# LCRF Layer 2 Formalism v1.2: Gauging the U(1) Symmetry of the GA Field ## 1. Objective The LCRF Layer 2 formalism v1.1 [[0184_LCRF_Layer2_GA_Formalism_v1]], [[0188_LCRF_Layer2_GA_Quantization]] successfully described a quantized GA field `hat{Ψ}` with mass, charge, and spin-1/2 properties, satisfying Poincaré and global U(1) symmetry. However, it lacked interactions beyond self-interaction (`λ` term). To incorporate electromagnetism, the fundamental force associated with conserved charge, we must promote the global U(1) symmetry to a **local gauge symmetry**. This node details this gauging procedure, introducing the electromagnetic vector potential `A` and deriving the coupled dynamics. ## 2. Gauging Procedure 1. **Local U(1) Symmetry:** Require the Lagrangian `L` to be invariant under local phase transformations: `hat{Ψ}(x) → hat{Ψ}'(x) = hat{Ψ}(x) e^(i α(x))` where `i` is the pseudoscalar $I$ and `α(x)` is now a function of spacetime position `x`. 2. **Covariant Derivative:** The original kinetic term `⟨ ħ (∇hat{Ψ}) i γ_3 \tilde{hat{Ψ}} ⟩_S` is *not* invariant under local transformations because `∇(e^(iα)) ≠ (∇e^(iα))`. We replace the vector derivative `∇ = γ^μ ∂_μ` with a **gauge covariant derivative `D = γ^μ D_μ`**: `D_μ = ∂_μ - i (q/ħc) A_μ` where: * `A_μ` are the components of the **gauge field (vector potential)**, transforming as `A_μ → A'_μ = A_μ + (ħc/q) ∂_μ α` to compensate for the derivative acting on `α(x)`. * `q` is the coupling constant (electric charge associated with the `hat{Ψ}` field). * The term `i A_μ` involves the pseudoscalar `I`. *(Note: In standard QED with spinors, this is `i e A_μ`, where `i = sqrt(-1)`. The use of the pseudoscalar `I` here needs careful verification for consistency within GA QED formulations, but preserves the U(1) structure.)* Let's denote the coupling constant as `q` for now. The covariant derivative `D hat{Ψ}` transforms like `hat{Ψ}` itself under the gauge transformation: `(D hat{Ψ})' = (D hat{Ψ}) e^(iα)`. 3. **Gauged Lagrangian:** Replace `∇` with `D` in the original Lagrangian [[0184_LCRF_Layer2_GA_Formalism_v1]]. `L_Ψ = ⟨ ħ (D hat{Ψ}) i γ_3 \tilde{hat{Ψ}} - m c hat{Ψ} \tilde{hat{Ψ}} ⟩_S - (λ/2) (⟨hat{Ψ}\tilde{hat{Ψ}}⟩_S)^2` 4. **Gauge Field Dynamics:** Add the kinetic term for the gauge field `A` itself, which is gauge invariant: `L_A = - (1/4) F_{μν} F^{μν}` where `F_{μν} = ∂_μ A_ν - ∂_ν A_μ` is the electromagnetic field strength tensor. 5. **Total Lagrangian:** `L_{total} = L_Ψ + L_A` `L_{total} = ⟨ ħ (γ^μ (∂_μ - i(q/ħc)A_μ) hat{Ψ}) i γ_3 \tilde{hat{Ψ}} - m c hat{Ψ} \tilde{hat{Ψ}} ⟩_S - V(⟨hat{Ψ}\tilde{hat{Ψ}⟩_S) - (1/4) F_{μν} F^{μν}` (Assuming `V(ρ) = (λ/2) ρ^2`). ## 3. Derived Equations of Motion Applying the Euler-Lagrange equations to `L_{total}` with respect to `hat{Ψ}` (or `\tilde{hat{Ψ}}`) and `A_μ`: 1. **Equation for `hat{Ψ}` (Gauged Non-Linear Dirac-Hestenes):** Varying w.r.t. `\tilde{hat{Ψ}}` yields (schematically): `ħ γ^μ D_μ hat{Ψ} i γ_3 - m c hat{Ψ} - ∂V/∂ρ * hat{Ψ} = 0` `ħ γ^μ (∂_μ - i(q/ħc)A_μ) hat{Ψ} i γ_3 - m c hat{Ψ} - λ ⟨hat{Ψ}\tilde{hat{Ψ}}⟩_S hat{Ψ} = 0` This is the non-linear Dirac-Hestenes equation minimally coupled to the gauge field `A_μ`. It describes how the charged GA field `hat{Ψ}` evolves under its own dynamics and the influence of the electromagnetic potential `A_μ`. 