Geometric algebra as the input language of collider foundation models

Abstract

A hard hadron-collider event is represented here as a single geometric object: the kinematics and object-type labels of all reconstructed final-state particles in one multivector $\evMV$, one $\Cl(1,3)\otimes\Vflav$ slot per object, whose higher grades the algebra generates from the measured four-momenta --- rather than as a list of four-momenta with label fields attached. The setting is geometric algebra, whose grade decomposition organises essentially every observable in current use: invariants at grade zero, four-momenta at grade one, decay-plane bivectors at grade two, oriented three-volumes at grade three, the CP-odd pseudoscalar at grade four. The high-level invariants, the low-level recipe and the equivariant-network inputs are recovered as projections onto specific grades. A per-grade dictionary of $32$ classical observables and an inventory of the symmetries acting on $\evMV$ are provided. Theorem settles which Lorentz invariants the higher grades unlock: none beyond $\{p_i\!\cdot\!p_j,\,m_i^2\}$, the CP-odd sign of the pseudoscalar being the one genuine channel. The representation is intended as a uniform input layer for foundation models of collider physics, and the realisations demonstrated here occupy two rungs of a ladder: the higher grades are formed inside the network from the per-object slots, or selected grade-two and grade-three objects are supplied at the input layer. It is demonstrated on the resonance-topology separation of $pp\!\to\!tWb$ with a Lorentz-equivariant multivector transformer of L-GATr type. An ablation over eight input representations separates what supplying algebraic content explicitly can and cannot buy: content reconstructible from the measured momenta unlocks no new invariant, by the theorem, whereas a fixed reference blade encoding the beam plane carries structure the momenta do not contain.