The products depend on the observation frame

Choose the frame before assigning numbers

A reference frame specifies how an observer assigns times, positions, energies, and momenta. The center-of-momentum frame is the one in which the total incoming spatial momentum vanishes. For a two-photon final state in that frame, the photons have equal energy and opposite momentum. A laboratory frame coincides with it only for particular incoming conditions.

For equal-energy electron and positron beams colliding head-on, their momenta cancel and the laboratory can be the center-of-momentum frame. For a moving positron striking an electron approximately at rest, the total incoming momentum is nonzero and the center-of-momentum frame moves through the laboratory. The reaction itself has not changed when described in either frame. What changes are the energy and directional coordinates assigned to its products. Stating the frame prevents the 511-keV near-rest example from being mistaken for a universal detector signature.

Four-momentum gives a common language

Write a photon's four-momentum as (E/c,p), with |p| = E/c. For a boost along the z direction, transverse momentum components are unchanged while energy and longitudinal momentum mix. Using a convention in which the final system moves at speed βc in the laboratory, E = γ(E*+βcpz*), where starred quantities refer to the center-of-momentum frame and γ = 1/√(1−β²).

The sign of the boost depends on which frame is being transformed into which, so define it before using the formula. For a photon emitted at angle θ* to the positive z direction, pz* = E*cosθ*/c and E = γE*(1+βcosθ*). The partner photon has the opposite rest-frame direction and energy γE*(1−βcosθ*). These relations provide a direct calculated prediction for the energy asymmetry without introducing any failure of conservation.

A collinear example with a large energy asymmetry

Take an illustrative near-rest pair whose two center-of-momentum photons each have E* = 511 kiloelectronvolts. Let the whole system move at β = 0.6 along z in the laboratory, so γ = 1.25. If one photon is emitted along the boost direction and the other opposite it, their laboratory energies are 1.25 × 511 × 1.6 = 1,022 kiloelectronvolts and 1.25 × 511 × 0.4 = 255.5 kiloelectronvolts.

The total laboratory energy is 1,277.5 kiloelectronvolts, greater than the rest-frame total because the system has overall motion. The photons remain collinear and opposite in this special geometry, but their momenta no longer cancel. Their net momentum is (1,022−255.5) keV/c = 766.5 keV/c. That equals β times total energy divided by c, as required for the boosted system. Unequal photon energies are therefore compatible with the same underlying two-photon reaction.

A transverse example changes the opening angle

Now let the two center-of-momentum photons emerge perpendicular to the boost direction, with opposite transverse momenta. Each laboratory energy is γE* = 638.75 kiloelectronvolts for the same β = 0.6 example. Both acquire positive longitudinal momentum, while their transverse momenta remain opposite.

The aberration formula gives cosθ = (cosθ*+β)/(1+βcosθ*). With θ* = 90 degrees, cosθ = 0.6, so each photon lies about 53.13 degrees from the forward axis. Their azimuths differ by 180 degrees, giving a laboratory opening angle of about 106.26 degrees rather than 180 degrees. The total transverse momentum still cancels, and the longitudinal momenta add to 766.5 keV/c. Equal photon energies do not by themselves prove that the pair was at rest in the laboratory. The measured directions supply the missing information.

The two-photon invariant checks both examples

For two photons with laboratory energies E1 and E2 and opening angle ψ, the squared invariant rest energy of their combined system is M²c⁴ = 2E1E2(1−cosψ). This follows from adding their four-momenta and using each photon's zero invariant mass. It is unchanged by a frame transformation.

For equal opposite 511-keV photons, the formula gives Mc² = 1.022 megaelectronvolts. For the collinear boosted example, the unequal energies still give the same value because ψ remains 180 degrees. For the transverse boosted example, the larger equal energies are compensated by the smaller opening angle. A reconstructed invariant mass is therefore a stronger consistency test than demanding that every photon individually have 511 keV. It combines energy and geometry in the way required by relativity, though detector uncertainties must be propagated through the nonlinear formula.

A fixed-target collision has its own center-of-momentum speed

For an incoming positron with total laboratory energy E and momentum p striking an electron at rest, the total energy is E+mec² and total momentum is p. The center-of-momentum speed satisfies βcm = pc/(E+mec²). In units with c = 1, the invariant collision energy squared is s = 2me²+2meE.

These relations show why one cannot simply divide all laboratory energy equally between two photons while also assuming opposite directions. The final state must carry the initial net momentum. A chosen emission angle in the center-of-momentum frame transforms into a corresponding laboratory energy and direction. The angular distribution itself requires the reaction amplitude and initial spin conditions; kinematics alone gives the mapping, not the probability of every angle. A complete prediction therefore combines a frame transformation with the appropriate dynamics and detector acceptance.

