Clifford Algebra Note 5 TOMONAGA's Super Multi-time Theory. PDF Text added
Note 5
TOMONAGA’s Super Multi-time Theory
1 <Schrödinger equation>
State vector ψ
Time t
Electromagnetic field A
Hamiltonian H
i 
ψ( t) = Hψ( t), ψ(0) = ψ (1)
ψ( t) = Hψ( t), ψ(0) = ψ (1)2 <Dirac’s paraphrase of Schrödinger equation >
Coordinate x
Momentum p
Electron N in number
Electromagnetic field A
H-em Electromagnetic field Hamiltonian
[ H-em + 
Hn ( x n, p n, A ( x n) ) + 
] ψ( t) = 0 (2)
Hn ( x n, p n, A ( x n) ) + ] ψ( t) = 0 (2)3 <Representation by unitary transformation>
u( t) = exp{ 
H-emt }
H-emt }A ( x n, t) = u( t) A ( x n) u( t) -1
Φ( t) = u( t) ψ( t)
[ Hn ( x n, p n, A ( x n, t) ) + ] Φ( t) = 0 (3)
Hn ( x n, p n, A ( x n, t) ) + ] Φ( t) = 0 (3)4 < Dirac’s multi-time theory- Time variant in number N >
[ Hn ( x n, p n, A ( x n, tn) ) + ] Φ( x1, t 1; … ; xN, tN ) = 0 (4)
] Φ( x1, t 1; … ; xN, tN ) = 0 (4)5 <Tomonaga’s representation of electromagnetic field>
Unitary transformation
U ( t) = exp { ( H 1 + H 2 ) t }
( H 1 + H 2 ) t } Schrödinger equation
[ H 1 + H 2 + H1 2+ ] ψ( t) = 0
] ψ( t) = 0 Independent time variant txyz at each point in space
[ H 12 ( x, y, z, txyz ) + ] Φ( t) = 0 (5)
] Φ( t) = 0 (5)6 < Tomonaga’s super multi-time theory>
Super curved surface C
Point on C P
4-dimensional volume’s transformation of C 
CP
CPInfinite small variation of state vector Φ[ C] = Φ[ Txyz] Φ[ C]
Φ[ C][ H 12 ( P ) + ] Φ[ C] = 0 (6)
] Φ[ C] = 0 (6)[References]
<Past work on multi-time themes>
Aurora Theory / Dictoron, Time and Symmetry <Language Multi-Time Conjecture> / Tokyo October 6, 2006
<For more details>
[Note]
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The perfect text is now uploaded by PDF at the next.
Clifford Algebra Note 5 TOMONAGA's Super Multi-time Theory
Tokyo
27 February 2018
SRFL Lab
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