion precise without assigning a probability interpretation to each mode.

Required background. The Klein–Gordon Equation supplies the mass shell and Cauchy data. Helpful background. Relativistic Phase Space explains how invariant amplitudes differ from ordinary Fourier coefficients.

Fourier modes and two independent amplitudes

Take $m>0$ and sufficiently regular free data on $\mathbb R^3$. Write $E_{\mathbf p}=\sqrt{m^2c^4+c^2\mathbf p^2}>0$ and use the unitary spatial Fourier convention. The general complex solution is

$$\phi(t,\mathbf x)=\int\frac{d^3p}{(2\pi\hbar)^{3/2}} e^{i\mathbf p\cdot\mathbf x/\hbar} \left[a(\mathbf p)e^{-iE_{\mathbf p}t/\hbar} +b(\mathbf p)e^{+iE_{\mathbf p}t/\hbar}\right].$$

Both terms use the same spatial momentum label. The first has eigenvalue $+E_{\mathbf p}$ under $i\hbar\partial_t$; the second has $-E_{\mathbf p}$. Some field expansions instead write the second mode as $e^{+ip\cdot x/\hbar}$ with $p^0>0$; its spatial label is then the negative of the one used here. Mixing these conventions changes reality and orthogonality formulas.

Each Fourier mode obeys an oscillator equation,

$$\partial_t^2\widetilde\phi+ \frac{E_{\mathbf p}^2}{\hbar^2}\widetilde\phi=0.$$

Let $f(\mathbf x)=\phi(0,\mathbf x)$ and $h(\mathbf x)=\partial_t\phi(0,\mathbf x)$. At zero time,

$$\widetilde f=a+b,\qquad \widetilde h=-\frac{iE_{\mathbf p}}{\hbar}(a-b).$$

Solving these two equations gives

$$a=\frac12\left(\widetilde f+ \frac{i\hbar}{E_{\mathbf p}}\widetilde h\right),\qquad b=\frac12\left(\widetilde f- \frac{i\hbar}{E_{\mathbf p}}\widetilde h\right).$$

Thus a single initial shape does not determine the frequency sector. Equivalently, the exact initial-value solution is

$$\widetilde\phi(t,\mathbf p)= \widetilde f\cos\frac{E_{\mathbf p}t}{\hbar} +\frac{\hbar\widetilde h}{E_{\mathbf p}} \sin\frac{E_{\mathbf p}t}{\hbar}.$$

This is a useful benchmark for spectral evolution codes; no dispersion approximation has been made.

Reality and positive frequency impose different conditions

A real solution requires real $f,h$, so their Fourier transforms obey $\widetilde f(-\mathbf p)=\widetilde f(\mathbf p)^*$ and the analogous condition for $h$. Consequently

$$b(\mathbf p)=a(-\mathbf p)^*.$$

A nonzero real solution cannot contain positive frequencies alone. A positive-frequency complex solution instead has $b=0$, requiring

$$\widetilde h=-\frac{iE_{\mathbf p}}{\hbar}\widetilde f, \qquad h=-\frac{i}{\hbar}H_0f.$$

The Square-Root Hamiltonian defines this nonlocal $H_0$. Arbitrary compactly supported $f,h$ generally fail this constraint. Causal propagation of the full Cauchy data cannot be applied to a positive-frequency packet by checking the support of $f$ alone.

For example, $h=0$ gives $a=b=\widetilde f/2$. Even if $\widetilde f$ is narrowly centered at nonzero momentum, both sectors occur. Vanishing initial time derivative is a standing-oscillation preparation, not a positive-energy preparation.

Phase, group, and front velocities

For a narrow packet on branch $\sigma=\pm1$, stationary phase gives

$$\mathbf v_g=\nabla_{\mathbf p}(\sigma E_{\mathbf p}) =\sigma\frac{c^2\mathbf p}{E_{\mathbf p}}.$$

Its magnitude is below $c$ for $m>0$. For a positive-frequency plane wave with nonzero momentum, the phase speed along $\mathbf p$ is $v_{\rm ph}=E_{\mathbf p}/|\mathbf p|>c$, and $v_{\rm ph}|\mathbf v_g|=c^2$. A plane wave has no localized front, so its phase speed does not measure information transport.

Packet distortion is governed by the Hessian

$$\frac{\partial^2 E}{\partial p_i\partial p_j} =\frac{c^2}{E}\delta_{ij}-\frac{c^4p_ip_j}{E^3}.$$

At nonzero central momentum its longitudinal eigenvalue is $m^2c^6/E^3$ and its two transverse eigenvalues are $c^2/E$. Ultrarelativistic longitudinal dispersion is therefore suppressed while transverse angular spreading remains. For massless modes away from $\mathbf p=0$, longitudinal curvature vanishes but transverse curvature need not. A group-speed bound alone does not prove causal support; that follows from the full hyperbolic initial-value equation.

Mode normalization is a separate choice

A continuum plane wave is not square integrable. One may use delta-normalized modes, box-normalized modes, or ordinary Fourier coefficients as above. Inserting $1/\sqrt{2E_{\mathbf p}}$ into a mode definition requires the inverse factor in its coefficient. The invariant mass-shell measure supplies another equivalent convention. These rewritings leave the field unchanged but change the displayed amplitude norm. An unqualified $|a(\mathbf p)|^2$ is not automatically a Born probability; specify the solution-space inner product first.

Exercises

1. A Fourier component has $\widetilde f=F$ and $\widetilde h=-i\Omega F$, with real $\Omega$. Find its sector amplitudes and the positive-frequency condition.

Solution

They are $a=(1+\hbar\Omega/E)F/2$ and $b=(1-\hbar\Omega/E)F/2$. Positive frequency requires $\Omega=E/\hbar$ at every occupied momentum, not one fixed frequency for a packet with a range of momentum magnitudes.

2. Construct the real solution with $f=\cos(\mathbf k\cdot\mathbf x)$ and $h=0$. Identify its four plane-wave components.

Solution

The solution is $\cos(\mathbf k\cdot\mathbf x)\cos(\omega_{\mathbf k}t)$, where $\omega_{\mathbf k}^2=c^2|\mathbf k|^2+m^2c^4/\hbar^2$. It contains both spatial momenta $\pm\hbar\mathbf k$ and both frequencies $\pm\omega_{\mathbf k}$. It is a generalized mode; finite-energy data require a packet or a finite box.

3. Evaluate the dispersion Hessian at rest and for $|\mathbf p|\gg mc$.

Solution

At rest it is $\delta_{ij}/m$, agreeing with $\mathbf p^2/(2m)$. At high momentum its transverse eigenvalues approach $c/|\mathbf p|$ and its longitudinal eigenvalue approaches $m^2c^3/|\mathbf p|^3$. These different scalings describe directional dispersion, not different limiting propagation speeds.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964 — scalar relativistic modes.
• W. Greiner, Relativistic Quantum Mechanics: Wave Equations, 3rd ed., Springer, 2000, doi:10.1007/978-3-662-04275-5 — frequency branches and wave mechanics.
• D. Tong, Quantum Field Theory, University of Cambridge lecture notes, 2006–2007, §2.1, Free Fields — Fourier decomposition into independent oscillator modes.