Download Pauli`s Exclusion Principle in Spinor Coordinate Space

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