Wavepackets, Modes, Frames, and Localization

A relativistic message is prepared and recovered by finite apparatuses, so its mode, bandwidth, energy, localization error, and receiver overlap must be part of the channel definition. Plane waves are useful basis vectors but are not localized signals. This page develops a Klein–Gordon-normalized packet, then uses an exactly solvable rapidity-Gaussian benchmark to distinguish a genuine change of frame from a receiver that selects the wrong mode.

Required background. Field communication defines the sender–field–receiver channel whose mode matching is studied here.

Helpful background. Detector responses and field observables separates supported detector readings from global particle labels.

Chapter map. The overview task map, common protocol comparison, and localization and failure controls show where packet matching enters the complete communication claim.

Klein–Gordon-normalized packet modes

Consider a real massive Klein–Gordon field in $1+1$ dimensional Minkowski spacetime, with natural units $c=\hbar=1$. For complex classical solutions $h$ and $g$, define

$$(h,g)_{\mathrm{KG}} =i\int_\Sigma d\Sigma^\mu \left(h^*\partial_\mu g-g\partial_\mu h^*\right).$$

Current conservation makes this inner product independent of the Cauchy surface $\Sigma$ when the solutions and boundary behavior lie in the stated domain. It is positive on the positive-frequency solution space, indefinite on the full complex solution space, and antilinear in its first argument. On a constant-$t$ slice, let

$$u_k(t,x)=\frac{e^{-i\omega_k t+ikx}}{\sqrt{4\pi\omega_k}}, \qquad \omega_k=\sqrt{k^2+m^2}.$$

Direct substitution gives $(u_k,u_{k'})_{\mathrm{KG}}=\delta(k-k')$. A packet

$$h_f=\int_{-\infty}^{\infty}dk\,f(k)u_k$$

therefore obeys

$$(h_f,h_g)_{\mathrm{KG}}=\int dk\,f(k)^*g(k), \qquad [a(f),a^\dagger(g)]=(h_f,h_g)_{\mathrm{KG}}.$$

Thus $\int dk\,|f(k)|^2=1$ is not an optional probability convention: it is exactly the condition that $a^\dagger(f)|0\rangle$ have unit norm. Other common plane-wave normalizations move factors of $2\omega_k$ between the basis and coefficient measure; mixing those conventions gives a wrong commutator and wrong receiver efficiency.

For several proposed channels, form the Gram matrix $G_{ij}=(h_i,h_j)_{\mathrm{KG}}$. Only an orthonormal family has independent canonical mode operators. If $G_{12}\ne0$, writing the two labels as separate tensor factors double-counts an overlapping degree of freedom. Orthogonalize the packet family or retain the full Gram matrix in the channel model.

Momentum width and a stated localization diagnostic

A normalized Gaussian coefficient is

$$f_{k_0,\sigma_k,x_0}(k) =\frac{1}{(2\pi\sigma_k^2)^{1/4}} \exp\!\left[-\frac{(k-k_0)^2}{4\sigma_k^2}-ikx_0\right].$$

Its Fourier coefficient amplitude

$$\psi_f(x)=\frac{1}{\sqrt{2\pi}}\int dk\,f(k)e^{ikx}$$

has $|\psi_f(x)|^2$ Gaussian with standard deviation $\sigma_x=1/(2\sigma_k)$. The coefficient-space tail outside the interval $|x-x_0|\le L$ is therefore

$$\epsilon_{\mathrm{tail}}(L) =\int_{|x-x_0|>L}dx\,|\psi_f(x)|^2 =\operatorname{erfc}(\sqrt2\,\sigma_k L).$$

For $\sigma_k=0.5$ and $L=3$ in consistent inverse-length units, $\sigma_x=1$ and $\epsilon_{\mathrm{tail}}=0.0026997961$. The Fourier transform of the delta-normalized one-particle coefficients is the Newton–Wigner representation for this massive spin-zero example; see Newton and Wigner 1949, pp. 400–406. It is a reproducible localization diagnostic, not a covariant position probability and not the support of the detector coupling. The Klein–Gordon solution contains the additional $1/\sqrt{\omega_k}$ basis weight and disperses; its arrival-time tail must be evaluated from the propagated solution. Exact apparatus locality should instead be imposed through a compactly supported switching and smearing function.

