QFT² Visualisations

Interactive diagrams of quantum field theory from the bilateral crossing geometry

These diagrams show the dynamical structure of QFT² — particles being created, propagating, and annihilating as bilateral crossing records. Each diagram corresponds to a section of the paper. The vacuum is \(\infty_0\). Particles are crossing records. The Feynman propagator connects egress to ingress. The \(i\varepsilon\) prescription is the arrow of time. Bosons close at \(2\pi\); fermions close at \(\pi\). Drag to rotate where applicable. Click to start animations.

I. The Vacuum as \(\infty_0\)

The quantum vacuum is not empty. It is \(\infty_0\) — the pre-crossing ground state that is simultaneously zero (egress face, no crossing records) and infinity (ingress face, all potential crossings). Vacuum fluctuations are ingress-face potential crossings that have not completed: they emerge from \(\infty_0\), reach a maximum displacement, and return before a crossing record is written. In the static vacuum mode the ground state is shown — \(\infty_0\) at the centre, the scale structure radiating outward as powers of 2 (the 2-chain). In the fluctuations mode, potential crossings appear and dissolve continuously, never completing their egress.

Colour	Element	Physical meaning
Gold	\(\infty_0\) — the origin	The pre-crossing ground state. Zero from the egress face; infinite potential from the ingress face. The unique source of all crossings.
Blue rings	2-chain scale ladder	The bilateral scale structure \(\{2^k : k \geq 0\}\). Each ring is one rung — one power of 2. All structural numbers must connect to this ladder.
Purple	Ingress-face potential	Potential crossings on the ingress face — unwritten, not yet actualised. In fluctuation mode these flicker in and out without completing. The Casimir effect is the constraint on which fluctuations fit between two plates.
Red	Egress-face actuality	Completed crossing records. In the static vacuum no red is visible — the vacuum carries no completed records.

Static vacuum The ground state of \(\infty_0\). The gold centre is the origin — zero from the outside, infinite potential from within. The blue rings show the 2-chain scale ladder, the bilateral structure that all existing particles must connect to. The background is not empty: it is the ingress face of \(\infty_0\), filled with potential crossings that have not been initiated.

Fluctuations Vacuum fluctuations as ingress-face potential crossings. Each purple arc emerges from \(\infty_0\), reaches a maximum, and dissolves back — a crossing record that was never written. These are not particles. They are the potential of \(\infty_0\) probing the scale structure without completing. The rate at which they flicker follows the bilateral scale: fluctuations at small radius (large scale) are faster; those at large radius (small scale) are slower. This is the UV/IR correspondence.

II. Creation and Annihilation as Crossing Events

The creation operator \(a^\dagger(\mathbf{k})\) initiates a bilateral egress crossing from \(\infty_0\): a particle record departs from the origin and propagates outward. The annihilation operator \(a(\mathbf{k})\) reverses this: the crossing record traces back toward \(\infty_0\) and dissolves. The commutation relation \([a, a^\dagger] = 1\) is the additive identity \(N + 0 = N\) in operator form — adding a crossing record then removing it returns to the original state, contributing exactly one unit. The create-then-destroy sequence and the destroy-then-create sequence differ by exactly this unit, visible in the two paths below.

Colour	Element	Physical meaning
Gold	\(\infty_0\)	The vacuum ground state from which all crossings depart.
Red	Egress crossing (particle)	A completed crossing record propagating outward from \(\infty_0\). The creation operator \(a^\dagger\) initiates this.
Blue	Return crossing	The annihilation path — the crossing record retracing toward \(\infty_0\). The annihilation operator \(a\) applies this.
Green	The unit difference	In commutator mode: the single unit of difference between \(a\,a^\dagger\) and \(a^\dagger\,a\). This is the \(+1\) in \([a, a^\dagger] = 1\) — the additive identity \(N + 0 = N\) made visible.

Create \(a^\dagger|0\rangle = |1\rangle\). A crossing record departs from \(\infty_0\) along the egress face. The red arc shows the crossing reaching its maximum displacement before the next operation. The particle exists as a completed bilateral record.

Annihilate \(a|1\rangle = |0\rangle\). The crossing record traces back toward \(\infty_0\). The blue arc shows the return path. When the crossing reaches \(\infty_0\), the record is dissolved. The state returns to the vacuum.

