One initial state. Two outgoing photons.
Center-of-momentum frame: the incoming momenta cancel. The two photons carry equal energy in opposite directions.
The arrows show momenta. This event picture represents allowed kinematics, not literal microscopic trajectories.
Annihilation is particle conversion through an interaction. It is distinct from forming a condensate or creating a stable bound state.
Where do the photons go?
The finite-electron-mass, unpolarized angular distribution changes with the collision energy.
Both photons are identical. This full-sphere distribution includes the event-counting factor ½, so its integral counts each event once. The displayed angle labels one photon; swapping the two gives the same event.
Two diagrams. One amplitude.
The photons can attach to the electron line in either order. Add the two probability amplitudes before squaring.
|𝓜|² = |𝓜t|² + |𝓜u|² + 2 Re(𝓜t𝓜u*)
A Feynman diagram is a term in a perturbative calculation. The internal electron line is a propagator, not a measured particle trajectory.
Change the description. Preserve the physics.
A common phase rotation changes the spinor’s phase but leaves ψ̄ψ invariant. Local gauge transformations also change the potential Aμ; the field strength Fμν stays unchanged.
ψ̄ψ → ψ̄ψ
The phase drawing illustrates the global part of U(1), not a numerical proof of local gauge invariance.
The stronger check: Ward identities
Replace a photon polarization vector ε by its momentum k. The full on-shell amplitude must vanish. The two diagrams cancel; either diagram alone generally fails this test.
ε₂ → k₂ ⇒ 𝓜t + 𝓜u = 0
Dμ = ∂μ + iqAμ, q = −e
This is the reference action being tested. Its electron mass and electromagnetic coupling are inputs here.
A validated process is one rung of the ladder.
This lab establishes a reference calculation. Deriving its fields and coefficients from a selected MTFT vacuum is a separate task.
Z(χ) = exp(χ / MP)
geff / g₀ = exp(−χ / 2MP)
The uniform breathing reduction supplies this gauge–geometry coupling. The effective coupling relation assumes a constant breathing background. Coupled gauge–fermion–breathing evolution has not been simulated here.
A null Maxwell plane wave has FμνFμν = 0 while carrying positive energy. Vanishing F² does not remove its gravitational stress-energy.
| Ingredient | Status | What this milestone means |
|---|
Charge and photons
Check the quantum amplitude, conservation, polarizations, and normalization.
Weak and Higgs sectors
Specify chiral multiplets, symmetry breaking, neutrino masses, and anomaly cancellation.
Quarks to nucleons
Retain unbroken color, match the strong coupling, and calculate nonperturbative bound states.
These gauge sectors coexist in the Standard Model. The ladder is our order of investigation, not a sequence in which Nature constructs the groups.