Wavepackets, Modes, Frames, and Localization

Relativistic communication uses localized wavepackets and localized detectors, not abstract plane-wave labels. Mode choice determines normalization, energy, localization tails, and receiver overlap; an observer transformation can change a mode decomposition while leaving predictions for fixed local detector observables covariant. Basis-dependent mode entanglement is therefore not automatically a frame-independent resource.

Required background. Field communication defines the sender and receiver channel whose modes must be physically matched.

Helpful background. Detector responses and field observables separates local operational readings from global particle language.

Normalized packet modes

For a free scalar field, choose a positive-frequency solution $h$ normalized in the Klein–Gordon inner product,

$$(h,h)_{\mathrm{KG}}=1, \qquad a(h)=(h,\phi)_{\mathrm{KG}}.$$

A Gaussian momentum envelope may be written

$$\widetilde h(\mathbf k)=\mathcal N \exp\!\left[-\frac{(\mathbf k-\mathbf k_0)^2}{4\sigma_k^2}\right] e^{-i\mathbf k\cdot\mathbf x_0},$$

with $\mathcal N$ fixed by the invariant one-particle measure. Its position-space profile has tails. Narrow momentum width improves frequency selectivity but broadens spatial localization, so bandwidth and support error cannot be optimized independently.

The mode is part of both encoding and receiver matching. A propagating field solution becomes a communication channel only after those localized operations and their constraints are supplied. The diagram is schematic.

Matched and mismatched receivers

Let the emitted packet be $h$ and the receiver select a normalized packet $g$. In an ideal lossless free model, the accessible amplitude is controlled by

$$\alpha=(g,h)_{\mathrm{KG}}, \qquad 0\le|\alpha|^2\le1.$$

Propagation, switching, and detector response modify this overlap into an effective channel coefficient. Compare a matched receiver $g=h$ at the reception hypersurface with a detuned central momentum, shifted arrival time, or relatively boosted receiver. Report the resulting loss and noise together with the packet’s tail norm outside the apparatus worldtube.

A Lorentz transformation maps the field solution and detector trajectory together. If the entire experimental arrangement is transformed, probabilities agree. If only the mode basis is changed while the detector is held fixed, mode occupation and mode entanglement can change because a different observable has been chosen.

Positive frequency and localization

Positive frequency is defined relative to a time flow. In Minkowski inertial frames the positive-energy mass shell is invariant, but accelerated or nonstationary observers use different mode decompositions. Moreover, exact compact support on a time slice is incompatible with several ideal spectral restrictions. Treat practical localization through a supported coupling or an explicit tail bound rather than claiming that a positive-frequency plane wave is a local signal.

The invariant quantities for a fixed protocol are local outcome probabilities, intervention contrasts, and information measures computed from the induced channel. A number such as entanglement between two arbitrarily selected global modes is meaningful only with that selection and its accessible operations stated. Concrete sender–receiver packet channels with localized detector couplings are derived by Cliche and Kempf 2010, §§ III–V and Tjoa and Gallock-Yoshimura 2022, §§ III–VI.

Frame mismatch and mode mismatch can reduce a receiver channel without violating covariance. The invariant comparison holds the localized physical operations fixed and changes only coordinates or representation. The map is schematic.

Exercises

Suppose $g$ and $h$ are normalized and $g=\alpha h+\sqrt{1-|\alpha|^2}\,h_\perp$. What fraction of a one-particle state $a^\dagger(h)|0\rangle$ is detected by an ideal number measurement in mode $g$?

Solution

The expectation of $a^\dagger(g)a(g)$ is $|(g,h)_{\mathrm{KG}}|^2=|\alpha|^2$. The remaining probability occupies the orthogonal mode. A localized detector adds switching, smearing, and noise, so this is the ideal mode-overlap benchmark rather than a complete apparatus prediction.

References

• Cliche, M., and Kempf, A. (2010). “The Relativistic Quantum Channel of Communication through Field Quanta.” Physical Review A 81, 012330. DOI. Open PDF.
• Tjoa, E., and Gallock-Yoshimura, K. (2022). “Channel Capacity of Relativistic Quantum Communication with Rapid Interaction.” Physical Review D 105, 085011. DOI. Open PDF.