Fractal Spacetime Quantum Field Theory: Rigorous Formulation and Experimental Tests

This paper establishes a rigorous mathematical framework for Fractal QuantumField Theory (FQFT) in fractal spacetime, achieving theoretical breakthroughsthrough the following innovations:1. Construction of Sierpinski Sponge Manifold: Using Hutchinson’s iter?ated function system, we prove that its Hausdorff measure satisfies HDH (M) =limδ→0 inf P(diam Ui)DH , ensuring mathematical rigor in fractal spacetime.2. Caputo-Type Fractional Derivative: We construct Dαf(x) = 1Γ(1−α)R x0f′(t)(x−t)α dt,which preserves causality and gauge symmetry.3. Fractal Renormalization Group Equations: We derive β(gi) = (4 −DH)gi +PCijk(DH)gjgk, revealing the relationship between the dimensionparameter DH and the Standard Model coupling constants.Experimental predictions show:• For DH = 3.98, the top quark mass correction ∆mt/mt = (0.3 ± 0.1)% agreeswith LHC data.• The Higgs phase transition strength α = 0.4 predicts gravitational wave sig?nals within the LISA detection window.Fractal lattice field theory simulations confirm theoretical self-consistency with er?rors < 5%. This study provides a new paradigm for unifying quantum gravity andhigh-energy physics.Keywords: Fractal Quantum Field Theory; Hausdorff Dimension; FractionalDerivative; Renormalization Group; LISA Gravitational Waves