Positive energy places a genuine limit on particle localization. Under broad hypotheses, a state with Hamiltonian bounded below cannot remain strictly confined to a bounded region in the naive particle-localization sense; Hegerfeldt 1974, pp. 3320–3321 gives the short causality argument. This does not make local QFT acausal. Local commutators and compact interaction supports control signaling, while a positive-frequency packet coefficient generally has tails.

Matched and mismatched moving receivers

Rapidity makes the inertial-frame calculation exact. Write the positive mass shell as

$$k=m\sinh\theta, \qquad \omega=m\cosh\theta,$$

and choose basis modes $u_\theta=\sqrt{m\cosh\theta}\,u_k$, so $(u_\theta,u_{\theta'})_{\mathrm{KG}}=\delta(\theta-\theta')$. The normalized sender envelope is

$$F_{\theta_0,\sigma}(\theta) =\frac{1}{(2\pi\sigma^2)^{1/4}} \exp\!\left[-\frac{(\theta-\theta_0)^2}{4\sigma^2}\right].$$

Its mean four-momentum is finite:

$$\langle H\rangle =m e^{\sigma^2/2}\cosh\theta_0, \qquad \langle P\rangle =m e^{\sigma^2/2}\sinh\theta_0.$$

Take the benchmark $m=1$, $\theta_0=1.5$, $\sigma=0.25$, and a receiver boosted by rapidity $\eta=0.4$ relative to the sender. The packet center in receiver coordinates is $\theta_0-\eta$. If the physical receiver is matched to this transformed packet, its mode $g$ equals the transformed $h$ and $|(g,h)_{\mathrm{KG}}|^2=1$ after both are expressed on the same Cauchy surface.

Now introduce the adversarial mismatch: keep the receiver’s numerical center at $\theta_0$ instead of transforming it. Two equal-width rapidity Gaussians separated by $\eta$ have

$$\alpha=(g,h)_{\mathrm{KG}} =\exp\!\left[-\frac{\eta^2}{8\sigma^2}\right], \qquad |\alpha|^2 =\exp\!\left[-\frac{\eta^2}{4\sigma^2}\right].$$

For the declared values,

$$\langle H\rangle=2.4270831123, \qquad \langle P\rangle=2.1968700405, \qquad \alpha=0.7261490371, \qquad |\alpha|^2=0.5272924240.$$

These are analytic values rounded to ten decimal places, so they have no sampling uncertainty; the last displayed digit is the only numerical rounding shown. An ideal one-particle number measurement in the mismatched receiver mode consequently accesses only $52.73\%$ of the excitation. The remainder is not destroyed by a coordinate transformation; it lies in modes orthogonal to the receiver filter. Propagation, finite switching, detector response, loss, and environmental noise can reduce the apparatus channel further and require their own uncertainty bounds. Explicit sender–field–receiver channels of this kind are developed by Cliche and Kempf 2010, §§ III–V and, for rapid detector interactions, by Tjoa and Gallock-Yoshimura 2022, §§ III–VI.

Which frame statements are invariant

A proper orthochronous Lorentz transformation maps the future mass shell to itself. It therefore mixes inertial positive-frequency modes only with other positive-frequency modes: there is no positive/negative-frequency Bogoliubov mixing merely because two inertial observers use different coordinates. If the state, packet, detector worldline, compact coupling, and recorded outcome are transformed together, the probability distribution is unchanged.

Accelerated or nonstationary time flows are a different question. Positive frequency is defined relative to a chosen time evolution, and inequivalent choices can mix creation and annihilation operators. Fulling’s Rindler construction is a foundational example; see Fulling 1973, pp. 2850–2862. Even there, a local detector response is obtained from its trajectory, switching, smearing, and field state—not by declaring a global particle number to be an invariant observable.