2. **Equation for `A_μ` (Maxwell's Equations with Source):** Varying w.r.t. `A_μ` yields: `∂_ν (∂L / ∂(∂_ν A_μ)) - (∂L / ∂A_μ) = 0`. * `∂L / ∂(∂_ν A_μ) = -F^{νμ}` * `∂L / ∂A_μ = ∂L_Ψ / ∂A_μ = ⟨ ħ (γ^μ (-i q/ħc) hat{Ψ}) i γ_3 \tilde{hat{Ψ}} ⟩_S = ⟨ (q/c) γ^μ hat{Ψ} i γ_3 \tilde{hat{Ψ}} ⟩_S` This term `⟨ (q/c) γ^μ hat{Ψ} i γ_3 \tilde{hat{Ψ}} ⟩_S` should correspond to the U(1) Noether current `(q/c) J^μ` derived previously (up to factors). Let `J^μ = c ⟨ hat{Ψ} γ^μ i γ_3 \tilde{hat{Ψ}} ⟩_S` (needs verification based on precise Noether derivation for this Lagrangian). The equation becomes: `∂_ν F^{νμ} = (q/c) J^μ` (using `J^μ` definition from standard Dirac-Hestenes theory, `J^μ = <Ψγ^μ\tilde{Ψ}>_V`). This is **Maxwell's equations** sourced by the conserved U(1) current `J^μ` associated with the `hat{Ψ}` field. ## 4. Interpretation * **Electromagnetism:** The formalism now includes the electromagnetic force mediated by the gauge field `A` (whose quanta are photons upon quantization of `L_A`). * **Interaction:** The coupling term `i q A_μ hat{Ψ}` in the covariant derivative describes the interaction between the charged `hat{Ψ}` field (fermions) and the electromagnetic field `A` (photons). * **Conserved Current:** The conserved U(1) current `J^μ` acts as the source for the electromagnetic field, consistent with charge conservation (A6). ## 5. Status within LCRF Layer 2 (v1.2) * **Progress:** Successfully incorporates U(1) gauge symmetry and the electromagnetic interaction in a way consistent with established physics (QED analogue) and the GA structure. Describes interaction between spin-1/2 fermions and spin-1 photons (quanta of `A`). * **Remaining Deficiencies:** * Still lacks other forces/symmetries (Weak, Strong - requires non-Abelian gauge theory, e.g., SU(2), SU(3)). * Still lacks spin-0 fields (Higgs analogue for mass generation beyond the `m` parameter). * Gravity and dynamic spacetime are absent. * Explicit IO principles (Η, Θ, M, K beyond basic interaction) are not yet integrated. * Measurement problem remains unresolved at this layer. ## 6. Conclusion and Next Steps LCRF Layer 2 formalism v1.2 successfully extends the GA field theory by incorporating U(1) gauge symmetry, yielding coupled equations for the `hat{Ψ}` field and the electromagnetic field `A` that resemble non-linear Dirac-Hestenes QED. This demonstrates the framework's capacity to include fundamental forces via the gauge principle. **Next Steps:** 1. **Analyze Coupled System:** Study the properties of this coupled system, including interactions (scattering) and potential bound states. 2. **Incorporate Other Symmetries:** Explore extending the formalism to non-Abelian gauge groups (SU(2), SU(3)) to incorporate Weak and Strong forces. 3. **Add Scalar Fields:** Introduce scalar fields (`Φ`) potentially responsible for mass generation (Higgs mechanism analogue) via coupling to `hat{Ψ}`. 4. **Integrate IO Principles:** Revisit how Η, Θ, M, K could modify this gauged QFT framework [[0189_LCRF_Layer2_GA_IO_Integration]]. 5. **Re-evaluate URFE:** Provide updated Layer 2 URFE responses based on this gauged GA QFT formalism (v1.2). Priority should likely be given to Step 5 (URFE re-evaluation) to assess the current standing before further complex extensions like non-Abelian groups or scalar fields.