Small motion produces measurable broadening

Even when positrons have slowed substantially, electrons in a material have momentum distributions and the annihilating pair need not be exactly at rest. The resulting photon energy distribution can broaden around the near-rest value, and the opening angle can deviate slightly from exact opposition. Binding and recoil can add further corrections.

A precision experiment should therefore distinguish intrinsic kinematic broadening from detector energy resolution and scattering in surrounding material. Measuring both photons' energies and directions can constrain the total pair momentum more strongly than one energy spectrum alone. The exact interpretation depends on the material and state preparation, so the present discussion supplies the kinematic framework rather than a universal numerical line width. A broad peak near 511 keV is not automatically a conservation violation; it may contain ordinary motion, interaction, and instrument contributions that must be modeled separately.

Detector acceptance changes the observed sample

A detector covers a finite angular region and accepts events within energy and timing windows. A boost changes how the physical distribution maps into those windows. Selecting photons near one laboratory direction can therefore select a particular energy range and bias the observed partner distribution.

Simulations or analytic acceptance calculations should use the same frame conventions as the reconstruction. An apparently missing low-energy partner may simply lie outside the aperture or threshold. Conversely, selecting only nearly opposite detections favors events with particular pair momenta and emission geometries. Report the selection before comparing observed spectra with an ideal center-of-momentum prediction. The selected sample can be perfectly consistent with the reaction while differing from the full distribution. Detector geometry is not an afterthought when frame-dependent angles and energies are the observables used to identify the process.

Changing the frame does not change the particle record

All observers describing the same isolated event agree on the species and on whether a proposed complete final state satisfies the covariant conservation equations. They can disagree on individual energies, spatial angles, and coordinate time intervals without disagreeing about the reaction. A photon remains a photon under the Lorentz transformation.

This is different from changing the physical incoming state. Accelerating one beam while holding the other fixed changes the collision's invariant energy as well as its center-of-momentum motion. Merely describing a fixed event from another frame changes neither. Keeping those operations separate prevents an observer transformation from being treated as a mechanism that creates or removes reaction energy. The boost examples are alternative descriptions of a chosen final state, not additional physical amplification stages and not operations that alter the underlying quantum state by measurement.

Report frame-dependent and invariant quantities together

A useful event description gives the laboratory energies and directions, the reconstructed total four-momentum, and the invariant mass or center-of-momentum energy. It states assumptions about unmeasured products, initial binding, and detector response. For a two-photon reconstruction, the opening angle is essential to the invariant check.

The familiar 511-keV back-to-back picture remains a valuable limiting case. It becomes more useful when readers can see how it changes under motion and why the conservation account survives. A mismatched energy pair is not automatically evidence of a new reaction, and an equal pair is not automatically proof of a stationary source. The combined kinematic information decides. This framework also clarifies the meaning of disappearance: the incoming particle inventory changes, while a frame-consistent outgoing energy-momentum state remains available for measurement.

Timing comparisons require the same care. Two detections simultaneous in one coordinate frame need not be simultaneous in another when they occur at different positions. That change of coordinate assignment does not alter the local detector records or allow usable signals to outrun light. The reaction kinematics and the causal propagation to each detector remain parts of one relativistic description.

Go to the original work.

These links identify the sources behind specific claims. Verification scope varies by source and is stated below.

1. 1 / Official researchKinematics — Particle Data Group 2026, Jackson; reviewed by Miller and Tovey ↗

   Four-momentum, Lorentz transformations, center-of-momentum energy and two-body kinematics

   Verification scope: Official 2026 PDF inspected, particularly sections 49.1, 49.2 and 49.4; review text itself reviewed August 2021

2. 2 / University teachingDavid Tong — Quantum Field Theory, section 6: Quantum Electrodynamics ↗

   Electron-positron annihilation into photons and other allowed QED channels

   Verification scope: Author-hosted indexed section text inspected; direct page fetch intermittently failed

3. 3 / Official researchParticle Detectors at Accelerators — Particle Data Group 2026 ↗

   Scintillation, semiconductor response, timing, energy calibration and finite detector resolution

   Verification scope: Official PDF detector sections and timing/calibration passages inspected; not a claim to have read all 128 pages

4. 4 / Primary experimentPrecise measurement of positronium — Progress of Theoretical and Experimental Physics (2012) ↗

   Experimental positronium lifetimes and spin-dependent photon multiplicity

   Verification scope: Author research account and lifetime section inspected

5. 5 / University teachingDavid Tong — Quantum Field Theory, section 3: Interacting Fields ↗

   Unitary scattering operator and four-momentum conservation in scattering amplitudes

   Verification scope: Author-hosted indexed scattering passages inspected; direct page fetch intermittently failed