Lorentz-Invariance and Modified-Dispersion Tests

Lorentz tests constrain explicit operator coefficients, species assignments, helicity structures, and preferred-frame assumptions. They do not constrain quantum gravity as a single parameter. The central calculation maps a symmetry-breaking or deformed-kinematics ansatz into propagation, birefringence, or reaction-threshold observables while fitting source physics and testing radiative stability.

Required background. Quantum-Gravity Observables and Test Taxonomy fixes the inference chain, and Lorentz Field Representations and Poincaré Particle Representations supplies the undeformed symmetry structure. Helpful background. Validity, Unitarity, and Breakdown controls EFT use, Lorentz-Invariant Phase Space supplies the conventional baseline, and Modified Dispersion, Analogue Horizons, and Universality separates robustness tests from microscopic identification.

Operator basis and observables

In a preferred cosmological frame, a convenient ultrarelativistic parameterization is

$$E^2=p^2+m^2+\sum_{n\ge1}\eta_n^{(a,h)} \frac{p^{n+2}}{M^n},$$

where $a$ labels species, $h$ helicity, and $M$ is a reference scale. At leading order,

$$v_g=\frac{\partial E}{\partial p} \simeq 1-\frac{m^2}{2E^2} +\frac{n+1}{2}\eta_n^{(a,h)} \left(\frac{E}{M}\right)^n .$$

An isotropic dispersion coefficient can therefore produce an energy-dependent flight time. Helicity-odd coefficients rotate linear polarization because the two circular polarizations acquire different phases. Species-dependent coefficients shift thresholds for pair production, photon decay, vacuum Čerenkov emission, and other reactions.

This parameterization is not a complete theory. Gauge invariance, rotational symmetry, CPT, locality, and energy–momentum conservation determine which operators are allowed and how coefficients correlate. Lower-dimension Lorentz-violating operators are generally more dangerous. A proposed Planck-suppressed coefficient must either be protected by a symmetry or accompanied by a credible account of radiative mixing Myers and Pospelov 2003.

First application: redshifted time of flight

For two observed photon energies $E_h$ and $E_l$ from redshift $z_s$, a leading propagation delay is

$$\Delta t_{\mathrm{LIV}} =\frac{n+1}{2}\eta_n \frac{E_h^n-E_l^n}{M^n} \int_0^{z_s}\frac{(1+z)^n}{H(z)}\,dz .$$

The observed delay is $\Delta t_{\mathrm{obs}}=\Delta t_{\mathrm{LIV}}+\Delta t_{\mathrm{int}}+\Delta t_{\mathrm{det}}$. A single transient cannot generally separate propagation from an intrinsic emission lag. A hierarchical analysis across redshifts can exploit their different scaling, but only if source evolution and selection effects are modeled.

As a dated example, LHAASO’s analysis of GRB 221009A found no significant energy-dependent delay and reported, at 95% confidence, a linear suppression scale above roughly ten Planck energies and a quadratic scale above roughly $6\times10^{-8}$ Planck energies LHAASO Collaboration 2024. These are strong constraints on the tested photon dispersion templates. They are not a measurement of a fundamental scale, and the intrinsic-lag model remains part of the result.

Birefringence can be more sensitive because a tiny helicity splitting accumulates a phase

$$\Delta\phi(E)\simeq \frac{E^{n+1}}{M^n} \int_0^{z_s}\frac{(1+z)^n}{H(z)}\,dz\, \Delta\eta_n .$$

Broadband polarization surviving propagation constrains rapid phase variation across the band. The inference requires a source-polarization model and a calibrated detector response.

Thresholds are coupled-system tests

For a reaction $a+b\to c+d$, the threshold follows from the full dispersion and conservation laws of every participant. Modifying only the incoming photon is not neutral: it is a particular species model. For head-on photon pair production, the Lorentz-invariant scale is $E_\gamma\epsilon\sim m_e^2$. A schematic correction becomes important when $\eta_n E_\gamma^{n+2}/M^n$ competes with $m_e^2$. Observing a photon beyond a putative decay threshold can sharply exclude a coefficient, but only after uncertainties in energy reconstruction, background, magnetic fields, and source opacity are included.

Deformed-relativity models may preserve a relativity principle while changing both dispersion and composition laws. Bounds derived assuming ordinary additive four-momentum do not transfer automatically to them.

Adversarial control: let the source imitate propagation

Fit the transient population with an intrinsic lag

$$\Delta t_{\mathrm{int}}=A_s+B_s \left(\frac{E}{E_0}\right)^\beta,$$

allowing $(A_s,B_s,\beta)$ to vary hierarchically with source class. Then fit the same data with and without $\eta_n$. If the propagation coefficient moves substantially when the lag family changes, the data do not identify propagation. A robust analysis also releases leave-one-source-out results so that one exceptional transient cannot dominate a population claim.

The strongest current conclusion is a network of null constraints on many photon, matter, and gravity-sector coefficients. It is scientifically important because it excludes large regions of specific EFT parameter spaces. It neither proves exact Lorentz invariance at all scales nor falsifies UV programs whose low-energy limits preserve the symmetry.

Common pitfalls

Writing a quantum-gravity scale without a coefficient convention. Experiments constrain combinations such as $M/\lvert\eta_n\rvert^{1/n}$. Different normalizations cannot be compared until the operator basis, sign, helicity, and species assignments match.

Ignoring naturalness. A dimension-five term cannot be considered in isolation if loops generate already excluded lower-dimension terms. State the protecting symmetry or treat radiative stability as an unresolved assumption.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• LHAASO Collaboration. “Stringent Tests of Lorentz Invariance Violation from LHAASO Observations of GRB 221009A.” Physical Review Letters 133, 071501 (2024). DOI.
• Liberati, S. “Tests of Lorentz Invariance: A 2013 Update.” Classical and Quantum Gravity 30, 133001 (2013). DOI.
• Myers, R. C., and M. Pospelov. “Ultraviolet Modifications of Dispersion Relations in Effective Field Theory.” Physical Review Letters 90, 211601 (2003). DOI.