Wavefunction²

The bilateral wave function — amplitude, phase, and the Born rule as geometry

Standard quantum mechanics reads one face of the wave function: the amplitude \(|\psi|\), squared to give probability. The phase \(e^{i\phi}\) is treated as a calculational convenience — real only in interference. Wavefunction² reads both faces simultaneously. The amplitude \(|\psi|\) is the egress face — actual, observable, past. The phase \(e^{i\phi}\) is the ingress face — potential, unobserved, future. Together they form the complete bilateral object \(\psi = |\psi|e^{i\phi}\), a section of the tautological line bundle \(\gamma^1\) over \(\mathbb{CP}^\infty\).

The Born rule \(P = |\langle\psi|\phi\rangle|^2\) is not an axiom. It is the natural probability measure on \(\mathbb{CP}^\infty\) — the Fubini–Study transition probability. The probability of a quantum measurement is \(\cos^2\) of the angle between the state and the measurement direction, because \(\mathbb{CP}^\infty\) is the space of all crossing directions and the Fubini–Study metric is its unique unitarily invariant measure.

I. The Bloch Sphere — \(\mathbb{CP}^1\)

ColourElementPhysical meaning
Blue Egress face — amplitude \(|\psi|\) The actualised face of the wave function. The observable part. Corresponds to the north hemisphere of the Bloch sphere — the \(|0\rangle\) side. In bilateral language: what has become real at the crossing.
Red Ingress face — phase \(e^{i\phi}\) The potential face. The phase, which standard QM treats as non-observable. Corresponds to the south hemisphere — the \(|1\rangle\) side. In bilateral language: what remains potential, approaching the crossing from the future.
Gold The equator — the crossing The boundary between egress and ingress. On the Bloch sphere: the equatorial great circle where \(|\psi|\) and \(e^{i\phi}\) are in equal superposition. The Present — the bilateral crossing point \(\tau_0\).
Purple State vector / measurement axis The current quantum state as a point on \(\mathbb{CP}^1\). In the Born rule mode: the two state vectors whose angle gives the transition probability. In measurement mode: the projection axis.
Teal Probability arc The angle \(\theta\) between two states. The transition probability is \(P = \cos^2\theta\). This is the Fubini–Study metric — the Born rule as geometry, not axiom.
Wave function mode The Bloch sphere as the space \(\mathbb{CP}^1\) of all single-qubit states. The north pole is \(|0\rangle\) (egress, blue — the actualised state). The south pole is \(|1\rangle\) (ingress, red — the potential state). The equator is the bilateral crossing — equal superposition of egress and ingress. The purple arrow is the current state vector. The grey translucent sphere makes the geometry readable from any angle. Drag to rotate; scroll to zoom.
Born rule mode Two state vectors (purple and teal) on the Bloch sphere. The angle \(\theta\) between them is shown by the arc. The transition probability \(P = \cos^2\theta\) is displayed. Drag to rotate. The Born rule is not an axiom — it is the Fubini–Study distance on \(\mathbb{CP}^1\). When the two states are identical (\(\theta=0\)), \(P=1\). When orthogonal (\(\theta=90°\)), \(P=0\). Animate to watch the states evolve under unitary flow.
Measurement mode A state (purple) is measured along an axis (gold). The measurement projects the state onto the egress face (blue, north) or the ingress face (red, south) with probabilities \(\cos^2(\theta/2)\) and \(\sin^2(\theta/2)\) respectively. The bilateral interpretation: measurement is face selection — it collapses the bilateral state onto one face of the crossing. The animate button shows the state precessing under a Hamiltonian and the measurement probability oscillating.

II. The Complete Bilateral Statement

The wave function is a section of the tautological line bundle \(\gamma^1\) over \(\mathbb{CP}^\infty\). Every quantum state is a crossing direction of \(\infty_0\) — a ray in the infinite-dimensional complex projective space. The amplitude is the length of the vector in the fibre; the phase is the angle within the fibre.

\[ \psi = |\psi|\,e^{i\phi} \in \Gamma(\gamma^1 \to \mathbb{CP}^\infty). \]

The Born rule follows from the Fubini–Study metric on \(\mathbb{CP}^\infty\): the probability of transitioning from state \(|\psi\rangle\) to state \(|\phi\rangle\) is \(\cos^2\) of the geodesic distance between them. The Heisenberg commutator \([x,p]=i\hbar\) follows from the curvature of \(\gamma^1\): the symplectic form \(\omega_\mathrm{FS}\) evaluated on the egress (position) and ingress (momentum) directions gives \(\omega_\mathrm{FS} = \hbar\). Neither is an axiom. Both are geometry.