Post on 12-Sep-2021
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 dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Outline
1 Geometry and Quantum Mechanics
2 Spinor Coordinates
3 Two or more Electrons
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
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
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor 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
∂xa∂xa= R = 0
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Quantum field equation.
In five dimensions.
1√
−g(i~
∂
∂xµ− eAµ)
√
−ggµν(i~∂
∂xν− eAν)ψ =
[m2 +3
16(R − e2
4m2 FαβFαβ)]ψ
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
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
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Second quantization of photons and gravitons
Aµ = Aµ(ret.) + Aµ(adv.)
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Second quantization of electrons
Specific heat of a monatomic gas, spectroscopy
{bα,bα′} = 0
{bα,b†α′} = δαα′
{b†α,b
†α′} = 0
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
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†
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
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
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
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†
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Transformation theory of interaction
12{γm, γn} ≡
12(γmγn + γnγm) =
γmn ≡
(
gµν −AµAν −Aµ
−Aν −1
)
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
An identified pair
e−
e−
e− e−
e−
e−
e−e− e−
e−
e−
e−
e+e+e+
e−
e−
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Parallel electrons
4−D 8−D
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Anti-parallel electrons
4−D
4−D
8−D
8−D
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Spinor wave propagation
1 2
21
1 2
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
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
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Multiple electrons in spinor space
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Multiple electrons in spinor space
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor Coordinates
Two or more Electrons
Ongoing considerations
Questions and problemsCalculational advantagesRelativistic formalism, Feynman exchangeInterparticle interaction/self-interactionOperatorsOther FermionsDirac-Thirring paradox, rotation in G.R.Newton’s bucketAharonov-Casher
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor 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 equivalenceprinciple as a property of differential equations.
D. Galehouse dcg@uakron.edu PEP in spinor space
Geometry and Quantum MechanicsSpinor 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, 411-416at www.iop.org/EJ/toc/1742-6596/33/1
D. Galehouse dcg@uakron.edu PEP in spinor space