QFT² — Quantum Field Theory from a Bilateral Crossing Geometry

Abstract. The foundational structure of quantum field theory is derived from the bilateral crossing geometry. The vacuum is \(\infty_0\) — the pre-crossing ground state. Creation and annihilation operators are egress and return crossings; their commutation relation \([a, a^\dagger] = 1\) is the additive identity \(N + 0 = N\) in operator form. The Feynman propagator is the bilateral transition amplitude between two spacetime points; the \(i\varepsilon\) prescription is the bilateral arrow of time — the monotonic increase of \(\tau\). The spin-statistics theorem follows from crossing closure: fermions close via the half-cycle \(e^{i\pi} = -1\) (requiring the 2-chain), bosons via the full cycle \(e^{2\pi i} = 1\). UV divergences are attempts to place infinities inside the finite bilateral system — impossible by the same argument that excludes singularities. Renormalisation is bilateral scale running. The path integral is the sum over all crossing histories. The S-matrix is the complete bilateral crossing record.

I. The Three Axioms

The Three Axioms

A1 — Existence is relational. No object exists independently of all others. Every state is defined by its intersections.

A2 — No intersection is preferred. The labelling of any intersection is arbitrary; the structure is invariant under relabelling.

A3 — The Present is the locus where Future meets Past. There exists a distinguished crossing point \(\tau_0\) at which potential (ingress) and actual (egress) states are identified.

These three axioms force the internal geometry \(S^3 \times \mathbb{CP}^2\) and the Standard Model gauge group \(\mathrm{SU}(3)_c \times \mathrm{SU}(2)_L \times \mathrm{U}(1)_Y\). Quantum field theory is the dynamical theory of bilateral crossings within this geometry. Every particle is a completed crossing record. Every interaction is a crossing intersection. Every field is the aggregate crossing structure over a region of spacetime.

II. The Vacuum as ∞₀

Definition — The Bilateral Vacuum

The quantum vacuum \(|0\rangle\) is identified with \(\infty_0\): the pre-crossing ground state, prior to all bilateral crossings, prior to all particle records. It is the unique state with zero crossing records — the origin from which all crossings depart and to which all crossings return.

Proposition — Properties of the Bilateral Vacuum

Uniqueness: \(\infty_0\) is the unique pre-crossing state. There is one vacuum.
Zero particle number: \(\hat{N}|0\rangle = 0\). No completed crossing records present.
Lowest energy: All crossing records carry positive energy. The vacuum, carrying none, has minimum energy.
Source of all crossings: Every particle state is obtained from \(|0\rangle\) by initiating bilateral crossings from \(\infty_0\).

Vacuum fluctuations. Vacuum fluctuations are ingress-face potential crossings that have not completed — unwritten crossing records not yet terminated. \(\infty_0\) is simultaneously zero (egress face, no crossings) and infinity (ingress face, all potential crossings). Vacuum fluctuations are the ingress-face content of \(\infty_0\): real potential, not yet actualised. The observable cosmological constant is the bilateral balance between the two faces — not the sum of all zero-point energies, which diverges.

III. Creation and Annihilation as Bilateral Crossings

Definition — Creation and Annihilation

For a particle of momentum \(\mathbf{k}\):

The creation operator \(a^\dagger(\mathbf{k})\) is a bilateral egress crossing: it initiates a completed crossing record at momentum \(\mathbf{k}\), adding one particle to the state.

The annihilation operator \(a(\mathbf{k})\) is the bilateral return: it removes one crossing record at momentum \(\mathbf{k}\), tracing the crossing back toward \(\infty_0\).

Proposition — Commutation Relations from the Additive Identity

The bosonic commutation relation \[ [a(\mathbf{k}),\, a^\dagger(\mathbf{k}')] = \delta^3(\mathbf{k} - \mathbf{k}') \] is the bilateral statement of the additive identity \(N + 0 = N\) in operator form: adding one crossing record and then removing it returns to the original state, contributing exactly one unit of the identity.