Commutator \([a, a^\dagger] = a\,a^\dagger - a^\dagger\,a = 1\). The two sequences are shown simultaneously — create-then-destroy (red then blue) and destroy-then-create (blue then red). They differ by exactly one unit: the green arc. This is \(N + 0 = N\): the crossing that was added and removed left exactly one unit of identity behind. The commutation relation is the additive identity in operator form.

III. The Feynman Propagator as Bilateral Thread

The Feynman propagator \(\Delta_F(x-y) = \langle 0|T\{\hat\phi(x)\hat\phi(y)\}|0\rangle\) is the bilateral amplitude for a crossing to depart from the egress face at \(y\) and arrive at the ingress face at \(x\). It is a thread connecting two spacetime points — the complete bilateral transition amplitude. The \(i\varepsilon\) prescription is the bilateral arrow of time: it ensures that positive-energy crossings travel forward in \(\tau\) (egress direction) and antiparticle crossings travel backward (ingress direction). Without \(i\varepsilon\), forward and backward \(\tau\) propagation would mix, violating the bilateral monotonicity of becoming-time.

Colour	Element	Physical meaning
Red	Egress point \(y\)	The source point — egress crossing, actual, past. The particle departs from here.
Blue	Ingress point \(x\)	The destination point — ingress crossing, potential, future. The particle arrives here.
Gold	Propagator thread	The bilateral amplitude \(\Delta_F(x-y)\) connecting the two points. The slight curvature encodes the \(i\varepsilon\) prescription — the tilt of the thread into the complex plane that enforces the arrow of time.
Purple	Antiparticle thread	The backward-\(\tau\) propagator: an antiparticle crossing from \(x\) to \(y\) in the reverse direction. Equivalent to a particle crossing from \(y\) to \(x\) with CPT inversion.
Green axis	The \(\tau\) axis	The bilateral becoming-time axis. The \(i\varepsilon\) prescription ensures all propagation is \(\tau\)-ordered — the arrow of time is not imposed externally but follows from the bilateral monotonicity of \(\tau\) (Axiom A3).

Particle (forward \(\tau\)) The particle propagator: \(x^0 > y^0\). The crossing departs the egress face at \(y\) and arrives at the ingress face at \(x\), traveling forward in becoming-time. The gold thread tilts slightly upward — the \(+i\varepsilon\) shift places the pole below the real axis, enforcing forward causality.

Antiparticle (backward \(\tau\)) The antiparticle propagator: \(y^0 > x^0\). Equivalently, a particle crossing from \(x\) to \(y\) with CPT inversion — egress and ingress faces swapped. The purple thread travels in the opposite \(\tau\) direction. The \(-i\varepsilon\) shift places the pole above the real axis, enforcing backward causality for the antiparticle.

Both (time-ordering) The full Feynman propagator: the time-ordered sum \(T\{\hat\phi(x)\hat\phi(y)\}\). When \(x^0 > y^0\) the gold thread is active; when \(y^0 > x^0\) the purple thread is active. The \(\Theta\)-function structure of the propagator selects the correct bilateral thread automatically. This is the bilateral realisation of causal propagation.

IV. Bosons and Fermions — Crossing Closure

The spin-statistics theorem follows from the closure phase of the bilateral crossing. A particle's crossing record completes a bilateral loop. For a boson the loop closes via a full cycle \(e^{2\pi i} = 1\): the winding number is an integer, no division by 2 is required, and the phase is trivially \(+1\). For a fermion the loop closes via a half-cycle \(e^{i\pi} = -1\): the winding number is a half-integer, division by 2 (the 2-chain) is required, and the phase is \(-1\). Under exchange of two identical fermions, this phase appears: \(\Psi(2,1) = -\Psi(1,2)\). If two fermions occupy the same state, \(\Psi = -\Psi\) forces \(\Psi = 0\). The Pauli exclusion principle is not postulated — it is the geometric consequence of \(e^{i\pi} \neq 1\).

