QM II — QM II: Unified Field Structure

Reader summary

If QM I shows that the Schrödinger equation can be derived rather than postulated, QM II asks a much bigger question. Can the rest of physics, gravity, particles, the Standard Model gauge group, the Higgs, also fall out of the same four phase fields? This paper shows that they can. The gradients of the phase fields define a tetrad and metric whose dynamics reproduce general relativity. Stable knots in the phase fields behave like particles, and they are forced to be fermions by a topological argument. The internal structure of those knots carries exactly the symmetry of the Standard Model. Scalar wobbles of the knots reproduce the Higgs. One ontology, one set of fields, one consistent picture.

Abstract

Phase Differential Theory (PDT) proposes that spacetime geometry and particle physics arise as effective structures associated with a deeper dynamical phase manifold defined by four scalar phase fields. This paper develops a controlled effective-field-theory framework in which gravitational dynamics, fermionic matter, gauge symmetries, and the Higgs sector emerge from the same underlying phase substrate. Gradients of the phase fields define an effective tetrad and metric whose long-wavelength dynamics reproduce an Einstein–Hilbert term. Topologically stabilised defect configurations behave as particle-like excitations, with fermionic statistics arising from Finkelstein–Rubinstein constraints on the defect configuration space. The internal Hilbert bundle associated with these defects admits an automorphism group isomorphic to SU(3)×SU(2)×U(1), and scalar bundle excitations yield an effective Higgs doublet. Hypercharge assignments follow from Yukawa invariance and anomaly cancellation within the effective bundle construction. Gauge couplings arise from normalisation integrals evaluated on defect backgrounds. All results are derived within a low-energy effective regime under explicit structural assumptions. Detailed coefficient-level mapping, renormalisation-group flow, and full microscopic derivations are deferred to subsequent work.

Falsifiable predictions

01
Stable particle-like excitations are topological defects classified by π₃(SU(2)) = ℤ. No fundamental excitations exist outside this classification.
02
Fermionic statistics follow from the Finkelstein–Rubinstein constraint on the defect configuration space; no bosonic alternative is allowed for the defect sector.
03
The internal symmetry group of defect excitations reduces to SU(3) × SU(2) × U(1). Any additional fundamental gauge factor at the same scale falsifies the bundle construction.
04
Gravity emerges from the same locked phase background as the gauge sector. The induced Planck scale must be set by microscopic stiffness parameters and not by independent tuning.

Full paper

1. Introduction

Modern physics rests on two highly successful frameworks: General Relativity, describing gravity through spacetime geometry, and the Standard Model, describing matter and interactions through gauge fields and fermionic degrees of freedom. Despite their empirical success, these theories rely on distinct ontological foundations and lack a common microscopic origin.

PDT offers a unified starting point. Rather than postulating geometry, gauge symmetry, or fermionic matter independently, the framework constructs these structures as emergent descriptions of a deeper phase manifold governed by four scalar phase fields. QM I showed that quantum mechanics arises as the effective envelope dynamics of the collective phase mode. The present paper extends this to the emergence of spacetime geometry, topological defect sectors, fermionic statistics, gauge symmetries, and the Higgs mechanism.

Assumptions

- **A1. Locked background:** the phase fields admit a background with approximately constant gradients. - **A2. Defect sector:** finite-energy configurations admit topologically stabilised defects classified by π₃(SU(2)). - **A3. Bundle structure:** internal defect excitations span a finite-dimensional Hilbert bundle with well-defined automorphisms. - **A4. Low-energy truncation:** only leading-order terms in the derivative expansion are retained. - **A5. Effective quantisation:** collective coordinates may be quantised semiclassically.

2. Phase manifold and microscopic action

The microscopic degrees of freedom are four scalar phase fields Φ_A(x). The action takes the form S = ∫ d⁴x [(1/2) G_AB ∂_μ Φ_A ∂^μ Φ_B − V_lock(Φ)], where G_AB is a constant internal metric and V_lock stabilises a locked background with approximately constant phase gradients.

3. Emergent spacetime geometry

Define the effective tetrad e_μ^A = ∂_μ Φ_A and induced metric g_μν = η_AB e_μ^A e_ν^B. Expanding the microscopic action about the locked background generates an effective Einstein–Hilbert term S_grav = (M_P²/2) ∫ d⁴x √(−g) R, with the Planck scale determined by microscopic stiffness parameters.

4. Topological structure

Define the normalised field n_A = Φ_A / |Φ| and construct U(x) = exp[i n_a(x) σ_a]. Finite-energy configurations satisfy U(x) → I at infinity and are classified by π₃(SU(2)) = ℤ, with a standard topological charge integral.

5. Hedgehog defect

A minimal spherically symmetric defect ansatz U(x) = exp[i F(r) x̂·σ] with boundary conditions F(0)=π, F(∞)=0, stabilised by a Skyrme-type term, yields a finite-energy soliton.

6. Collective coordinate quantisation

Rotational degrees of freedom A(t) ∈ SU(2) give an effective Lagrangian L_rot = (1/2) I Tr(Ω²) with Ω = A⁻¹ Ȧ. Canonical quantisation yields rotational levels E_J = J(J+1)/(2I).

7. Fermionic statistics

The configuration space of the defect is multiply connected, π₁(C) = ℤ_2. Imposing the Finkelstein–Rubinstein constraint requires ψ → −ψ under 2π rotation. Allowed representations correspond to half-integer spin.

8–10. Internal Hilbert space and gauge group

The local Hilbert space factorises as H = H_rot ⊗ H_int ⊗ H_vib. Internal defect degrees of freedom span H_int ≃ ℂ³ ⊗ ℂ². The automorphism group reduces to SU(3) × SU(2) × U(1).

11–14. Higgs, hypercharge, anomalies, gauge couplings

Scalar vibrational modes give an effective complex doublet H(x) ∈ ℂ² with Lagrangian L_H = |D_μ H|² − λ(|H|² − v²/2)². Hypercharges follow from Yukawa invariance and anomaly cancellation. Gauge couplings arise from normalisation integrals over defect currents.

15–16. Limitations and conclusion

The derivation is effective, not UV-complete. Representation content, chirality, and RG flow are deferred. Within PDT, spacetime geometry, fermionic matter, gauge symmetries, and the Higgs mechanism arise as manifestations of a single underlying phase manifold.

BibTeX

@article{fincham_hilton_qm_ii,
  author  = {Fincham, Graham and Hilton, Daniel},
  title   = {Unified Field Structure of Phase Differential Theory: Emergence of Fermions, Gauge Symmetries, and Gravity from a Phase Manifold},
  journal = {Phase Differential Theory Series, QM II},
  year    = {2025}
}