Pauli's Exclusion Principle in Spinor Coordinate Space

Geometry and Quantum MechanicsSpinor Coordinates

Two or more Electrons

Pauli’s Exclusion Principle in SpinorCoordinate Space

Daniel Galehouse

University of Akron

Theoretical and Experimental aspects of the Spin Statisticsconnections and related symmetries, 2008

D. Galehouse [email protected] PEP in spinor space



Outline

1 Geometry and Quantum Mechanics

2 Spinor Coordinates

3 Two or more Electrons




The problem of derivatives.

Matrix mechanics

pq − qp = −i~

Wave mechanics

∂

∂qq − q

∂

∂q= 1

General relativity

DjΦi = Φi

;j =∂Φi

∂x j + ΓijkΦk




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

∂xa∂xa= R = 0




Quantum field equation.

In five dimensions.

1√

−g(i~

∂

∂xµ− eAµ)

√

−ggµν(i~∂

∂xν− eAν)ψ =

[m2 +3

16(R − e2

4m2 FαβFαβ)]ψ




Interaction mechanism

Conformal mediation

Rij(ωγmn) = 0 → Rij(γ

mn) = Tij

Gravitational source equations

Rαβ = 8πκ[

FαµFµβ+m|ψ|2 e2

m2 AαAβ+m|ψ|2 1−(e2/m2)A2

2−(e2/m2)A2 gαβ

]

Electromagnetic source equation

Fβµ|µ = 4πe|ψ|2Aβ




Second quantization of photons and gravitons

Aµ = Aµ(ret.) + Aµ(adv.)




Second quantization of electrons

Specific heat of a monatomic gas, spectroscopy

{bα,bα′} = 0

{bα,b†α′} = δαα′

{b†α,b

†α′} = 0




Local definition of spinor coordinates.

ξA = ξAr + iξA

i , ξA = ξAr − iξA

i , A = 1 · · ·4

ǫAB = ǫAB = diag(1,1,−1,−1)

dxm = ζAγm BA dξCǫCB + dξAγmB

AζCǫCB ≡ ζγmdξ† + dξγ†mζ†




Conformal Waves in spinor space

Using, for the Dirac wave function,

ΨB =∂Ψ

∂ξB

if Ψ is a function in extended space-time, the conformal wave

0 = Ψ ≡ ǫAB ∂

∂ξA

∂

∂ξB Ψ ≡ ǫAB ∂ΨB

∂ξA

gives according to the chain rule, the Dirac equation

ζD[

γm BD

∂ΨB

∂xm

]

= 0




Local Dirac electron

A plane wave in five space

Ψ = ei(~k~x−ωt−mτ) ≡ eikmxm, km = (~k , ω,m)

becomes after differentiation in spinor space

ΨA ≡∂Ψ

∂ξA = Ψikm∂xm

∂ξA ⇒

iΨkmγ†mζ† = iΨ

k0 0 im − k3 −k1 + ik2

0 k0 −k1 − ik2 im + k3

im + k3 k1 − ik2 −k0 0k1 + ik2 im − k3 0 −k0

ζ†




Transformation theory of interaction

12{γm, γn} ≡

12(γmγn + γnγm) =

γmn ≡

(

gµν −AµAν −Aµ

−Aν −1

)




An identified pair

e−

e−

e− e−

e−

e−

e−e− e−

e−

e−

e−

e+e+e+

e−

e−




Parallel electrons

4−D 8−D




Anti-parallel electrons

4−D

4−D

8−D

8−D




Spinor wave propagation

1 2

21

1 2




Boundary development

1 2

21

++

− −

Standard boundary condi-tons:

ψ′(1) = a[ψ(1) − ψ(2)]

ψ′(2) = a[ψ(2) − ψ(1)]

Spinor coordinate boundarycondition:

ΨA =∂Ψ

∂ξA

Ψ = 0




Multiple electrons in spinor space




Multiple electrons in spinor space




Ongoing considerations

Questions and problemsCalculational advantagesRelativistic formalism, Feynman exchangeInterparticle interaction/self-interactionOperatorsOther FermionsDirac-Thirring paradox, rotation in G.R.Newton’s bucketAharonov-Casher




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 equivalenceprinciple as a property of differential equations.




References

D. Galehouse, The Geometry of Quantum Mechanics,in preparation.

D. Galehouse, J. Phys., 2(1):50–100, 2000.Conf. Ser. Vol 33, 411-416at www.iop.org/EJ/toc/1742-6596/33/1


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Pauli's Exclusion Principle in Spinor Coordinate Space