Colour	Element	Physical meaning
Blue	Crossing loop	The bilateral crossing record completing its closure loop. The winding traces the particle's full bilateral cycle from \(\infty_0\) and back.
Gold	Closure point	The point where the loop closes. For bosons: at \(2\pi\), phase \(+1\). For fermions: at \(\pi\), phase \(-1\). The 2-chain is required for the half-cycle.
Red	Second fermion loop	In Pauli mode: the second fermion attempting to occupy the same state. The two red loops are identical — they must produce the same crossing record. But each carries phase \(-1\), so together they cancel to zero.
Green	Phase indicator	The running phase of the closure. For bosons it returns to \(+1\) (green stays above axis). For fermions it returns to \(-1\) (green crosses axis at \(\pi\)). The crossing of the axis is division by 2 — the 2-chain step.

Boson (\(2\pi\) closure) The boson crossing loop completes a full \(2\pi\) rotation and returns to its starting point with phase \(e^{2\pi i} = +1\). No division by 2 is required. The winding number is integer. Under exchange of two identical bosons: \(\Psi(2,1) = +1 \cdot \Psi(1,2)\). Multiple bosons may occupy the same state. Bose-Einstein statistics.

Fermion (\(\pi\) closure) The fermion crossing loop closes at the half-cycle \(\pi\) — division by 2, the 2-chain step. The phase is \(e^{i\pi} = -1\). The winding number is half-integer. Under exchange of two identical fermions: \(\Psi(2,1) = -1 \cdot \Psi(1,2)\). The state is antisymmetric. Watch the green phase indicator cross the axis at exactly \(\pi\) — this is the 2-chain in action.

Pauli exclusion Two identical fermions attempting to occupy the same quantum state. Each carries the half-cycle phase \(-1\). Their combined state satisfies \(\Psi(1,2) = \Psi(2,1) = -\Psi(1,2)\), which forces \(\Psi(1,2) = 0\). The two loops cancel — the state is identically zero. Watch the two red loops arrive at the same closure point and annihilate. The exclusion principle is not a separate postulate: it is the geometric consequence of \(e^{i\pi} = -1 \neq +1\).

V. The Feynman Diagram as Crossing Intersection

A Feynman diagram is a network of bilateral crossing records and their intersections. Each vertex is a point where crossing records meet — where particles interact. Each propagator line is a bilateral thread connecting two vertices, the amplitude for a crossing to travel between two spacetime points. The complete diagram is the amplitude for an initial crossing configuration to evolve to a final one. Here: electron-positron scattering \(e^+e^- \to \mu^+\mu^-\) via a virtual photon. The electron and positron are egress-face crossings (red); the muon pair emerges on the right; the photon propagator is the bilateral thread connecting the two vertices (gold).

Colour	Element	Physical meaning
Red	Electron / muon lines	Fermion crossing records — half-cycle closure, spinor structure. The arrows indicate the direction of the crossing record in \(\tau\).
Purple	Positron / antimuon	Antifermion crossing records — the ingress-face crossings propagating backward in \(\tau\) (equivalently, fermions propagating forward with CPT inversion).
Gold	Photon propagator	The virtual photon bilateral thread — the gauge boson connecting the two interaction vertices. Its amplitude is \(-ig_{\mu\nu}/(k^2 + i\varepsilon)\). It is off-shell: \(k^2 \neq 0\) because it carries the momentum transferred between the fermion lines.
Green	Vertices	The two crossing intersections where the electromagnetic coupling \(e\) occurs. Each vertex contributes a factor of \(ie\gamma^\mu\) to the amplitude. The vertex is the point where a fermion crossing record emits or absorbs a gauge crossing.

Tree level The leading-order amplitude for \(e^+e^- \to \mu^+\mu^-\). The incoming electron (red, left) and positron (purple, left) annihilate at the left vertex, producing a virtual photon (gold thread). The photon propagates to the right vertex, where it creates a muon (red, right) and antimuon (purple, right). This single diagram gives the leading cross-section. The crossing intersection at each vertex is the electromagnetic coupling constant \(e\).

One loop The next-order correction: a virtual electron loop on the photon propagator. The loop is a fermion crossing record that forms a closed cycle — an internal crossing that does not appear in the initial or final states. In the bilateral framework this is a closed crossing record contributing to the photon's self-energy. The loop integral runs over all possible momenta of the internal crossing, weighted by the bilateral amplitude. UV divergences from this loop are regularised by the bilateral cutoff — the scale at which the crossing structure becomes unresolvable.

