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 ( xn, pn, A (xn) ) + 
] ψ(t) = 0 (2)
Hn ( xn, pn, A (xn) ) + ] ψ(t) = 0 (2)
3 <Representation by unitary transformation>
u(t) = exp{ 
H-emt }
H-emt }
A (xn, t) = u(t) A (xn) u(t)-1
Φ(t) = u(t) ψ(t)
[ Hn ( xn, pn, A (xn, t) ) + ] Φ(t) = 0 (3)
Hn ( xn, pn, A (xn, t) ) + ] Φ(t) = 0 (3)
4 < Dirac’s multi-time theory- Time variant in number N >
[Hn ( xn, pn, A (xn, tn) ) + ] Φ( x1, t1; … ; xN, tN ) = 0 (4)
] Φ( x1, t1; … ; xN, tN ) = 0 (4)
5 <Tomonaga’s representation of electromagnetic field>
Unitary transformation
U (t) = exp { (H1 + H2 ) t }
(H1 + H2 ) t }
Schrödinger equation
[H1 + H2 + H12+ ] ψ(t) = 0
] ψ(t) = 0
Independent time variant txyz at each point in space
[ H12 (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
CP
Infinite small variation of state vectorΦ[C] = Φ[Txyz] Φ[C]
Φ[C]
[ H12 ( 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>
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