Proof. \(a^\dagger(\mathbf{k})\) adds a crossing record at \(\mathbf{k}\). \(a(\mathbf{k})\) removes one at \(\mathbf{k}\). The composition \(a(\mathbf{k})a^\dagger(\mathbf{k}')\) adds at \(\mathbf{k}'\) then removes at \(\mathbf{k}\). The difference from the reversed order is: did the record at \(\mathbf{k}\) exist before removal? If \(\mathbf{k} = \mathbf{k}'\), the normal-ordered difference is exactly \(\delta^3(\mathbf{k}-\mathbf{k}')\) — one unit contributed by the record that was added and removed. This is \(N + 0 = N\). ∎

For fermions, the half-cycle closure phase \(e^{i\pi} = -1\) makes crossing records antisymmetric under permutation, forcing anticommutation: \[\{a(\mathbf{k}),\, a^\dagger(\mathbf{k}')\} = \delta^3(\mathbf{k} - \mathbf{k}').\]

Fock space as crossing record space. The Fock space \(\mathcal{F} = \bigoplus_{n=0}^\infty \mathcal{H}^{\otimes_s n}\) (symmetrised for bosons, antisymmetrised for fermions) is the space of all completed crossing records of definite particle number. The \(n\)-particle sector is the space of \(n\) completed bilateral crossings. The vacuum \(|0\rangle\) is the \(n=0\) sector — \(\infty_0\) with no crossing records initiated.

IV. The Quantum Field as Bilateral Crossing Amplitude

Definition — The Quantum Field

The quantum field \(\hat{\phi}(x)\) at spacetime point \(x\) is the sum of all bilateral crossing amplitudes at \(x\): \[ \hat{\phi}(x) = \int \frac{d^3k}{(2\pi)^3 2\omega_k} \left[ a(\mathbf{k})\, e^{-ik\cdot x} + a^\dagger(\mathbf{k})\, e^{+ik\cdot x} \right]. \] The positive-frequency part \(a(\mathbf{k})e^{-ik\cdot x}\) is the egress amplitude: an actual crossing record propagating forward in \(\tau\). The negative-frequency part \(a^\dagger(\mathbf{k})e^{+ik\cdot x}\) is the ingress amplitude: a potential crossing, the pre-crossing ingress face.

Particle–antiparticle from bilateral faces. For a complex field, the positive-frequency part contains \(a(\mathbf{k})\) (particle annihilation, egress crossing) and the negative-frequency part contains \(b^\dagger(\mathbf{k})\) (antiparticle creation, ingress crossing). Particles are egress-face crossings (actualised, past). Antiparticles are ingress-face crossings (potential, future). CPT symmetry is the full bilateral inversion, which is a symmetry of the crossing structure by Axiom A2.

V. The Feynman Propagator as Bilateral Amplitude

Definition — The Feynman Propagator

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 \(x\). In momentum space: \[ \tilde{\Delta}_F(k) = \frac{i}{k^2 - m^2 + i\varepsilon}. \]

Proposition — The \(i\varepsilon\) Prescription is the Bilateral Arrow of Time

The \(i\varepsilon\) prescription is not an analytic regularisation trick. It is the bilateral arrow of time: the monotonic increase of \(\tau\) (Axiom A3). The shift \(+i\varepsilon\) ensures that positive-energy particles propagate forward in \(\tau\) (egress direction) and antiparticles propagate backward (ingress direction). The time-ordering operator \(T\) selects \(\tau\)-ordered bilateral crossings. Without \(i\varepsilon\), forward and backward \(\tau\) propagation would mix, violating the bilateral monotonicity condition.

Proof. In position space: \[ \Delta_F(x-y) = \theta(x^0-y^0)\,\Delta^+(x-y) + \theta(y^0-x^0)\,\Delta^+(y-x). \] The \(\theta\) functions select \(\tau\)-ordered contributions: forward-propagating when \(x^0 > y^0\), backward when \(y^0 > x^0\). This is bilateral monotonicity of \(\tau\): causal propagation in one direction only. The \(i\varepsilon\) in momentum space implements the \(\theta\) function structure via the residue theorem. ∎

VI. Spin-Statistics

Theorem — Spin-Statistics from Bilateral Closure

Under exchange of two identical particles, the state acquires the closure phase:

Integer spin (bosons): closure phase \(e^{2\pi i} = 1\). Symmetric states. Bose–Einstein statistics. Commutation relations.
Half-integer spin (fermions): closure phase \(e^{i\pi} = -1\). Antisymmetric states. Two identical fermions in the same state forces \(\Psi = 0\). Pauli exclusion. Fermi–Dirac statistics. Anticommutation relations.

