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Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Pauli’s Exclusion Principle in Spinor Coordinate Space Daniel Galehouse University of Akron Theoretical and Experimental aspects of the Spin Statistics connections and related symmetries, 2008 D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Outline 1 Geometry and Quantum Mechanics 2 Spinor Coordinates 3 Two or more Electrons D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons The problem of derivatives. Matrix mechanics pq − qp = −i~ Wave mechanics ∂ ∂ q−q =1 ∂q ∂q General relativity Dj Φi = Φi;j = D. Galehouse [email protected] ∂Φi + Γijk Φk ∂x j PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Conformal waves Wave equations from the Riemann tensor. Let the conformal factor be Ψp with p = 4/(n − 2). Ψ obeys a linear wave equation in n dimensions. ∂2ψ =R=0 ∂x a ∂xa D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Quantum field equation. In five dimensions. p ∂ ∂ 1 p (i~ µ − eAµ ) −ġg µν (i~ ν − eAν )ψ = ∂x ∂x −ġ [m2 + 3 (Ṙ − 16 D. Galehouse [email protected] e2 F F αβ )]ψ 4m2 αβ PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Interaction mechanism Conformal mediation Rij (ωγ mn ) = 0 → Rij (γ mn ) = Tij Gravitational source equations 1−(e2 /m2 )A2 αβ αβ α µβ 2 e2 α β 2 R = 8πκ F µ F +m|ψ| m2 A A +m|ψ| 2−(e2 /m2 )A2 g Electromagnetic source equation F βµ |µ = 4πe|ψ|2 Aβ D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Second quantization of photons and gravitons Aµ = Aµ (ret.) + Aµ (adv.) D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Second quantization of electrons Specific heat of a monatomic gas, spectroscopy {bα , bα′ } = 0 {bα , bα† ′ } = δαα′ {bα† , bα† ′ } = 0 D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Local definition of spinor coordinates. ξ A = ξrA + iξiA , ξ Ā = ξrĀ − iξiA , A = 1···4 ǫAB̄ = ǫAB̄ = diag(1, 1, −1, −1) dx m = ζ A γ mA B dξ C̄ ǫC̄B + dξ A γ̄ mBA ζ C̄ ǫC̄B ≡ ζγ m dξ † + dξγ †m ζ † D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Conformal Waves in spinor space Using, for the Dirac wave function, ΨB = ∂Ψ ∂ξ B if Ψ is a function in extended space-time, the conformal wave 0= Ψ ≡ ǫĀB ∂ ∂ξ Ā ∂ΨB ∂ Ψ ≡ ǫĀB B ∂ξ ∂ξ Ā gives according to the chain rule, the Dirac equation m B ∂ΨB D =0 ζ γ D ∂x m D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Local Dirac electron A plane wave in five space ~ m Ψ = ei(k ~x −ωt−mτ ) ≡ eikm x , km = (~k , ω, m) becomes after differentiation in spinor space ΨA ≡ ∂Ψ ∂x m = Ψik m ∂ξ A ∂ξ A ⇒ k0 0 im − k3 −k1 + ik2 0 k0 −k1 − ik2 im + k3 ζ† iΨkm γ †m ζ † = iΨ im + k3 k1 − ik2 −k0 0 k1 + ik2 im − k3 0 −k0 D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Transformation theory of interaction 1 1 m n {γ , γ } ≡ (γ m γ n + γ n γ m ) = 2 2 γ mn gµν − Aµ Aν ≡ −Aν D. Galehouse [email protected] −Aµ −1 PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons An identified pair e− e− e− e− e− e− e− e− e− e+ e+ e+ e− e− D. Galehouse [email protected] e− PEP in spinor space e− e− Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Parallel electrons 8−D 4−D D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Anti-parallel electrons 4−D 4−D D. Galehouse [email protected] 8−D 8−D PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Spinor wave propagation 2 1 2 1 1 2 D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Boundary development 2 1 − + 1 Standard tons: boundary condi- ψ ′ (1) = a[ψ(1) − ψ(2)] − + ψ ′ (2) = a[ψ(2) − ψ(1)] 2 Spinor coordinate boundary condition: ΨA = ∂Ψ ∂ξ A Ψ=0 D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Multiple electrons in spinor space D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Multiple electrons in spinor space D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Ongoing considerations Questions and problems Calculational advantages Relativistic formalism, Feynman exchange Interparticle interaction/self-interaction Operators Other Fermions Dirac-Thirring paradox, rotation in G.R. Newton’s bucket Aharonov-Casher D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons Geometry of the Pauli Equivalence Principle The geometrical description of fundamental physics. The natural relevance of spinor coordiantes for electrons. The elementary description of the Pauli equivalence principle as a property of differential equations. D. Galehouse [email protected] PEP in spinor space Geometry and Quantum Mechanics Spinor Coordinates Two or more Electrons References D. Galehouse, The Geometry of Quantum Mechanics, in preparation. D. Galehouse, J. Phys., 2(1):50–100, 2000. Conf. Ser. Vol 33, 411-416 at www.iop.org/EJ/toc/1742-6596/33/1 D. Galehouse [email protected] PEP in spinor space