The failure control is consequently simple. Perform two calculations: first transform the complete experiment and require identical outcome probabilities; then hold the localized detector fixed and replace only its selected mode by the untransformed numerical label. The first is a covariance test. The second is a physically different receiver and may show the $|\alpha|^2$ loss above. Calling that loss “frame dependence of the same experiment” confuses a passive description change with an active mode mismatch.

Limitations

The numerical benchmark uses a free massive scalar field, one spatial dimension, ideal KG modes, and an ideal modal number readout. Its rapidity Gaussian is not compactly supported. A laboratory claim must add a supported detector coupling, propagate the packet to the receiving worldtube, bound the tail and mode-overlap errors there, and state energy and bandwidth constraints. Curved spacetime, acceleration, dispersion, interactions, and finite detector size can all change the effective channel, but none permits replacing an operational detector calculation by basis-dependent particle language.

Exercises

Verify the plane-wave Klein–Gordon normalization used on this page.

Solution

On a constant-$t$ slice,

$$(u_k,u_{k'})_{\mathrm{KG}} =i\int dx\left(u_k^*\partial_tu_{k'}-u_{k'}\partial_tu_k^*\right).$$

The time derivatives supply $-i\omega_{k'}$ and $+i\omega_k$, while the spatial integral gives $2\pi\delta(k-k')$. On the delta-function support $\omega_{k'}=\omega_k$, so the prefactor is

$$\frac{\omega_k+\omega_{k'}}{4\pi\sqrt{\omega_k\omega_{k'}}} 2\pi=1.$$

Hence $(u_k,u_{k'})_{\mathrm{KG}}=\delta(k-k')$ and $(h_f,h_f)_{\mathrm{KG}}=\int dk\,|f(k)|^2$.

Derive the rapidity-Gaussian mismatch and reproduce the numerical efficiency.

Solution

Multiplying two normalized equal-width envelopes centered at $a$ and $b$ gives a Gaussian centered at $(a+b)/2$ and a constant factor $\exp[-(a-b)^2/(8\sigma^2)]$. Integration of the centered Gaussian leaves that factor, so $\alpha=\exp[-\eta^2/(8\sigma^2)]$. With $\eta=0.4$ and $\sigma=0.25$, $\alpha=e^{-0.32}=0.7261490371$ and $|\alpha|^2=e^{-0.64}=0.5272924240$. A matched transformed receiver has $a=b$ and unit overlap.

Compute the Gaussian coefficient tail for $\sigma_k=0.5$ and $L=3$, and explain why it is not a compact-support claim.

Solution

The Fourier density has standard deviation $\sigma_x=1/(2\sigma_k)=1$. Therefore the interval $|x-x_0|\le3$ covers three standard deviations and

$$\epsilon_{\mathrm{tail}} =\operatorname{erfc}(\sqrt2\,\sigma_kL) =\operatorname{erfc}(3/\sqrt2) =0.0026997961.$$

The nonzero result already rules out exact support for this diagnostic amplitude. Moreover, the physical KG solution has its own frequency weight and time evolution. Exact operational support belongs to the detector’s spacetime test function; the packet tail must be carried as a separate approximation error.

References

• Cliche, M., and Kempf, A. (2010). “The Relativistic Quantum Channel of Communication through Field Quanta.” Physical Review A 81, 012330. DOI. Open PDF.
• Fulling, S. A. (1973). “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time.” Physical Review D 7, 2850–2862. DOI.
• Hegerfeldt, G. C. (1974). “Remark on Causality and Particle Localization.” Physical Review D 10, 3320–3321. DOI.
• Newton, T. D., and Wigner, E. P. (1949). “Localized States for Elementary Systems.” Reviews of Modern Physics 21, 400–406. DOI.
• Tjoa, E., and Gallock-Yoshimura, K. (2022). “Channel Capacity of Relativistic Quantum Communication with Rapid Interaction.” Physical Review D 105, 085011. DOI. Open PDF.