VI. Renormalisation as Bilateral Scale Running

The three Standard Model gauge couplings \(\alpha_1, \alpha_2, \alpha_3\) run with energy scale \(\mu\) according to the renormalisation group equations. In the bilateral framework, this running is the dependence of crossing capacity on the bilateral scale — the energy at which the crossing structure is probed. At the unification scale, all three couplings converge to \(\alpha_U = 1/42\), the crossing capacity of \(S^3 \times \mathbb{CP}^2\) per degree of freedom. The prime number theorem governs the distribution of irreducible crossings (primes) across the scale ladder: the logarithmic running of couplings has the same form as \(\pi(x) \sim x/\log x\).

Colour	Element	Physical meaning
Red — \(\alpha_3\)	Strong coupling (QCD)	Runs fastest — decreasing from large values at low energies to \(\alpha_U = 1/42\) at unification. Asymptotic freedom: the strong force weakens at high energy. The largest crossing capacity per degree of freedom at low scales.
Blue — \(\alpha_2\)	Weak coupling	The electroweak \(\mathrm{SU}(2)\) coupling. Runs more slowly than \(\alpha_3\). Crosses \(\alpha_1\) near the electroweak scale where \(\mathrm{SU}(2)\times\mathrm{U}(1)\) breaks to \(\mathrm{U}(1)_\mathrm{em}\).
Green — \(\alpha_1\)	Hypercharge coupling	The \(\mathrm{U}(1)_Y\) coupling. Runs upward (increases with energy) — unlike the non-Abelian couplings which decrease. Converges to \(\alpha_U = 1/42\) at the unification scale from below.
Gold	Unification point \(\alpha_U = 1/42\)	The bilateral crossing capacity of \(S^3\times\mathbb{CP}^2\) per degree of freedom. All three couplings converge here: \(\mathrm{Vol}(\mathbb{CP}^2)/\pi^2 \div (N_\mathrm{gen} \cdot \dim M) = (1/2)/(3\times7) = 1/42\).
Purple	Prime ladder rungs	The irreducible crossing scale positions — primes on the bilateral scale ladder. The spacing follows the prime number theorem: gaps grow logarithmically. The RGE beta functions and \(\pi(x)\sim x/\log x\) are the same bilateral structure.

SM running The three SM gauge couplings running with \(\log\mu\) from the electroweak scale (\(\sim 100\) GeV, left) to the GUT scale (\(\sim 10^{16}\) GeV, right). The strong coupling (red) decreases rapidly — asymptotic freedom. The weak coupling (blue) decreases more slowly. The hypercharge coupling (green) increases. All three meet at \(\alpha_U = 1/42\) (gold dot). This is the bilateral scale running converging to the crossing capacity of \(S^3\times\mathbb{CP}^2\).

Prime ladder The scale ladder as a prime structure. Each purple mark on the vertical axis is a prime — an irreducible crossing scale. The spacing between marks grows logarithmically, following the prime number theorem \(\pi(x) \sim x/\log x\). The gauge coupling running follows the same logarithmic structure: \(g^{-2}(\mu) \sim \log(\mu/\Lambda)\). The renormalisation group running and the prime number theorem are the same bilateral scale structure at different levels.

Unification The convergence of all three couplings to \(\alpha_U = 1/42\) — the crossing capacity of the internal geometry \(S^3\times\mathbb{CP}^2\). The formula is: \(\alpha_U = \mathrm{Vol}(\mathbb{CP}^2)/\pi^2 \div (N_\mathrm{gen}\cdot\dim M) = (1/2)/(3\times 7) = 1/42\). There are no UV divergences at the unification scale — the bilateral framework excludes infinities inside the finite system by the same argument that excludes singularities and odd perfect numbers. The unification point is a geometric fact about \(S^3\times\mathbb{CP}^2\), not a coincidence.

Technical note. All six visualisations are pure canvas — no WebGL, no external 3D libraries. Colour palette consistent with the framework's existing visualisations: blue #4a7fc1 (ingress/quantum), red #c1614a (egress/actual), gold #e8c84a (the Present / \(\infty_0\)), purple #a07acc (Möbius phase / antiparticle), green #7a9e7a (structural elements). All animations use requestAnimationFrame. Touch supported on all canvases.