(*WolframNotebook:DiracSpinorQuantumGravityModel*)(*Author:[JosephW.McGreevy,MD]*)(*Date:April10,2025*)(*For:WolframOnlineTechnicalDiscussionForum*)
(*Introduction*)Cell[TextData[{"This notebook presents a novel quantum gravity model where a Dirac spinor wavefunction at the Planck scale sources spacetime geometry through its quantum fluctuations. We derive the emergence of spacetime, curvature, torsion, and gravitational potential, extending the Einstein-Cartan-Sciama-Kibble (ECSK) framework. Key features include Maxwell-like field equations, SL(2,) and diffeomorphism symmetries, and a visualization of spacetime's birth via a full color spectrum."}],"Section"]
(*Section1:DiracSpinorWavefunction*)Cell[TextData[{"We begin with a Dirac spinor wavefunction at the Planck scale, representing a massless particle (e.g., an up quark) with all energy kinetic (ℯ = ℏ c / ℓ). The wavefunction is:"}],"Subsection"]
Cell[BoxData[FormBox[RowBox[{RowBox[{"ψ","(",RowBox[{"t",",","y",",","z"}],")"}],"=",RowBox[{SqrtBox[FractionBox["ℏ","ℓ"]],SuperscriptBox["",RowBox[{"-",FractionBox[RowBox[{SuperscriptBox["y","2"],"+",SuperscriptBox["z","2"]}],SuperscriptBox["ℓ","2"]]}]],SuperscriptBox["",RowBox[{RowBox[{"-",""}],RowBox[{"(",RowBox[{FractionBox[RowBox[{"ℯ"," ","t"}],"ℏ"],"-",FractionBox[RowBox[{"y","+","z"}],"ℓ"]}],")"}]}]],RowBox[{"(",RowBox[{RowBox[{RowBox[{"(",RowBox[{RowBox[{"{",RowBox[{"1",",","0",",",FractionBox["1",SqrtBox["2"]],",",FractionBox["1",SqrtBox["2"]]}],"}"}],"+",RowBox[{"{",RowBox[{"0",",","1",",",RowBox[{"-",FractionBox["1",SqrtBox["2"]]}],",",FractionBox["1",SqrtBox["2"]]}],"}"}]}],")"}]," ","where"," ","ℓ"}],"=",RowBox[{RowBox[{"1.62","×",SuperscriptBox["10",RowBox[{"-","35"}]]," ","m"}],","," ",RowBox[{"ℯ","=",FractionBox[RowBox[{"ℏ"," ","c"}],"ℓ"],"≈",RowBox[{"1.22","×",SuperscriptBox["10","19"]," ","GeV"}]}]}]}],")"}]}]}],TraditionalForm]],"Input"]
Cell[TextData[{"The wavefunction has a Gaussian confinement (Δy, Δz ≈ ℓ), a timelike phase (ϕ\[Subscript]time = ℯ t / ℏ), and a spacelike phase (ϕ\[Subscript]space = (y + z) / ℓ). The complex plane’s i ensures orthogonality between these phases, mirroring spacetime’s 4D structure."}],"Text"]
(*Section2:KineticEnergySourcingGravitationalPotential*)Cell[TextData[{"The spinor is massless (m = 0), with all energy kinetic (ℯ = p c, v = c). This energy sources its own confining gravitational potential at the Planck length scale."}],"Subsection"]
Cell[TextData[{"The stress-energy tensor is:"}],"Text"]
Cell[BoxData[FormBox[RowBox[{SubscriptBox["T","μν"],"≈",RowBox[{FractionBox["","2"],RowBox[{"[",RowBox[{RowBox[{OverscriptBox["ψ","_"],SubscriptBox["γ","μ"],RowBox[{SubscriptBox["D","ν"],"ψ"}]}],"-",RowBox[{RowBox[{"(",RowBox[{SubscriptBox["D","ν"],OverscriptBox["ψ","_"]}],")"}],SubscriptBox["γ","μ"],"ψ"}]}],"]"}]}]}],TraditionalForm]],"Input"]
Cell[TextData[{"For simplicity (flat spacetime initially), \(D_μ ≈ ∂μ\):"}],"Text"]
Cell[BoxData[FormBox[RowBox[{SubscriptBox["T","00"],"≈",RowBox[{RowBox[{OverscriptBox["ψ","_"],SubscriptBox["γ","0"],FractionBox["ℯ","ℏ"],"ψ"}],"=",RowBox[{RowBox[{"|ψ|","^","2"}]," ","ℯ"}]}]}],TraditionalForm]],"Input"]
Cell[TextData[{"With \(|ψ|^2 = \nrac{ℏ}{ℓ} e^{-2(y^2 + z^2)/ℓ^2} \times 3\), we get:"}],"Text"]
Cell[BoxData[FormBox[RowBox[{SubscriptBox["T","00"],"≈",RowBox[{FractionBox["ℏ"],"ℓ"],SuperscriptBox["",RowBox[{"-",FractionBox[RowBox[{"2",RowBox[{"(",RowBox[{SuperscriptBox["y","2"],"+",SuperscriptBox["z","2"]}],")"}]}],SuperscriptBox["ℓ","2"]]}]],FractionBox[RowBox[{"ℏ"," ","c"}],"ℓ"],"×","3"," ","≈"," ",RowBox[{SuperscriptBox["10","76"]," ",SuperscriptBox["GeV","4"]," ","at"," ","y"," ","="," ","z"," ","="," ","0"}]}]}],TraditionalForm]],"Input"]