The distinction is whether the 2-chain is required for crossing closure. Fermions require it (half-cycle); bosons do not (full cycle). The Pauli exclusion principle follows from \(e^{i\pi} = -1 \neq 1\): the half-cycle phase is never trivial.

The full derivation is given in the companion paper The Spin-Statistics Theorem from Bilateral Crossing Closure.

VII. The Feynman Rules from Crossing Geometry

Definition — Crossing Vertices

A Feynman vertex is a bilateral crossing intersection: a spacetime point where \(n\) crossing records meet. The vertex factor is determined by the gauge coupling structure, fixed by the gauge group \(\mathrm{SU}(3)\times\mathrm{SU}(2)\times \mathrm{U}(1)\) and the crossing geometry of \(S^3\times\mathbb{CP}^2\).

Proposition — Propagators from Bilateral Closure Type

Each particle type has a Feynman propagator determined by its bilateral crossing: \begin{align*} \text{Scalar:} &\quad \dfrac{i}{k^2 - m^2 + i\varepsilon} \\[8pt] \text{Fermion:} &\quad \dfrac{i(\not{k} + m)}{k^2 - m^2 + i\varepsilon} \\[8pt] \text{Gauge boson:} &\quad \dfrac{-i}{k^2+i\varepsilon} \!\left(g^{\mu\nu} - (1-\xi)\dfrac{k^\mu k^\nu}{k^2}\right) \end{align*} The scalar numerator is \(1\) (full-cycle closure). The fermion numerator \(\not{k}+m\) carries spinor indices from the 2-chain half-cycle closure. The gauge boson numerator has the Lorentz metric (vector crossing), with the \(\xi\)-freedom from Axiom A2.

Gauge invariance from Axiom A2. The gauge freedom — the \(\xi\)-dependence in the gauge boson propagator — is Axiom A2 in operator form: the labelling of intersections is arbitrary. Physical S-matrix elements are \(\xi\)-independent. The Ward identities \(k_\mu \mathcal{M}^\mu = 0\) express the same invariance.

VIII. Renormalisation as Bilateral Scale Running

Proposition — UV Divergences Cannot Exist in the Bilateral System

Ultraviolet divergences — loop integrals diverging as loop momentum \(\ell \to \infty\) — are attempts to place an infinity inside the finite bilateral system. By the Write Completion axiom of structural mathematics, a completed crossing record is finite. An infinite loop integral would require an infinite crossing record inside the finite system. This is the same impossibility as a singularity, an odd perfect number, or the axiom of infinity: attempting to place \(\infty_0\) inside the system it generates. UV divergences are not physical. They arise from extending the theory beyond its domain of validity.

Renormalisation as scale running. The renormalisation group equation \[\mu \frac{dg_i}{d\mu} = \beta_i(g_1, g_2, g_3)\] expresses the running of crossing capacity with the bilateral scale \(\mu\). The logarithmic running of gauge couplings \(g^{-2}(\mu) \sim \log(\mu/\Lambda)\) has the same form as the prime counting function \(\pi(x) \sim x/\log x\). The RGE running and the prime number theorem are the same bilateral scale structure read at different levels. The unification of the three SM couplings at \(\alpha_U = 1/42\) is the bilateral scale running converging to the single crossing capacity of \(S^3 \times \mathbb{CP}^2\) at the unification scale.

IX. The S-Matrix as Complete Crossing Record

Definition — The S-Matrix

The S-matrix element \(\langle f|S|i\rangle\) is the bilateral amplitude for an initial crossing configuration \(|i\rangle\) to evolve to a final configuration \(|f\rangle\): the complete bilateral crossing record of an interaction, summing all possible crossing histories between the initial and final states.

Proposition — The Path Integral as Sum over Crossing Histories

The Feynman path integral \[ Z = \int \mathcal{D}\phi\; e^{iS[\phi]} \] is the sum over all possible bilateral crossing histories \(\phi\), weighted by the phase \(e^{iS[\phi]}\). The stationary phase condition \(\delta S = 0\) selects the actual crossing history — the classical path — from the ingress-face superposition of all possible crossings. The ingress face contains all potential crossing histories; the egress face contains the actual crossing (stationary phase, classical trajectory).

LSZ reduction as asymptotic crossing extraction. The LSZ formula extracts the S-matrix element from the time-ordered correlation function by applying Klein-Gordon operators \((\square + m^2)\) to project onto on-shell crossing records. On-shell means the crossing is complete: \(k^2 = m^2\) is the condition that the crossing record is a written, finite, completed object.

X. The Hierarchy Problem

Proposition — The Hierarchy Problem is Resolved by Scale Structure

The hierarchy problem asks why \(m_H \approx 125\) GeV is so much smaller than \(M_\mathrm{Pl} \approx 10^{19}\) GeV, given that quantum corrections to \(m_H^2\) are quadratically sensitive to the UV cutoff. In the bilateral framework: UV divergences do not exist as physical quantities. The Higgs mass is determined by the bilateral crossing structure of \(S^3\times\mathbb{CP}^2\), not by a UV cutoff. The top Yukawa condition \(y_t(\tau_0) = 1\) at the crossing point gives \(v = m_t\sqrt{2}\), fixing the Higgs VEV without reference to the Planck scale. There is no hierarchy problem because there is no quadratic sensitivity: the bilateral framework does not generate it. The ratio \(m_H/M_\mathrm{Pl}\) is a geometric ratio between two crossing scales, not a fine-tuning.

XI. Open Problems

1. The Lagrangian from the crossing geometry. The Standard Model Lagrangian should be derivable from the bilateral crossing action on \(S^3\times\mathbb{CP}^2\). The gauge kinetic terms, Yukawa couplings, and Higgs potential each follow from the crossing structure, but a complete explicit derivation has not been done.

2. Loop calculations. The Feynman rules are identified structurally. Explicit loop calculations — showing that the bilateral cutoff reproduces the standard renormalised results — have not been performed.

3. Non-perturbative effects. Instantons, confinement, and the QCD vacuum are non-perturbative bilateral crossing effects. The instanton is a crossing tunnelling between topologically distinct winding-number sectors of \(\infty_0\). The framework identifies these as distinct crossing configurations but has not computed the instanton amplitude from first principles.

4. Gravity. The bilateral crossing geometry provides a natural setting for quantum gravity: the metric is the bilateral crossing density at each point. The Einstein equations follow from the condition that the crossing density is consistent with the bilateral causality structure. This is a direction, not yet a derivation.

XII. Summary

The vacuum is \(\infty_0\).
Particles are completed crossing records.
Creation is a crossing departure. Annihilation is a return.
\([a, a^\dagger] = 1\) is \(N + 0 = N\).
The propagator connects egress to ingress.
\(i\varepsilon\) is the arrow of time.
Fermions require the 2-chain. Bosons do not.
UV divergences are infinities inside a finite system.
The path integral sums all crossing histories.
The S-matrix is the complete crossing record.

QFT concept	Bilateral reading
Vacuum \(\|0\rangle\)	\(\infty_0\): pre-crossing ground state
Creation \(a^\dagger\)	Egress crossing: adding a record
Annihilation \(a\)	Return crossing: removing a record
\([a, a^\dagger] = 1\)	Additive identity \(N + 0 = N\)
Fock space	Space of completed crossing records
Quantum field \(\hat\phi(x)\)	Bilateral crossing amplitude at \(x\)
Feynman propagator	Bilateral transition amplitude
\(i\varepsilon\) prescription	Bilateral arrow of time (\(\tau\) monotone)
Bosons	Full-cycle closure \(e^{2\pi i} = 1\)
Fermions	Half-cycle closure \(e^{i\pi} = -1\) (2-chain)
Pauli exclusion	\(e^{i\pi} \neq 1\): 2-chain not trivial
Gauge invariance	Axiom A2: no preferred intersection
UV divergences	Infinities inside finite system (impossible)
RGE running	Bilateral scale running on prime ladder
Path integral	Sum over crossing histories
S-matrix	Complete bilateral crossing record
On-shell \(k^2 = m^2\)	Completed, written crossing record